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ALGEBRAIC AND TOPOLOGICAL PROPERTIES OF C(X) AND THE j TOPOLOGY By WARREN WILLIAM MCGOVERN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA Copyright 1998 by Warren William McGovern This dissertation is dedicated to those people who have played a part in cre ating me as a whole. First, and foremost, I dedicate this to my abuelo. Me enseiiaste que los dos aspects mas importantes en esta vida son la educaci6n y la pelota. To my mother, who has always been there for me. You taught me how to fight for what I believe in. To Stephanie... what can I say except that you are the love of my life (and you are beautiful when you are angry.) And finally, to Socky and Sydney. ACKNOWLEDGEMENTS I would like to take the time and thank those persons who have helped shape my mathematical mind. I thank Dr. Alexandre Turull, who, way back when, encouraged me to learn analysis before I even knew what algebra was. He threw me the most wickedest curveball during my oral examination, and that I will never forget. I thank Dr. Paul Robinson, who inspired me to learn differential geometry. He also gave me an appreciation for Bob Newhart. I thank Dr. James Brooks, who taught me everything I know about analysis, measure theory, and functional analysis. For this I am indebted. I thank Dr. Robert Hatch, who, when I was very young, suggested I read Kuhn's "Scientific Revolution," which has altered my state of mind forever. And finally, to Dr. Jorge Martinez, who has probably heard all of this be fore, but now I get to say it formally. He is the one who inspired me to become a mathematician. Over the last five years he and I developed more than just the usual advisorstudent relationship. I was allowed, encouraged, and urged to study that material which was needed for the completion of this dissertation. He also became my friend; someone who I could go to and either, just talk about sports (or what is a sport?), or simply chat about life. Gracias Sefior Martinez. TABLE OF CONTENTS ACKNOWLEDGEMENTS ............................ iv ABSTRACT .................................... vi CHAPTERS 1 PRELIMINARIES ............................... 1 L Introduction ................ ............... 1 2 Topology . . 3 3 C(X), The Ring of Realvalued Continuous Functions on X .... 8 4 Cardinal Functions; Terminology and Notation .............. 12 5 StoneDuality .................... ............ 15 6 Rings Of Quotients .................... ......... 18 2 COUNTABLE SETS IN TOPOLOGICAL SPACES ............. 24 1. Generalized Metric Spaces ................... ..... 24 2 wDensity ..................... ............ 28 a Weak PaSpaces .................... ........... 31 3 COMPLETENESS .................... ........... 36 1. NonDiscrete Linear PSpaces ................. ..... 36 2 Completions and the Chinese Remainder Theorem ............ 42 3 Completeness of C(X) and C'(X) ................. ... 45 4 The wChinese Remainder Theorem .................... 53 5 j and its Relations to Other Topologies ................. 56 4 CARDINAL FUNCTIONS .................... ....... 60 1. Cardinal Functions and C,(X) ................... 60 2 Special Topological Properties of Cy(X) .............. ... 64 5 FILTERS OF IDEALS ................ ............ 71 L Gabriel Filters of Ideals and Rings of Continuous Functions ........ 71 2 5, The Fixed Filter .................... ......... 78 3 0Ring of Quotients .................... ........ 82 4 C*spaces .................... .............. 90 5 Almost C*covers .................... .......... 98 6 The Filter .................... ............ 101 REFERENCES ........ ............ .... .......... 104 BIOGRAPHICAL SKETCH .................. .. ....... .. 107 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ALGEBRAIC AND TOPOLOGICAL PROPERTIES OF C(X) AND THE 3 TOPOLOGY By Warren William McGovern May 1998 Chairman: Dr. Jorge Martinez Major Department: Mathematics In this dissertation we define a to be the filter of ideals of C(X), the ring of realvalued continuous functions on a topological space X, generated by the fixed maximal ideals. as denotes the filter of ideals generated by countable intersections of fixed maximal ideals. We investigate the topological and algebraic properties arising from 5 and 5s. Regarding topological properties, we let Cy,(X) denote C(X) under the istopology. It is shown that C, (X) is complete if and only if the strict functional tightness of X is countable. We define the class of vanishing spaces and show Ca(X) is of countable tightness if and only if X is a vanishing space. In particular, if X is compact, then X is a vanishing space if and only if X is scattered. Regarding algebraic properties of 5, it is shown that 5 is a Gabriel filter. We prove that for a space with no isolated points, C(X) is sclosed if and only if X is a C*space and no point of X is a Gs. We investigate the class of C*spaces. CHAPTER 1 PRELIMINARIES 1. Introduction Over the last sixty years there has been an enormous amount of research on C(X), the ring of continuous functions on an arbitrary topological space X. This research lies in many different branches of mathematics, including algebra, functional analysis, topology, settheory, latticeordered groups, and function rings. For exam ple, in the mid1930s, Stone and (ech, independently, introduced a compactification of the space X by topologizing the set of ultrafilters of zerosets of continuous functions. In the early 1940s, Stone and Nakano, independently, investigated latticetheoretic properties of C(X) and described these in terms of the topology. Also, in 1956, Gill man and Henriksen [GH] compared and characterized idealtheoretic properties of the ring C(X) in terms of topological properties of X. Then in 1960, Gillman and Jerrison published the book Rings of Continuous Functions [GJ], a nowstandard ref erence for any person planning to do research on the subject. Since this publication, research into the theory of C(X) has blossomed, and with this have come some re markable ideas. Fine, Gillman, and Lambek introduced the idea of studying filters of dense open sets of X and their relationship to rings of quotients of C(X). Later, in the early 1980s, researchers began looking at C(X) in a different light. As C(X) is a subset of the full product of copies of the reals, and the latter has a natural topological structure, topologists began to study C(X) as a topological space and aimed to relate topological properties of C(X) to topological properties of X. The versatility of C(X) is what makes studying the subject so beautiful and interesting. Therefore, the main purpose of this dissertation is to continue in the long line of investigations which relate some mathematical property of C(X) to the topological space X. Specifically, we consider 3, the filter of ideals of C(X) generated by the fixed maximal ideals, and discuss two main themes. First, we investigate C(X) as a topological space under the topology induced by 3. Second, we show that 3 is a Gabriel filter and then investigate the ring of quotients and the various concepts that arise from this ring of quotients. Chapters 1 and 2 are devoted to preliminary background. Chapter 1 dis cusses those topics from the different branches of mathematics, which will be needed throughout the dissertation. Chapter 2 addresses the classes of topological spaces which we will be investigating. In particular, we are interested in spaces whose topologies are in some way, described by countable sets, or those spaces which are linked to their Stonetech Compactification by countable sets. In Chapter 3 we begin a detailed study of specific topological ring structures on C(X), using particular filters of ideals. For example, we investigate the completeness of these topological structures. Chapter 4 investigates the relationship between the cardinal invariants, topological ring structure on C(X), and the topological space X. In Chapter 5 we explore rings of quotients which arise from our filter of ideals, 5'. In this chapter, we investigate some classes of topological spaces, which come from our rings of quotients. Finally, we assume the reader is familiar with basic notation from the different branches of mathematics. For example, from settheory, we use [X]<" to denote the collection of all finite subsets of X. (The reader should be aware that we carelessly use the ordinal w to denote the least infinite cardinal Ro.) From algebra, we use v'7 to denote the radical of an ideal; i.e., the set of elements for which some power lies in I. The following convention will be used. If we refer to a statement within the chapter we shall refer to it as to a statement in a different chapter then we shall write number>. 2. Topology We use this section to describe the classes of spaces which we shall consider throughout the dissertation. Unless otherwise noted we assume that all spaces considered are Hausdorff. The following material is taken from Gillman and Jerrison [GJ]. Definition 2.1 Let X be a topological space. C(X) denotes the ring of realvalued continuous functions on X under pointwise addition and multiplication. A zeroset of X is a set of the form f'(0) for some f E C(X). The complement of a zeroset is called a cozeroset. Observe that by continuity zerosets are closed and cozerosets are open. Definition 2.2 We call a topological space X Tychonoff if it is Hausdorff and it satisfies the following property: given an arbitrary point p and a disjoint closed set K there is an f E C(X) for which f(p) = 0 and f(x) = 1 for all x E K. Observe that this is equivalent to saying given a point and a disjoint closed set there are disjoint zerosets containing them. (In the literature, this last condition is usually termed completely regular but we shall solely use the term Tychonof.) We remark that every Tychonoff space is regular, and that every normal space, is Tychonoff. These inclusions are strict. The following proposition may be found in Gillman and Jerrison [GJ], Theorem 3.2. Proposition 2.3 Let X be a Hausdorff space. Then X is Tychonoff if and only if the collection of zerosets of X forms a base for the closed sets of X. Tychonoff spaces have the property that they have a "largest" compactifica tion. (Recall that a compactification of X is compact space in which X is embedded densely.) We discuss this presently. First, we need a definition. Definition 2.4 Let S C X. We say S is C*embedded in X if every bounded contin uous realvalued function on S may be extended to a bounded continuous realvalued function on X. We define S to be Cembedded in X if the same can be done as above except for arbitrary continuous realvalued functions on S and X. The following theorem is one of the most important achievements in the theory of rings of continuous functions. The proofs of the equivalences of items (i), (ii), and (iv) date back to Stone and Cech in the year 1937. The theorem, as stated, may be found in Gillman and Jerrison [GJ] Theorem 6.5. Theorem 2.5 Let X be a Tychonoff space. Then there exists a compactification OX of X satisfying the following equivalent conditions. (i) X is a dense C*embedded subspace of 3X; (ii) disjoint zerosets of X have disjoint closures in iX; (iii) for any two zerosets Z1, Z2 of X , clpx(Zi n Z2) = clpxZi n cldxZ2; (iv) every continuous function T : X + K from X into a compact space K has a continuous extension 7r : fX + K. Furthermore, /X is unique in the sense that if K is a compactification of X satisfying the above then there is a homeomorphism from K to fX acting as the identity on X. 3X is called the StoneCech compactification of X and 7T is called the Stone extension of r. We now make our comment prior to Definition 1.4 complete. Definition 2.6 Let X be a Tychonoff space and Ki, K2 be compactifications of X. We call Ki (projectively) larger than K2 if there is a continuous map 0 : KI * K2 which is the identity on X. We sometimes write K1i > K2, or K2 < K1. Observe that as these spaces are compact 0 is necessarily surjective. Also, the above defined relation is in fact a partial order modulo equivalence via a homeomorphism, which restricts to the identity on X. The existence of the StoneCech compactification shown in Theorem 2.5 is one of the reasons we shall restrict our attention to Tychonoff space; as they are precisely the class of spaces which may be embedded densely into a compact Hausdorff space. Another reason is tied to the relationship between a space X and its ring of real valued continuous functions C(X). The following theorem, which is Theorem 3.9 in Gillman and Jerrison [GJ], explains why. Theorem 2.7 Let Y be a topological space. Then there exists a Tychonoff space X and a continuous surjection 4 : Y + X, such that the mapping f 4 f o 4 is a ring isomorphism of C(X) onto C(Y). Thus, from now on we shall assume, unless otherwise noted, that all spaces are Tychonoff. We continue our discussion about those classes of spaces which shall play an important role in our development. Again our primary reference is Gillman and Jer rison [GJ]. The first few definitions deal with what the AMS has classified "peculiar spaces." If at all possible we localize our definitions to points. Most of these local ized definitions were invented, historically, after the more global concepts. We give references when possible and examples in 2.15. Observe that we do not describe any ringtheoretic characterizations of these spaces. The next section is reserved for these connections. Definition 2.8 (i) A point p E X is said to be a Ppoint if every zeroset containing p is a neighbourhood of p. (ii) We say X is a Pspace if every point is a Ppoint; equivalently, if the intersection of any denumerable sequence of open sets is again open, i.e., the topology on X is closed under countable intersections. Another characterization is that every zeroset of X is open, whence clopen. An explicit example of a nondiscrete Pspace is given in Example 2.15(iii). In general, if X is a space, then one obtains a Pspace from X by taking as base for a new topology on X the collection of all countable intersections of open sets from X. This process is called Pifying X and is denoted by Xs. We will also call Xs the Pification of X. Definition 2.9 (i) A point p E X is said to be an almost Ppoint if every zeroset containing p has nonempty interior. (ii) We say X is an almost Pspace if every point is an almost Ppoint; equivalently, every nonempty GS has nonempty interior. Note that X is an almost Pspace if and only if there are no proper dense cozerosets. This last equivalence can be found in Levy [Le]. An example of a nondiscrete almost Pspace is given in Example 2.15(i). Definition 2.10 (i) Let p E X. We say p is an Fpoint if p is not in the closure of two disjoint cozerosets. (ii) We say X is an Fspace if every point is an Fpoint; equivalently, every cozeroset is C*embedded (see [GJ] 14.25.) (iii) We say X is a quasi Fspace if every dense cozeroset is C*embedded. Definition 2.11 (i) A point p E X is said to be a BDpoint if p is not in the closure of two disjoint open sets, one of which is a cozeroset. (ii) We call X a basically disconnected space if every point is a BDpoint; equivalently, if every cozeroset has clopen closure. Definition 2.12 (i) A point p E X is said to be an EDpoint ifp is not in the closure of two disjoint open sets. This particular definition is due to van Douwen (see [vD].) (ii) We say X is extremally disconnected if every point is an EDpoint. This is equivalent to saying that the closure of an arbitrary open set is again open. Observe that all Pspaces are almost Pspaces as well as basically discon nected. Extremally disconnected implies basically disconnected implies Fspace. (In general, basically disconnected spaces are zerodimensional; that is, they have a base of clopen sets.) Before we give examples of the above classes of spaces we recall two classes which generalize the compact ones. Definition 2.13 We call X locally compact if every point has a compact neigbour hood. It is known that X is locally compact precisely when X is an open subspace of 3X (see 6.9 (d) [GJ].) Definition 2.14 We say X is a Lindel5f space if every open cover has a countable subcover. Example 2.15 (i) Let X = ON N. By exercises 6S.8 and 6W.4 in Gillman and Jerrison [GJ], X is a compact almost Pspace which is not basically disconnected. By Theorem 14.27 in Gillman and Jerrison [GJ], X is also an Fspace. Hence the space OIN N realizes that the class of Pspaces is strictly contained in the class of almost Pspaces, and that the class of basically disconnected spaces is strictly contained in the class of Fspaces. (ii) Consider the space E = NU{p} where p E fNN. E is an extremally disconnected space, and hence basically disconnected, which is not a Pspace. Hence the class of P spaces is strictly contained in the class of basically disconnected spaces. (A description of E may be found in Gillman and Jerrison [GJ], 4M.) (iii) Let D be an uncountable discrete space and A a point not in D. AD is the space obtained by adjoining A to D, where all the points of D are isolated and a set containing A is open if and only if its complement is countable. Then AD is a Lindel6f, Pspace, hence basically disconnected, which is not extremally disconnected, whereby we see that the class of basically disconnected spaces strictly contains the class of extremally disconnected spaces. (A description of AD may be found in Gillman and Jerrison [GJ], 4N, though they call the space S. We have opted to employ the notation AD as it has become more or less standard in the literature.) 3. C(X). The Ring of Realvalued Continuous Functions on X The objective of this section is to give pertinent background information concerning C(X), the ring of realvalued continuous functions on the topological space X. Recall that all spaces are Tychonoff. One of the main theorems concerning C(X) is the theorem of Gelfand Kolmogorof. To appropriately state the theorem we need a few definitions, which will also be used throughout the dissertation. Again, our main reference for this section is Gillman and Jerrison [GJ]. Definition 3.1 Let A be a collection of subsets of a set X. We say A is a filter if (1) for each A1,A2 E A then A1 n A2 E A, and (2) if A, C A2 and Ai E A, then A2 A. A is an ultrafilter if it is maximal. Zorn's Lemma implies that every filter is contained in an ultrafilter. An ultrafilter is called fixed if its intersection is nonempty, otherwise we call it free. iX is the collection of all zeroset ultrafilters. /X is made into a compact Hausdorff space by taking as a base for the closed sets, the collection of sets of the form Z = {p E pX : Z E p}, where Z is a zeroset of X. To each x E X corresponds a unique point in OX, namely the zeroset ultrafilter consisting of those zerosets of X containing x. (In the previous section we saw what properties PX has.) As for notation we shall use the following convention. If p E OX, then the corresponding ultrafilter is denoted AP. The difference between AP and p is simply notational. Definition 3.2 Let Max(C(X)) denote the collection of maximal ideals of C(X). If M E Max(C(X)) then let Z[M] = {Z(f) : f E M}. Z[M] is an ultrafilter on the collection of zerosets of X. A maximal ideal M is called fized if nZ[M] is nonempty, otherwise we call M free. We view Max(C(X)) as a topological space under the Zariski (hullkernel) topology. Max(C(X)) is a compact Hausdorff space. Theorem 3.3 [GelfandKolmogoroff] Let M E Max(C(X)). Then there is some p E pX such that M = MP = {f E C(X) : p clpxZ(f)}. For a point p E X let Mp = {f E C(X) : f(p) = 0}. Then Mp is a fixed maximal ideal and Mp = MP. The map MP 4 p is a homeomorphism between Max(C(X)) and fX. The restriction of this map to the fixed maximal ideals is a homeomorphism onto X. Definition 3.4 Let p E fX. Define Op = {f E C(X) :cloxZ(f) is a neighbourhood of p}. If p E X, then OP = Op = {f E C(X) : f vanishes on a neighbourhood of p}. Observe that OP < MP. Proposition 3.5 [Theorem 7.15 [GJ]] Let P be a prime ideal of C(X). Then there exists a unique maximal ideal say MP of C(X) containing P. Moreover, OP < P < MP. Proposition 3.6 [Theorem 14.3(c) [GJ]] The collection of prime ideals of C(X) forms a root system, i.e., the collection of primes containing a given one forms a chain. We will soon give a list of examples of spaces whose properties are intrinsically tied in with the ideal structure of the ring. But for now we find it useful to describe the maximal ideals of C*(X), the ring of bounded continuous real valued functions on X. First, observe that as a result of Theorem 2.5, C*(X) is ringisomorphic to C(P3X). It is natural to wonder if there is a correspondence between the maximal ideals of C*(X) and the maximal ideals of C(X). For example, is it true that M n C'(X) is maximal for M E Max(C(X)) and does this constitute all of the maximal ideals of C*(X)? Alas, this is not the case. It is true that the contraction of a maximal ideal is prime but it is not necessarily maximal. To fully do the job we need a different distinction between the maximal ideals of C(X). Definition 3.7 Let M E Max(C(X)). Observe that C(X)/M is a field containing a copy of R. We call M a real maximal ideal if C(X)/M is precisely R. Otherwise, we say M is a hyperreal maximal ideal. Observe that every fixed maximal ideal is a real ideal, but reverse is not true in general. For example, consider W the set of countable ordinals under the interval topology; i.e., W= {o: < } It is known (see 5.12, [GJ]) that /W is precisely the set of ordinals less than or equal to the first uncountable ordinal, denoted W*; i.e., W* = { : a < i)}. A quick check produces that in C(W) the free maximal ideal M1' is a real maximal ideal. Definition 3.8 We say the space X is realcompact if the fixed maximal ideals are precisely the real ideals. We let vX denote the subspace of PX consisting of all real maximal ideals. vX is called the (Hewitt) realcompactification of X. The following theorem is a useful characterization of realcompact spaces. It may be found in Walker [W] Theorem 1.53, where it is attributed to Hewitt. Theorem 3.9 The following are equivalent for a Tychonoff space: (i) X is realcompact; (ii) Every free maximal ideal of C(X) is hyperreal; (iii) if p e fX X, then there is a zeroset of3 X disjoint from X containing p. Returning to describe the maximal ideals of C*(X) we have the following theorem. Theorem 3.10 [Theorem 7.2 [GJ]] The maximal ideals of C*(X) are precisely the ideals of the form M*P = {f E C*(X) : fo(p) = 0}. The question then becomes when is MnOC*(X) = M*'? We have the following results. Proposition 3.11 [Theorem 7.9(c) [GJ]] For p E fX we have MP n C*(X) = M*P if and only if MP is a real maximal ideal. Proposition 3.12 [Corollary 1.58 [W]] For a space X, vX = PX ifand only C*(X) = C(X). In this case we call X pseudocompact. We now finish this section by relating certain topological properties of points of X, or even X itself, to ringtheoretic properties of C(X). Some of these ring theoretic conditions will not be discussed in the dissertation, but we include them in this section because they are useful in illustrating the type of theorems found in the literature. These connections are the main reason for the author's interest in this area. Proposition 3.13 The following statements are true for the space X and p E X. (i) p is a Ppoint of X if and only if O, = M,. (ii) X is a Pspace if and only if Op = Mp for all p E X (see 7L.1 [GJ]). (iii) X is a Pspace if and only C(X) is a von Neumann regular ring, i.e., every prime ideal is maximal, or, equivalently, every ideal is semiprime. (iv) p is an Fpoint if and only if Op is a prime ideal. (v) X is an Fspace if and only if C(X) is a Bdzout ring; i.e., every finitely generated ideal in C(X) is principal (see 14.25 [GJ].) (vi) X is basically disconnected if and only if C(X) is a coherent ring (see [Gz].) (vii) X is an almost Pspace if and only if C(X) coincides with its classical ring of quotients (see [FGL].) (viii) X is extremally disconnected if and only if every family of continuous functions which has an upper bound in C(X) has a supremum in C(X). 4. Cardinal Functions: Terminology and Notation The purpose of this section is to define those cardinal functions which we shall use in this dissertation. We also include useful results, usually comparing these different invariants. Our approach is to define the cardinal invariants "locally"; i.e., via points of the space, and then obtain a "global" invariant for the whole space. A nice reference for this section is Engleking [Eng]. Definition 4.1 Let x E X. (i) The weight of X is defined to be the infimum of the cardinalities of bases for X and is denoted by w(X). A space is called second countable if its weight is countable. (ii) The character of x, denoted X(x, X) is the smallest cardinal for which there is a base of neighborhoods at x of this size. The character of X is defined as X(X) = sup X(x, X) xEX If x(X) = w then we call X firstcountable. (iii) The pseudocharacter of x, denoted O(x, X) is the smallest cardinal of the form U[ where U is a collection of open sets whose intersection is the singleton x. The pseudocharacter of X is defined as 4(X) = sup )(X, X). xEX If the pseudocharacter of x is countable, then we say x is a Gspoint. Observe that as a compact G6 in a Tychonoff space is a zeroset it follows that the pseudocharacter of x is Ro if and only if the singleton x is zeroset. Finally, if O(X) = w, then we say X has nice pseudocharacter. (iv) The cellularity of X, c(X), is the supremum of cardinalities of the form JUl where U is a collection of pairwise disjoint open sets of X. (v) The Lindelof degree of X is defined to be the smallest cardinal n such that every open cover contains a subcover of size < K. (v) The rcharacter of x, denoted 7r(x,X) is the smallest cardinal of the form jUl where U is 7rbase for X at x. (A rbase for X at x is a collection, U, of open sets such that if O is an open neighbourhood of x then there is some U E U for which U C O.) The rcharacter of X is defined as r(X) = sup r(x, X). xEX (vi) The density of X, d(X), is the minimal cardinality of a dense subset of X. If d(X) = w, then we call X separable. The character, pseudocharacter, and ircharacter of a space X were all defined locally at a point and then the cardinal function was taken to be the supremum over all points of X. We may do this (as well as for the infimum) for an arbitrary cardinal function. Since we shall have cause to use these other cardinal functions we define them now. Definition 4.2 Let 4 be a locally defined cardinal function; i.e., for every space X and x E X, r(x,X) exists. Then let *V(X) = sup $(z, X) xEX and Q.(X) = inf Q(X, X). xEX Then Q* and 1. are welldefined cardinal functions. Finally, we would like to conclude this section by comparing some of the cardi nal invariants defined in Definition 4.1 as well as giving some fundamental examples. We begin with some examples. Example 4.3 (1) Let X be a metric space. Then x(X) = I(X) = Ro. Also, c(X) = d(X) = w(X). (2) Let x E X be a nonisolated almost Ppoint. Then O(x, X) > Ro. Hence, if X is a nondiscrete almost Pspace, then i(X) > R0. (3) Let p E ,X X. Then O(X) > No (see 9.6, [GJ].) (4) Suppose X is a space with precisely one nonisolated point. Then c(X) = d(X) = w(X) = IX. Lemma 4.4 For any Tychonoff space c(X) < d(X) < w(X) and I(X) < x(X) (see 1.7.12 [Eng].) Proposition 4.5 Let X be a compact space. Then O(X) = X(X) < w(X) (see 3.1.F [Eng]. In fact, ?k(x,X) = X(x,X) for all x E X.) Definition 4.6 Let X be a Tychonoff space. Suppose there exists a partial order < on X such that for every x, y E X either x < y or y < x. Then we call this order a total order. If the collection of all sets of the form (x,y)= {z E X: < z < y} or (x,oo)= {z E X: < z} or (00,) )= {z X:z < x} together with X form a base the topology on X, then we call X a linearly ordered space. Proposition 4.7 Suppose X is a linearly ordered space. Then x(X) < c(X), X(X) = I(X) < c(X), 1(X) < c(X), and either d(X) = c(X) or d(X) = c(X)+ (see 3.12.4 [Eng].) Proposition 4.8 [50 [GJ]] Suppose X is a linearly ordered space. Then X is a P space if and only if no point of X is the limit of either a decreasing or increasing sequence of points of X. 5. StoneDuality Extremally disconnected spaces are a wellstudied class of spaces. The class has ties to rings of continuous functions (see Proposition 3.13(viii)) as well as to the theory of Boolean algebras. We use this section as an introduction to the latter. Our main goal in this section is to recite Stone's Representation Theorem (or also, called Stone duality.) Our main references for this section are Porter and Woods [PW] and Walker [W]. Definition 5.1 A Boolean algebra is a complemented distributive lattice. We denote the greatest and least element by 1 and 0, respectively. Note: to rule out the trivial case, we shall assume that 0 5 1. If every subset of B has a supremum and infimum, then we call B a complete Boolean algebra. Similarly, if every countable subset has a supremum and infimum, then we call B a acomplete Boolean algebra. Example 5.2 (i) Let I be any set. We denote the power set by P(I). P(I) is partially ordered under inclusion. Taking intersection, union, and settheoretic complement as our operations, we obtain that P(I) is a complete Boolean algebra. (ii) Let X be a topological space and A C X. We call A regular closed if A = clxintxA. We denote the family of all regular closed sets of X by R(X). 7R(X) is partially ordered under inclusion. Consider the following operations: (i) Vi,l Ai = clx(U.ejintxAi); (ii) AiE Ai = clx(intx(ni,, Ai)); (iii) A' = clx(X A). These operations make R(X) into a complete Boolean algebra (see Proposition 2.3 [W]). (iii) Let X be a topological space. Denote the collection of all clopen subsets of X by B(X). B(X) is a subalgebra of 1R(X). (Observe, though, that B(X) is not, in general, a complete subalgebra.) Definition 5.3 Let B be a Boolean algebra. A filter on B is a set a such that if a,b E a, then a A b E a and if a < b with a E a, then it follows that b e a. An ultrafilter on B is a maximal filter on B. Set S(B) = {a : a is a an ultrafilter on B}. For a E B, let U(a) = {a E S(B) : a ~ a}. Proposition 5.4 [Proposition 3.2(b) [PW]] Let B be a Boolean algebra and a, b E B. Then : (i) U(O) = S(B) and U(1) = 0; (ii) U(a A b) = U(a) U U(b); (iii) U(a V b) = U(a) n U(b); (iv) U(a') = S(B) U(a). As a result of the previous proposition, we may equip S(B) with a topology; taking the set of all U(a) as a base for the open sets. S(B) is called the Stone space of B. We now recite Stone's Representation Theorems. Theorem 5.5 [Theorem 3.2(d) [PW]] Let B be a Boolean algebra. Then (i) S(B) is a compact zerodimensional space; (ii) {U(a) :ae}= B il ;f iL  (iii) the map U : B > B(S(B)) is a Boolean algebra isomorphism. Theorem 5.6 Let X be a compact zerodimensional space. As we noted before B(X) is a Boolean algebra. Then S(B(X)) is homeomorphic to X under the map a + na, where a e S(B(X)). The next propositions show the usefulness of Stone duality: characterizing a compact zerodimensional space via its Boolean algebra of clopen sets. Proposition 5.7 [Proposition 2.5 [W]] Let X be a Tychonoff space. Then B(X) is a complete Boolean algebra if and only if X is extremally disconnected. Proposition 5.8 Let X be a Tychonoff space. Then B(X) is a crcomplete Boolean algebra if and only if X is basically disconnected. 6. Rings Of Quotients As we will spend a considerable amount of time in this dissertation discussing com mutative rings we take time to supply the reader with the appropriate background information. Thus, we use this section to define certain elementary ringtheoretic notions as well as rings of quotients. Unless otherwise noted all rings are commu tative with identity different than 0. Given our description of maximal ideals of C(X) it is easy to see that the intersection of Max(C(X)) equals 0. This is not always the case. Our first two definitions of this section are concerned when things of this nature occur. Definition 6.1 For a ring A, we denote the collection of all maximal ideals of A by Max(A). The intersection of all maximal ideals is called the Jacobson radical of A and we denote it by f(A). If f(A) = 0 we call A semiprimitive. Definition 6.2 For a ring A, we denote the collection of all prime ideals of A by Spec(A). The intersection of all prime ideals is called the radical of A and we denote it by n(A). Observe that n(A) consists of all nilpotent elements of A. We call A semiprime if its radical is 0; equivalently, if A has no nonzero nilpotent elements. Definition 6.3 For a ring A we let Min(A) denote the collection of all minimal prime ideals of A. Observe that if A is semiprime, then a prime ideal P is minimal if for every a E P there exists a b P for which ab = 0. The following is an easy application of the definitions of prime ideals. Its proof may be found in Chapter 2 of Lambek [L]. Proposition 6.4 The ring A is semiprime if and only if it is isomorphic to a subring of a direct product of fields. Since we are discussing prime ideals it seems fitting to recall the Chinese Remainder Theorem, especially as it will rear its head later on in the dissertation. Its proof may be found in practically any undergraduate/graduate textbook on modern algebra. Theorem 6.5 [Chinese Remainder Theorem] Let P, P2,* ,Pn be a collection of pairwise comaximal primes of the ring A; this means that P. + Pj = A for all i 9 j. Then the natural map 4: A + A/Pi x x A/P, is surjective and ker(O) = Pi n n P,. Next, we discuss the hullkernel topology. This is also often termed the Zariski topology. The following information may be found in Lambek [L]. Let A be a ring and suppose II is a collection of prime ideals of A. We make II into a topological space by taking as open sets all sets of the form U(S) = {P E H: S P} where S is any subset of A. U(S) is called the hull of S. In particular, if S = {a}, then we write U(a) and the collection of these as a runs over all elements of A forms a basis for the open sets of this topology on II. The collection of sets of the form V(S) = {PE I : S C_ P} is precisely the collection of closed sets in II. V(S) is called the kernel of S. Example 6.6 (i) If II = Spec(A), then we call it the prime spectrum of A. The Zariski topology on Spec(A) is compact but rarely Hausdorff. (ii) If II = Max(A) then we call it the maximal ideal space of A. As in (i), Max(A) is also compact. (iii) Let II = Min(A). Then under the Zariski topology Min(A) is a zerodimensional Hausdorff space (see [HJ].) We now turn our attention to rings of quotients. Rings of quotients were originally defined by considering equivalence classes of fractions whose denominators came from a fixed multiplicative set. For an integral domain A this method using the multiplicative set of nonzero elements produces Q, the field of quotients of A. In general, the classical ring of quotients, denoted by q(A), is formed by this method using the multiplicative set of nonzero divisors. Then in the 1950's, Utumi and John son independently observed that these fractions could be viewed as homomorphisms from specific ideals of A into A; i.e., as partial endomorphisms on A. Utumi, in 1956, was able to generalize this method and defined a ring of quotients of A. We now give that definition. Definition 6.7 Suppose A is a subring of the ring R. We call R a ring of quotients of A if and only if for all s, t E R, t f 0 implies there exists an a E A such that sa E A and ta o 0. As this definition is very abstract our intention is to give an equivalent char acterization of rings of quotients. In fact, we will construct the "maximal ring of quotients," Q(A). It will then follow that every ring of quotients of A lies between A and Q(A). To this end we need the formal definition of a dense ideal, domain, and fraction. Definition 6.8 Let I < A be an ideal of A. We call I dense if whenever rl = 0 it follows that r = 0. Let us denote the collection of all dense ideals of A by 0. Observe 0 is not dense. Also, 3 is a filter of ideals which happens to be closed under products (see 2.3 [L];) i.e., if I, J E 0, then I n J, IJ E 2, and if I < I', then I' E Z. For a dense ideal I E 3, a a fraction with domain I is any Ahomomorphism f : I + A. Hom(lI, A) is the collection of all fractions with domain I. We call f a fraction if there exists some dense ideal I for which f is a fraction with domain I. Definition 6.9 The maximal ring of quotients of A, denoted Q(A), is defined to be the collection of all fractions modulo the following equivalence relation: suppose fi, f2 are fractions with domains 11, 12, respectively. We say fi = f2 if they agree on the intersection of their domains. As such Q(A) is a commutative ring with identity. Observe that the above equivalence relation is precisely the definition of direct limit. In other words, we have Q(A) = limHomA(I,D) + where the limit runs over all dense ideals. The following theorem demonstrates why we call Q(A) the maximal ring of quotients of A. For its proof, we suggest Proposition 6 of Chapter 2.3 in Lambek [L]. Theorem 6.10 Let A be a subring of R. Then R is a ring of quotients of A if and only if there is an embedding of R into Q(A) which is the identity on A. The following corollary is wellknown, but as its proof is a simple application of our definitions we include it for completeness sake. Corollary 6.11 The classical ring of quotients q(A) satisfies Definition 6.7. More over, we may view A < q(A) Q(A). Proof: The second statement follows from the first and the preceding theorem. To see that q(A) satisfies Definition 6.7 observe that any classical fraction ', where b is a nonzero divisor, may be viewed as a fraction. To see this, first, note that as b is a nonzero divisor the principal ideal (b) is dense. Next, multiplication on the left by a may be viewed as a fraction with domain (b); i.e., I(rb) = ar. This correspondence in fact embeds q(A) < Q(A) while fixing elements of A.N We would like to finish this section with some results about C(X) and its classical and maximal ring of quotients. We denote the classical and maximal rings of quotients of C(X) by q(X) and Q(X), respectively. For proofs the reader may consult Fine, Gillman, and Lambek [FGL]. Theorem 6.12 [Representation Theorems, 2.6 [FGL]] For any Tychonoff space X we have q(X) = limC(O); 0 E ((X) and Q(X) = limC(); OE O(X) where (E(X), (resp., S(X)) is the collection of dense cozero sets (resp., dense open sets) of X We now consider certain qualities among the rings C(X), q(X),and Q(X). Proposition 6.13 [3.3 [F(CI For a Tychonoff space X, q(X) = Q(X) if and only if every realvalued continuous function defned on a dense open set can be continuously defined on a dense cozero set. Corollary 6.14 Let X be a perfectly normal space; i.e., every open set is a cozeroset. Then q(X) = Q(X). In particular, if X is a metric space, then q(X) = Q(X). Proposition 6.15 [3.4 [FGL]] For a Tychonoff space X, C(X) = q(X) if and only if X has no proper dense cozerosets; i.e., X is an almost Pspace. Proposition 6.16 [3.5 [FGL]] For a Tychonoff space X, C(X) = Q(X) if and only if X is an extremally disconnected Pspace. 23 Remark 6.17 To see that there are nondiscrete extremally disconnected Pspaces, the reader should refer to 12H in Gillman and Jerrison [GJ] and the discussion on measurable cardinals. CHAPTER 2 COUNTABLE SETS IN TOPOLOGICAL SPACES 1. Generalized Metric Spaces At this point in our development we wish to consider certain classes of Tychonoff spaces. The importance of these classes will become evident in the next chapter, when we consider the completeness of the uniform space C(X) under the topological ring uniformity given by certain ideal topologies, (yet to be defined.) We ask the readers to please bear with us, as our intention is to provide background which will help the discussion to follow. We recommend Engleking [Eng] as an appropriate source for this section. Definition 1.1 If {u;} is a sequence of a space X we say that the sequence converges to a point x E X and write x = limui, if for every neighbourhood O of x, there is some natural number No for which u, E 0 for all n > No. Definition 1.2 For a subset U C X, we let s(U) = {zx X1 there exists a sequence {ui} C U such that u,  x}; s(U) is the collection of all sequential limits of U. Notice that in general, s(U) C s(s(U)). We Ir,......I .. 1. define s"(U) = s(s"l(U)) for each natural number n. A set U is called sequentially closed if s(U) = U. Definition 1.3 For a subset U C X, we let [U]f = {x E X1 there exists a sequence {u,} C U such that x E clx(u,)}. If X is understood then we shall use ]. Observe that .l I'  = [U]X; henceforth, we shall call the set [U],, the wclosure of U. We say U is wclosed if [U], = U. In general, for a cardinal r, we let [U]X = {x E XI there exists a subset V of U such that IVI < r and x E clxV}. [U]x is called the rclosure of U and say U is Tclosed if U = [U]X. A quick check gives us the following relations which we state as our first lemma. As the proof follows straight from the definitions we leave it out. Lemma 1.4 For subsets U, V C X with U C V and an arbitrary infinite cardinal r the following hold. (a) s(U) C s(V); (b) [U], C [V],; (c) U C s(U) C s.(U) C ... s"(U) C... C [U], C clx(U). Definition 1.5 A space X is called sequentialif every sequentially closed set is closed. This is equivalent to saying that a subset U is closed if and only if the closure of the intersection of U with any convergent sequence is a subset of U. A more conventional way of defining a sequential space is for each U, which is not closed in X, there is a sequence in U which converges to a point outside of U. The definition of wclosure was introduced by Moore and Mr6wka in 1964 in connection with sequential spaces. For a good source on sequential spaces refer to Franklin [Frl] and Franklin [Fr2]. A finer class of spaces is exhibited by the following definition. Definition 1.6 A space X is said to be a FrechetUrysohn space if for every subset U C X, we have that s(U) = clxU. Example 1.7 (i) Every first countable space is Frech6tUrysohn and every Frechet Urysohn space is sequential. However, there are sequential spaces which are not FrechetUrysohn, and similarly, FrechttUrysohn spaces which are not first countable (see [Eng], Example 1.6.19.) (ii) Recall from Gillman and Jerrison [GJ], 14N.2, that an Fspace has the property that no point is a limit of a sequence of distinct points. It follows that every set is sequentially closed. Thus, an Fspace is sequential if and only if it is discrete. A more general class of spaces is given by the following definition, which was introduced by Arkhangel'skii. Definition 1.8 A space X is said to have countable tightness if every wclosed set is closed. Notice that this is equivalent to I 1. = clx(U) for all subsets U C X. As in Definition 1.5 we also have an equivalent condition for a space to have countable tightness: a subset U is closed precisely when the closure of any countable subset of U is again a subset of U. In general, we say a space has tightness no greater then n if every Kclosed set is closed. There is a cardinal function which represents this definition, namely t(X) = min{K : every Kclosed set is closed}. Example 1.9 The existence of a countably tight space which is not sequential is known. We wish to give an example. Consider E from 1.2.15(ii). We recall some facts about E which may be found in Gillman and Jerrison [GJ], 4M. First, a subset of N is clopen in E if and only if it is not an element of the ultrafilter p. Second, if U C E and contains p then U is closed. Finally, if U E p then clzU = U U {p}. It follows that given any nonclosed subset U of E, [U], = U U {p} = cdU, and so E has countable tightness. We now suppose by way of contradiction that E is sequential. Then choose U C N with U E p. Then there exists a sequence ui E U such that p = limui. Then it follows that {ui,}N E p. Now, let Ui = {u}id=2n and U2 = {ui}i=2n+i. As p is an ultrafilter it follows that one and only one of the U, is in p, say U1. Then p $ limi=2n+l ui but this contradicts the simple fact that every subsequence of a convergent sequence is convergent and converges to the same point. Thus, E is a space of countable tightness which is not sequential. In fact, any countable space which is not sequential will do. In contrast to Example 2.7(ii), E is also an example of a nondiscrete basically disconnected space of countable tightness. To find a compact countably tight space which is not sequential is the heart of the MooreMr6wka problem. The answer turns out to be independent of ZFC. We suggest HuSek and van Mill [HvM], Chapter 4. Remark 1.10 Observe that given an arbitrary space X, it is easy to construct se quential spaces and spaces of countable tightness. This is achieved by enlarging the topology (of closed sets) on the space to include all sequentially closed, or respectively, all wclosed sets. In Engleking [Eng], Proposition 1.6.15, it is proved that if X is sequential and we have a function f : X + Y then f is continuous if and only if whenever we have x = limx; then f(x) = limf(xi). We now show the analogous condition for spaces of countable tightness. Proposition 1.11 Let t(X) = e and f : X + Y. Then f is continuous if and only if x E clx{x}),E implies that f(x) E cly{f(Ma))}ae Proof: The necessity is obvious. As for the sufficiency, let B be closed in Y. We shall show that f'(B) is closed in X. Let x E clxf (B). Since t(X) = K we may find a collection of points from f'(B), say {z(,ae such that x E clx{xo},. Then f(x) E cly{f(x,)}oe,. Since B is assumed to be closed we conclude that f(x) E B and so x E f'(B).M In the next section we will turn our attention to those spaces X having a "nice" wclosure inside an arbitrary compactification. 2. wDensity As the title of this chapter suggests, countable sets are of interest to us. We stated at the end of the last section, that we wish to devote some time to discuss those spaces whose wclosure inside a compactification is the whole of the compactification. This class of spaces will play a pivotal role in the next chapter. Definition 2.1 Let X be a space and K be a compactification of X. X is said to be wdense in K if [X]f = K. If K = OX, and X is wdense in K, then we call X wdense. We soon show why we reserve wdensity to fX. But first, a remark and then some examples concerning Pspaces. Remark 2.2 Our choice to call these spaces wdense should be obvious. The only alternative in the literature is found in the exercises of Chapter 5 of Porter and Woods [PW]. There the authors call this class of spaces the class of Ppseudocompact spaces where P is the property of being Robounded. As this does not pertain to the present discussion we disregard and employ the name wdense. Example 2.3 (i) Let D be a discrete space. Then D is wdense if and only if D is countable. This follows from the fact that if D has cardinality a and a > n > w then there exists an ultrafilter, say p E /D for which p is in the closure of some subset E < D of cardinality n and does not lie in the closure of any subset of D with cardinality less than n (see [CN].) (ii) Consider AD already defined in 1.2.15(iii) It is a nonseparable, noncompact w dense space. To see this, first recall that the zerosets, i.e., the clopen sets of AD are the countable subsets of D and the cocountable sets containing our distinguished point, say A. Then the only Zultrafilter on AD containing only cocountable subsets is the one corresponding to the point A. Thus, for each point p E 3AD AD there exists some countable clopen set say Z C AD for which p E clpDZ. Whence p E [AD], and AD is therefore wdense. Remark 2.4 To recap, we have that the class of wdense spaces contains the class of separable spaces, compact spaces, and spaces whose StoneCech compactification is of countable tightness. Also, AD, as well as T, the Tychonoff plank (see [GJ] 8.20), are wdense spaces. For the readers' information, we note that there are wdense spaces which have no compactifications of countable tightness. The example we have in mind is the FrechetUrysohn wfan V,. First, we note that aN is the onepoint compactification of N. We denote the point at infinity by oo. Next, V, is the resulting quotient space of N x aN obtained by identifying the points whose aN coordinate is oo. Clearly, V. is wdense as it is separable. In Arkhangel'skii [Arl], it is shown that V. has no compactification of countable tightness. A natural question at this time is whether wdensity is invariant under certain topological operations. First, following from Example 2.3, we see that there are sub spaces of wdense spaces which are not wdense. Next, we investigate the relationship between wdense spaces and their products. Proposition 2.5 Let X be wdense in K and suppose f : X + Y is a dense contin uous map; i.e., the image of X under f is a dense subspace of Y, then Y is wdense in any compactification which is a surjective image of K so that the diagram below commutes. Proof: Suppose the following is a commutative diagram. X K Y * Z If p E Z, then choose x E K so that Of(x) = p. By our assumptions, we may find a sequence of points in X, say {xi} such that x E cl{,i}. Let pi = f(z,). Now, bf (cl{pi}) is a closed set containing the zi and so it contains x, i.e., p E cl{pi}, whence Y is wdense in K.B Corollary 2.6 Suppose X is wdense and f : X 4 Y is a dense continuous map, then Y is wdense. Proof: This follows from the fact that the following is a commutative diagram. X + fiX Y 4+ lY Corollary 2.7 Suppose X is wdense in a compactification K. Then X is wdense in every compactification L with L < K. Corollary 2.8 Suppose X is wdense. Then X is wdense in K for every compacti fication, K, of X. Proposition 2.9 The spaces X,, i E N, are wdense precisely when HiNXi is w dense in flE3Xi. Proof: If IIiENX, is wdense in niN/fXi, then as a result of Proposition 2.6 we have each Xi is wdense, because each 3X, is a continuous image of TIINXi, as required. Conversely, if (pi) E IliENXi then for each natural j, we may choose sequences {zxi,} so that pj E cl{xi,,}=1. Then it is clear that (Pi) E cl{(xi),,i{i,j} C NxN}.E Proposition 2.10 The topological sum of a finite number of wdense spaces is again wdense. Proof: Recall that under a finite indexing set I, fl(( Xi) = fD X,. The conclusion then follows.E Note that since an uncountable discrete space is not wdense it follows that an arbitrary topological sum of wdense is not wdense. At this point we are not sure whether wdensity of two spaces implies wdensity of their product space. If the spaces are compact or separable, then it is clearly true. It is also true if the resulting product space is pseudocompact. This follows from Proposition 2.9 and Glicksburg's theorem on the StoneCech compactification of products (see [Gl] or [W].) 3. Weak P.Spaces In this section we consider the class of weak Paspaces. Our first priority is to give some basic facts concerning them. As we shall see this class of spaces contains (prop erly) the class of P.spaces. (Here a is fixed, and the class of Paspaces consists of those spaces whose topologies are closed under all intersections of size < a.) The simi larity between Paspaces and weak Paspaces will then be illustrated. The importance of weak Paspaces will become evident once we develop the theory of completions of C(X) under the .Fstopology (yet to be defined). With that said let us begin. Definition 3.1 Let X be a Tychonoff space and x be a nonisolated point of X. We define the norm of x in X as 11rx1 = min{SI : x S C X, and x E clxS}. Observe that [llxl < IXI. If x is isolated we denote ilx = IX\. We now define the norm of X as IIX I = :TIri lli[ : x e X}. Note that this is a welldefined cardinal function on the space X. The above norm was defined by Williams and Zhou in connection with mono tonically normal compact spaces (see [WZ] or [J].) Example 3.2 We point out via an example that it is not the case that if I \ = jXI then X is discrete. Consider the nondiscrete space E introduced in 1.2.15(ii). E has cardinality w. All but one point of E is isolated, namely p. Thus, IIS = p[I = w. Definition 3.3 Let X be a Tychonoff space and p E X. Let a be an infinite cardinal. We say x is a weak P,point if for every S C X with ISI < a and x 0 S then x clxS. Equivalently, we say x is a weak Papoint if x > a. If a = wi, then we call x a weak Ppoint. We let wPa(X) be the subcollection of X consisting of all weak Papoints. If wP,, (X) = X, then we say X is a weak Pspace; i.e., X is a weak Pspace precisely when no point is in the closure of a countable set disjoint from it. If wP,(X) = X, then we call X a weak Paspace. We now prove a useful theorem characterizing the class of weak P.spaces for an arbitrary cardinal a. First, we define the coa topology. Definition 3.4 Let X be a set. The coa topology on the set X is the topology (of open sets) consisting of X, 0, and those sets which are coa; i.e., whose complements have cardinality < a. Observe that this topology is not Hausdorff. Proposition 3.5 Let X be a Tychonoff space and a a cardinal. Then the following are equivalent. (a) X is a weak P,space; (b) 1IX > a; (c) every subset of X whose cardinality is less than a is closed (and hence discrete); (d) the topology on X is strictly finer than the coa topology; (e) for every subset U of X we have [U], = U. Proof: (a) <= (b) The fact that these two are equivalent follows straight from the definitions. (a) => (c) Let X be a weak P.space and S some subset of X of cardinality less than a. If x E clxS then since x is a weak P,point we have that x E S whence S is closed. (c) # (d) Again, this follows straight from the definitions. (c) = (e) Recall that [U], = {x E Xx E clxS for a subset S C U of cardinality less than a}. (c) implies that each S above is closed and so [U], C U. As the reverse containment always holds (e) follows. (e) =t (a) Let x E X. If S has cardinality less than a and x e clxS then x E [S], and so x E S. It follows that x is a weak P,point.I Remark 3.6 It follows from Proposition 3.5 (e) and Definition 1.8 that if X is a weak Paspace of with t(X) < a, then X is discrete. To see this, let U be an arbitrary subset of X. Then U = [U], = clxU, where the first equality follows from 3.5 (e) and the second from 1.8. In the introduction of this section we hinted at certain similarities between the class of P.spaces and weak P,spaces. The fact of the matter is that P,spaces are weak P.spaces. In fact, P.points are weak P,points. This is a straightforward application of the definitions. In the case that X is a totally ordered space with the interval topology we know that the converse of this is true (see the exercises of Chapter 5 in C1]; I that is, every weak Papoint of a totally ordered space is a P point. We now consider two examples which demonstrate our comment about the class of Paspaces being a proper subclass of the weak P,spaces. Example 3.7 (i) Let X = 8, U {oo} where the topology on X consists of each point in 1, being open and the neighborhoods about oo are those sets whose complement have cardinality at most Nk, for some finite k. Notice that oo is a nondiscrete Gs point so that X is not a Pspace. As in Example 3.2, I.\'1 = ]oo = 8, so that X is a weak PR,space for each finite k (yet not a weak PR.space.) Thus, we have an example of a zerodimensional, weak P,space, for each k < w, which is not a Pspace. (ii) Let D be a realcompact discrete space and choose p E 3D D with the norm of p in pX, IIp > w. That such a point exists follows from 12.I of Gillman and Jerrison [GJ]. Let X = D U {p} with the subspace topology inherited from PX. Then by Theorem 5.14 of Gillman and Jerrison [GJ] we obtain a denumerable sequence of sets in the ultrafilter AP whose intersection is empty. This translates to saying there exists a countable collection of open neighborhoods of p in X which meet at x. It follows that p is a Gspoint of X, whence X is not a Pspace. Next, as the norm of p in #X is uncountable it follows that p is a weak Ppoint of X. Thus, X is a weak Pspace which is not a Pspace. It is well known that a Pspace is compact precisely when it is finite, (see, for example, exercise 4AG [PW].) We now show that this is a property of weak Pspaces generalizing the wellknown theorem for Pspaces. Later, we shall use this propo sition in connection with a strengthened version of the classical Chinese Remainder Theorem. Proposition 3.8 Let X be a compact weak Pspace. Then X is finite. Proof: If X were infinite then choose a countable subset S of X. By Proposition 3.5 we have that S is closed and discrete. Now, S is compact, as closed subsets of compact spaces are compact. But this is impossible since we cannot have an infinite discrete compact space.E Corollary 3.9 If X is a locally compact, weak Pspace, then X is discrete. We would like to conclude this section with a proposition which ties together all of our definitions so far. It is known (see 9.6 [GJ]) that if X is not compact, then X(fX) > w. In Remark 2.4 we remarked upon noncompact spaces whose StoneCech compactification are of countable tightness. The existence of weak Ppoints in N N affords us a short proof of the following fact. Proposition 3.10 If 3X is countably tight, then X is pseudocompact. 35 Proof: First, observe that in a space of countable tightness if x is a weak Ppoint, then considering the set X {x} we obtain that x is isolated. Now, in Kunen [K1], Kunen observed that /N N has nonisolated weak Ppoints and therefore is not countably tight. If X is not pseudocompact then vX $ fiX. A well knownfact (see 9.4 [GJ]) is that vX / fiX implies that inside PX X there exists a copy of ONN N. Next, any closed subspace of a countably tight space is again countably tight. Thus, if X is not pseudocompact, then /X is not countably tight, otherwise the copy of ON N inside pX X would also be countably tight.E Reznichenko has exhibited an example of a pseudocompact space X whose 3X is sequential, hence countably tight. We urge the reader to consult Arkhangel'skii's article in HuSek and van Mill[HvM]. CHAPTER 3 COMPLETENESS 1. NonDiscrete Linear PSpaces Throughout this section A will constitute a commutative ring with identity and will denote a filter of ideals of A. As we are interested in certain topological rings, specifically, the topology on A induced by the given filter 0, we shall begin this section with some background information on topological rings and linear topologies. For excellent sources on topological groups or rings, please refer to Bourbaki [B2], Fuchs [Fu], Golan [Go], Stenstrom [St], or Warner [Wn]. Each is distinct from the others yet they all cover the preliminaries. A topological ring is a ring A together with a topology so that the ring oper ations (a,b) a + b, a + a, (a, b) + ab are continuous functions. The following theorem is pivotal in determining when a ring A is a topological ring. Theorem 1.1 [Theorem 3.5 [Wn]] Let A be a ring and U be a filter of subsets of A, each containing zero. Then there exists a unique topology on A for which i is a system of neighborhoods of 0, if i satisfies the following conditions: (1) For each U E il, there exists V E Ui such that V + V C U; (2) U E iA implies U E il; (3) for each a E A and U Ui, there exists a V E such that aV C U; (4) for each U E il there exists a V E il such that VV C U. Conversely, if A is a topological ring, then the neighbourhood system of 0 satisfies the above four conditions. Observe that for the filter 6 of ideals of A, the filter of subsets of A generated by ( satisfies the above conditions. The topology on A guaranteed by the preceding theorem then is called the linear topology on A with respect to 0, or simply the 0topology. Our main focus will be linear topological rings. There is a connection between the theory of Pspaces and topological groups. It is known that the Pification of a topological group is again a topological group. In particular, Comfort [Co] has results tying the Pification of special classes of topolog ical groups with pseudocompactness and the StoneCech compactification. In this section we wish to show that the Pification of a linear topology is again a linear topology. The idea will be to start with a given ring A and a specified filter of ideals 6. Considering the filter as a base of open sets about the identity 0, we obtain con ditions on 6 which show that A with its linear topology is a nondiscrete Pspace. We shall mainly concern ourselves with filters which make the topology on A Haus dorff, which is equivalent to the condition ( = 0. In this case, observe that A is a zerodimensional Hausdorff space; i.e., there is a collection of clopen sets which form a base for the open sets of A. (To see this, note that the complement of an open ideal is a union of costs, each of which is open, hence the ideal is also closed.) We shall use (A) to denote the trivial filter of all ideals; i.e., the discrete topology on A. We now consider a number of filters on a ring A which are, in general, nontrivial. Please be aware that the phrase "linear Pspace" connotes a topological ring with a filter of ideals such that the underlying topology is that of a Pspace. We do not mean a totally ordered space. Definition 1.2 a) Let 0 denote the filter of ideals generated by the collection of all maximal ideals; i.e., I E 0 if and only if there is a finite subset N < Max(A) such that nNA < I. b) Let T3 denote the filter of ideals generated by the collection of all minimal prime ideals. c) In general, if Ui is a collection of ideals then the Utopology is the topology induced on A by the filter of ideals generated by it. d) If 6 is a filter of ideals then 6s shall denote a new filter of ideals obtained by closing the filter up under countable intersections; i.e., I E 06 if and only if there is a countable subset NA C 6 such that nN C I. f) In the case that A = C(X), we let 5 denote the filter of ideals generated by the collection of all fixed maximal ideals; i.e., I E 5 if and only if there is a finite subset F C X such that nlEFM, C I. Ca(X) shall denote the ring with the topology induced by j. We shall also call this the 3topology on C(X). Remark 1.3 We use the the notation 0 above, following Warfield [Wa]. The nota tion originated in connection with pureinjective modules and Noetherian rings. We now show that the Pification of a linear topology is again a linear topology. Its proof is probably known though we were unable to find it, and so we give our own. Theorem 1.4 Let A be a ring with a Hausdorff filter (. The topology obtained on A by Pifying the linear topology given by ( is again linear and its corresponding filter of ideals is precisely sa. Proof: Let r be the topology on A corresponding to the filter 6. Notice first that A is a Pspace under the (O, thus, the topology rT obtained via the process of Pifying A is contained in the topology ro,. We wish to show the reverse containment. Let a E A and U E 0s. Consider the basic open set a + U E re,. Let Gi E 0, i = 1,2,... so that nGi < U. Pick an arbitrary x E a + U. Now, x + Gi Er for all natural i, whence n x + Gi E T8. But a quick check shows that nx+Gi =x+nG, x+U=a+U. What we have shown is that for each x E a + U there is some neighbourhood V E T{ of x which lies inside a + U; i.e., a + U E 7T.0 The following proposition can be found in the literature; it easily follows from the definitions. Recall that a ring is semiprimitive if the intersection of its maximal ideals is 0. Proposition 1.5 1) The ring A is semiprimitive if and only if the topology on A given by f is Hausdorff. 2) The ring A is semiprime if and only if the topology on A given by q3 is Hausdorff. In connection with Pspaces, we have the following characterizations of when a ring A with a given filter 6 is a Pspace. Theorem 1.6 The following are equivalent for a ring A and filter of ideals 6. (i) A is a Pspace; (ii) A is an almost Pspace; (iii) 0 is an almost Ppoint of A; (iv) 0 is a Ppoint of A; (v) 6 is closed under countable intersections. Proof: By homogeneity and the definitions, i) 4 ii) = iii) is clear. That iv) =4 v) =. i) follows from the fact that 6 is a neighbourhood base of clopen sets about 0 and Theorem 1.4. Thus, all that we have left is to show that iii) = v) (as iv) and v) are equivalent.) Suppose 0 is an almost Ppoint. Let F, F2, ... E 6 and denote their meet by F. We now show that 0 E intAF. Since 0 is an almost Ppoint, F has nonempty interior, whence there is an 0 E 6 and a b E A such that b + O C F. It is evident that b E F from which it follows that O < F. Since 6 is a filter we conclude that F E 0, i.e., 0 is closed under countable intersections.E Remark 1.7 Observe that the above theorem is also true for topological groups, in general. This follows from the fact that every open set V about 0 contains an open set U for which U = U and U+ U C V. We now give some basic results on the application of 6 to a filter of ideals. We then apply what we know to C(X). To some extent we are generalizing results on filtrations (see [Bl]), a topic not pertinent to this dissertation. The following proposition will be pivotal. Proposition 1.8 a) Assume A is a semiprimitive ring. Then A is discrete under the fs topology if and only if Max(A) is separable. b) Suppose A is a semiprime ring. Then A is discrete under the 3a topology if and only if Min(A) is separable. Proof: a) Suppose A discrete under the fs topology. Then there are countably many maximal ideals, say MI, M, ... whose intersection is 0. Consider the set S = {M, M2,...} C Max(A). Since Max(A) has the hullkernel topology, we may find an ideal I < A such that clMz(A)S = V(I), where V(I) = {M E Max(A)II < M). Thus, I < Mi, for all natural i. This forces I = 0, whence clMx(A)S = V(I) = V(0) = Max(A) and so Max(A) has a countable dense subset; i.e., Max(A) is separable. As for the sufficiency, suppose there is a countable dense set S C Max(A), say S = {M1, M2, .... Then clMas(A)S = V(0). Denote the intersection of S by the ideal I < A. Then as above S C V(I) whence V(I) = Max(A). We then apply the assumption that A is semiprimitive and arrive at the conclusion that I = 0. Thus, 0 = I E s and so A is discrete. The proof of b) follows in a similar fashion.E We may derive a little bit more from the preceding proof. In general, if it C Max(A) for A, a commutative ring with identity, then the it topology on A is Hausdorff if and only if it is a dense subspace of Max(A). We also have the following corollary to Proposition 1.8. Corollary 1.9 a) For a Tychonoff space X, C, (X) is a nondiscrete linear Pspace if and only fX is not separable. b) For a Tychonoff space X, Ca,(X) is a nondiscrete linear Pspace if and only X is not separable. Recall that if X is locally compact, then X is separable if and only if 3X is separable. Thus, as a special case of Corollary 1.9 we have: Corollary 1.10 If X is locally compact, then Ca,(X) is a nondiscrete linear Pspace if and only X is not separable. Example 1.11 Let W be the set of all countable ordinals. As is well known W is a zerodimensional space with loW = /W = W+, a nonseparable space. Thus, we have that C(W) is an example of nondiscrete linear Pspace. Observe that Theorem 1.6 showed that for linear topologies Pspaces and almost Pspaces coincide. We conclude this section with an example showing that this is not the case for weak Pspaces. Example 1.12 Let D be the discrete space of cardinality R,. Let g be the filter base of ideals of C(X) consisting of those ideals of the form MT = {f E C(X) : f(t) = 0 for all t E T}, where T ranges over all sets of cardinality Rk for k finite. Then Cg(X) is a weak Pspace which is not a Pspace. To see that Cq(X) is not a Pspace observe that 0(O,C(X)) = Ro. Now, let S = {fi,f2, '} be a collection of nonzero functions. Let xi E coz(fi) and T = {X},eN. Then MT n S = 0, so that 0 i clS, whence Cg(X) is a weak P space. 2. Completions and the Chinese Remainder Theorem In this section, we combine information on completions of Hausdorff spaces with our knowledge of linear Pspaces, and highlight what happens with &s and fs. We begin with some preliminaries taken from Bourbaki [B2] and Fuchs [Fu]. Definition 2.1 Let A be a commutative ring with identity and 6 be a Hausdorff filter of ideals. Label 05 by an index set I, so that I is a directed set under reverse inclusion. A net in A is a set of elements {ai,},i in A indexed by I. The net {ai}iel is said to converge if there is an a E A such that for every i E I there is a j E I for which ak a E Fi for all k > j. The net {a,}ier is said to be Cauchy if to each i E I there is a j ] I such that ak ak, E Fi whenever k, k' > j. The ring A is called complete if every Cauchy net converges. Proposition 2.2 Let A be as in Definition 2.1. Then A is isomorphic to a dense subring of a complete Hausdorff ring A0, which is determined up to isomorphism. Ae is called the completion of A with respect to 6. (If it is clear which filter is being considered then we shall use A for the completion of A.) The substance of Proposition 2.2 is one example of the natural phenomenon that takes place in studying the relationship between topology and algebra. It is a common notion in mathematics to take an algebraic object with a set theoretic property or a topological or lattice structure and apply a "completion" to obtain an algebraic object of the same type. If we let 0 = 0, then we may describe the completion of A explicitly. This was done more generally in a paper by W. Brandal. We now recite his theorem. For the necessary definitions we refer the reader to Brandal [Br]. Theorem 2.3 [Brandal] Let {On},eN be a family of pairwise comaximal filters of ideals of A; i.e., each 0. is a filter so that if Ii E Oi and Ij E 0j with i f j, then I, + Ij = A. Let 0 be the filter generated by the the union of the .6. For each n let Ae,, be the completion of (A, ,). Then with respect to 6, the completion of A is HAe,. under the product topology. In Brandal [Br], we find that in an example prior to the above theorem the author shows that if 0 is a filter with = M (a nonHausdorff topology) then the completion of A with respect to 0 is in fact AIM with the discrete topology. (We wish to point out that Brandal does consider the completion of a nonHausdorff topology, a situation in which one loses the embedding of A in its completion.) Applying Brandal's Theorem, we have the following corollaries. Corollary 2.4 Suppose that A is semiprimitive. Then the completion of A with respect to 9 is HMEMaM(A)A/M with the product topology. Remark 2.5 The proof of Theorem 2.3 follows from the Chinese Remainder Theo rem. It allows us to obtain A as a dense subring of HIA,,. That HIA, is complete follows from the fact that a product of complete spaces is again complete. The rest of this section will be concerned with the task of explicitly describing the completion of A with respect to the Is and &stopologies. Accordingly, we shall only concern ourselves with semiprimitive rings A; i.e., s is Hausdorff. When we consider A with the Qf topology we lose our topological embed ding of A in IHMEM,(A)A/M with its product topology. But if we consider the finer topology on THMEMa(A)A/M whose base is the collection of sets of the form rneNPn (Mn/Mn), (for arbitrary collections {M,}n<,), where each p. is the projec tion onto the nthcoordinate, then in fact we have that A < IIMEM.a(A)A/M. (This topology is often called the wproduct topology.) Notice that II~MeMa(A)A/M with this new finer topology is a Pspace; (in fact, it is the Pification of the product topology hence we denote the ring with its Pspace topology by (IIMEMa(A)A/M)s.) Next, we apply Corollary 1, p. 250, of Bourbaki B13. which states that a topology 7, which is finer than a complete topology and whose open neighborhoods of 0 are closed with respect to the coarser topology, is also complete. Therefore, (nIMEMa(A)A/M)s is complete, whence the completion of A with the &s topology is in fact the closure of A inside (lIMEMz(A)A/M)s. Thus, it is natural to ask whether the full product is the completion of A. As noted before, for the Otopology, the fact that the embedding of A is dense relies on the Chinese Remainder Theorem. We now discuss a strengthened version of the Chinese Remainder Theorem. Definition 2.6 A ring A is said to satisfy the wChinese Remainder Theorem (or wCRT) if whenever there are elements al,02,... E A and distinct maximal ideals M, M2, ... there is an element a E A such that a ai E Mi for all natural i. More generally, let 1 C Max(A) be dense. Then we say A satisfies the wCRT for t if the above property holds for each arbitrary countable collection of maximal ideals in it. Theorem 2.7 Suppose A is a semiprimitive ring. Then A satisfies the wChinese Remainder Theorem if and only if the completion of A with respect to the Qs topology is (IIMEMa(A)A/M)s. Proof: The necessity is just a generalization of the proof of Theorem 2.3. Thus, we suppose that (IIMEMax(A)A/M)s is the completion of A. What this entails is that A < (IIMEMa(A)A/M), is dense. Now, let a1,02, ... E A and Mi, M2,... be a pairwise disjoint collection of maximal ideals. Consider the basic open set (rM) + N P' (M, M) nEN of (IIrMEM(A)A/M)s, where rM, = a, and 0 otherwise. As A is dense in (IMeMaz(A)A/M)s there is some a E A such that (a) E (rM) + n P(M1/M,). nEN It follows that a rM, E M, for all natural i; i.e., a ai E M, for all natural i.E Remark 2.8 The above proof may be generalized further to obtain the statement: A satisfies the wCRT for it if and only if the completion of A with respect to the it topology is (HMEuA/M)s. The natural question at this time is: when does a ring A satisfy the wChinese Remainder Theorem? We leave the answer to this question until the next section where we will discuss completion points of Max(A). The author does admit that the answer is somewhat surprising. 3. Completeness of C(X) and C*(X) In Chapter 2, we discussed the class of wdense spaces. In this section we show its importance in the study of C(X) as a topological ring. From Remark 2.8, we know that we may view Cn,(X) as a subspace of (tlpE1xC(X)/MP)6. We shall demonstrate that in fact we may represent C(X) as a topologicall) subspace of (IIlexC(X)/M,)s = (Rx)s if and only if X is wdense. In turn we have that Cna(X) = Ca,(X). Thus, we find that the class of wdense spaces is a wellbehaved class deserving further investigation. For the secondhalf of this section we will con sider only the astopology and determine certain properties of C(X). We begin by investigating certain topological properties of the bounded subring C*(X). Recall our Proposition 1.3.11 regarding the equation MP n C*(X) = M*". If X is not pseudocompact, it should be clear that the subspace topology on C*(X) inherited from Ca(X) is strictly finer than Cn(fX). (In fact, this condition char acterizes the class of nonpseudocompact spaces.) As for the tstopology things are slightly different and more difficult. For some particular classes of spaces we have been able to characterize those spaces (inside the particular ones) which satisfy the following equation (,) C',(X) = CQ(IX). (We shall use the convention that C, (X) denotes C*(X) with the subspace topology inherited from Cn,(X). Cn,(fX) denotes the fstopology on C(f3X). Thus, the above equality means the subspace topology equals the Qstopology on C(PX).) We start with an easy proposition. Proposition 3.1 IfvX is wdense, then X satisfies (*). Proof: If vX is wdense, then for each p E PX vX there is a sequence of points of vX, say {pi}, for which p E cldx{p,}. This translates to nMp, < MP. Thus, we have the following string of inequalities nMP'* = nMP' n C*(X) < MP n C*(X) < Mp*, which shows that the Qftopology on C(/X) is finer, hence equal to the subspace topology on C*(X).M We are interested in when the converse of the above proposition is true. Note that as X satisfies (*) if and only if vX satisfies (*) we may assume that X is realcompact. We have the following results for realcompact spaces. Recall that a acompact space is necessarily realcompact (see 8.2, [GJ].) Proposition 3.2 Suppose X is a locally compact, acompact space. Then X satisfies (*) if and only if X is wdense. Proof: Recall from the exercises of Chapter 1 in Walker [W], that a space is locally compact and ocompact if and only X is a cozeroset of OX. Suppose X satisfies (*) yet X is not wdense. Let p E fX X witness this. Since X satisfies (*) it follows that there is a countable subset of 3X, say T, for which ntETMt* < MP n C'(X). It follows that p E clxT but by our choice of p we may as sume that T is disjoint from X. Next, choose f E C*(X) for which Z(f ) = fXXX. Then f E ntETMt*, whence f e MP n C*(X); i.e., p E c1,xZ(f). But Z(f) is empty; a contradiction. Therefore, X must be wdense.E The next two propositions follow from the proof of Proposition 3.2. Proposition 3.3 Suppose X is realcompact and satisfies (*). Then every weak P point of 3X X lies in the fXclosure of some countable subset of X. Proof: Suppose X satisfies (*), and let p E fX X be a weak Ppoint of iX X. Since X satisfies (*) it follows that there is a countable subset of fX, say T, for which nteTM'* < MP n C*(X). It follows that p E clpxT. Since p is a weak Ppoint of fIX X, we conclude that T C X.0 Corollary 3.4 Suppose X is real compact and PX X is a weak Pspace. Then X satisfies (*) if and only if X is wdense. Carefully dissecting the proof of Proposition 3.2 we see that we may relax the conditions of X being a locally compact, ocompact space. All we need is that X satisfy the following condition. Definition 3.5 We call a space yrealcompact if it satisfies the following property (7) For each T C_ fX X, T countable, there is a zeroset of fX disjoint from X containing T. From Theorem 1.3.9 (iii) we observe that a yrealcompact space is necessarily real compact. Theorem 3.6 Suppose X is 7realcompact. Then X satisfies (*) if and only if X is wdense. As the title of this section suggests we are interested in the completeness of C(X) and C*(X) and how they relate. To that end we begin with a simple lemma. Lemma 3.7 For an arbitrary Tychonoff space X, C6,(X) < Ca,(X) is a closed subring. Proof. Let f E C(X) such that if p,p2,... E fIX then there exists g E C*(X) for which g f E MP' for all i; i.e., f lies in the closure of C*(X). Since the clo sure of C*(X) in C(X) is a subring of C(X) we may assume that f > 0. Suppose that f 0 C*(X). Then we may choose a countable subset S C X, enumerated as S = {xi,X,...} so that f(x,) > n for each natural n. But this contradicts the as sumption that f is in the closure of C*(X) since no bounded continuous function on X may agree with f on S. Thus, we conclude that C*(X) < C(X) is closed.E Since closed subsets of complete subspaces are complete we have the following corollary to 3.7. Corollary 3.8 If Cn,(X) is complete then so is C~,(X). Recall that Ca,(X) is complete if and only if C(X) < (HVnExC(X)/MP)s is a closed subspace. But the closure is relatively easy to describe. It is the collec tion of points r E (lpeixC(X)/MP)s for which S < fX a countable subset im plies there is a continuous function fs E C(X) so that r(MP) + MP = f, + MP for all p E S. In studying this problem for compact spaces we first notice that (Illp3xC(X)/MP)s = (nIIexC(X)/Mz)s = (RX)b, hence the closure of C(X) is a col lection of realvalued functions on the space. The problem then, in general, is one of continuity and topological properties. We investigate this phenomenon further, and so we consider those Tychonoff spaces X for which the representation of C(X) occurs as realvalued functions. To this end, as hinted in the introduction of this section, we consider the application of wdensity to the class of commutative rings with identity A. Recall Definition 2.1.3. Theorem 3.9 For a dense subset it C Max(A), we have that A under the II ll , topology is (topologically) a subspace of (IHMeIA/M)s. Conversely, the topology induced on A as a subspace of (IInMEA/M)s is the ([ilU])stopology. Proof: This follows from the following theorem which may be found in Warner [Wn] 5.18. If is a continuous homomorphism from a topological group G to a topological group H, and 93 is a fundamental system of neighborhoods of 0 E G, then f is open if and only if for each V E 3, f(V) is a neighbourhood of 0 E H. Now let f : A + (IIMEuA/M)s be the natural continuous homomorphic in jection. Let A be the image of A under f with the subspace topology induced from (HIMEA/M)s and view f : A + A. What needs to be shown is that f is a homeo morphism; i.e., that f is open. So let M < A be a countable intersection of maximal ideals from [U],. By the preceding, we need to show that f(M) is a neighborhood of 0. But this is equivalent to showing that there is an intersection, say M', of a countable number of maximal ideals in it such that M' < M, which then follows from the definition ofit and [1t],.I Corollary 3.10 Cnf(X) is a topological subspace of (IExC(X)/M,)s = (RX) if and only if X is wdense. In this case Cn,(X) = C, (X). Corollary 3.11 Suppose that i is a dense subset of Max(A) which is wdense in Max(A). Then with respect to the i8topology, the completion of A is the closure of A inside (IIHMeA/M)s. Theorem 3.9 enables us to represent A in a much smaller product space. In principle, this should be useful, considering our task is to describe certain completions. For the case C(X), we shall see that this is in fact what occurs. We concern ourselves with the wdense spaces X, for which we obtain an immediate (partial) converse to Corollary 3.8. Theorem 3.12 Suppose that X is wdense. Then the following statements are equiv alent: (i) Ca,(X) is complete; (ii) Cy,(X) is complete; (iii) if r Rx such that for each countable subset S C X there exists some contin uous function which agrees with r on S then r is continuous; (iv) C ,(X) is complete; (v) Ca,(3X) is complete. Proof: By Corollary 3.10 we know that (i) and (ii) are equivalent. As X is wdense, Proposition 3.1 implies (iv) and (v) are equivalent. A simple calculation shows that (ii) and (iii) are equivalent. Corollary 3.8 says that (i) implies (iv). Thus, all we are left with is (iv) implies (i). We prove it by contradiction. Suppose C(X) is not complete. Let r E nIexC(X)/M, = Rx have the property that given any S = {xi,x2,...} C X, there is an fs E C(X) for which fs and r agree on S, but r C C(X). As previously noted, we may assume that r > 0. Now, r C(X) is equivalent to saying for some x E X we have positive real numbers a < r(x) b and consider r A m. This is of course not continuous at x and yet for each countable subset S the bounded continuous function f A m agrees with r A m on S. Whence, C'(X) is not complete.I Proposition 3.13 Let X be an wdense space of countable tightness. Then Cn,(X) is complete. Proof: Let r E C(X). We wold like to show that r is a continuous function from X into R. By Proposition 2.1.11, it suffices to show that if x E clx{x,} then f(x) E clR{f(xi)}. Letting S = {x,} U {x} we obtain a continuous function fs which agrees with r on S. Since fs is continuous, r(x) = fs(x) E cla{fs(x,)} = dcl{r(xi)} whence r E C(X) and C(X) is complete.I We illustrate the beauty of Theorem 3.12 with the following examples. In both cases we have compact spaces so that .l, :; t is trivially satisfied. Example 3.14 C(W*) is not complete under the istopology. This follows by con sidering the characteristic function on the set {wi}, say X. If S is a countable sub set of W* then if {wl} S we have that 0 agrees with X on S. Otherwise, let r = sup(S {wi}). As S is countable we have that r < wi, and as is well known, we are able to find some clopen set separating S {wl} from wl. The characteristic function on this clopen set (or its complement) then agrees with X on S. But X is certainly not continuous on W*. By iii) of Theorem 3.12 we have that our claim is true.I Example 3.15 C(aD) is complete under the fstopology. Suppose r : aD + R has the property described in iii) of Theorem 3.12. Without loss of generality, we assume that r > 0 and that r(a) = 0. Recall that r will be continuous if and only if r l(, oo) is finite for all natural n. Thus, if r is not continuous we may choose an n and a countable subset S such that S C r'(!,oo). But then choosing f E C(aD) which agrees with r on S U c, we arrive at a contradiction. Note that aD is in fact a compact sequential space hence the conclusion of this example follows from Proposition 3.13.E Our goal is to completely describe the class of wdense spaces for which Cn,(X) is complete. In fact, we will do more. Using a cardinal function defined by Arkhangel'skii we shall name the class of spaces for which Cg,(X) is complete. The cardinal function is called the strict functional tightness and is denoted by tR(X). Arkhangel'skii and others used the strict functional tightness to determine when C(X) under the topology of pointwise convergence, denoted C,(X), is realcompact. (We shall discuss Cp(X) in later sections and in greater detail but for now the topology of pointwise convergence on C(X) is the topology induced on C(X) as a subspace of Rx under the product topology.) Combining theorems of Arkhangel'skii and Us penskii, it is true that the topology of pointwise convergence is realcompact if and only if tR(X) = w (see [Ar3]). Our Theorem 3.17 adds to what is already known. We continue with the promised definition. Definition 3.16 Let r be a fixed cardinal. A function f E Rx is called strictly r continuous if for every subspace A C X of cardinality at most r there is a continuous function g : X + Y such that glA = flA. The strict functional tightness, denoted tR(X), is the least cardinal r such that every strictly rcontinuous function is continuous. Applying Definition 3.16 together with Theorem 3.12, we have the following theorem. First, it is useful to note that (ii) and (iii) of 3.12 are equivalent for an arbitrary Tychonoff space. Theorem 3.17 Suppose that X is Tychonoff. Then the following statements are equivalent: (i) C,(X) is complete; (ii) tR(X) = w; (iii) Cp(X) is realcompact. Corollary 3.18 Suppose X is wdense. Then tR(X) = w if and only if tR(OX) = w. We shall see in the next section that there are non wdense spaces for which tR(X) = U, yet tR(3X) 7 w. 4. The wChinese Remainder Theorem In Example 3.14 we saw that the characteristic function on the singleton {wl} is not in C(W*) whence C(W*) is not complete. This event provides us with a path to follow in investigating theorems concerning the noncompleteness of C(X). We presently expand on this theme. Definition 4.1 Let A be a semiprimitive ring with 1 and it C Max(A) be dense. Let M E U. Consider the representation of A < NIINA/N. Let XM E IINeuA/N represent the function defined by x(N) = M, ifM=N; xM(N) 1, otherwise. We say M is a completion point of A with respect to Ui if XM E A14. Recall that f E Aj, precisely when given an arbitrary countable subset S of iL there is an a E A such that a agrees with f on it (in IINEuA/N.) If A = C(X) for some Tychonoff space X and it = fX, then we distin guish two cases. If p E PX is a completion point which happens to lie in the real compactification, vX, of X then we shall call p a real completion point. Otherwise, we call p a hyperreal completion point. Later on we will address the question of whether there are hyperreal completion points. It is evident that if Cu,(X) is complete, then the completion points of X are precisely the isolated points of U. We now describe the completion points of it in our next proposition. Proposition 4.2 Let A be a semiprimitive ring with 1 and Ui C Max(A) be dense. Then M E i is a completion point of A with respect to U if and only if M is a weak Ppoint of i. Proof: Necessity: Suppose M is a completion point of A. Let S = {Mi, } be a subset of 1 disjoint from M. As M is a completion point of A there is an a E A which agrees with XM on S U {p}. Then U(a) n U (of the hullkernel topology) is an open set of it containing M which is disjoint from S and so M is a weak Ppoint of l. Sufficiency: Suppose M is a weak Ppoint of U and let S be a countable subset of U. We want to find an a E A such that a agrees with XM on S. Without loss of generality we assume that M is disjoint from S. As M is a weak Ppoint it follows that there is a U(a) containing M yet disjoint from S. Now, since A/M is a field and a + M f M it follows that there is an r E A such that ra + M = 1 + M. Thus, since ra + Mi = Mi we conclude that XM lies in the completion of A with respect to Us, i.e., M is a completion point of A with respect to it.E The rest of this section is dedicated to answering two questions which have been raised so far, as well as one posed by Mel Henriksen (see [H].) The first is: are there any semiprimitive commutative rings A with 1 which satisfy wCRT; i.e., the completion of A under the ns topology is IIMMEMa(A)A/M. Secondly, does there exist a Tychonoff space X with a hyperreal completion point? Lastly, in Henriksen [H] the author questions if there is a way of characterizing weak Ppoints via C(X). We have attempted to answer these questions. The first is a corollary to Proposition 4.2. Corollary 4.3 Let X be a Tychonoff space. Then p E X is a weak Ppoint if and only if the characteristic function on p is in the completion of C&,(X). Theorem 4.4 Let A be a semiprimitive commutative ring with identity. Then the following are equivalent. (i) nMEMea(A)A/M is the completion of An,; (ii) A satisfies the wCRT; (iii) Max(A) is finite. Proof: The equivalence of (i) and (ii) was shown in Theorem 2.7. Vacuously the condition "Max(A) finite" implies (i). As for (i) implies (iii), Proposition 4.2 implies that every point of Max(A) is a weak Ppoint. But then Proposition 2.3.8 finishes the proof.E Remark 4.5 As to the question of whether there are hyperreal completion points of a space X we need look no further than Kunen [K2] to give an affirmative answer. Since there are nonisolated weak Ppoints of fa (for uncountable a,) we see that there do exist hyperreal completion points. Example 4.6 It is not true that the completion of C(X) under the f~ topology is necessarily a ring of realvalued functions. Consider an uncountable cardinal a. Let p be a nonprincipal weak Ppoint of 3a. Then by Proposition 4.2 the characteristic function on the singleton set {p}, X is in the completion of C(a). Since p 0 vX, C(a)/MP there exist infinitely large elements (see 5.7 [GJ].) Thus, choose f in C(a) for which f + MP is infinitely large. Finally, observe that fx E C(a) and fx is infinitely large, so that the completion of C(X) contains elements which may not be represented as realvalued functions. 5. 1 and its Relations to Other Topologies This section is devoted to demonstrating how Cr(X) relates to other topologies on the algebra C(X), which have been investigated in the literature. In particular, we consider the uniform norm topology, the mtopology, and the topology of pointwise convergence. Basic information on the first two may be found in the exercises of Chapter 2 in Gillman and Jerrison [GJ]. As for the topology of pointwise convergence we suggest Arkhangel'skii [Ar3]. We define these presently. The topology of uniform convergence is defined on C(X) (resp. C*(X)) by taking as a base for the neighbourhood system at g all sets of the form {f: If(x) g(x)l < e, for all x E X}, where c > 0 and f E C(X) (resp. C*(X)). Note that C(X) will be a topological ring or a topological vector space precisely when X is pseudocompact, in which case C(X) = C*(X) is a Banach algebra defined by the norm: lf =L su PexIf(x)1 (see 2M [GJ].) Cu(X) (resp. C (X)) shall denote C(X) (resp. C*(X)) with the topology of uniform convergence. The mtopology on C(X) is defined on C(X) by taking as a base for the neighbourhood system at g all sets of the form {f E C(X) : If g u}, where u is a positive invertible element of C(X) (resp. C*(X)). Under the m topology, C(X) is a topological ring. It is a topological vector space if and only if X is pseudocompact, in which case the mtopology and the uniform norm topology coincide. If X is not pseudocompact, then it is simple to show that the subspace of constants is discrete so that C(X) is not a topological vector space. In either case, the mtopology on C(X) is finer than the uniform norm topology. As above, we use C,n(X) (resp. C0,(X)) to denote C(X) (resp. C*(X)) under the mtopology. The topology of pointwise convergence on C(X) (resp. C*(X)) is defined by taking as a base for the neighbourhood system at g all sets of the form {f: If(xi) g(x)l < c,, 1 < i where f E C(X) (resp. C'(X)). Cp(X) (resp. C*(X)) denotes C(X) (resp. C'(X)) under the topology of pointwise convergence. Observe that the topology of pointwise convergence may, equivalently, be obtained as the subspace topology of C(X) < RX where RX is equipped with the product topology. Both Cp(X) and C,(X) are topological algebras. The following theorems show how the above topologies relate to each other as well as to the 3topology. For our purposes here, (X, r) < (X,7r') for topological spaces, means that r' is finer than r. We use the symbol < to denote strictly finer. The proof of the next theorem follows straight from the definitions. Theorem 5.1 For a Tychonoff space X, Cp(X) 5 Cu(X) < Cm(X) and C;(X) < CY(X) _< C*(X). Theorem 5.2 For a Tychonoff space, Cm(X) is never finer than Ca(X). Also, Ca(X) is finer than Cm(X) if and only ifX is finite. In general, C,(X) < CI(X), and equality holds if and only if X is finite. Proof: Suppose C,(X) were finer than Cy(X). Choose an element x E X. Then, since the maximal ideal M, is an open neighbourhood about 0, it follows that there is a positive unit u such that {f E C(X) : 1fl < u} C M., whence u E M., contradicting the fact that u is invertible. As for the second statement, the ;jth.:,r.: is clear since the discrete topology is the finest topology on any space. So suppose CI(X) is finer than Cm(X) and assume by way of contradiction that X is infinite. Consider the open set (in the mtopology) of functions which lie between 1 and 1. Our assumption gives rise to a finite set of elements in X, say {xl,... ,x}, such that if f(xz) = 0 then Ifl I 1. Choose an x {zi,...,z, }. Since X is Tychonoff we may find g E C(X) for which g(xi) = 0 but g(x) = 7. This contradicts that Igl < 1. We now show that C,(X) < CW(X). Let g E C(X), F = {1,... x,} E [X]< and 1,..., > 0. Let U= {f: ff(x,) jr,I <5i foralli= 1,..,n} be a basic open set (in the topology of pointwise convergence) about g. Let f E U. Then f + nle Mx C U, from which we conclude that U is an open set in 1 and so Cp(X) < Ca(X). The fact that Cp(X) is strictly finer than Cy(X) follows once it is observed that Cp(X) is always connected.E Though Cp(X) < Ca(X), the topologies are very closely related. What we mean by this is stated in the following proposition. Proposition 5.3 For an arbitrary space X we have C,(X) = (Cp(X))8 Proof: To see this observe that Ca,(X) is precisely the topology obtained by viewing C(X) as a subspace of IlExR under the wtopology, and where each copy of R is discrete. But this is also how (C,(X))s is obtained.E As Cy(X) is zerodimensional a natural question is whether it is a Pspace. We answer this now. Proposition 5.4 Under the Jtopology, the following are equivalent for an arbitrary Tychonoff space X. (i) Cy(X) is an almost Pspace; (ii) CW(X) is a Pspace; (iii) CI(X) is a weak Pspace; (iv) X is finite. Proof: That (i) (ii) is Theorem 1.5; (ii) = (iii) and (iv) 4 (i) are both clear. What we have left to prove is that (iii) = (iv). For that we need a lemma. Lemma 5.5 The Tychonoff space X is finite if and only f(X) is finite for every fE C(X). Proof: Suppose f(X) is finite for every f E C(X). It is evident that X is zero dimensional, in fact, a Pspace. If X were infinite then we may choose an infinite clopen partition, say {U,}. Choose a countable subset S C R. Without loss of generality, we assume that {U,} is enumerated by the elements of S. For each s E S, choose x E U, and f, E C(X) such that f,(x,) = s whereas f,(U,) = 0. Then f = E s f. E C(X) and If(X)I = R. Continuing with the proof of 5.4: suppose that Cy(X) is a weak Pspace but X is infinite. By the above lemma, there exists some f E C(X) such that If(X)I > Ro. Let 3 be the collection of finite unions of closed intervals with rational endpoints, and 3f = {J E 3 : f(J) 7 X}. Let Z be the collection of inverse images of the elements in 3f. Now, each element Z E Z is a zeroset, say Z(fz) = Z. Let S = {fz : Z E Z}. Observe that given a finite subset F of X there is some Z E Z which contains F, so that fz E MR. This implies 0 E clc,(x)S. Since S is countable, we contradict that Cy(X) is a weak Pspace. The proof of Proposition 5.4 is now complete.I We point out that the proof of the previous lemma was used in Levy and Rice [LR] and attributed to A.W. Hager. There the authors used a similar argument to characterize when a Pspace has the property that If(X)I < w for all f E C(X). CHAPTER 4 CARDINAL FUNCTIONS 1. Cardinal Functions and Ca(X) In this chapter we wish to compute some common cardinal invariants of Ca(X). Theorem 1.1 For a Tychonoff space X, we have the following qualities: (i) w(C,(X))= c IX; (ii) x(Cw(X)) = rx(CA(X)) = XI; (iii) c(Cy(X)) = c; (iv) O(Ca(X)) = d(X); Proof: (i) Recall that the collection, B, of sets of the form f + nfeFMX where f E C(X) and F E [X]<", is a base for the topology. There is a naturally induced 11 correspondence 4 from B onto [R]<" x [X]<", where 4S(f + nfEFMr) = ({f(x) : x E F}, F) Observe that the reason 4 is onto is that X is Tychonoff. It follows that w(Cy(X)) < ]J(B) = cIXl. Now, suppose D is a base for the atopology on C(X). Without loss of generality, we assume that V is a subcollection of B. Let 7rl be the projection of [R]<" x [X]<" onto [R]<" and ~r = 71r o0. Similarly, let 7r2 be the projection of [R]<" x [X]<" onto [X]<' and s2 = tr2 o Next, let R = UDEZ 7ri(D) and F = UDeV 7r2(D). If IJD < 1B then we consider two cases: Case 1: Suppose IXI > c. Then there is some x E X F. Hence, 1 + M. V. Since V) is a base, though, we may find an f E C(X) and a finite subset G C X for which f + nfEGMy C 1 + M. and, in particular, x G. But, again, we use that X is Tychonoff and find a function g E C(X) such that g(y) = f(y) for each y E G and g(x) = 0. It follows that g E f + nyEGMy yet g 1 + Ms; a contradiction. Thus, \DI = IBI. Case 2: Suppose IXI < c. It should be clear that now we may choose an r E R R. Choose an arbitrary x E X and obtain an f + nEGM, E D such that f + nyeGMy C r + M,. Now, it is apparent from Case 1 that z E G. But f(x) E R implies that f(x) Z r, contradicting the above containment. Thus, again we have IDI = IBl and so w(Cy(X)) = c XI which concludes the proof of (i). (ii) Recall that since CI(X) is homogeneous x(Cy(X)) = x(0,Ca(X)) = inf{ BI : B is a base for 0}. Since the collection, B, of finite intersections of maximal ideals form a base at 0 and the natural map $ from B onto [X]<" is a 11 correspondence it follows that x(C((X)) < I' I = IXI. As above, we let D be a subcollection of B forming a base of neighborhoods about 0. Let F = UDEV O(D). If the cardinality of V is strictly smaller than the cardinality of B then choose an x E X F and find a G E [X]<" for which nylGMy E D and nVeGMy C M,. Now, because X is Tychonoff, this cannot happen, unless x E G, which we have chosen otherwise. Thus, every base of neighborhoods about 0 has cardinality no smaller than IXI. The fact that x(Ca(X)) = 7r(Ca(X)) follows from a similar proof to the one in Theorem 3.1.6. In fact we may conclude that for a linear topology the character always equals the 7rcharacter. (iii) This is true by a personal communication with Mary Ellen Rudin. (iv) Suppose we have a collection U of ideals in 5 whose meet is 0. Let V= U F MFEU Now, if f(x) = 0 for all x E V, then f = 0. By the Tychonoff property, we obtain that V is dense in X, otherwise we could choose x E X clxV and then a nonzero f E C(X) which vanishes on V. As each ideal in U corresponds to a finite set, a simple calculation reveals that IVI = IUI, and so d(X) < Vi(Cy(X). That the converse is true follows from the fact that nfEsMr = 0 for any dense subset S C X.0 Next, we would like to calculate the tightness of Cy(X). The importance of this cardinal invariant will be seen in the next section as it is the most clear instance in which the the interplay between X and Ca(X) is drastically different than the interplay between X and Cp(X). To calculate the tightness of Cy(X) we need some definitions. For one, we shall define the notion of a vanishing space. We begin. Definition 1.2 A zcover of X is a collection Z of zerosets with the property that given any F E [X]<" there is a zeroset in Z E Z with F C Z. If Z is a zcover of X then a zsubcover is a subcollection of Z which also satisfies the above property. As a zcover is in fact a cover of the space (by zerosets) we define a useful cardinal function. Define the zeroset degree of X (or when our slanguage is clear, the vanishing degree of X) as z(X) = min{K : every zcover of X has a zsubcover of size no greater than n} If z(X) = w, then we call X a countably vanishing space (or simply a vanishing space.) Observe that z(X) is always infinite, unless X is finite. Moreover, z(X) < IXI. Remark 1.3 The reason for defining the vanishing degree of a space should be ap parent considering we are interested in the tightness of Cy(X). In particular, if Z is a zcover of X then by choosing an fz E C(X) for each Z E Z, (such that Z(fz) = Z), we then have 0 E clc,(x){fz : Z E Z}. Conversely, if 0 E clc,(x)S then {Z(f) : f E S} forms a zcover. This relationship then implies the following result. Proposition 1.4 For any Tychonoff space X, we have z(X) < a if and only if t(Ca(X)) < K; i.e., z(X) = t(Ca(X)). Corollary 1.5 For a space X, CW(X) is countably tight if and only ifX is a vanishing space. Corollary 1.6 If i(X) = w, then z(X) = IX(. Furthermore, a vanishing space X has nice pseudocharacter if and only if it is countable. Proof: Recall that X has nice pseudocharacter is equivalent to saying t(X) = w. This implies that [X]<" is a zcover of X with no proper zsubcover. Since I[X]<" = \X, the result follows.M We would like to conclude this section by considering one more difference between C,(X)theory and Cg(X). We shall return to the vanishing degree and, in fact, in the next section we will characterize the class of compact vanishing spaces. For an arbitrary space X, it is true that X is a subspace of C,(C,(X)). A familiar class of spaces characterizes this property for Ca(CT(X)). The analogous theorem for C,(X) may be found in Okuyama and Terada [OT]. Theorem 1.7 Let p: X * C(Cy(X)) be defined by p(x)(f) = f(x). Then p is an embedding of X into CT(CW(X)) if and only if X is a Pspace. Proof: Let x E X. First, we show that p(x) is continuous; that is p(x) E C(Cy(X)). Let a < b and f E p(x)'((a,b)). Next, let r = f(x) E (a,b). If g is continuous on X and f(x) = g(x) then p(x)(g) = g(x) = r E (a,b), whence we have that p(x)(f + M,) C (a,b) and so p(x) is continuous on Ca(X). It is clear that p is injective, hence all we need is that p be continuous and open. First, let us assume that X is a Pspace. Let iP + F, M, be a basic open set of Ca(Cj(X)), where G E [CZ(X)]<". Suppose p(x) E v + fneG M5, i.e., g(x) = p(x)(g) = O(g) for each g E G. Denote r, = g(x) for each g E G. Since g is continuous at x and X is a Pspace, the set O = g'(rg) is an open neighbourhood of x. We now designate O = n9eGO,, an open neighbourhood of x, and find that if y E 0 then p(y)(g) = g(y) = g(x) = V(g), i.e., p(y) E v + ngEG M., which shows that p is continuous. As for p being open, let O be an open subset of X. Choose x E O and an f E C(X) for which f(x) = 1 and f(X 0) = 0. We now wish to demonstrate that (p(x) + Mf) n p(X) C p(O) from which we will conclude that p is an embedding. If p(y) E p(x) + Mf then 0 = p(y)(f) p(x)(f) = f(y) f(x), so f(y) = 1. Thus, y 0 X O, i.e., y E 0 whence p(y) E p(O). Conversely, suppose that p is a continuous map. Let Z(g) be a zeroset of X. Then M, is a basic open set of Cy(Cy(X)). As p is continuous we have that p'(M,) is an open subset of X. But p'(M,) = {x EX: p(x)(g) = 0} = {x E X: g(x) = 0} = Z(g) This shows that every zeroset of X is open, which is precisely one of the many char acterizations of a Pspace.I 2. Special Topological Properties of Ca(X) As stated in the previous section, we aim to characterize the class of compact vanishing spaces. This is equivalent to determining when Cy(X) is countably tight, for compact X. In addition, we classify those X for which CI(X) is first countable, Frech6t Urysohn, sequential, etc. The motivation comes from two sources: Arkhangel'skii [Ar3] and Okuyama and Terada [OT]; in the first reference the author is interested in Cp(X), and in the second the authors investigate Ck(X), the compact open topology. The reader will note that the corresponding results in our context are much different. We begin with a few results which follow from the preceding section. For the first one recall that for topological groups, "first countable" is equivalent to being "metrizable" (see 38C [Wi].) We need some definitions. Definition 2.1 (i) A space X is said to be of point countable type if every element of X is contained in some compact subset which has a countable base of neighborhoods. Observe that any first countable space is of point countable type. (ii) A space X is called a qspace if for every element x E X there is a sequence of neighborhoods of x, say {U,} such that each sequence {(,} with x, E U, has a cluster point in X. Note that any space of point countable type is a qspace. Proposition 2.2 The following are equivalent for a space X. (i) Ca(X) is metrizable; (ii) Cy(X) is first countable; (iii) Cy(X) is of point countable type; (iv) Cy(X) is a qspace; (v) X is countable. Proof: Clearly (i) + (ii) + (iii) + (iv) and (v) + (i). Suppose (iv) and assume, by way of contradiction, that X is uncountable. By assumption there is a countable sequence of finite subsets of X, say {F,} and F. C_ F,+1, such that {MF,} satisfies the hypothesis of a qspace. Let F = UF, and choose x E X F. For each n choose fn E C(X) such that f,(x) = n and f,(F) = {0}. Then f, E MF, but because no continuous function can agree with infinitely many f,'s at x, it follows that the sequence {f,} does not have a cluster point in Ca(X).I We should observe that the above is also true if we replace Ca(X) by C,(X). The equivalences of (i), (ii), and (v) are shown for C,(X) in Theorem I.1.1 of Arkhangel'skii [Ar3]. A natural generalization of a metric space is that of a submetrizable space. We now classify those X for which Ca(X) satisfies this property. First, we have formally: Definition 2.3 A space X is called submetrizable if there exists a continuous bijec tion of X onto a metrizable space. Theorem 2.4 The following are equivalent for a space X. (i) Cy(X) is submetrizable; (ii) every compact subset of Cy(X) is a Gsset; (iii) 0(f, C,(X)) = No for every f E C(X); (iv) X is separable. Proof: By Theorem 1.1 (iv), we know that (iii) and (iv) are equivalent. Also, it is obvious that (ii)  (iii). Suppose Cy(X) is submetrizable and let p : Cy(X)  M witness this fact. Let K C Cy(X) be compact. As p(K) is compact in a metric space, it follows that it is a Gsset of M, whence K is a Gsset. This shows that (i) implies (ii). Suppose X is separable. Let Y C X be a countable dense subset of X. As Y is dense the map 0 : Cy(X) + Ca(Y) is a continuous injection. By Proposition 2.2, the latter is a metric space, whence Ca(X) is submetrizable.M We could continue this line of investigation; i.e., showing similarities between certain topological properties of Cp(X), Ca(X), and X, but we prefer to shed some light on the differences between Cp(X) and Ca(X). In the last section we saw that Cy(X) is countably tight if and only if X is a vanishing space. As for Cp(X) we have the following theorem which may be found in Arkhangel'skii [Ar3], Theorem II.1.1. The u.tl.ji. r... is due to Arkhangel'skii and the necessity is due to Pytkeev. Recall that 1(X) is the Lindel6f degree of X. Theorem 2.5 [Arkhangel'skii, Pytkeev] t(Cp(X)) r if and only I(X") < 7 for every n E N. By the above theorem it follows that for every compact space X, Cp(X) is countably tight. ,Inir!,,I for every Lindel6f Pspace X, C,(X) is countably tight. We would like to show by example that at least for the class of compact spaces this is not the case for Ca(X). Our example should, of course, be uncountable. Observe that to calculate the tightness of Cg(X) there is, in principle, less work involved than calculating the tightness of C,(X) as we need not worry about the vanishing degree of all the finite products of X. Example 2.6 Let X be any uncountable metric space. (In fact, any uncountable space of nice pseudocharacter will do.) Then by Corollary 1.6 we see that X is not a vanishing space and hence Cy(X) is not countably tight. The above example brings up the natural question of whether we may compare the cardinal invariants 1(X) and z(X). If X is a compact metric space then 1(X) < z(X) and equality holds precisely when X is countable. We now demonstrate that this inequality holds for all spaces. Theorem 2.7 For any Tychonoff space X, we have the following inequality: l(X) < z(X). Proof: Let z(X) = a. Let {Oi}ier be an open cover of the space X with II = 1(X). Let F E [X]<`. There is a GF E [I]<" such that F C UiEGF,0. Let HG, = X UiEGFOi; note that HG, is closed. There is some zeroset ZF for which F C ZF C UiEGFOi. Let Z = {ZF : F E [X]<}. Then Z is a zcover of X. So there is a zsubcover Z' of size a. Let I' = U{GF : ZF E Z'}. Observe that I'l < a as each GF is finite. By construction, {O,},Ig is an open subcover, whence 1(X) < a. Corollary 2.8 Every vanishing space is Lindel6f. Hence every vanishing space is normal. At this point there are a number of natural questions; for example: (1) are there any uncountable compact vanishing spaces; (2) how do the following compare for Cy(X) when X is compact: metrizability, first countability, and the properties of being FrechttUrysohn, sequential, and countably tight? First we show that the vanishing degree of aD, the onepoint compactification of an uncountable discrete set D, is w, whence oD is a vanishing space. This answers (1). Example 2.9 aD is an uncountable compact vanishing space. Proof: Let Z be a zcover of aD. Without loss of generality we assume that Z consists of zerosets which contain the point at infinity. It follows that all the zerosets in Z are cocountable. To each Z E Z choose g E C(aD) for which Z(g) = Z and let S be the join of the g. Select an arbitrary f E S. Then coz(f) is countable. We proceed by induction. To each F E [coz(f)]<', choose fF E S such that fF(F) = 0. Let S = {fF E SIF [coz(f)]<} U {f} and AU = U coz(g). gESI Since S1 is countable we obtain that As is countable. Next, for every F E _,] we choose fF E S such that fF(F) = 0. We let 2 = {fF E SIF E [Ai]<} U S and A2 = U coz(g). ses, The induction step is now clear and so we have the following sets: Si,S2,... and A, Az,.... Let S = Ui S, and A = Ugs coz(g). S is countable, whence so is A. Let = {Z(g): gE A}. Then Z is countable. Finally, we would like to show that Z is a zsubcover of Z. Let F E [aD]<. Write F = Fi U F2 where Fi = F n a and F2 = F A. Observe that every gE S vanishes on F2. Next, since S is the union of the increasing collection of sets S, it follows that there is a natural number n such that F1 C An. Then choose f E Sn+1 for which f(F1) = {0}. As we said above f(F2) = {0}, whence f(F) = {0} and we are done.0 Remark 2.10 Observe that the above argument shows that the vanishing degree of W* is w. Next, we are after when Ca(X) is FrechdtUrysohn or sequential. The motiva tion for this problem again is obtained from C,theory, namely, the following result due, independently, to Gerlits and Pytkeev, (see [Ge], [Py], or [Ar3].) Recall that a space is scattered if every subset contains an isolated point. Theorem 2.11 Let X be compact. Then the following are equivalent: (i) Cp(X) is a FrechdtUrysohn space; (ii) C,(X) is a sequential space; (iii) X is scattered. Now, observe that for Cp(X), tightness is not a strong enough property to distinguish between compact spaces, as Cp(X) is countably tight for all compact X. As for Ca(X), the tightness does reveal distinctions between compact spaces. The next theorem demonstrates these differences. The idea for the proof of the theorem is taken from the proof of Theorem 2.11; we suggest Arkhangel'skii [Ar3]. Theorem 2.12 Let X be compact. Then the following are equivalent: (i) Ca(X) is countably tight; (ii) X is a vanishing space; (iii) X is scattered. Proof: That (i) and (ii) are equivalent follows from Corollary 1.5. Suppose X is a compact vanishing space. By Example 2.6, we obtain that Cy([0, 1]) is not countably tight. Since tightness is hereditary, it follows that X cannot be mapped continuously onto [0,1]. Otherwise, C 1\ I. 111 + Ca(X). But this is one of the many characterizations of scattered spaces. Hence, (ii) + (iii). Suppose X is a scattered compact space. V. V. Uspenskii has shown that the Pification of a scattered compact space is a Lindelof Pspace (see Lemma II.7.14 [Ar3].) Consider the embedding Xs + X. This induces an embedding of Cy(X) + Ca(X6). Since Ca(Xs) is countably tight and tightness is hereditary, it follows that CW(X) is countably tight.E As of yet we are not able to determine whether countable tightness, Frechtt Urysohn, and sequential are equivalent conditions for Ca(X). CHAPTER 5 FILTERS OF IDEALS 1. Gabriel Filters of Ideals and Rings of Continuous Functions The previous chapters have focused on topological properties (some based on cardinal invariants) of CW(X). We now turn our attention to a more algebraic point of view concerning the filter 5. Our basic goal is to study a particular ring of quotients of C(X) which arises naturally from our filter J. For the ease of the reader we shall take the time to review some basic facts concerning filters of ideals, in particular, the socalled Gabriel filters. Throughout, A will denote a commutative ring with identity. Our basic reference is Stenstrom [St]. Recall that for a E A and J an ideal of A, a'J = {bE A:abE J} and a1 = {b E A : ab = 0}. Also, AMod denotes the class of all left Amodules. Definition 1.1 Let 6 be a collection of ideals of A. We say 0 is a topologizing filter of ideals (or simply a filter) if it satisfies the following criterion: (i) if I < J and I E 6, then J E O; (ii) if I, J E then I n J E . If 6 happens to satisfy the following condition then we call 6 a multiplicative filter. (iii) If I, J E then IJ E 6. Lastly, 0 is called a Gabriel filter if it satisfies (iv) if J is an ideal of A for which there is an ideal I E 0 such that for each a E I a'J E 0, then J E 6. In the case that 0 is a filter, we define the 0pretorsion radical as follows. to(A) = {x E A: x E 0}. If 0 happens to be a Gabriel filter, then we call to the torsion radical. For additional information on torsion radicals we suggest the reader consult Stenstrom [St]. Let us remark that, in the noncommutative context there is a distinction between "filter of ideals" and "topologizing filter of ideals." We need not be concerned about that here, as the concepts coincide for commutative rings (see [St].) Moving on, we would like to show a few simple ways of constructing new filters from old ones. Proposition 1.2 Let 0 be a given filter of ideals on the ring A. Define the radical of 6 as follows: V = {I Then v6 is a filter of ideals of A. Moreover, if 6 is a multiplicative filter (resp. Gabriel filter), then V/ is a multiplicative filter (resp. Gabriel filter). In case 6 = v/, we say 0 is a radical filter. Proof: That V' is a topologizing filter, follows from elementary facts concerning the radical of an ideal; we suggest Atiyah and MacDonald [AM]. As for the radical respecting the multiplicative property this is also trivial. We now show the radical respects the Gabriel property. Let J, I be ideals such that for every a E I, a'J E vrO. Let us assume that J < I. We want to demonstrate that J E vl; i.e., that VJ E 6. This is demonstrated once we show that for every f E VI, f/Jj E 6. Now, if f E vJ, then f" E I for some natural n. Next, we obtain that (f")'J E V. Since, (f')'J = {a E A: f'a E J} < {a E A: (fa)" E J} = f'1J, and the ideal on the left is in the filter v/, it follows that the ideal on the right lies in V .l Let 6 be a filter. In case Q5 is not a Gabriel filter then a natural question is whether there is a way of obtaining a Gabriel filter canonically from (. The answer is affirmative, and from Chapter VI.5.4 of Stenstrom [St], we have the following description of the Gabriel filter generated by (, denoted G((). Theorem 1.3 Let ( be a filter of ideals on A. Then G(Q5) = {I < A : for every I < J,J A, there exists a E J such that a'J E 6}. We now give a more constructive way of obtaining G(6). Definition 1.4 Let 6 be a filter. Define a collection of ideals by transfinite induction. Let ( = J'""' ) For a nonlimit ordinal a let J(')(0) = {I < A : 3/ < a, 3J E J()(0) such that for each a E J3, < a for which a'I E J(7")(6)}. A quick check shows that for each nonlimit ordinal a, J(')(6) is a filter. As such, for each limit ordinal define JM)(() = U J()(0), I Finally, let J(5) = U, J(')(6). It is clear that J(() is a welldefined set. We now show that J(O) is a Gabriel filter. Lemma 1.5 Let 5 be a filter on the ring A. Then J(O) is a Gabriel filter. Moreover, J(O) = G(O). Proof: It should be clear that if J(6) is a Gabriel filter, then J(6) = G(O). First, observe that by properties of ordinals J() = U J')(0), where a = II(A)I. Also, J(O) is a filter. Next, suppose J satisfies the Gabriel con dition relative to J(O). Then there is an ordinal a < a and a I E J5")() such that for each a E I there is an ordinal y, a for which a'J E J(')(0). Let T = sup{y : a E I}. Then r < a and we have that J E J(T+)((), whence J E J(6) and so J(O) is a Gabriel filter.I Corollary 1.6 Let 6 be a filter. Then 0 is a Gabriel filter if and only if 6 = J(')(). For a filter 6 we call J(O) the Gabrielization of 6. This makes sense as J(O) = G(0); i.e., J(() is the smallest Gabriel filter containing 6. Recall the following theorem. Theorem 1.7 [St, VI. (Thm. 5.1) Prop. 4.1] Let A be a ring. There is a onetoone correspondence between the following classes. (i) (Gabriel) Filters of ideals on A. (ii) Classes of hereditary (torsion) pretorsion classes of modules. We now give a sketch of the correspondence referred to in the theorem. Definition 1.8 A hereditary pretorsion class is a class of modules which is closed under subobjects, quotient objects, and direct sums. Similarly, a hereditary torsion class is a hereditary pretorsion class which is also closed under extensions; that is, if L, N are in the class and 0+L M+ M+N0 is an exact sequence, then M is in the class. If 0 is a (Gabriel) filter of ideals then the class of modules T= {M E AMod : E6 for each x M} is a hereditary (torsion) pretorsion class. Conversely, if T is a hereditary (torsion) pretorsion class, then = {I Since multiplicative filters fall between ordinary filters and Gabriel filters a natural question is whether there is a way of characterizing them in terms of hereditary pretorsion classes. First, a lemma, which is known, but whose proof we give anyway for the sake of completeness. Lemma 1.9 Let 0 be a filter on the ring A. Then 0 is a multiplicative filter if and only if whenever I E 6, then 12 E 6. Proof: The necessity is clear. As for the sufficiency, let I, J E (. We wish to show that IJ E (. As 5 is a filter it follows that In J E 6. Observe that by our assump tion (In J)2 E 6. Since (In J)2 < IJ, it follows that IJ E 6.M. Suppose J is an ideal of A, I E and for each a E I we have I, < a'J for some ideal I, E (. Then the Gabriel condition implies that J E 6. The natural thing to try is to impose the restriction that we can actually take I. = I. What does it say about 6 if J is still pulled into 0? This is the context of the following discussion. It is in the same vein as Definition 1.4. Definition 1.10 Suppose 0 is a filter. Define M(0)(6) = (. We define inductively the following sets for each natural number n. 1'". "'i = {J E (A) : there is an I E M(")(S) such that I < a'J, Va E I} It is easy to show that each M(n)(6) is a filter. Let M()(6) = U,eM(")(O). A quick check shows that M(')(6) is a filter. In fact, we have more than that. We have the following lemma. Lemma 1.11 Let A and 6 be as above. Then M(")(6) is a multiplicative filter. Proof: Recall that all we need to show is that if I E M(')(6) then I2 E M(")(0). So we suppose that I belongs to the filter in question. Then I E M(n)(0) for some natural n. Now, since I it follows that 12 E M("n+)(6), whence M(")(6) is a multiplicative filter.I As before, it is a simple exercise to show that the multiplicative filter con structed in the above definition is in fact the multiplicative filter generated by 6; i.e, M(")(6) = {J E I(A) : I" < J, for some n E N and I E 6} We are now able to state the following corollary. Corollary 1.12 Let A be a ring and 6 be a filter of ideals of A. Then 6 is a multiplicative filter if and only if 6 = .I' il = M(")(O). With the above construction in hand, we now set out to demonstrate there is a correspondence between multiplicative filters on a ring and certain classes of its modules. We know that Gabriel filters correspond to torsion classes. Since the multiplicative condition is weaker than the Gabriel condition, we aim for a weaker condition placed on extensions of torsion modules. Theorem 1.13 Let A be a ring, 6 a filter and T its corresponding pretorsion class of Amodules. Then 6 is a multiplicative filter if and only if whenever there is an exact sequence 0 L M + N + 0, with L, N E T and M satisfies the following: for arbitrary xl, x2 E A, m E M, xlm, z2m E L imply X1X2m = 0, then M E T. Proof: Suppose 6 is a multiplicative filter and L, M, N are as above. We wish to show that M E T; i.e., m1 E 6 for every m E M (recall the correspondence from Theorem 1.7.) Equivalently, using Corollary 1.12, we hope to find an I E 6 such that I < al'm for all a E I. We use the fact that N = M/L. Let m E M. As N is torsion, miL = (m+ L)' E 6. By our hypothesis, if xim, 2m E L then xix2m = 0. This shows that m1L < x'lm for all x E m'L, whence m1 E 0. Conversely, let I E 6. We want to show that 12 E 6. The following is an exact sequence: 0 + I/I2 + A/I2 + A/I + 0. Since I E 6, A/I E T. It is clear that 1/12 E T. Next, observe that for any m E A, it is true that m1I < Xi(miI2), whenever xl E mtI. This is equivalent to saying that xlm,x2m E I imply that XpX2m E 72, which is the desired condition on A/I2, so that the hypothesis implies that A E I2 E T, whence 12 E 0.0 2. J, The Fixed Filter In the study of C(X) there are natural ways of using the space X to obtain filters of ideals on C(X). Let X be a Tychonoff space and KC a collection of subsets of X which is directed upwards, i.e., given F1, F2 E K there is an F E K such that F1 U F2 C F. Let 5c be the collection of ideals of the form CneF M for some F E 1C. Then c forms a filter base of ideals. Indeed, the filter generated by C is multiplicative as the maximal ideals of C(X) are idempotent. In particular, if KC = [X]<", then we denote Sc by 5, as introduced earlier in this dissertation. We shall use the following convention: if F C X, then we define XF = X F. In particular, if p E X, then Xp = X {p}. Similarly, for a subset S C X we let Ms denote the ideal of C(X) consisting of those continuous functions which vanish on S and Os be the ideal consisting of those continuous functions which vanish on a neighbourhood of S. Definition 2.1 Let I be an ideal of an arbitrary ring A. We let Vs(I) = {P E Spec(A) : I < P} and V(I) = {M E Max(A) : I < M}. If A = C(X), then define V = {M E V(I) : M = M. for some x E X} and I = nl{M : I < Mp}. Note that V1 is precisely the set of fixed maximal ideals containing I. Recall that under the Zariski topology on Spec(A) (resp. Max(A)), Vs(I) (resp. V(I)) is a closed set. Similarly, V1 is viewed as a closed set in X. Proposition 2.2 Let I be an ideal of C(X). Then I = clc,(x)I. Proof: First, observe that since I is an intersection of fixed maximal ideals, it is closed. Therefore, let g E I and F E [X]<'. For each x E F, let r, = g(x). Let Fi = F n Vi and F2 = F VI. Observe that our g vanishes on Fl. To each x E F2, we may find an h, E I such that h,(x) f 0. Then choose a k, E C(X) for which k,(F{x}) = {0} and k,(x) = r,(h,(x))1 (which we may do by complete regularity.) Let h = EEeF2 krh E I. Then h(x) = r, for all x E F.M Lemma 2.3 The open ideals in the a filter are in onetoone correspondence with the collection of finite subsets of X. Furthermore, they are precisely of the form MF, where F E [X]< . Proof: Use the Chinese Remainder Theorem and the fact that the ideals of R are of the above form.E Definition 2.4 We now apply the Gabriel condition to the fixed filter of ideals of C(X). (Recall condition (iv) of Definition 1.1.) is a Gabriel filter if the elements of j are precisely those ideals which satisfy (*) I is an ideal for which there is a finite subset F of X, such that whenever f E C(X) vanishes on F then f'I = MFf for some finite subset Ff. Our next two lemmas shows that 1) the only ideals which can possibly satisfy (*) are those which do not lie in free maximal ideals; i.e., for which V(I) = Vi; and 2) if I does satisfy (*) then V, is finite. Lemma 2.5 Let I be an ideal of C(X) and suppose M E V(I) Vi. Then I does not satisfy (*). Proof: Suppose that I does satisfy (*) and let F be a finite subset of X which witnesses this fact. Pick an f E ME M. Then there is a finite subset Ff of X (in fact a subset of Vi) such that {g : fg E I} = MFf. By choice of M, we have that if fg E I then fg E M. Since f M, a prime ideal, we have that MF,. < M. But this cannot be.I Lemma 2.6 If I satisfies (*) then V, is finite. Proof: Let F be a finite subset of X which exhibits that X satisfies (*). Suppose \VIj > w and let {M,, : i E N} be a countable subset of V, which is disjoint from F. To each i < w choose f, E C(X) for which s, E coz(f,) and F C Z(f,). Recall that the cozerosets of X are closed under countable unions (1.14,[GJ]), whence there is a cozeroset containing each si yet is still disjoint from F. Choose an arbitrary f E C(X) whose cozeroset equals the one in question. Then consider f'l; we have that {g : fg E I} = MG, for some G E [X]<". Now, if fg E I then fg E M,, for every natural i. But by our choice of f we have that g E niEfNM,, i.e., MG < niNMi. From this contradiction we conclude that VI is finite.E The following lemma is straightforward, but we shall have cause to use it on more than one occasion. Lemma 2.7 Suppose I satisfies (*) and I < J then J satisfies (*). Proof: This is part of what goes in to showing that to each filter of ideals on a ring there is a smallest Gabriel filter consisting of those ideals which satisfy (*). U Corollary 2.8 If I satisfies (*) then so does I. It is easily shown that if X is a Pspace then 5 is a Gabriel filter since every ideal of C(X) is an intersection of maximal ideals. The natural question then is to consider whether this may be generalized to arbitrary Tychonoff spaces. First, it should be noted that the only prime ideals which satisfy (*) are the fixed maximal ideals. To see this observe that if the nonmaximal prime P satisfied (*) and MF witnesses that, then without loss of generality we may assume that p E F, where Mp is the unique maximal ideal containing P. Next, select an f E MF P and then, since f'P = MG for some finite G, it follows that we may choose a g E MG P. But this means that fg E P even though neither f nor g is in P, contradicting primeness. We continue the above argument and elaborate on this matter in the next theorem, which is the main result of this section. Theorem 2.9 Let X be a Tychonoff space. Then g is a Gabriel filter. Proof: What we are aiming to show is that if I is an ideal which satisfies (*) then I is in fact an intersection of a finite number of maximal ideals. We shall have cause to use the following description of ideals. This can be found in Gillman and Kohls [GK], Proposition 5.2. For an arbitrary ideal I, we have I= n (, OP), and note that if I Z MP, then (I, OP) = C(X). Thus, if I is the ideal in question we actually get I = (I, p). Now, if for each p E VI we have (I, Op) = Mp, then the proof of the theorem is done. Thus, we may assume that for some p E Vi, (I, Op) < Mp. By Lemma 2.7, (I, Op) satisfies (*). Observe that Op < (I,Op) < Mp. What we have just done is reduce the question to looking at an ideal J satisfying (*), where O, < J < Mp. We aim to show that J = Mp. So suppose Op < J < Mp and J satisfies (*). Now, let MF, for F E [X]<' witness this fact. Without loss of generality, we assume that p E F. We claim that, if g E Mp, then g'J > Mp. Once the claim is shown observe that for arbitrary f,g E Mp, g E {h : hf E J}; i.e., fg E J. This then implies that M2 < J < Mp. But as shown in Gillman and Jerrison [GJ], exercise 2B, M, = Mp and so J = Mp. Thus, to finish off the proof of the theorem all we need is to prove the claim; i.e., g E Mp implies g'J > Mp. Let g E M,. Enumerate F = {p,xi,...,x,,,} and let r, = g(x,) for each 1 < i < m. By the Tychonoff property, it follows that there is a k, E Op, hence ki E J, for which k,(xj) = 5,jri. Let k ki l Observe that k E J and that g k E MF. Finally, we have the following qualities: {h : hg E J} = {h : h(g k + k) E } = {h : h(g k) E J}. But since g k vanishes on F we get that g'J E 5 for all g E Mp. Since J < g1J and V(J) = {Mp} it follows that Mp < g'J. N 3. gRing of Quotients Recall that in the sixth section of Chapter 1, we discussed Utumi rings of quotients of a given ring A. Due to Theorem 1.6.10, these are precisely the ring extensions of A which lie canonically inside Q(A). Both q(A) and Q(A) were constructed via direct limits of fractions over particular filters of ideals. Recall that Q(A) is constructed using the multiplicative filter of dense ideals. In fact, it is true that 0), the collection of all dense ideals of A is a Gabriel filter. Moreover, if D' is a Gabriel filter consisting of dense ideals then A < A,, < Q(A), and hence a ring of quotients in the sense of Utumi. Thus, the construction of a Gabriel ring of quotients is in some sense a generalization of the process we saw in the first chapter. Throughout this section 05 will be an arbitrary Gabriel filter on a ring A, and as such we may define a ring of quotients of A with respect to 0. We remind the reader of the construction. First, recall that the torsion radical associated to the Gabriel filter S is te(A) = {a E A: a' E 6}. Next, we define the ring of quotients with respect to 8, Ae = lim HomA(I, A/t(A)), IE6 and give it a ring structure in the natural way. We denote the canonical homomor phism from A into AO by OA (see Ch. IX, [St].) From here on out, by "a ring of quotients" we shall mean a subring of Q(A) containing A canonically; otherwise, we shall say a ring of quotients with respect to the filter 6. Recall, from Chapter 1, the following example. If 0o denotes the Gabriel (sub)filter of i) whose elements are the regular ideals, we then obtain An, = q(A), the classical ring of quotients of A (recall that an ideal is said to be regular if it contains a regular element; i.e., a non zerodivisor.) We briefly turn our investigation into the fixed filter to questions that typically arise when dealing with Gabriel filters on an arbitrary ring. The first deals with localizations. Recall that if P is a prime ideal then the complement of P in A forms a multiplicative set in which case one may form the localization Ap. There is a corresponding Gabriel filter, namely, Op ={I The ring of quotients with respect to Op is in fact the localization at P. In general, if one has a collection of prime ideals, say P C Spec(A), then one forms the Gabriel filter Op = {I < A : Vs(I) n P = 0}. We determine when we may view the fixed filter on C(X) as such. We shall have use for the following proposition. Proposition 3.1 (Proposition 6.13, [St]) Let 0 be a Gabriel filter on the ring A. Then the following are equivalent: (i) 0 is equal to Op for some P C Spec(A); (ii) 0 = OD( ), where D(O) = {P E Spec(A) : P 1}; (iii) For every ideal I with I 0 6, there is a prime P containing I such that P 6. Applying this to the fixed filter on C(X), we first observe that D(M) = Spec(C(X)) {M, : x E X} and so D() = {I < C(X) : Vs(I) = V1}. As an example, if X is discrete; i.e., j has a basis consisting of finitely generated (equiva lently, principal) ideals, then 5 = 3D(a). In fact, we get more, but first a remark. Remark 3.2 Recently, Mulero (see [M]) showed that for C(X) whenever the radical of an ideal is a zideal, then the ideal is semiprime. (Recall that an ideal I of C(X) is called a zideal whenever Z(f) = Z(g) and f E I, then g E I.) Specifically, if V7 = M, is a maximal ideal, then I = v i = M. We use this fact now. Theorem 3.3 The fixed filter j is of the form Jp for some P C Spec(5). Proof: By the observation made prior to the remark, it is evident that we wish to show = SD(a). It is clear that 3 C oD(a). So, let I E ID( ). It follows that Vs(I) = V(I) = Vi. Thus, the radical of I is the intersection of the fixed maximal ideals containing I. By our remark, we have I = Mv,. Since V(I) = Vi it follows that VI is closed in /X, whence is compact in X. We wish to show that VI is finite, so suppose not. The idea then, is to find a nonmaximal prime ideal P for which I < P, contradicting our choice of I E 1D(). Recall the description of ideals in C(X) given in Theorem 2.9. Namely, I= n(I, O). If there is no nonmaximal prime above I, then (I, Op) = Mp, for each p E V1. Here, we are using Mulero's result as well as the fact that the sum of zideals is again a zideal ([GJ],14.8). Since V1 is compact and infinite we may choose a point p E Vi which is not a Ppoint in X. Next, pick an f E C(VI) such that f(p) = 0, yet f does not vanish on a neighbourhood of p; i.e., f E M O, relative to C(Vi). Since compact subspaces are C*embedded, we may choose an extension, say g, of f to all of X. As, g vanishes at p and Mp = (I, Op), there are fi E I and f2 E Op for which f = f\ + f2. Now, there exists an open neighbourhood, 0, of p such that f2(O) = 0. Re stricting to V, we get that f = gly = f2, since f1 vanishes on Vi. Also, f21i vanishes on O n VI, so that f2 E Op (relative to V1.) But this contradicts our choice of f2, whence we conclude that Vi is finite.E Putting Theorems 2.9 and 3.3 together along with the fact that every ideal in Sis semiprime we obtain the following corollary. Corollary 3.4 Let X be a Tychonoff space. Then the fixed filter on C(X) is a radical Gabriel filter, which is of the form 3 = iD(W)* We shall spend the rest of this section describing those spaces for which C(X)W is a ring of quotients of C(X); i.e., C(X)a < C(X)s, and also investigate when C(X) <_ C(X)so. This is tantamount to asking when 5 is a subGabriel filter of 0 and 3o, for if I E 3 and I is not dense then letting f e I' we see that VA(f) = 0, so that iA is not a monomorphism. We can easily characterize those spaces X for which C(X)g is a ring of quo tients, i.e., an extension of C(X) contained in Q(X). Lemma 3.5 C(X)a is a ring of quotients of C(X) if and only X has no isolated points. Proof: Recall that an ideal I of C(X) is dense if and only if coz(I) = UfI coz(f) is a dense subspace of X. Thus, since coz(nfeFMz) = F we obtain our result.E Lemma 3.5 shows that C(X)a is a ring of quotients of C(X) if and only if j C 3. As for characterizing those spaces X for which C(X)y < C(X)so, the situation is not as trivial. There are examples of spaces X for which C(X) <: C(X)so yet 3 Z Do (we shall present these shortly.) We do have the following result, though. Recall that a crowded space is one which has no isolated points. The term is due to van Douwen, and we are particularly fond of it. Proposition 3.6 For a crowded Tychonoff space X, the Gabriel filter 3 is a sub Gabriel filter of Zo if and only if X has no almost Ppoints. Proof: If 3 C o, then each M, contains a regular element; i.e., a continuous func tion whose cozeroset is dense, or, equivalently, whose zeroset has empty interior. It then follows that each x is not an almost Ppoint. The converse follows by applying the above in reverse, noting that the set of regular elements forms a multiplicative set, whence for each F E [X]<', MF is a regular ideal.E Remark 3.7 The class of crowded spaces with no almost Ppoints contains all metric spaces, or more generally those X which have nice pseudocharacter, and all compact crowded spaces whose cellularity is countable. There is also an example of a compact . ,: II. disconnected space which is not extremally disconnected and has no almost Ppoints (see [HM] after Question 2.6.7.) For the definition of fraction dense space we suggest the reader consult Hager and Martinez [HM]. If X is crowded then the question of describing the ring of quotients C(X)W is a natural one. We know from Fine, Gillman, and Lambek [FGL] the following inequality: C(X)y < limC(XF), where the F's range over all finite subsets of X. This then leads us to determining when the following qualities hold: (1) C(X) = C(X)F, (or, as Stenstrom [St] puts it, C(X) is Jclosed,) (2) C(X) = lim, C(XF), (3) C(X) = lim, C(XF). We borrow notation from Ball and Hager [BH2] and write C[] = limC(XF). Now, if each XF is Cembedded in X, that is C(X) = C(XF), then it is clear that C(X) = C[J]. The converse is also true. The following lemma shows another characterization of this phenomenon and answers (3) above. First, a useful definition. Definition 3.8 Let X be a Tychonoff space and p E X. We call p a C*point (resp., a Cpoint) if Xp is a C*embedded (resp., Cembedded) subspace of X. If every point of X is a C*point (resp., a Cpoint) then we call X a C*space (resp., a Cspace.) Lemma 3.9 The following are equivalent for any crowded Tychonoff space X. (i) C() = C[]; (ii) C(X) C(XF) by restriction for each finite set F; (iii) each XF is Cembedded in X; (iv) X is a Cspace; (v) each XF is C*embedded in X and O(x, X) > No for every x E X; (vi) X is a C*space and 0.(X) > 0o. Proof: That (i) + (ii) 4 (iii) 4 (iv) 4 (i) follows straight from the definitions. That (v) and (vi) are equivalent is obvious. Thus, we prove (iii) and (iv) are equivalent. First, suppose that each XF is C*embedded in X and O(x, X) > No for every x E X. If XF is not Cembedded in X then there is an unbounded f E C(XF) with no continuous extension to all of X. Without loss of generality, we may assume f is positive and (by translation) f > 1. Now, f1 exists in C*(XF). By our assumption, we may choose g an extension of f' to all of X. It follows that Z(g) C F. Since no point of X is a zeroset we have that Z(g) = 0, i.e., g is invertible. Finally, a quick check shows that g' extends f continuously to all of X, a contradiction. Therefore, each XF is Cembedded in X For the reverse direction, first observe that Cembedding implies C*embedding. Next, let x E X and suppose r(x, X) = No; that is, there is some g E C(X) for which Z(g) = {x}. Then letting f be the restriction of g to Xp, we obtain an invertible element of C(XF). Finally observe that f is an unbounded continuous function on XF which has no continuous extension to all of X.E Remark 3.10 Observe that (iv) and (v) of Lemma 3.9 are equivalent for an arbitrary Tychonoff space, except that in (v) all that is needed is that #(x,X) > No for every nonisolated x E X. Example 3.11 Let X be a compact extremally disconnected space. As X is ex tremally disconnected, each XF is C*embedded, in fact every open set is C*embedded. Recall from Proposition 1.4.5, that for compact spaces #(x,X) = (x,X) for all x E X. Since X is extremally disconnected, hence an Fspace, we conclude that each point x with X(x,X) = 0i(x,X) = w is isolated, whence X satisfies the condi tions of Lemma 3.9 (see [GJ], 14.N.2.) Thus, every compact extremally disconnected space is a Cspace. By similar arguments, every extremally disconnected space is a C*space. The space E (see [GJ]) is an example of an noncompact extremally discon nected which is not a Cspace. In E, the unique nonisolated point a has countable pseudocharacter. In the next section we shall discuss C*spaces at length; giving examples and exhibiting techniques to construct nonextremally disconnected spaces which are C spaces (or C*spaces.) We will also demonstrate that C*spaces (resp., Cspaces) have properties which are similar to those of extremally disconnected spaces. For the rest of this section, we continue our discussion of the coincidence of rings of quotients in relation to Pspaces. Recently, in unpublished work we have been able to show that for any nowhere dense closed subset V of X (*) Homc(x)(Ov, C(X)) = C(Xv). This has the following implications. Proposition 3.12 Let X be a crowded Pspace. Then C(X)y = C[S]. Furthermore, C(X) = C(X)w if and only if X is a C*space. Proof: The first statement follows from (*) and the fact that O, = Mp for all p e X. The second statement is just an application of Lemma 3.9 to Pspaces. Xp being Cembedded in X is equivalent to Xp being C*embedded in X, since the pseu docharacter of a nonisolated Ppoint is at least ,i.M As yet we have no examples of Pspaces with nonmeasurable cardinality which satisfy the conditions of Proposition 3.11. For a nonisolated C*point in a Pspace, let Y be a Pspace which is not realcompact. Then let x E vY Y and define X = {x} U Y. Then x is a C*point of the Pspace X. Next, here is an example of a compact, almost Pspace which is a C*space and hence a Cspace, albeit under certain stringent settheoretic conditions. This example also exhibits a space for which C(X) <5 C(X),, yet 3 Z ZDo (see the discussion prior to Proposition 3.6.) As of yet we have no example strictly in ZFC which makes this work. Example 3.13 Under a certain form of Martin's axiom, (we urge the reader to con sult van Douwen, Kunen, and van Mill [vDKvM] 3.6 for the exact model) it is true that every point of ON N is a nonisolated C*point. Thus, /N N is an almost Pspace and a C*space. Thus, applying Lemma 3.9 and Proposition 1.6.15 to 3NN we obtain that C(X)= C(X)a = C(X)s,. That 5 O So follows from Proposition 3.6. It turns out that the statement "PN N is a C*space" is in fact independent of ZFC; it is shown in Fine and Gillman [FG] that, assuming the continuum hypothesis, no point of 3N N is a C*point. 4. C*spaces We now turn our attention to those spaces whose ring of quotients C[3] coincides with C(X). More generally, we are interested in the class of C*spaces. Recall from Definition 3.8 that X is a C*space if and only if each Xp C X is C*embedded. This is equivalent to saying that X C O3Xp, for all p E X. We would like to generate a basic theory of this class of spaces. As we saw in Example 3.11, every extremally disconnected space is a C*space (in fact, van Douwen showed that every EDpoint is a C*point.) Thus, some of our results shall parallel existing ones for EDspaces. We would also like to accomplish two goals: 1) give a technique for constructing nonED C'spaces, and 2) create a C*cover. We commence by first tackling (1) and save (2) for the next section. Remark 4.1 If X is a noncompact Tychonoff space then we denote its growth by X*; i.e., X* = fX X. Similarly, if Z C X is a noncompact zeroset of X we let Z* = clfxZ Z. The butterfly space of X, denoted Abx, or if X is understood we simply write (b, is the space obtained by taking the topological sum of two disjoint copies of /pX and identifying a point in X* with its counterpart in the other copy. We leave the other points alone. The corona or growth of the butterfly space is simply the copy of 3X X under the identification. It will be understood that the copies of X are labeled X1 and X2, respectively. If f, E C(P3X),i = 1,2 and their restrictions to the growth of X agree, then there is a unique function on dbx which agree on the X,'s. We denote this function by fi U f2. Butterfly spaces are used in showing that there are spaces of finite rank whose ring of continuous functions are not quasinormal. For the appropriate definitions we suggest Larson [Ln]. In current dissertation work, Chawne Kimber has elaborated on this paper. Definition 4.2 Consider a Tychonoff space X. We assume that X is noncompact. Let p E X*. Recall, from Chapter 1, that Z E AP if and only if p e Z*. We call p an almost compact point of X if there is a Z E AP for which Z* = {p} and p E intpxclpxZ. Recall that a space is called almost compact if X* is a singleton. Hence, if X is almost compact then p E X* is an almost compact point of X. This also holds if X* is finite. Almost compact points are useful in the study of butterfly spaces and their place in the class of C*spaces. We will shortly show that if X has no almost compact points then every point of the growth of its butterfly space db is a C*point. We have the following which characterizes almost compact points. Lemma 4.3 Let p E X*. Then p is an almost compact point of X if and only if p is an isolated point of X*. Proof: Suppose Z E AP, Z* = {p}, and p E intZ*. It follows that intZ* n X = {p}, whence p is an isolated point. If p is an isolated point of X*, then there is an open set O of PX for which O n X* = {p}. Choose a zeroset neighbourhood Z E Z[13X] of p such that Z C O. It follows that Z n X* = {p}.m Theorem 4.4 Let X be a noncompact Tychonoff space with no almost compact points. Then every point of the growth of the butterfly space of X is a C*point. If X* happens to be C*embedded in fX then the converse holds. Proof: Let p E X* and f E C'*(d6). We would like to extend f to all of db. Now, if we let gi = fix,, i = 1,2, then, by density, there are unique continuous extensions f, to all of 3Xi. Observe that both fi and f2 agree on all points of the corona except, possibly, for p. We claim that fi(p) = f2(p), so that by the Pasting Lemma we get fi U f2 as a continuous extension of f to all of d(. We proceed to prove the claim. Now, by restricting fi and f2 to the corona we have two continuous functions on X* which agree everywhere except, possibly, for p. If they were to disagree at p then it would follow that p is an isolated point of X*, but our assumption says there are no isolated points of the corona. Thus, it follows that fi(p) = f2(p) and our claim is proven. Next, assume that X* is C*embedded in fiX. Suppose p is an isolated point of the corona. Extend the zero function on the corona to fi and the characteristic function on {p} to f2 on fX. Then by deleting p we have a continuous function fi U f2 on dbp which can not be extended continuously to p.0 Example 4.5 Let X be a locally compact, noncompact extremally disconnected space (one may take X to be crowded if so desired) and consider Ax. dbx is a C'space if and only if X contains no clopen subspaces which are almost compact; e.g., N. Suppose Qbx is a C* space. By Theorem 4.4, since we are assuming that X is locally compact and hence X* is C*embedded in 3X, it follows that X has no almost compact points. Suppose S C X is a clopen subspace which is almost compact. Let {p} = S*. As S is an open subspace of an extremally disconnected it is C*embedded and so cldxS = 3S, whence p E X*. Therefore, there is a zeroset S of X with {p} = S* and claxS is open. This means that p is an almost compact point, a contradiction. Conversely, suppose bx is a not a C*space. Again, by Theorem 4.4, this means there is a p E X, which is an almost compact point. Let Z E Z[X] with p E intoxclpxZ and clpxZ n X* = {p}. If we let S be the closure of the intpxdclxZ, as fX is extremally disconnected, it follows that S is a clopen subset of PX (hence a zeroset) with the same properties stated above for Z. Similarly, it can be shown that Sp is a clopen subspace of X with the same properties. Finally, observe that the desired properties imply that S is an almost compact space. Thus, Example 4.5 shows that there are (compact) C*spaces which are not extremally disconnected (not even Fspaces.) Nonetheless, the two classes are cer tainly similar. We conclude this section by characterizing Cspaces and C*spaces in two ways that are similar to the various characterizations of EDspaces. Proposition 4.6 A space X is a C*space (resp., a Cspace) if and only if X (resp., vX) is a C*space (resp., a Cspace). Proof: We shall demonstrate for Cspaces. The respective proof for C*spaces is similar. First, suppose that X is a Cspace. Let p E vX and f E C((vX)p). If p E vX X then we have X C (vX), C vX and so f may be extended to all of vX. If p E X then we have two cases. (i). Suppose p is an isolated point of X, then it follows that vX = vXp U {p} and so (vX)p = vXp. So p is an isolated point of vX whence a Cpoint of vX. (ii) Suppose p is not an isolated point of X. Let g be the restriction of f to X,. Since X is a Cspace let h be the extension of g to all of X and finally k = h". By density it follows that k is an extension of f. For the converse we assume that vX is a Cspace. Let p E X and f E C(X,). By exercise 8G.2 of Gillman and Jerrison [GJ], it follows that Xp is Cembedded in (vX)p. Thus we may continuously extend f to (vX),. Finally, since vX is a Cspace we may extend f to p.M The proof of the next proposition is straightforward; therefore, we leave it out. Proposition 4.7 Let X, be a C*space (resp., a Cspace) for each A E A. Then (G]EA X, is a C*space (resp., a Cspace). Recall that an extremally disconnected space is characterized by the property that X is extremally disconnected if and only if every open set is C*embedded (see [GJ].) To obtain the analogous characterization for a C*space we need the following definition. Definition 4.8 Let X be a space and A C X. We call A an almost open subset of X if A intxA is a finite set. ,i,,l.lrl:,, we call A an almost closed subset of X, if clxA A is finite. For brevity, we shall use a.o. and a.c. for almost open and almost closed sets of X, respectively. Rfj,(X) denotes the collection of all a.o. regular closed subsets of X. Notice that a dense a.c. subset is necessarily open and of the form XF for some finite F. Theorem 4.9 The following are equivalent for a Tychonoff space X. (1) X is a C*space; (ii) each dense a.c. subset of X is C*embedded; 
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