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## Material Information- Title:
- Relations on spaces
- Creator:
- Lin, Shwu-Yeng Tzeng, 1934-
- Publication Date:
- 1965
- Language:
- English
- Physical Description:
- iii, 51 leaves : illus. ; 28 cm.
## Subjects- Subjects / Keywords:
- Distance functions ( jstor )
Hausdorff spaces ( jstor ) Homomorphisms ( jstor ) Integers ( jstor ) Mathematical theorems ( jstor ) Mathematics ( jstor ) Semigroups ( jstor ) Topological spaces ( jstor ) Topological theorems ( jstor ) Topology ( jstor ) Dissertations, Academic -- Mathematics -- UF Generalized spaces ( lcsh ) Mathematics thesis Ph. D Topology ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis - University of Florida.
- Bibliography:
- Bibliography: leaves 48-50.
- General Note:
- Manuscript copy.
- General Note:
- Vita.
## Record Information- Source Institution:
- University of Florida
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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- 000554490 ( ALEPH )
13406797 ( OCLC ) ACX9333 ( NOTIS ) AA00004945_00001 ( sobekcm )
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RELATIONS ON SPACES By SHWU-YENG TZENG LIN A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA August, 1965 ACKNOWLEDGMENTS The author wishes to acknowledge her indebtedness and express her sincere gratitude to Dr. A. D. Wallace, Professor of the Department of Mathematics and Chairman of her Supervisory Committee, for his guidance and sugges tions throughout the preparation of this dissertation. She wishes to express her deep appreciation to Professor A. R. Bednarek, who read the entire manuscript in detail and made numerous corrections and improvements. Thanks are due also to all members of the Supervisory Committee for their encouragements. The author is grateful to Dr. D. C. Rose for correcting the language. She is also grateful to her husband Dr. Y.-F. Lin, who was always patient in discus sing mathematics with her and who rendered many valuable suggestions. And she extends her thanks to Mrs. K. P. Grady for her excellent work in typing the manuscript. ii TABLE OP CONTENTS Page ACKNOWLEDGMENTS ii Chapter I.PRELIMINARIES 1 II.A CHARACTERIZATION OF THE OUTPOINT ORDER ON A TREE 11 III.FIXED POINTS AND MINIMAL ELEMENTS 25 IV.FIXED POINT PROPERTIES AND INVERSE LIMIT SPACES 38 BIBLIOGRAPHY 48 BIOGRAPHICAL SKETCH 51 iii CHAPTER I PRELIMINARIES A relation R on a topological space X is_ a_ subset of X x X. If A is a subset of X and if R is a relation on X, let AR = p2 ((AXX)^R) and RA = Pl ((X x A) r\ R) where p^, p2 are respectively the first and the second projections. For x in X we shall write simply xR for { x } R. Let R (-D = { (y, x) | (x, y) e R ) , and AR^ { x | xR C A } , R["1]A = { y | Ry C A } . In all that follows, X will always denote a Hausdorff space, and R a relation on X. The closure of a set A will be denoted by A* and the interior of A by A . Definition 1.1. A relation R on X is said to be lower semi-continuous (abbreviated l.s.c.) upper semi-continuous (abbreviated u.s.c.) 1 2 at x in X if and only if x in (vr(-1)) > l for all V = Vo C X. R is said to be (W 0 J .l.s.c. , on X if and only if R is , l.S.C. \ at u.s.c. / * U.S.C. i x for all x in X. R is said to be continuous if and only if R is both l.s.c. and u.s.c. Definition 1.2. A relation R on X is , upper semiclosed (u. s .cl.) -j V if and only if n \ lower semiclosed (l.s.cl.)J ^ Rx -J is closed for any x in X. Definition 1.3* A relation R on X is said to be a partial order1 if and only if the following conditions are satisfied: (a) (reflexivity): (x, x) Â£ R for all x in X, (b) (antisymmetry): (x, y) 6 R and (y, x) Â£ R imply x = y, (c) (transitivity): (x, y) 6 R and (y, z) Â£ R imply (x, z) Â£ R. VR VR (-D [-1] - implies x in 1. A relation R is said to be a quasi-order if (a) and (c) are satisfied. 3 We here state some theorems to which we will refer throughout this work. The proofs of these theorems may be found in [19]. Theorem 1.1. If R is closed in X x x and if A is compact, then both AR and RA are closed; moreover, if V is an open set then R^-^ V and VR^-1^ are open. Theorem 1.2. The following statements are equi valent : (a) R is l.s.c. on X. (b) RA Â£ (RA) for all A C X. (c) A* R C (AR) for all AC X. (d) RV is open for all open VC X. (e) p^ | R is open, where p^ | R is the restriction of p^ to R. Theorem 1.3- The following statements are equi valent : (a) R is u.s.c. (b) VR^"1^ is open for all V C X. (c) AR^-1) is closed for all closed A C X. (d) A R^"1^ C (AR[_1]) for any A C X. (e) (AR^_1^)^C A* r(-1) for any A C X. Theorem 1.4. If X is a compact Hausdorff space, 4 then R = R* if and only if (a) R is u.s.c. on X, and (b) R is u.s.cl. on X. Definition 1.4. A subset C of X is an R-chain if and only if C x c C R U R^-1^. As a consequence of the well-known Hausdorff maximality principle, there is a maximal R-chain for any relation R on any space X. A useful result of Wallace [14] is the following. Theorem 1.5- If R is both u.s.cl. and l.s.cl. quasi-order, then every maximal R-chain is closed. Definition 1.5- An element a in X is R-minimal (R-maximal), whenever (x, a) e R ((a, x) e R) implies (a, x) e R ((x, a) e r). The following fundamental theorem first proved in [14] will be used repeatedly. Theorem 1.6. If R is a u.s.cl. (l.s.cl.) relation on a compact space X, and if A is a non-void closed subset of X, then A contains an R-maximal (R-minimal) element a of A. The Alexander-Kolmogoroff cohomology groups will he used as developed in [21]. In what follows the coef ficient group is fixed and therefore will not be mentioned. We record here some useful theorems from [21]. Theorem 1.7. If B C A C X, then H(X,A) -X H(X,B) H(A,B) > H1(X, A) -^H1(X,B) ^>H1(A,B) * 6 ^ Hn(X, A) J > is an exact sequence, where i* and j* are the induced homomorphisms of the inclusion maps i and j from A to X and X to X, respectively, and 6 is the coboundary operator for the triple (X, A, B). Theorem 1.8. For the space X and any connected set A of X the homomorphism i* of H(X) -X H( A) X H1(X, A) is an epimorphism and hence 6 is 0. For simplicity we denote by f:(X, A) - (Y, B) that f is a_ function from X to_ Y and A is. Â£ subset of X such that f(A) C B C Y. Furthermore, if i:(X, A) -> (X, B) is_ an inclusion map, we will write 6 i:(X, A) C (X, B). Theorem 1.9. continuous, and if u: v: w: If f: (X, A, B) -> (X' A' (X, A) (X' A') , (X, B) -> (X', B' ) , (A, B) -> (A', B') B' ) is are defined by u(x) = v(x) = w(x) = f(x), then the ladder Hp(X, A) HP(X, B) u1 -^->HP(A, B) I w -^->HP(X, A)^HP(X, B)^HP(A, B) 3-- is analytic, that is, each rectangle of the ladder is analytic. We state here a special case of The Mayer-Vietoris Sequence which will be sufficient in what follows for our purpose. Theorem 1.10. If X is a compact Hausdorff space, and if X = X-^ Xg where X^ and Xg are closed subsets of X, then there exists an exact sequence A Hq_1(X1 r\ X2) -A> Hq(x) -i: Hq(X1) X Hq(Xg) Hq(xx r\ x2) -A, ... 7 where (j* i* 6 ) and (j* i* ) are the homo- v a a cr a or morphisms in the exact sequences for the triples (X, X ) and (Xa, Xx X2 , a ) respectively for k: (X2, X1 kh X2) c (x, x1) ; J* = x I* _ i* 1* . - 1 1 1 2 and A i* v* 1 T Jl k 2 As a consequence of the foregoing theorem we have the following Theorem 1.11. With the hypotheses of Theorem 1.10 and if X-^ r\ Xg is connected, the homomorphism A in h(x1 r\ x2) -A* h1(x) -A_> h1(x1) X h1(x2) is 0. Proof. Observe the following ladder H(x) 2+ H1(X, Xx) ^ H1(X) > u* k* 'r j j* -7- * h(x1 ^ x2) % h1(x2, x1 r\ x2) % hx(x2) ^ where u: Xx a X2 C X and where (j*, i*, 6]L) / (J*, Tg, ^) and k are as in Theorem 1.10. It follows from Theorem 1.8 that if h is in H(X^ f\ Xg) then there is an h' in H(X) such that u*(h') = h. Thus by Theorem 1.9., we have 8 k* -1 2(h) = k* k* 61(h/ ) which contains 1(h/). And A = j k* 6^ being a well-defined homomorphism, we see that for any a and 8 in k -1 6*2(h) j*(a) = j*(p). Therefore, A(h) = J* k* _1 2(h) = j^61(h/) = 0. The following notation is convenient. _If P is a_ subset of Q and if h e Hp(Q) then h | P denotes the image of h under the natural homomorphism induced by the inclusion map of P into Q. Theorem 1.12. (Reduction Theorem). If compact Hausdorff space, if A is closed, and if such that h | A = 0, then there is an open set U A such that h | U =0. Definition 1.6. (X, A) is a compact pair if and only if X is a compact Hausdorff space and A is a closed subset of X. X is a h e HP(X) containing Theorem 1.13 (Map Excision Theorem). If (X, A) and (Y, B) are compact pairs, and if f: (X, A) -> (Y, B) is a closed map such that f takes X A topologically onto Y B, then 9 f* : HP(Y, B) = HP(X, A). \ * r * Theorem 1.14. (Homotopy Lemma). If (X, A) is a compact pair and if T is a connected space, and if for each t in T At : (X, A) -> (X x t, A x t) is defined by A^(x, t) = (x, t), then A = A"'*' for r and s any elements of T. I? s Definition 1.7 A space X is unicoherent if and only if X is connected and X = A B with A and B closed and connected implies A rs B is connected. X 2 is hereditarily unicoherent if every subcontinuum of X is unicoherent. Theorem 1.15- If X is a continuum and if H1(X) = 0 for G Â£ 0, then X is unicoherent. Definition 1.8. If X is a space, if A C X and if h is a non-zero member of HP(A), then a closed set P C A is a floor for h if and only if h | F M while h | F' = 0 for any closed proper subset F' of F. space. 2. A continuum is a compact connected Hausdorff Theorem 1.16. (Floor Theorem). If (X, A) is a compact pair and If h is a non-zero member of HP(A), then h has a floor. Moreover, every floor is connected. Theorem 1.17 If (X, A) is a compact pair, if R = A x A U A where A= { (x, x) [ x e X } then HP(X, A) HP(X/R) for all p> 1. CHAPTER II A CHARACTERIZATION OF THE OUTPOINT ORDER ON A TREE A tree is a continuum such that every two distinct points are separated by the omission of a third point. Let X be a tree and let z be an arbitrary, but fixed element of X. Let Q(z) be the set of all such pairs (a, b) in X X such that at least one of the following three conditions is satisfied : (i) a z, (ii) a = b, or (iii) a separates z and b in X. It turns out that Q(z) is a continuous partial order on X, and with respect to this partial order z is the unique minimal element. We shall refer to Q(z), for any z in X, as a cutpoint-order [l] on the tree X. The purpose of this chapter is to give a characterization of the cut- point order on a tree (Theorem 2.4). We also obtain a new characterization of a tree from Relation-theoretic and Cohomological view-points. 11 12 Several characterizations of a tree have been given [2], [5], [23], and [24]. Perhaps the most useful of these characterizations is Lemma 2.1. [2], [5]- A continuum X is a tree if and only if it is locally connected and hereditarily unicoherent. An excellent proof of this lemma may be found in Ward [24]. Definition 2.1. A space X is said to be semi-locally-connected (abbreviated s.l.c.) at a point x of X provided for any open set U in X containing x there exists an open set V containing x such that V C U and that X V has only a finite number of components. If X is s.l.c. at each of its points, it is said to be s.l.c. In 1953> A. D. Wallace [15] proved that one- cod imensional"*" compact connected and locally connected topological semigroup with unit and zero is a tree. L. W. Anderson and L. E. Ward, Jr. in 1961 [l] modified Wallace's result by eliminating the necessity of hypothe sizing a unit. More precisely, they proved that if 1. For the definition and properties of Codimension, see Wallace [21], or Cohen [6]. 13 X is a compact connected, locally connected, one- codimensional topological semilattice, then X. is a tree. Wallace [17] improved this result by weakening the local connectedness of X to semilocal connectedness of X. These elegant results on Topological Algebra, motivated the following Lemma which bears a Relation-theoretic analogy. Lemma 2.2. If R is a relation on a compact Hausdorff space X with RX a one-codimensional semi- locally connected subspace, such that / \ * (i)the relation R is closed, i.e., R = R , (ii)H^Rx) = 0 for every x in X, (iii)the collection j Rx | x e X^ has the finite intersection property (abbreviated f.i.p.), and (iv)Ra r\ Rb is connected for each pair a, b in X, then RS is a tree for every closed subset S of X. The proof of Lemma 2.2. depends on the following: Lemma 2.3- If A, M and B are disjoint non-void closed subsets of a normal space X, and if A is either compact or consisting of finitely many components such that (i) M does not separate A and B in X, and (ii) for 14 any open set U containing M there is an open subset V of U containing M such that X V has only a finite number of components, then there exists a closed and connected subset N of X such that N <, X M and N meets both A and B. This lemma was first proved by G. T. Whyburn [26] for the particular case in which X was assumed to be a metric continuum and card A = card M = card B = 1. The non-metric case was implicit in a paper by Wallace [17] but without proof. We postpone the proof of Lemma 2.3* to the end of this chapter. Proof of Lemma 2.2. It follows from (i) and Theorem 1.1. that RX is closed and from (iii) and (iv) that RX = (Rx |x e X) is connected and thus RX is a continuum. Similarly RS is a continuum. Since every subcontinuum of a tree is itself a tree and since RS is a subcontinuum of RX, it is sufficient to show that RX is a tree. We first show H^(RX) =0. If there were a non-zero h e H1(RX), then there would be a maximal (non void) tower J of closed subsets A of X such that h i RA ? 0. Let Aq= r\ { A | A 7 j Then 15 h Â¡ RAq 0, for if h | RAq = 0, then by the Reduction Theorem (Chapter 1, Theorem 1.12.) there would,', be an open V 1 RA such that h I V* = 0. It would then follow * o 1 from Theorem 1.1. that R^-1^ V would be an open set containing AQ. If R^"^V is designated by U then R U C V so that there is an A in J with A < U and RA C RU C V*; therefore h [ RA = 0, a contradiction. Case 1. Card Aq = 1, i.e., Aq = { x } By (ii) H^(RA ) = 0, a contradiction. Case 2. Card Aq > 1. Write Aq = A^ U A0 where both A^ and A2 are proper closed subsets of A . We consider the following part of the Mayer-Vietoris exact sequence (Chapter I, Theorem 1.10.), h(ra1 n ra2) A, h1(rao) _Â£*, h1(ra1) x h1(ra2). Since by (iii) and (iv) RA1 r\ RAg = IJ | RA a Rb | (a, b) Â£ A^ x Ag | is connected, then A = 0 (Theorem 1.11.), and h RA e Ker J = ImA = 0, o a contradiction. RX is a continuum and H^RX) = 0 imply that RX is unicoherent (Theorem 1.15*) RX being of codimension one and H1(RX) = 0 imply that H^K) = 0 for every closed subset K of RX [21], and thus every subcontinuum of RX is unicoherent. 16 We now prove that every two points of RX are separated in RX by a third point. Suppose there were two points a and b such that no point separated a and b in RX. Then by Lemma 2.3-, for any p different from both a and b, there would be a continuum P which would be irreducible from a to b and which would not contain p. If q were an element of P distinct from a and b there would also be a continuum Q, irreducible from a to b and which would not contain q. But then P Q would be a subcontinuum of RX which would not be unicoherent, since P n Q by our selection of P and Q is obviously not connected. This contradiction completes the proof. Theorem 2.4. If X is a compact Hausdorff space, and if P is a relation on X, then the following conditions (i)X is of 1-codimension and s.l.c., (ii)P is a closed partial order, (iii)P is left monotone, i.e., Px is connected, and H^(Px) = 0 for every x in X, (iv){Px | x e X) has the f.i.p., and (v)P is right monotone, i.e., xP is connected for every x in X, are necessary and sufficient conditions that X be a 17 p tree, and that P be a cutpolnt-order. r Proof: We first prove the sufficiency. Conditions (ii), (iv) and the first half of (iii) imply that Pa Pb = U { Px I x e Pa r\ Pb } is connected, and thus Lemma 2.2. yields that X is a tree. Since X is compact and { Px I x Â£ X} has the f.i.p., then { Px | x e x } Â£ Indeed, it is a single point, the unique P-minimal element of X. Let us denote by 0 the set { Px I x e X } . We prove that P = Q(0) If (a, b) Â£ Q(0) such that a = 0 or a = b, then clearly (a, b) must be also in P. If a separates 0 and b in X, then since Pb is a continuum containing 0 and b, it must contain a, and we again conclude that (a, b) is in P. Thus Q(0) C P. Conversely, if (a, b) is in P, then since a is in aP r\ Pb, and since both aP and Pb are continua, then aP ^ Pb is a subcontinuum of the tree X, and therefore 2. Condition (v) is not necessary for X to be a tree. used. 3. Only reflexivity and transitivity of P are 18 by Lemma 2.1. it is unicoherent. Thus aP r\ Pb is also a continuum. Now, by virtue of the Hausdorff-Maximality- Principle, aP r\ Pb has a maximal P-chain, C, and (a) C is closed (Theorem 1.5)> and (b) C _is connected. For if C were not connected there would be two non-void disjoint closed sets A and B such that C = A U B and b Â£ B. The set A contains a maximal element m. Define A' and B; by the equations A/ = Pm o C and B/ = C Pm. B C mP, and since A C. A1 then B/ C B. Now A1 r\ B1 C Pm n (mP r\ B) = (Pm rN mP) r\ B = Q , therefore C = A' U b' is a separation. If bQ designates the minimal element in B', then by the maximality of C mP r\ Pb = { m, b } o o J which contradicts the connectedness of mP r\ Pb . o Therefore, any maximal P-chain in aP r\ Pb is connected. (c) aP r\ Pb has a unique maximal P-chain, which we denote by Cp(a> b). If C and C' were two distinct maximal P-chains in aP n Pb, then both C and C/ would contain a and b, and C U C' would then be connected, and hence C n C' would be connected. But for x Â£ C c' 19 C ^ c' = (Px r\ C C) u (xP C r\ C') = C ^ (Px U XP) r\ C1 Is obviously a separation, a contradiction. Since (0, b) Â£ Q(0) C P and since X is a tree, there is a unique connected Q-chain [22], [23] Cq(0, b) C Pb which contains both 0 and b. Pb must also have a connected P-chain containing both 0 and b and this P-chain must be unique. We denote by C (0, b) the unique connected P-chain in Pb containing 0 and b. Since a Q-chain is also a P-chain, then Cp( 0, b) = CQ(0, b) . Similarly, there is a unique connected P-chain Cp(0, a) in Pa containing both 0 and a. It is clear that Cp(0, a) V Cp(a, b) = Cp(0, b) = CQ(0, b). As a consequence, a Â£ Cq(0, b) and hence (a, b) Â£ Q which was to be proved. We next prove the necessity. Let X be a tree and let P be the outpoint order on X with respect to a point z in X. We prove that X and P satisfy the conditions (i), (ii), (iii), (iv) and (v) stated in the theorem: Proof of (i). By Ward [24] a tree is a compact connected commutative idempotent semigroup with zero, therefore it is acyclic [17]. Hence in particular 20 H^(X) =0. We now show H^A) = 0 for every A = A* C X and thus X is of one codimension, unless Xis degenerate. Suppose on the contrary that H (A) ^ 0 for some closed subset A of X. If h is a non-zero member of H^(A), then by the Floor Theorem (Theorem l.l6.) there is a floor F C A for h, which is connected. The set F being a subcontinuum of a tree is itself a tree and hence is acyclic. Therefore, H^(F) = 0 which contradicts the fact that F is a floor, and thus H^(A) = 0. The semilocal connectedness of X follows from the fact that X is compact and locally connected. Proof of (ii). This is proved in Ward [24]. Proof of (iii). The outpoint order P is order dense [23] and since by (ii) P = P* we have that every maximal P-chain in Px is connected [22], thus Px is connected. Indeed, Px itself is a tree and therefore as has been proved in (i), H^(Px) = 0. Proof of (iv). This is obvious, since P has the least element z. Proof of (v). Replacing Px by xP in the argument of (iii), we easily obtain the connectivity of xP. Proof of Lemma 2.3 (l) There is a_ component Aq of A such that M does not separate Aq and B in X. For if otherwise, to each component A^ of A there is a pair of disjoint open sets and T^ such that X M = U and } A^, T^ ^ B. Since A is either compact or consisting of finitely many .components, there is a finite subfamily { G^, G^, .., G \ of G-^s 21 such that A C m v { Gi 1 m i = 1, 2, .. ., m } Whence G = W G. and i*l 1 T = r\ i=l T. 1 are two disjoint open sets such that X M = G KJ t and G P A, T p B so that the hypothesis (i) is contradicted. Throughout the rest of the proof, let #[S] be the number of components of the space S. Let Xf = { | A be the collection of all open sets containing M but missing A such that #[X V^] is finite. For each Â£ 1/- let R^ be the component of X that contains the component Aq of A. Designate R ^ u{R^ A e A } (2) R i_s open. Let y e R, and let Â£ be such that R^ contains y. By the normality of X there is a V, e {/ with V C V* C V, Since X V, [X |X |X A [X has only a finite number of components and since (x y0 3 X vj 3 x Vv we have R, C R C R* Therefore R is open. A P (3) For each VQ e Â¡j- there is an R^ containing R such that a n (X va)] = #[rx (X vo)] 22 for all containing R^. Let C-^, Cg, .., be the n components of X V so that X Va = C-^ o Cg U where = R^, then for any contained in VQ we have X containing X Va, so that each Cf must be either totally contained in R^, the component of X contain ing Aq, or disjoint from R^. Thus #[Rp rs (X Va)] < # (X Va) = n. But, # [R6 rv (X va) ] < Â§ [RT r\ (x va) ] if R^ R^ } R Hence there is an R^ } R^ such that the number #[R^ r\ (X V^)] is the maximal so that #[RM. (4) For each VQ e there is an R^ such that n ^ (X va> = \ ^ (X va) for all R^ containing R^. By virtue of (3) above, there is an R, containing R such'that A Ct #t>V ~ (X Va) 3 = #[R* ^ (X Va)] for all R containing R^. We note, further, that *V ^ (X Va> \ ^ (X Va) for all R^ containing R^. For if as in (3) above, we express X Va = C1 u C2 u ... u cn where, without loss of generality, CL is contained in R^ 23 for 1 = 1,2, . ., k (k < n) and Ck r\ R^ = O for i = k + 1, . ., n, then it follows from R .containing [X R^ and #[R,, ^ (X VQ)] = # [R* ^ (X Vj] cr that C. r\ R = Q for i = k + 1, ..., n. And therefore, 1 [X Rp. ^ = RA ^ k (3) R = R* M; that is, R _is closed in X M. If y is a point of R M then there exists a Va in V- missing y. By the normality of X there is a \k in If such that C V^* C VQ. By (4) there is an R^ containing R^ such that A (X Vp) = R* A (X Vp) for all R^ containing R^. Furthermore, if U is an open set about y, then U ^ R^ 41 O For, if we designate the set U r\ (X V^*) by W then y is in both R M and W so that W ^ R 4= O and there is an R^ such that W n R =p Without loss of generality we may assume that R, contains R, Then o A D 4= W n R = W ^ Ra C (X Vp) = W r\ R^ r\ (X Vp) = W A R . Thus, U ri R^ } W A R^ 4: so that y is in R^ = R^. Therefore, R is closed in X M. It is to be noted that R meets B and hence R T r\ B ^ Q for some r ; because otherwise. X M = R U(X-M-R) would be a separation, of X M between Aq and that (l) would be contradicted. We now conclude result by taking N = R^ . , so the CHAPTER III FIXED POINTS AND MINIMAL ELEMENTS In the Symposium of General Topology and its Relations to Modern Analysis and Algebra (Prague 196l), Professor A. D. Wallace announced [18] among other things the following fixed point theorem. Theorem 3-1 [18, Theorem 5]* If X is a continuum, if P is a closed left monotone partial order on X such that pa"* c ( pa)*" for each A C X, and if z separates Pa and Pb in X, then Pz = z. Wallace applied this theorem to prove Theorem 3 2 [18, Theorem 6]. If X is a con tinuum and if P is a closed left monotone partial order on A such that PA*C (PA)* for each A ( X, then the set K of P-minimal elements is connected. However, in [18] Wallace gave no proof of the fixed point theorem. For the sake of completeness, we will give in this chapter a proof of Theorem 3*1 by first proving Theorem 3-2 and using this result to establish Theorem 31 We will also show that under certain conditions the state ments (i) If z separates Pa and Pb in X then 25 26 Pz = z. r (ii) The set of P-minimal elements 'is connected, are equivalent (Theorem 3**0- Furthermore, if P is a partial order on the *??' o continuum X and K is the P-minimal elements in X, we will prove in Theorems 3-5^ 3*6 and 3-7 that the cohomology groups of X and those of K are isomorphic for all non-negative dimensions for certain classes of X with suitable choice of P. The following lemma will be used in the proofs of foregoing theorems. Lemma 33 [cf. l6]. If X is a compact Haus- dorff space and if P is a. lower semi-closed partial order on X such that PA* < (PA)* for each A ( X, then the set K of P-minimal elements is closed.' Proof. Suppose by way of contradiction that there is an x in K* K. Then since Px is closed and X is compact, Px has a minimal element which must be a P-minimal element in X, and thus Px r\ K \ Q . Let y be any element in PxaK. Obviously x % y, so that there is an open set U containing x whose closure excludes y. If V = U r\ K, then PV = V, and x is in V*. Since 27 PV*C (pvf = V*, then PV* C U* and hence y e Px C PV* C U*. This contra dicts the fact that y ^ U*. Therefore the set K is closed. Proof of Theorem 3*2. Let us assume contrary to the conclusion of the theorem that K is not connected. By Lemma 3-3 there are two disjoint non-void closed sets A and B such that K = A ^ B. Since P is closed, then AP and BP are closed and X = AP w BP. Furthermore, the connectedness of X yields AP os BP % Thus AP rs BP has a minimal element, say t. Since Pt = (Pt r\ AP) V (Pt BP) , then (Pt AP) r\ (Pt BP) = Pt rs (AP r\ BP) = t. If we designate C = (Pt r\ AP) t and D = (Pt/^ BP) t, then both C and D are open in Pt. Furthermore, C and D are both non-void. For if C is void, then Pt r\ AP = t, and so t must be in A. This implies that t is not in BP which contradicts the fact that t is a minimal element in AP r\ BP. Similarly, D is not void. Consequently, Pt t = C U D is a separation. The connectedness of Pt yields C* = Pt r\ AP and D* = Pt r\ BP. Now, for each x in Pt t we have Px C Pt t and the connectedness of Px then implies that either Px C C or Px C D. Thus, 28 for each x in C we see that Px C C, and hence PC C C, and it follows that C* 3 (PC)* b PC = P(C ^ t) ) Pt 5 D which is a contradiction. Therefore K is connected. Proof of Theorem 3-1- Let X z = A ^ B where A and B are non-void separated sets and aP ( A and bP C B. Let K be the set of P-minimal elements. Now, if Pz ^ z, then K C X z and K = (K r\ A) V (Kn B) is a separation for K which contradicts the connectedness of K. Theorem 34. If X is a continuum and P is a closed left monotone partial order on X such that the set of P-minimal elements K is closed, then the following statements are equivalent: (i) If z separates Pa and Pb in X then Pz = z. (ii) The set K is connected. Proof. (i) implies (ii). Suppose the set K is the union of two disjoint non-void closed sets A and B. Since K is the set of P-minimal elements then X = KP = (A ^ B)P = AP U BP. And the hypotheses that X is connected and P is closed yield AP r\ BP 4= Let z be a P-minimal element of AP BP. Then there are elements a e a and b^B such 29 that {a, b}< Pz z. The equalities Pz = (Pz ^ AP) b (Pz n BP) and Pz a (AP r\ BP) = z show that Pz z = (Pz r\ AP z) ^ (Pz A AP z) is a separation, that is, z separates a = Pa and b = Pb in the continuum Pz, and so by (i) z is a minimal element which is neither in A nor B. This contradiction estab lishes the connected ness of K. (ii) implies (i). The proof as given for Theorem 3-1 applies here as well. We now present an example to motivate Theorem 3*3. Example. Let X = { (a, b) 1 a2 4- b2 = 1 ^ U {(0, b) | 1 <_ b <_ 2} for reals a and b. Let X be endowed with the Euclidean topology. Let P = { ((0, b1), (0, b2)) [ 1 < bx < b2 < 2j U A where A = | (x,x) | x Â£ X } Then P = P* is a both left and right monotone partial order on X such that PA C (PA) for each A ( X and K = { (a, b) l a2 + b2 = l} = S1 (1-sphere) is the set of P-minimal elements. It is to be noted in this example that H^Px) = 0 for each x e X while H^(K) ^ H^(S) + 0 for any non trivial coefficient group. However the following equality 30 Hp(x) = HP(K) holds for all non-negative dimensions. . It is interesting to seek conditions on X and a relation P which imply the equality HP(X) HP(K) for all non-negative dimensions. The purpose of the next theorem is to take a small step in this direction. Theorem 3-5- If X is a continuum, and if P is an upper semiclosed partial order on X such that (i)PA* C (PA)* for all A C X, (ii)there exists an element u in X such that X K C Pu where K = {x| Px = x }, and (iii)the quotient space X/K x K u A modulo K X K U A is a topological semilattice under the natural partial order induced by P, where A { (x, x) I x e X } , then, Hp(x) s H^K) for all integers p > 0. Proof. For simplicity in notation we write Y = X/KXKUA. Since every compact topological semilattice has a (unique) zero, we write z for the zero of the semi lattice Y. By the hypothesis (ii), Y has a (unique) unit which will be denoted by u also since no confusion is likely to occur. We will accomplish the proof in three steps. (a) H^(X, K) = HP(Y, z) for all integers p >_ 0. Let f : (X, K) -> (Y, z) be the natural map. Since (X, K) is a compact pair, f is a closed map, and moreover, f 31 takes X K topologically onto Y z. Thus-by the Map r * Excision Theorem (Theorem 1.13) f*- : HP(Y, z) -> HP(X, K) is an isomorphism. (b) HP(X, K) = 0 for all integers p > 0. We establish this by showing HP(Y, z) = 0 for all integers p >_ 0. Define ffc : (Y,z) -> (Y,z) by f (y) = t a y for all t e Y, where a is the semilattice operation on Y. If for each t Â£ Y define : (Y, z) -> (YX Y, z xY) by At(y) = (y, t) then f^_ = A o Since by the Homotopy Lemma (Theorem 1.14) 7\* = Au*, thus f *= A*. A*= A*. A*= f *. z z u u Since f^ is the identity map, so is fz. Let i : (z,z) C (Y, z) be an inclusion map and let h : (Y, z) (z, z) be defined by h(y) = z for all y e Y. Since f = i h, the following diagram Cj is analytic, for all non-negative integers p, that is fj = f*= h** i*. Since Hp(z, z) = 0 for all integers p > 0 and HP(Y, z) = fJ(Hp(Y, z)) C h*(Hp(z, z)) for all 32 integers p >_ 0, then HP(Y, z) = 0 for all integers p >_ 0 as desired. (c) HP(X) = H^K) for all integers p >_ 0. By combining the above results with the exact sequence for the triple (X, K, ) (Theorem 1.7)* we obtain the exact sequence 0 = HP(X, K) -> HP(x) > HP(K) -> HP+ 1(X, K) = 0 for p = 0, 1, 2, ... Consequently, Hp(x) and HP(K) are isomorphic for all non-negative dimensions. We now center our attention on the case in which X is of codimension one. Theorem 36. If P is a closed partial order on a continuum X and if ir : X^X/KXKUA designates the natural map, such that (i)PA*C (PA)* for all A C X, (ii)P is left monotone and H^(7r(Px)) = 0 for all x e x, (iii)X is of codimension one, then Hp(x) HP(K) for all p > 0. Proof. Denote X = X/KX A and P = { (tt(x), ir(y)) J (x, y) p J then since X is compact and ir continuous, P is closed and left monotone; indeed we have P7r(x) = tt(Px) By virtue of Theorem 1.17* we have HP(X, K) = HP(X) for all 33 p >_ l. Since X is of codimension one we hav. HP(X, K) = 0 for p >_ 2 [21] We show H^X, K) = 0 by proving H^(X) = 0 Using the same argument employed in the proof of Lemma 2.2 one sees that H^(PS) = 0 for all closed sets "I ^ S in X; in particular we have H (X) = 0 We have noted in part (c) of the proof for Theorem 3*5 that if HP(X, K) = 0 for all p > 1 then HP(X) s HP(K) for all p > 1. The equality H^(x) = H(K) follows from the fact that both X and K are connected [Theorem 3*2]. We remark that the hypothesis (iii) in Theorem 3-6 may be weakened to : (iii ) cd (X K)* = 1, and this may even be dropped completely if each Px is a chain, as will be seen in the following Theorem 3 7 If P is a closed partial order on a continuum X such that (i) PA* C (PA)*, and (ii) each Px is a connected chain, then Hp(x) Â£ HP(K) for all p > 0. Proof. Let X, 7r and P be defined as in Theorem 3-6. Since P is closed and 7rPx is a connected chain P is closed and Ptt(x) is a connected chain. It follows then that each P7r(x) is a generalized arc and hence Hp(P7t(x)) = 0 for all p >_ 1. As it has been noted 3^ in the proof of Theorem 3*6, H1(PS) = 0 for 11 closed sets S in X. We now show H^(X) = 0 for all p >_ 1 by proving HP(PS) = 0 for all p >_ 1 and for each closed set S in X. If there were a least integer n such that Hn(PA) i= 0 for some closed set A in X, then n would have to be greater than 1, and A =f= Let h be a non zero member of Hn(PA), then h | Pa = 0 for each a e A and hence by the Reduction Theorem there is an open set V containing Pa such that h I V =0; then by Theorem 1.1 there is an open set U containing a such that PU C V. Thus the collection a of all open subsets U of A such that h J PU = 0 forms an open cover of A. Also Q. is closed under finite union, for if and Ug are in a denote hQ = h | PU* U PU*. Since P^ U U2)* = P(U* ^ U*) = PU-^ V PU2 to show GL is closed under finite union it suffices to show hQ = 0. In the following part of Mayer- Vietoris exact sequence : Hn-1(PU*^ PU*) * Hn(PU*^PU*) Hn (PU *) x Hn(PU*) since J*(hQ) = (h I PU*, h I PU*) = (0, 0) hQ is in the image of A. But Hn_1(PU*^ PU*) = Hn_1(PS) where S = PU*r\?uJf is a closed subset of X, and hence Hn-'L(PS) = 0 by the minimality of n. This proves hQ = 0 so that a is closed under finite union. Since A is compact, A is a union of some finitely many elements of (X Thus A must be in (X and so h = 0, a contradiction. This 35 together with Theorem 1.17 implies HP(X, K) = Hp(x) = 0. for all p >_ 1. It then follows from the exact sequence for the triple (X, K, O) [Theorem 1.7] and the connectedness of X and K [Theorem 3-2] that Hp(x) = HP(K) for all p > 0. Remark. Theorem 3-7 may be stated more generally by repacing the hypothesis (ii) by (ii') each 7r(Px) is acyclic. We conclude this chapter by exhibiting an example which answers a question in Topological Semigroups. A clan is a compact connected topological semigroup with unit [20]. Let S be a clan and let R = { (x, y) | (x, y) e S X S, xS uSx c yS u Sy } then R is a closed quasi-order on S and the set of R-minimal elements is also the minimal ideal of S. The question to be answered is: if. a. clan S is. a. tree, is its minimal ideal an arc or a. point 9 The answer is affirmative if S is abelian (or normal: xS = Sx for all x e S), but it is negative in general. Example. Let S = X ^ I be the subset of Euclidean 3-space such that X = { (x,0,0) | -1 < x < 1 } U {(0,y,0) | -1 < y < 1} I = { (0,0,t) f 0 < t < 1 } ; 36 let S be endowed with the Euclidean topology and let the multiplication o on S be defined below (the usual multiplication of reals is denoted by juxtaposition) : i. Aop = ii. (I, o) unit interval; i.e., A for all A Â£ X and for all p e S is the usual semigroup of the real (0,0,tx) o(0;0,t2) = ( 0, 0, t-^g) , iii. For each (0,0,t) e I and for any (0,y,0)ex, (0,0,t)o(x,0,0) = (tx,0,0) and (0,0,t)o(0,y,0) = (0,ty,0). Then, (l) (S, o) is associative. (2) o : S S -> S is continuous. (3) (S, o) is a clan with (0,0,l) as unit. (4) The minimal ideal of (S, o) is X which is neither an arc nor a point. It is interesting to observe that the semigroup S given in the above example may be realized as a semi group of matrices by the following one-to-one correspond ence : t x + yf-1 (x,y,t) + 0 1 37 where -l<_x, y <_ 1, 0 < t < 1 and (tx)2 + (:xy)2 + (yt)2 = 0. The correspondence is indeed an isomorphism. Prom this the assertions (l), (2) and (3) in the example are self-evident. / CHAPTER IV FIXED POINT PROPERTIES AND INVERSE LIMIT SPACES A space X is said to have th _f. jd. jd. (fixed point property) if, for every continuous function f: X -> X there exists some x in X such that x = f(x). Hamilton [8] has proved that the chainable metric continua have the f. p p . A space X is said to have the F. jd. jd. (fixed point property for multifunctions) if every continuous multifunction'*' F : X -> X has a fixed point, i.e., there exists a point x in S such that x e F(x). Obviously if X has the F. p. p. then it has the f. p. p., but the converse need not be true. Strother [13] has exhibited two continua X and Y both have the F. p. p. but their Cartesian product X xY fails to have the F. p. p. Borsuk [3] has constructed a decreasing sequence of three-cells whose intersection does not have the f. p. p. The inter section is the inverse limit, the bonding maps being in jections. As a counter theorem to the results of Strother and of Borsuk, we prove that if (X-^, ir^ /\ ) is an 1. Following Strother [12], a multifunction F : X -* Y is continuous if, and only if, F(x) is closed for each x, and F ^(A) is open (closed) if A is open (closed). 38 39 inverse system of compact spaces such that each X^ has the F. p. p. then the inverse limit space has the F. p. p. As a corollary to this, we obtain Ward's generalization [25] of the Hamilton theorem [8] that every chainable metric con tinuum has the F. p. p. Our result is, indeed, stronger than the Ward's, since it includes some of the non-metric chainable continua as well. Definition 4.1. The collection (X-^, tt^ -A ) is as inverse system of spaces if: (i)A is a directed set, (ii)A in A implies that X^ is a Hausdorff space, (iii)whenever A > p. there is a continuous function ir-. : X, -> X, , Ap, A p. (iv)if A > p. and p, > v, then tt^ = The function v^ is called a bonding map. If A is in A let be the subset of the Cartesian product P{ X^ | A e A } defined by | x | if A > p. then tt-^x(A) = x(p.)} , where x(A) denotes the A-th coordinate of x. Definition 4.2. The inverse' limit space Xq, of the inverse system of spaces (X-^, tt^ A ) is defined to be = r\ { A A e A } 40 endowed with the relative topology inherited fpom the product topology for P { | A e A } ; in notation X - J (XA- A ) We write p^ : P { X^ | A e A } * X^ for the A-th projection of P [ X^ | A e A } i.e., p^(x) = x(A) for all xin P|X^| A^A}; the restriction p^ j X^, will be de noted by 7which will be called a projection map. It is readily seen from the definition that an element x of P { XA | A e A] is in X^ if and only if tt^^tt^x) = tt (x) whenever A > p. A more detailed account of inverse limit space may be found in Lefschetz [10], Eilenberg and Steenrod [7], Capel [4] and Mardesic [11]. The following known results (see, e.g., [4], [10]) will be used. Lemma 4.3 (i) The collection Â¡ A Â£ A and is an open subset of X^ j forms a basis for the topology of X. (ii)The inverse limit space X^, is Hausdorff; if A e A is a closed subset of P { X^ [ A e A} so that X^ is closed in P { X^ | A e A } (iii)If X^ is compact for each A in A then X^ is compact; if, in addition, each X^ is non-void then X is non-void. (iv)If X^ is a continuum for each A ^A then the inverse limit space is a continuum. Lemma 4.4. If A is a compact subset of X^ and if 7= ir^ | then (tt^(A), ir^ A) is an inverse system of spaces such that A = 11m (tta(a), tt^, A) and each bonding map tt^ is onto. In the sequel, since we are only interested in compact spaces, each projection map twill be assumed to_ be_ onto; for if otherwise, by virtue of Lemma 4.4, each may be replaced by tt^X*.) without disturbing the resulting inverse limit space. We are now ready to state our main result. Theorem 4.5. Let (X^, tta A) be an inverse system of compact spaces such that each X^ has the P. p. p., then the inverse limit space X*, also has the F. p. p. We divide the proof of this theorem into the following steps. In Lemmas 4.6, 4.7 and 4.8 X,*, will be assumed to be the inverse limit space of the inverse system (X^, tt^, A ) of compact spaces. Lemma 4.6. If F : X# -> X^ is a continuous multifunction, define F^ : X^ -> X^ by F^ = m^F tt^ for each A. Then F, is a continuous multifunction. A Proof. (i) If t is in X^, then since F : X^ -* X is a continuous multifunction, by Theorem 1.1 and Theorem 1.4 of Chapter I, Fm^^t) is a closed subset of X^. Thus, F^it) is closed for every t in X, . (ii) If CA is a closed subset of XA, then is closed. It is readily seen that F ^tta^(Ca) Is closed in Xm and hence compact; therefore ,ir^F~'Sr^ (C^) = F^(C^) is compact and hence closed. (iii) If UA is open in XA, then F^(U^) is open. 7rA and F being continuous, F 'Tr-^(U^) is an open subset of Xoo It follows then, by virtue of Lemma 4.3 (i), that 7r^F_17T^1(U^) = F^(U^) is open. Thus, by (i), (ii) and (iii) above, F^: X-^X-^ is continuous. Lemma 4.7 Let F : Xffl -> X, be a continuous multifunction, let F^ : X^ -> X^ be defined as in Lemma 4.6 Then, for each x in Xa, , (i) (F^ir^(x) ttA[X, A)2 and (ttaF(x) ttA[x, A ) are inverse systems of compact spaces, (ii) lim (FA7rA(x), 7tA[x, A) = lim (ttaF(x) ^,A) (iii) F(x) = lim (Fa7Ta(x), 7Ta^, A). Proof. Since each FA is continuous (Lemma 4.6) and each XA is compact, so is for all A e A . 2. For simplicity in symbolism, henceforth if A C lim (XA, 7r A^, A) then (u^A, tta A) will mean KA a ) 43 To show that (F^tt-^(x) tt^ A ) forms ah inverse system, it suffices to show 7T^^F^tt^(x) Â£ F^Tr^(x) whenever A > \i. To this end we first observe Vx> < h^irAlx)^(x) = ir^fx), sinc 7T-, TT-> = 7T From this we have Ap A [X ^AixVa^) C rAM.VV(x) = VV r1)rV(x) = V v\(x) = F 7T (x) , (1 [l by the definition of F^, F^ and the equality 7r^7r^ = The fact that (7r^F(x) tt-^, A) forms an inverse system follows from Lemma 4.4. (ii) For each A e A and any x e X^, we have x e Tr^7r^(x) and hence, n\F(x) C tt-vF tt'VAx) = (tt.F v~)rli(x) = F,7iu(x) -1> Thus, lim (7TaF(x), tt^, A)c^ (F^U), tt^, A). To prove the other inclusion, we show Xoo lim (ttaF(x), ttA[i, A) C Xm- lim (F^tt^x) ir^, A). Let y be in Xa lim (ttaF(x), /\ ) then, by Lemma 4.4 44 there exists a p e A such that ir^(y) tt^P(x) Let and V be two disjoint open sets in X such that P M- Vy) Â£ UH and V(x) C VH so that F c It follows then from Lemma 4.3 (i) and the continuity of F that there exists a 6 Â£ A and an open set in X^ such that x a ir^U^ and (*> fKX) c y\ Since A is directed, there is a Aq e A > ll and A > 6, we shall use this A o o proof of this lemma. If we write U, = Ao the equality Tr^1^ then ( ) A such that throughout the n\"\Uc and use A qO u may be rewritten as and hence C T-\ , F, U, = 7T, F 7T,1(U, ) / 7T, 7T 1V = 7T, (iT, TT, ) 1V Aq Aq Ao A0 Aq V Aq P P A0 AqP Aq p In particular, F, TT, (x) / IT, 1 V . Aq Aq V A0p, |1 Similarly, one obtains (y) 6 since tt (y) Â£ U and v = 7r, 77\ The fact that U P P P Aq p are disjoint implies '^AqI.^ij,rv'n'AoP^p = ^ and and 45 consequently 7r, (y) 4 F, tj\ (x) From this we conclude Ao Ao Ao y ^ lim (F^tt^(x), -it^ A ) as desired. (iii) This follows immediately from (ii) and Lemma 4.4. Lemma 4.8. Let F : X^, be a continuous multi function, let F^ : X, -> X^ be defined as in Lemma 4.6 Let EA = {eA I eA 6 XA ancl eA e VeA>} fchen forms an inverse system. Proof. It suffices to prove 7r^(_L41^ Â£ E^ whenever A > p. Let e^ e F-^(e^), then Â£ = Va^A^A* = VV c V(V1rH)r1(eA) : FnV1(eA) = WaK^a* = Â¡'u(1rA|X(eA))- -1, Thus, ir^E^ C E^ as as to be Proven. Proof of Theorem 4.5 Since each X^ has the F. p. p. and by Lemma 4.6 each F^ : X^ - X-^ is continu ous, each E^ is closed and non-void. By Lemma 4.8, (E^, tt^ A ) is an inverse system of compact spaces, so it has a non-void inverse limit space lim (E-^, tt^, A ) . We now conclude the proof by showing that each x in (Ea, ttA/U, A ) is a fixed point under F; i.e., xÂ£F(x). 46 If x is in lim (Ev 7rA^, A) then tta(x) for-all A G A ; i.e., tta(x) e for all A G A Conse quently, by Lemmas 4.4 and 4.7, we have x = lim (irA(x), tt^, A) g lim (P^(x) vA) = F(x). In fact, with the assumption of Theorem 4*5 and the notation of Lemma 4.8 together with the notation E = | x | x G F(x) J. we can make the following sharper assertion. Theorem 4.5* E = lim (E^, tt^, A ) . Proof. Prom the proof of Theorem 4.5, we have E ) lim (Ea, tt^, A) . It remains to prove that E C lim (Ea, tt^, A ) Let x be in E, then x G F(x) and therefore, for all A e A , tt^(x) g tt^F(x) C tTaFCit-VaJCx) = F-Jtt^x)). That is, tta(x) e for all A; consequently, by Lemma 4.4 E < lim (Ea, ir^ A ) A chain (U^, U^, , U ) is a finite sequence of sets U. such that U. r\ U. if and only if J | i j i < 1. A Hausdorff space X is said to be chainable 47 .if to each open cover of X there is a finite open cover /[X = (U^, Ug, .., Un) such that (i) U refines V"> (ii) u= ( Uf, Ug, .., Un) forms a chain. It follows that a chainable space is a continuum. It is implicit in a paper by Isbell [9] that each metrizable chainable continuum is the inverse limit space of a sequence of (real) arcs. This together with a theorem of Strother [13] that a bounded closed interval of the real numbers has the F. p. p. imply the following result of Ward [23] as a consequence of Theorem 4.5- Corollary 4.9. Each chainable metric continuum has the-F. p. p. Examples of inverse limit spaces of inverse systems of real arcs exist which are not metrizable; for instance, the "long line" is one such. Thus, Theorem 4.5 is a proper generalization of that of Ward's [25]* BIBLIOGRAPHY 1. Anderson, L. W. and Ward, L. E., Jr. One-dimensional topological semilattices. Ill. J. Math., 5 (1961), 182-186. , 2. Borsuk, K. lber die Abbildungen Raume auf die Kreislinie. der metrischen kompakten Fund. Math., 20 (1933), 224-231. 3. Sur un continu acyclique qui se laisse trans former topologiquement en lui meme sans points invariants. Fund. Math., 24 (1935), 51-58. 4. Capel, C. E. Inverse limit spaces. Duke Math. J., 21 (1954), 233-245. 5. Cech, E. Sur les continus Peaniens unicoherents. Fund. Math., 20 (1933), 232-243- 6. Cohen, H. A cohomological definition of dimension for locally compact Hausdorff. spaces Duke Math. J. , 21 (1954), 209-224. 7* Eilenberg, S. and Steenrod, N. Foundations of algebraic topology. Princeton University Press, Princeton 1952. 8. Hamilton, 0. H. A fixed point theorem for pseudo-arcs and certain other metric continua. Proc. Amer. Math. Soc., 2 (l95l), 173-174. 9. 10. 11. / 12. 14. Isbell, J. R. Embeddings of Inverse limits 4 Ann. of Math., 70 (1959), 73-84. Lefschetz, S, Algebraic topology. Amer. Math. Soc. Colloq. Publ., no. 27, New York 1942. Mardesic, S. On Inverse limits of compact spaces. Glasnik Mat. Flz. Astr., 13 (1958), 249-255- Strother, W. L. Continuity for multi-valued functions and some applications to topology. Doctoral Dissertation, Tulane University 1952. On ari open question concerning fixed points. Proc. Amer. Math. Soc., 4 (1953), 988-993- Wallace, A. D. A fixed point theorem. Bull. Amer. Math Soc., 51 (1945), 613-616. 15- Cohomology, dimension and mobs. Summa Brasil. Math., 3 (1953), 43-55- l6. Struct ideals. Proc. Amer. Math. Soc., 6 (1955), 634-638. 17 Acyclicity of compact connected semigroups. Fund. Math., 50 (l96l), 99-105- 18. Relations on topological spaces. Proc. Symp on General Topology and its Relations to Modern Analysis and Algebra. Prague 1961, 356-360. 19. Relation-theory, Lecture Notes. University of Florida, 1963-1964. 20. Topological semigroups, Lecture Notes. University of Florida, 1964-1965- 50 21. Wallace, A. D. Algebraic topology, Lecture Notes. University of Florida, 1964-65 22. Ward, L. E. Jr. Partially ordered topological spaces. Proc. Amer. Math. Soc., 5 (1954), l44-l6l. 23- _A note on dendrites and trees. Proc. Amer. Math. Soc., 5 (1954), 992-994. 24. Mobs, trees, fixed points. Proc. Amer. Math. Soc., 8 (1957), 798-804. 25- A_ fixed point theorem. Amer. Math. Monthly, 65 (1958), 271-272. 26. Whyburn, G. T. Analytic topology. Amer. Math. Soc., 1942. BIOGRAPHICAL SKETCH Shwu-yeng Tzeng Lin was born May 11, 193^ at Tainan, Formosa. In June, 1953, she was graduated from Tainan Girl's High School. In June, 1958> she received the degree of Bachelor of Sciences from Taiwan Normal University. She worked as a Research Assistant at the Mathematics Institute of Academia Sinica from 1958 to I960. In the fall of I960 she enrolled, and worked as a teaching assistant, in the Department of Mathematics, Graduate School of the Tulane University, New Orleans, Louisiana, and received the degree of Master of Sciences in May, 1963- From September, 1963, until the present time she has pursued her work toward the degree of Doctor of Philosophy at the University of Florida. Shwu-yeng Tzeng Lin is married to You-Feng Lin and is the mother of one child. 51 This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been approved by all members of that committee. It was submitted to the Dean of the College of Arts and Sciences and to the Graduate Council, and was approved as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August, 1965 Dean, College of Arts and Sciences Â£7, & UNIVERSITY OF FLORIDA V ^ 'i'1""",,,, I'll III' 3 1262 08556 7369 |

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36 relationsonspaceOOlins i RELATIONS ON SPACES By SHWU-YENG TZENG LIN A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA August, 1965 ACKNOWLEDGMENTS The author wishes to acknowledge her indebtedness and express her sincere gratitude to Dr. A. D. Wallace, Professor of the Department of Mathematics and Chairman of her Supervisory Committee, for his guidance and suggesÂ¬ tions throughout the preparation of this dissertation. She wishes to express her deep appreciation to Professor A. R. Bednarek, who read the entire manuscript in detail and made numerous corrections and improvements. Thanks are due also to all members of the Supervisory Committee for their encouragements. The author is grateful to Dr. D. C. Rose for correcting the language. She is also grateful to her husband Dr. Y.-F. Lin, who was always patient in discusÂ¬ sing mathematics with her and who rendered many valuable suggestions. And she extends her thanks to Mrs. K. P. Grady for her excellent work in typing the manuscript. ii TABLE OP CONTENTS Page ACKNOWLEDGMENTS ii Chapter I.PRELIMINARIES 1 II.A CHARACTERIZATION OF THE OUTPOINT ORDER ON A TREE 11 III.FIXED POINTS AND MINIMAL ELEMENTS 25 IV.FIXED POINT PROPERTIES AND INVERSE LIMIT SPACES 38 BIBLIOGRAPHY 48 BIOGRAPHICAL SKETCH 51 iii CHAPTER I PRELIMINARIES A relation R on a topological space X is_ a_ subset of X x X. If A is a subset of X and if R is a relation on X, let AR = p2 ((AXX)^R) and RA = P]_ ((X x a) r\ R) where p^, p2 are respectively the first and the second projections. For x in X we shall write simply xR for { x } R. Let R (-D _ = { (y, x) | (x, y) e R } , and AR^ { x | xR C A } , R['1]A= {y | Ry C A } . In all that follows, X will always denote a Hausdorff space, and R a relation on X. The closure of a set A will be denoted by A* and the interior of A by AÂ° . Definition 1.1. A relation R on X is said to be lower semi-continuous (abbreviated l.s.c.) upper semi-continuous (abbreviated u.s.c.) 1 2 at x in X if and only if x in (vr(-1))Â° > l for all V = Vo C X. R is said to be (W 0 J .l.s.c. , on X if and only if R is ,l.S.C* \ at u.s.c. / * U.S.C. i x for all x in X. R is said to be continuous if and only if R is both l.s.c. and u.s.c. Definition 1.2. A relation R on X is , upper semiclosed (u. s .cl.) -j V if and only if n \ lower semiclosed (l.s.cl.)J ^ Rx J is closed for any x in X. Definition 1.3* A relation R on X is said to be a partial order1 if and only if the following conditions are satisfied: (a) (reflexivity): (x, x) Â£ R for all x in X, (b) (antisymmetry): (x, y) 6 R and (y, x) Â£ R imply x = y, (c) (transitivity): (x, y) 6 R and (y, z) Â£ R imply (x, z) Â£ R. VR â– VR (-D [-1] â–º implies x in 1. A relation R is said to be a quasi-order if (a) and (c) are satisfied. 3 We here state some theorems to which we will refer throughout this work. The proofs of these theorems may be found in [19]. Theorem 1.1. If R is closed in X x x and if A is compact, then both AR and RA are closed; moreover, if V is an open set then R^-^ V and VR^-1^ are open. Theorem 1.2. The following statements are equiÂ¬ valent : (a) R is l.s.c. on X. (b) RAÂ° Â£ (RA)Â° for all A C X. (c) A* R C (AR) * for all AC X. (d) RV is open for all open VC X. (e) p^ | R is open, where p^ | R is the restriction of p^ to R. Theorem 1-3- The following statements are equiÂ¬ valent : (a) R is u.s.c. (b) VR^"1^ is open for all V C X. (c) AR^-1) is closed for all closed A C X. (d) AÂ° R^"1^ C (AR[_1])Â° for any A C X. (e) (AR^_1^)^C A* r(-1) for any A C X. Theorem 1.4. If X is a compact Hausdorff space, 4 then R = R* if and only if (a) R is u.s.c. on X, and (b) R is u.s.cl. on X. Definition 1.4. A subset C of X is an R-chain if and only if C x c C R U R^-1^. As a consequence of the well-known Hausdorff maximality principle, there is a maximal R-chain for any relation R on any space X. A useful result of Wallace [14] is the following. Theorem 1.5- If R is both u.s.cl. and l.s.cl. quasi-order, then every maximal R-chain is closed. Definition 1.5- An element a in X is R-minimal (R-maximal), whenever (x, a) e R ((a, x) e R) implies (a, x) e R ((x, a) 6 R). The following fundamental theorem first proved in [14] will be used repeatedly. Theorem 1.6. If R is a u.s.cl. (l.s.cl.) relation on a compact space X, and if A is a non-void closed subset of X, then A contains an R-maximal (R-minimal) element a of A. The Alexander-Kolmogoroff cohomology groups will he used as developed in [21]. In what follows the coefÂ¬ ficient group is fixed and therefore will not be mentioned. We record here some useful theorems from [21]. Theorem 1.7. If B C A C X, then HÂ°(X, A) -X HÂ°(X,B) HÂ°(A,B) â€” > H1(X, A) â€”Â¿->H1(X,B) â€”^>H1(A,B) â€”â€”* 6 ^ Hn(X, A) â€”J > is an exact sequence, where i* and j* are the induced homomorphisms of the inclusion maps i and j from A to X and X to X, respectively, and 6 is the coboundary operator for the triple (X, A, B). Theorem 1.8. For the space X and any connected set A of X the homomorphism i* of HÂ°(X) -X HÂ°( A) X H1(X, A) is an epimorphism and hence 6 is 0. For simplicity we denote by f:(X, A) -Â» (Y, B) that f is_ a_ function from X to_ Y and A is. Ãœ subset of X such that f(A) C B C Y. Furthermore, if i:(X, A) -> (X, B) is_ an inclusion map, we will write 6 i:(X, A) C (X, B). Theorem 1.9. If f :(X, A, B) continuous, and if u: (x, A) (x', A') v: (x, B) -> (X', B') w: (A, B) â– > (A', B') are defined by u(x) = v(x) = w(x) = f(x), then the ladder â€”â€”â€”Â»HP (X, A) â€”Â¿->HP(X, B) -^%HP(A, B) - 5 y â€”5~>HP(X, A)â€”^HP(X, B) tt^HP( A, B)â€”g->- is analytic, that is, each rectangle of the ladder is analytic. We state here a special case of The Mayer-Vietoris Sequence which will be sufficient in what follows for our purpose. Theorem 1.10. If X is a compact Hausdorff space, and if X = X-^ Xg where X^ and Xg are closed subsets of X, then there exists an exact sequence â€”A Hq_1(X1 r\ X2) -A> Hq(X) -iX Hq(X1) X Hq(Xg) â€” Hq(xx r\ x2) -A* ... 7 where (i* , i* , Ã³ ) and (j * , i* , Ã³ ) are the homo- v Â°a â€™ a â€™ or ' a â€™ or morphisms in the exact sequences for the triples (X, X , â–¡ ) and (Xa, Xx X2 , a ) respectively for k: (X2, X1 kh X2) c (x, x1) ; J* = x I* - 1 1 1 2 â€™ and A k Ã³2 â€˜ As a consequence of the foregoing theorem we have the following Theorem 1.11. With the hypotheses of Theorem 1.10 and if X-^ r\ Xg is connected, the homomorphism A in hÂ°(x1 r\ x2) -A* h1(x) -A_> h1(x1) X h1(x2) is 0. Proof. Observe the following ladder HÂ°(x) 4 H1(X, Xx) â€”^ H1(X) â€” > u* k* 'r -g- -r* -t-'H HÂ°(X1 ^ X2) â€”%â– H1(X2, X1 ^ X2) % HX(X2) â€”^ where u: Xx a X? C X and where (j*, i*, Â¿1) / (J*, Tg, Ã³^) and k are as in Theorem 1.10. It follows from Theorem 1.8 that if h is in HÂ°(X^ f\ Xg) then there is an h' in HÂ°(X) such that u*(h') = h. Thus by Theorem 1.9., we have 8 k* _1 Ã³g(h) = k* k* 61(h/ ) Â» â€¢ â€¢ which contains Â¿1(h/). And A = j k* 6^ being a well-defined homomorphism, we see that for any a and 8 in k * -1 6*2(h) , j*(a) = j*(8). Therefore, A(h) = j* k* _1 02(h) = j*<51(h/ ) = 0. The following notation is convenient. _If P is a_ subset of Q and if h e Hp(Q) then h | P denotes the image of h under the natural homomorphism induced by the inclusion map of P into Q. Theorem 1.12. (Reduction Theorem). If compact Hausdorff space, if A is closed, and if such that h | A = 0, then there is an open set U A such that h | U =0. Definition 1.6. (X, A) is a compact pair if and only if X is a compact Hausdorff space and A is a closed subset of X. X is a h e HP(X) containing Theorem 1.13Â» (Map Excision Theorem). If (X, A) and (Y, B) are compact pairs, and if f: (X, A) + (Y, B) is a closed map such that f takes X - A topologically onto Y - B, then 9 f* : HP(Y, B) = HP(X, A). x â€¢ t â€¢ Theorem 1.14. (Homotopy Lemma). If (X, A) is a compact pair and if T is a connected space, and if for each t in T At : (X, A) -> (X x t, A x t) is defined by A^(x, t) = (x, t), then A * = A"'*' for r and s any elements of T. I? s Definition 1.7Â» A space X is unicoherent if and only if X is connected and X = A B with A and B closed and connected implies A rs B is connected. X 2 is hereditarily unicoherent if every subcontinuum of X is unicoherent. Theorem 1.15- If X is a continuum and if H1(X) = 0 for G Â£ 0, then X is unicoherent. Definition 1.8. If X is a space, if A C X and if h is a non-zero member of HP(A), then a closed set P C A is a floor for h if and only if h | F M while h | F' = 0 for any closed proper subset F' of F. space. 2. A continuum is a compact connected Hausdorff Theorem 1.16. (Floor Theorem). If (X, A) is a compact pair and If h is a non-zero member of HP(A), then h has a floor. Moreover, every floor is connected. Theorem 1.17â€¢ If (X, A) is a compact pair, if R = A x A U A where A= { (x, x) [ x e X } then HP(X, A) es HP(X/R) for all p> 1. CHAPTER II A CHARACTERIZATION OF THE OUTPOINT ORDER ON A TREE A tree is a continuum such that every two distinct points are separated by the omission of a third point. Let X be a tree and let z be an arbitrary, but fixed element of X. Let Q(z) be the set of all such pairs (a, b) in X * X such that at least one of the following three conditions is satisfied : (i) a Â» z, (ii) a = b, or (iii) a separates z and b in X. It turns out that Q(z) is a continuous partial order on X, and with respect to this partial order z is the unique minimal element. We shall refer to Q(z), for any z in X, as a cutpoint-order [l] on the tree X. The purpose of this chapter is to give a characterization of the cut- point order on a tree (Theorem 2.4). We also obtain a new characterization of a tree from Relation-theoretic and Cohomological view-points. 11 12 Several characterizations of a tree have been given [2], [5], [23], and [24]. Perhaps the most useful of these characterizations is Lemma 2.1. [2], [5]- A continuum X is a tree if and only if it is locally connected and hereditarily unicoherent. An excellent proof of this lemma may be found in Ward [24]. Definition 2.1. A space X is said to be semi-locally-connected (abbreviated s.l.c.) at a point x of X provided for any open set U in X containing x there exists an open set V containing x such that V C U and that X - V has only a finite number of components. If X is s.l.c. at each of its points, it is said to be s.l.c. In 1933, A. D. Wallace [15] proved that one- cod imensional"*" compact connected and locally connected topological semigroup with unit and zero is a tree. L. W. Anderson and L. E. Ward, Jr. in 1961 [l] modified Wallace's result by eliminating the necessity of hypotheÂ¬ sizing a unit. More precisely, they proved that if 1. For the definition and properties of Codimension, see Wallace [21], or Cohen [6]. 13 X is a compact connected, locally connected, one- codimensional topological semilattice, then X. is a tree. Wallace [17] improved this result by weakening the local connectedness of X to semilocal connectedness of X. These elegant results on Topological Algebra, motivated the following Lemma which bears a Relation-theoretic analogy. Lemma 2.2. If R is a relation on a compact Hausdorff space X with RX a one-codimensional semi- locally connected subspace, such that / \ * (i)the relation R is closed, i.e., R = R , (ii)H^Rx) = 0 for every x in X, (iii)the collection j Rx | x e X^ has the finite intersection property (abbreviated f.i.p.), and (iv)Ra r\ Rb is connected for each pair a, b in X, then RS is a tree for every closed subset S of X. The proof of Lemma 2.2. depends on the following: Lemma 2.3- If A, M and B are disjoint non-void closed subsets of a normal space X, and if A is either compact or consisting of finitely many components such that (i) M does not separate A and B in X, and (ii) for 14 any open set U containing M there is an open subset V of U containing M such that X - V has only a finite number of components, then there exists a closed and connected subset N of X such that N <, X - M and N meets both A and B. This lemma was first proved by G. T. Whyburn [26] for the particular case in which X was assumed to be a metric continuum and card A = card M = card B = 1. The non-metric case was implicit in a paper by Wallace [17] but without proof. We postpone the proof of Lemma 2.3* to the end of this chapter. Proof of Lemma 2.2. It follows from (i) and Theorem 1.1. that RX is closed and from (iii) and (iv) that RX = J (Rx |x e X) is connected and thus RX is a continuum. Similarly RS is a continuum. Since every subcontinuum of a tree is itself a tree and since RS is a subcontinuum of RX, it is sufficient to show that RX is a tree. We first show H^(RX) =0. If there were a non-zero h e H1(RX), then there would be a maximal (nonÂ¬ void) tower J of closed subsets A of X such that h i RA f 0. Let Aq= n { A | A Â£ [[ ) Then 15 h Â¡ RAq 0, for if h | RAq = 0, then by the Reduction Theorem (Chapter 1, Theorem 1.12.) there would,', be an open V 1 RA such that h I V* = 0. It would then follow o 1 from Theorem 1.1. that R^-1^ V would be an open set containing AQ. If R^â€"^V is designated by U then R U < V so that there is an A in 7 with A < U and RA C RU C V*; therefore h [ RA = 0, a contradiction. Case 1. Card Aq = 1, i.e., Aq = { x } . By (ii) H^(RA ) = 0, a contradiction. Case 2. Card Aq > 1. Write Aq = A^ W A0 where both A^ and A2 are proper closed subsets of A . We consider the following part of the Mayer-Vietoris exact sequence (Chapter I, Theorem 1.10.), hÂ°(ra1 n ra2) A, h1(rao) _Â£* h1(ra1) x h1(ra2). Since by (iii) and (iv) RA^ RAg = IJ | RA a Rb | (a, b) Â£ A^ x Ag | is connected, then A = 0 (Theorem 1.11.), and h RA e Ker J = ImA = 0, o a contradiction. RX is a continuum and H^RX) = 0 imply that RX is unicoherent (Theorem 1.15*)â€¢ RX being of codimension one and H^(RX) = 0 imply that H^K) = 0 for every closed subset K of RX [21], and thus every subcontinuum of RX is unicoherent. 16 We now prove that every two points of RX are separated in RX by a third point. Suppose there were two points a and b such that no point separated a and b in RX. Then by Lemma 2.3-, for any p different from both a and b, there would be a continuum P which would be irreducible from a to b and which would not contain p. If q were an element of P distinct from a and b there would also be a continuum Q, irreducible from a to b and which would not contain q. But then P Q would be a subcontinuum of RX which would not be unicoherent, since P n Q by our selection of P and Q is obviously not connected. This contradiction completes the proof. Theorem 2.4. If X is a compact Hausdorff space, and if P is a relation on X, then the following conditions (i)X is of 1-codimension and s.l.c., (ii)P is a closed partial order, (iii)P is left monotone, i.e., Px is connected, and H^(Px) = 0 for every x in X, (iv){Px | x e X) has the f.i.p., and (v)P is right monotone, i.e., xP is connected for every x in X, are necessary and sufficient conditions that X be a 17 p tree, and that P be a cutpolnt-order. r â€¢ Proof: We first prove the sufficiency. Conditions (ii), (iv) and the first half of (iii) imply that Pa Pb = U { Px ( x e Pa n Pb } is connected, and thus Lemma 2.2. yields that X is a tree. Since X is compact and { Px I x Â£ X} has the f.i.p., then { Px | x e x } Â£ â–¡ Indeed, it is a single point, the unique P-minimal element of X. Let us denote by 0 the set { Px I x e x } . We prove that P = Q(0) . If (a, b) Â£ Q(o) such that a = 0 or a = b, then clearly (a, b) must be also in P. If a separates 0 and b in X, then since Pb is a continuum containing 0 and b, it must contain a, and we again conclude that (a, b) is in P. Thus Q(0) C P. Conversely, if (a, b) is in P, then since a is in aP r\ Pb, and since both aP and Pb are continua, then aP ^ Pb is a subcontinuum of the tree X, and therefore 2. Condition (v) is not necessary for X to be a tree. used. 3. Only reflexivity and transitivity of P are 18 by Lemma 2.1. it is unicoherent. Thus aP r\ Pb is also a continuum. Now, by virtue of the Hausdorff-Maximality- Principle, aP r\ Pb has a maximal P-chain, C, and (a) C is closed (Theorem 1.5Â«)> and (b) C _is connected. For if C were not connected there would be two non-void disjoint closed sets A and B such that C = A U B and b Â£ B. The set A contains a maximal element m. Define A' and B; by the equations A/ = Pm o C and B/ = C - Pm. B â€˜ C mP, and since A C. A1 , then B/ C B. Now A1 r\ B1 C Pm n (mP r\ B) = (Pm rN mP) r\ B = Q , therefore C = A' U b' is a separation. If bQ designates the minimal element in B', then by the maximality of C mP r\ Pb = { m, b } o ' o J which contradicts the connectedness of mP r\ Pb . o Therefore, any maximal P-chain in aP r\ Pb is connected. (c) aP r\ Pb has a unique maximal P-chain, which we denote by Cp(a> b). If C and C' were two distinct maximal P-chains in aP n Pb, then both C and C/ would contain a and b, and C U C' would then be connected, and hence C n C' would be connected. But for x Â£ C - c' 19 C ^ c' = (Px r\ C C) u (xP r\ C C') = C ^ (Px U xP) r\ C1 is obviously a separation, a contradiction. Since (0, b) Â£ Q(0) C P and since X is a tree, there is a unique connected Q-chain [22], [23] Cq(0, b) C Pb which contains both 0 and b. Pb must also have a connected P-chain containing both 0 and b and this P-chain must be unique. We denote by C (0, b) the unique connected P-chain in Pb containing 0 and b. Since a Q-chain is also a P-chain, then Cp( 0, b) = CQ(0, b) . Similarly, there is a unique connected P-chain Cp(0, a) in Pa containing both 0 and a. It is clear that Cp(0, a) V Cp(a, b) = Cp(0, b) = CQ(0, b). As a consequence, a Â£ Cq(0, b) and hence (a, b) Â£ Q which was to be proved. We next prove the necessity. Let X be a tree and let P be the cutpoint order on X with respect to a point z in X. We prove that X and P satisfy the conditions (i), (ii), (iii), (iv) and (v) stated in the theorem: Proof of (i). By Ward [24] a tree is a compact connected commutative idempotent semigroup with zero, therefore it is acyclic [17]. Hence in particular 20 H^(X) =0. We now show H^A) = 0 for every A = A* C X and thus X is of one codimension, unless Xis degenerate. Suppose on the contrary that H (A) ^ 0 for some closed subset A of X. If h is a non-zero member of H^(A), then by the Floor Theorem (Theorem l.l6.) there is a floor F C A for h, which is connected. The set F being a subcontinuum of a tree is itself a tree and hence is acyclic. Therefore, H^F) = 0 which contradicts the fact that F is a floor, and thus H^A) = 0. The semilocal connectedness of X follows from the fact that X is compact and locally connected. Proof of (ii). This is proved in Ward [24]. Proof of (iii). The outpoint order P is order dense [23] , and since by (ii) P = P* we have that every maximal P-chain in Px is connected [22], thus Px is connected. Indeed, Px itself is a tree and therefore as has been proved in (i), H^Px) = 0. Proof of (iv). This is obvious, since P has the least element z. Proof of (v). Replacing Px by xP in the argument of (iii), we easily obtain the connectivity of xP. Proof of Lemma 2.3Â» (l) There is a_ component Aq of A such that M does not separate Aq and B in X. For if otherwise, to each component A^ of A there is a pair of disjoint open sets and T^ such that X - M = U and } A^, T^ ^ B. Since A is either compact or consisting of finitely many .components, there is a finite subfamily { G^, G^, . .., G \ of G-^s 21 such that A C v { Gi 1 i = 1, 2, .. ., m } . Whence G = m G. and i=l 1 T = m r\ i=l T. 1 are two disjoint open sets such that X - M = G KJ t and G P A, T p B so that the hypothesis (i) is contradicted. Throughout the rest of the proof, let #[S] be the number of components of the space S. Let If = { | A be the collection of all open sets containing M but missing A such that #[X - V^] is finite. For each Â£ If let R^ be the component of X - that contains the component Aq of A. Designate R ^ u{R^ A e A } (2) R i_s open. Let y e R, and let Â£ If be such that R^ contains y. By the normality of X there is a V, e If with V C V* C V, . Since X - V, [X |X |X A [X has only a finite number of components and since (x - yÂ° 3 X - vj 3 x - Vv we have R, C RÂ° C R. Therefore R is open. A ^ P (3) For each VQ e If , there is an R^ containing R such that a #[>V n (X - Va)] = n 22 for all containing R^. Let C-^, Cg, . .., Cn be the n components of X - V , so that X - Va = C-^ o Cg â€¢ â€¢ â€¢ U where = R^, then for any contained in VQ we have X - containing X - Va, so that each Cf must be either totally contained in R^, the component of X - containÂ¬ ing Aq, or disjoint from R^. Thus #[Rp rs (X - Va)] < # (X - Va) = n. But, # [r6 rv (x - va) ] < Â§ [rt r\ (x - va) ] if R^ 2 R5 ^ R . Hence there is an R^ } R^ such that the number #[R^ r\ (X - VQ)] is the maximal so that #[ (4) For each VQ e there is an R^ such that Â«n for all R^ containing R^. By virtue of (3) above, there is an R, containing R such' that A Ct #t>V ~ (X - va>] = #tRA ^ *V ^ (X - Va> ^ (X - Va) for all R^ containing R^. For if as in (3) above, we express X - Va = ci U c2 U ... U Cn where, without loss of generality, CL is contained in R^ 23 for i = 1, 2, . . ., k (k < n) and Ck r\ R^ = O for i = k + 1, ..., n, then it follows from R .containing [X R-^ and that #[R^ ^ (X - VQ)] = # [Ra ^ (X - VQ)] C. r\ R = Q for i = k + 1, . .., n. And therefore, 1 Rp. ^ k = RA kN (X - Va) . (3) R = R* - M; that is, R _is closed in X - M. If y is a point of R - M then there exists a Va in V- missing y. By the normality of X there is a V, P in such that C V^* C VQ. By (4) there is an R^ containing R^ such that A (X - Vp) = R* A (X - Vp) for all R^ containing R^. Furthermore, if U is an open set about y, then U ^ R^ 41 O â€¢ For, if we designate the set U r\ (X - V^*) by W then y is in both R - M and W so that W ^ R 4= O , and there is an R^ such that W n RÂ¿ 0 â€¢ Without loss of generality we may assume that R, contains R, . Then o A â–¡ * W n RÃ³ = W ^ Ra C (X - Vp) = W r\ R^ r\ (X - Vp) = W A R . Thus, U ri R^ } W A R^ 4: G , so that y is in R^ = R^. Therefore, R is closed in X - M. It is to be noted that R meets B and hence R T r\ B ^ Q for some t ; because otherwise. X - M = R U(X-M-R) would be a separation, of X - M between Aq and B, so that (l) would be contradicted. We now conclude result by taking N = R^ . the CHAPTER III FIXED POINTS AND MINIMAL ELEMENTS In the Symposium of General Topology and its Relations to Modern Analysis and Algebra (Prague 196l), Professor A. D. Wallace announced [18] among other things the following fixed point theorem. Theorem 3-1Â» [18, Theorem 5]* If X is a continuum, if P is a closed left monotone partial order on X such that PA* C (PA)* for each A C X, and if z separates Pa and Pb in X, then Pz = z. Wallace applied this theorem to prove Theorem 3 - 2 . [18, Theorem 6]. If X is a conÂ¬ tinuum and if P is a closed left monotone partial order on A such that PA*C (PA)* for each A ( X, then the set K of P-minimal elements is connected. However, in [18] Wallace gave no proof of the fixed point theorem. For the sake of completeness, we will give in this chapter a proof of Theorem 3*1 by first proving Theorem 3-2 and using this result to establish Theorem 3â€¢1â€¢ We will also show that under certain conditions the stateÂ¬ ments (i) If z separates Pa and Pb in X then 25 26 Pz = z. r Â» (ii) The set of P-minimal elements 'is connected, are equivalent (Theorem 3**0- Furthermore, if P is a partial order on the continuum X and K is the P-minimal elements in X, we will prove in Theorems 3-5^ 3*6 and 3-7 that the cohomology groups of X and those of K are isomorphic for all non-negative dimensions for certain classes of X with suitable choice of P. The following lemma will be used in the proofs of foregoing theorems. Lemma 3â€¢3â€¢ [cf. l6]. If X is a compact Haus- dorff space and if P is a. lower semi-closed partial order on X such that PA* < (PA)* for each A ( X, then the set K of P-minimal elements is closed.' Proof. Suppose by way of contradiction that there is an x in K* - K. Then since Px is closed and X is compact, Px has a minimal element which must be a P-minimal element in X, and thus Px r\ K if Q . Let y be any element in PxnK. Obviously x ^ y, so that there is an open set U containing x whose closure excludes y. If V = U r\ K, then PV = V, and x is in V*. Since 27 PV*C (pvf = V*, then PV* C U* and hence y e Px C PV* C U*". This contraÂ¬ dicts the fact that y ^ U*. Therefore the set K is closed. Proof of Theorem 3*2. Let us assume contrary to the conclusion of the theorem that K is not connected. By Lemma 3-3 there are two disjoint non-void closed sets A and B such that K = A ^ B. Since P is closed, then AP and BP are closed and X = AP w BP. Furthermore, the connectedness of X yields AP os BP % â–¡ . Thus AP rs BP has a minimal element, say t. Since Pt = (Pt r\ AP) v (Pt BP) , then (Pt AP) rs (Pt BP) = Pt rs (AP rs BP) = t. If we designate C = (Pt r\ AP) - t and D = (Pt/^ BP) - t, then both C and D are open in Pt. Furthermore, C and D are both non-void. For if C is void, then Pt r\ AP = t, and so t must be in A. This implies that t is not in BP which contradicts the fact that t is a minimal element in AP r\ BP. Similarly, D is not void. Consequently, Pt - t = C U D is a separation. The connectedness of Pt yields C* = Pt r\ AP and D* = Pt r\ BP. Now, for each x in Pt - t we have Px C Pt - t and the connectedness of Px then implies that either Px C C or Px C D. Thus, 28 for each x in C we see that Px C C, and hence PC C C, and it follows that C* 3 (PC)* b PC * = P(C ^ t) ) Pt 5 D which is a contradiction. Therefore K is connected. Proof of Theorem 3-1- Let X - z = A ^ B where A and B are non-void separated sets and aP ( A and bP C B. Let K be the set of P-minimal elements. Now, if Pz ^ z, then K C X - z and K = (K r\ A) V (Kn B) is a separation for K which contradicts the connectedness of K. Theorem 3â€¢4. If X is a continuum and P is a closed left monotone partial order on X such that the set of P-minimal elements K is closed, then the following statements are equivalent: (i) If z separates Pa and Pb in X then Pz = z. (ii) The set K is connected. Proof. (i) implies (ii). Suppose the set K is the union of two disjoint non-void closed sets A and B. Since K is the set of P-minimal elements then X = KP = (A ^ B)P = AP U BP. And the hypotheses that X is connected and P is closed yield AP r\ BP 4= â–¡ â€¢ Let z be a P-minimal element of AP BP. Then there are elements a e a and b^B such 29 that {a, b}< Pz - z. The equalities Pz = (Pz ^ AP) b (Pz n BP) , and Pz a (AP r\ BP) = z show that Pz - z = (Pz r\ AP - z) ^ (Pz A AP - z) is a separation, that is, . z separates a = Pa and b = Pb in the continuum Pz, and so by (i) z is a minimal element which is neither in A nor B. This contradiction estabÂ¬ lishes the connected ness of K. (ii) implies (i). The proof as given for Theorem 3-1 applies here as well. We now present an example to motivate Theorem 3*3. Example. Let X = { (a, b) 1 a2 4- b2 = 1 ^ U {(0, b) | 1 <_ b <_ 2} for reals a and b. Let X be endowed with the Euclidean topology. Let P = { ((0, b1), (0, b2)) | 1 < bx < b2 < 2j U A where A = | (x,x) | x Â£ X } . Then P = P* is a both left and right monotone partial order on X such that PA C (PA) for each A ( X and K = { (a, b) l a2 + b2 = l} = S1 (l-sphere) is the set of P-minimal elements. It is to be noted in this example that H^Px) = 0 for each x e X while H^(K) = H^(S) + 0 for any nonÂ¬ trivial coefficient group. However the following equality 30 Hp(x) = HP(K) holds for all non-negative dimensions. . It is interesting to seek conditions on X and a relation P which imply the equality HP(X) - HP(K) for all non-negative dimensions. The purpose of the next theorem is to take a small step in this direction. Theorem 3-5- If X is a continuum, and if P is an upper semiclosed partial order on X such that (i)PA* C (PA)* for all A C X, (ii)there exists an element u in X such that X - K C Pu where K = {x| Px = x }, and (iii)the quotient space X/K x K u A modulo K X K u A is a topological semilattice under the natural partial order induced by P, where A - { (x, x) I x 6 X } , then, Hp(x) s H^(K) for all integers p > 0. Proof. For simplicity in notation we write Y = X/KXKUA. Since every compact topological semilattice has a (unique) zero, we write z for the zero of the semiÂ¬ lattice Y. By the hypothesis (ii), Y has a (unique) unit which will be denoted by u also since no confusion is likely to occur. We will accomplish the proof in three steps. (a) H^(X, K) = HP(Y, z) for all integers p >_ 0. Let f : (X, K) -> (Y, z) be the natural map. Since (X, K) is a compact pair, f is a closed map, and moreover, f 31 takes X - K topologically onto Y - z. Thus-by the Map Â» * Excision Theorem (Theorem 1.13) f*- : HP(Y, z) -> HP(X, K) is an isomorphism. (b) HP(X, K) = 0 for all integers p > 0. We establish this by showing HP(Y, z) = 0 for all integers p >_ 0. Define ffc : (Y,z) -Â» (Y,z) by f (y) = t a y for all t e Y, where a is the semilattice operation on Y. If for each t Â£ Y define A^: (Y, z) -> (YX Y, z xY) by A^(y) = (y, t) , then f = A o A^. Since by the Homotopy Lemma (Theorem 1.14) Az*= Au*, thus f *= A*. A*= A*. A*Â» f *. z z u u Since f^ is the identity map, so is f z. Let i : (z,z) C (Y, z) be an inclusion map and let h : (Y, z) (z, z) be defined by h(y) = z for all y e Y. Since f = i â€¢ h, the following diagram Ã¡j is analytic, for all non-negative integers p, that is fj = f *= h* * i* . Since Hp(z, z) = 0 for all integers p > 0 and HP(Y, z) = fJ(Hp(Y, z))Ch*(Hp(z, z)) for all 32 integers p >_ 0, then HP(Y, z) = 0 for all integers p >_ 0 as desired. (c) HP(X) = H^K) for all integers p >_ 0. By combining the above results with the exact sequence for the triple (X, K, â–¡ ) (Theorem 1.7)* we obtain the exact sequence 0 = HP(X, K) -Â» HP(x) â€¢> HP(K) -> HP+ 1(X, K) = 0 for p = 0, 1, 2, ... . Consequently, Hp(x) and HP(K) are isomorphic for all non-negative dimensions. We now center our attention on the case in which X is of codimension one. Theorem 3Â«6. If P is a closed partial order on a continuum X and if ir : X^X/KXKUA designates the natural map, such that (i)PA*C (PA)* for all A C X, (ii)P is left monotone and H^(7r(Px)) = 0 for all x e x, (iii)X is of codimension one, then Hp(x) Â« HP(K) for all p > 0. Proof. Denote X =X/KXK^A and P = { (tr(x), ir(y)) J (x, y) e p J then since X is compact and ir continuous, P is closed and left monotone; indeed we have P7r(x) = tt(Px) . By virtue of Theorem 1.17* we have HP(X, K) = HP(X) for all 33 p >_ l. Since X is of codimension one we hav.Ã© HP(X, K) = 0 for p >_ 2 [21] . We show H^X, K) = 0 by proving H^(X) = 0 . Using the same argument employed in the proof of Lemma 2.2 one sees that H^(PS) = 0 for all closed sets "1 ^ S in X; in particular we have H (X) = 0 . We have noted in part (c) of the proof for Theorem 3*5 that if HP(X, K) = 0 for all p > 1 then HP(X) s HP(K) for all p > 1. The equality H^(x) = HÂ°(K) follows from the fact that both X and K are connected [Theorem 3*2]. We remark that the hypothesis (iii) in Theorem 3-6 may be weakened to : (iii ) cd (X - K)* = 1, and this may even be dropped completely if each Px is a chain, as will be seen in the following Theorem 3 Â«7â€¢ If P is a closed partial order on a continuum X such that (i) PA* C (PA)*, and (ii) each Px is a connected chain, then Hp(x) Â£ HP(K) for all p > 0. Proof. Let X, 7r and P be defined as in Theorem 3-6. Since P is closed and 7rPx is a connected chain P is closed and Ptt(x) is a connected chain. It follows then that each P7r(x) is a generalized arc and hence Hp(P7t(x)) = 0 for all p >_ 1. As it has been noted 3^ in the proof of Theorem 3*6, H1(PS) = 0 for Â¿11 closed sets S in X. We now show H^X) = 0 for all p >_ 1 by proving HP(PS) = 0 for all p >_ 1 and for each closed set S in X. If there were a least integer n such that Hn(PA) i= 0 for some closed set A in X, then n would have to be greater than 1, and A Â± â–¡ . Let h be a nonÂ¬ zero member of Hn(PA), then h | Pa = 0 for each a Â£ A and hence by the Reduction Theorem there is an open set V containing Pa such that h I V =0; then by Theorem 1.1 there is an open set U containing a such that PU C V. Thus the collection a of all open subsets U of A such that h J PU = 0 forms an open cover of A. Also Q. is closed under finite union, for if and Ug are in a denote hQ = h | PU* U PU*. Since P^ U Ug)* = P(U* ^ U*) te PU-^ ^ PUg , to show G is closed under finite union it suffices to show hQ = 0. In the following part of Mayer- Vietoris exact sequence : Hn-1(PU*^ PU*) *Â» Hn(PU*^PU*) Ã Hn(PU*) x Hn(PUg) since J*(hQ) = (h I PU*, h I PUg) = (0, 0) , hQ is in the image of A. But Hn_1(PU*^ PU*) = Hn_1(PS) where S = PU^r\?uJf is a closed subset of X, and hence Hn-'L(PS) = 0 by the minimality of n. This proves hQ = 0 so that a is closed under finite union. Since A is compact, A is a union of some finitely many elements of (X â€¢ Thus A must be in G and so h = 0, a contradiction. This 35 together with Theorem 1.17 implies HP(X, K) = Hp(x) = 0. for all p >_ 1. It then follows from the exact sequence for the triple (X, K, O) [Theorem 1.7] and the connectedness of X and K [Theorem 3-2] that Hp(x) = HP(K) for all p > 0. Remark. Theorem 3-7 may be stated more generally by repacing the hypothesis (ii) by (ii') each 7r(Px) is acyclic. We conclude this chapter by exhibiting an example which answers a question in Topological Semigroups. A clan is a compact connected topological semigroup with unit [20]. Let S be a clan and let R = { (x, y) | (x, y) e S X S, xS uSx c yS u Sy } then R is a closed quasi-order on S and the set of R-minimal elements is also the minimal ideal of S. The question to be answered is: if. a. clan S is. a. tree, is its minimal ideal an arc or a. point 9 The answer is affirmative if S is abelian (or normal: xS = Sx for all x e S), but it is negative in general. Example. Let S = X ^ I be the subset of Euclidean 3-space such that X = { (x,0,0) | -1 < x < 1 } U {(0,y,0) | -1 < y < 1} I = { (0,0,t) | 0 < t < 1 } ; 36 let S be endowed with the Euclidean topology and let the multiplication o on S be defined below (the usual multiplication of reals is denoted by juxtaposition) : i.Aop = ii.(I, o) unit interval; i.e., A for all A Â£ X and for all p e S is the usual semigroup of the real (0,0,tx)o(0;0,t2) = (0,0,t1t2), iii.For each (0,0,t) e I and for any (0,y,0)ex, (0,0,t)o(x,0,0) = (tx,0,0) and (0,0,t)o(0,y,0) = (0,ty,0). Then, (l) (S, o) is associative. (2) o : S S -> S is continuous. (3) (S, o) is a clan with (0,0,l) as unit. (4) The minimal ideal of (S, o) is X which is neither an arc nor a point. It is interesting to observe that the semigroup S given in the above example may be realized as a semiÂ¬ group of matrices by the following one-to-one correspondÂ¬ ence : â– Â» t x + y/^1 (x,y,t) + 0 1 37 where -l<_x, y <_ 1, 0 < t < 1 and (tx)2 + (:xy)2 + (yt)2 = 0. The correspondence is indeed an isomorphism. Prom this the assertions (l), (2) and (3) in the example are self-evident. / CHAPTER IV FIXED POINT PROPERTIES AND INVERSE LIMIT SPACES A space X is said to have thÃ© _f. jd. jo. (fixed point property) if, for every continuous function f: X -> X there exists some x in X such that x = f(x). Hamilton [8] has proved that the chainable metric continua have the f. p . p . A space X is said to have the F. jo. jo. (fixed point property for multifunctions) if every continuous â€™multifunction'*' F : X -> X has a fixed point, i.e., there exists a point x in S such that x Â£ F(x). Obviously if X has the F. p. p. then it has the f. p. p., but the converse need not be true. Strother [13] has exhibited two continua X and Y both have the F. P- P- but their Cartesian product X x Y fails to have the F. P- p. Borsuk [3] has constructed a decreasing sequence of three-cells whose intersection does not have the f. p. p. The interÂ¬ section is the inverse limit, the bonding maps being inÂ¬ jections. As a counter theorem to the results of Strother and of Borsuk, we prove that if (X^, tt-^, /\ ) is an 1. Following Strother [12], a multifunction F : X -> Y is continuous if, and only if, F(x) is closed for each x, and F "*'(A) is open (closed) if A is open (closed). 38 39 inverse system of compact spaces such that each has the F. p. p. then the inverse limit space has the F. p. p. As a corollary to this, we obtain Ward's generalization [25] of the Hamilton theorem [8] that every chainable metric conÂ¬ tinuum has the F. p. p. Our result is, indeed, stronger than the Ward's, since it includes some of the non-metric chainable continua as well. Definition 4.1. The collection (X-^, tt^ , A- ) is as inverse system of spaces if: (i)A is a directed set, (ii)A in A implies that X^ is a Hausdorff space, (iii)whenever A > p there is a continuous function 7T-y : X, -> X, , Ap A [1 (iv)if A > p and p > v, then tt^ = â€¢ The function 7r^ is called a bonding map. If A is in A let be the subset of the Cartesian product P{ X^ | A e A } , defined by = | x j if A > p then tt-^x(A) = x(p)} , where x(A) denotes the A-th coordinate of x. Definition 4.2. The inverse' limit space X^ of the inverse system of spaces (X-^, tt^ , A ) is defined to be xâ€ž = r\ { sA i * 6 a } 40 endowed with the relative topology inherited fpom the product topology for P { | A e A } ; in notation X = Iim (X^, ir^ , A ) . We write p^ : P { X^ | A â‚¬ A } â– * X^ for the A-th projection of P [ X^ | A e A } , i.e., p^(x) = x(4) for aH xin P|X^| A^A}; the restriction p^ j X^ will be deÂ¬ noted by 7which will be called a projection map. It is readily seen from the definition that an element x of P { XA | A e A] is in X^ if and only if = tt (x) whenever A > p. A more detailed account of inverse limit space may be found in Lefschetz [10], Eilenberg and Steenrod [7], Capel [4] and Mardesic [11]. The following known results (see, e.g., [4], [10]) will be used. Lemma 4.3Â» (i) The collection ^ir^(U^) Â¡ A Â£ A and is an open subset of X^ j forms a basis for the topology of XÂ». (ii)The inverse limit space X^, is Hausdorff; if A â‚¬ A , is a closed subset of P { X^ [ A e A} so that X^ is closed in P â€¢{ X^ | A e A } â€¢ (iii)If X^ is compact for each A in A then X^ is compact; if, in addition, each X^ is non-void then Xjo is non-void. (iv)If X^ is a continuum for each A o A then the inverse limit space is a continuum. Lemma 4.4. If A is a compact subset of X^ and if t| , then (tt^(A), ir^' , A) is an inverse system of spaces such that A = 11m (ir^(A), tt^, A) and each bonding map ir^ is onto. In the sequel, since we are only interested in compact spaces, each projection map tt-^ will be assumed to_ be_ onto; for if otherwise, by virtue of Lemma 4.4, each X^ may be replaced by tt^(X*.) without disturbing the resulting inverse limit space. We are now ready to state our main result. Theorem 4.5Â» Let (X^, , A) be an inverse system of compact spaces such that each X-^ has the P. p. p., then the inverse limit space X*, also has the F. p. p. We divide the proof of this theorem into the following steps. In Lemmas 4.6, 4.7 and 4.8 will be assumed to be the inverse limit space of the inverse system (X^, tt-^, A ) of compact spaces. Lemma 4.6. If F : X^ -> X^ is a continuous multifunction, define F^ : X^ -> X^ by F^ = v-^F v^ for each A. Then F, is a continuous multifunction. A Proof. (i) If t is in X^, then since F : X^ -*â€¢ Xâ€ž is a continuous multifunction, by Theorem 1.1 and Theorem 1.4 of Chapter I, Fir~^(t) is a closed subset of X^. Thus, F4tJ is closed for every t in X, . (ii) If CA is a closed subset of XA, then is closed. It is readily seen that F ^tta^(Ca) Is closed in Xm and hence compact; therefore â€¢7r^F~'Sr^ (C^) = F^(C^) is compact and hence closed. (iii) If UA is open in XA, then F^(U^) is open. 7rA and F being continuous, F â– 'â€˜Tr-^(U^) is an open subset of Xoo â€¢ It follows then, by virtue of Lemma 4.3 (i), that ir^F-1TT^1(U^) = F^(U^) is open. Thus, by (i), (ii) and (iii) above, F^: X-^X-^ is continuous. Lemma 4.7Â» Let F : Xffl -> XÂ«, be a continuous multifunction, let F^ : X^ -> X^ be defined as in Lemma 4.6 Then, for each x in Xa, , (i) (F^rr^(x), ttA[x, A)2 and (ir^F(x) , v^, A ) are inverse systems of compact spaces, (ii) lim (FA7rA(x), 7tA[x, A) = 11m (ttaF(x) , tta , A) (iii) F(x) = lim (Fa7Ta(x), 7Ta^, A). Proof. Since each FA is continuous (Lemma 4.6) and each XA is compact, so is F-^ir-^(x) for all A e A . 2. For simplicity in symbolism, henceforth if A C lim (XA, 7r A^, A) then (u^A, tta , A) will mean KA> a ) â€¢ 43 To show that (F^tt-^(x) , 7r^(X> A ) forms ah inverse system, it suffices to show 7T^^F^tt^(x) Â£ F^7r^(x) whenever A > |x. To this end we first observe Vx> < K^aiAaM - nÂ£Vx)â€™ sincÃ© 7T-, 7T-> â€” ir â€¢ From this we have Ap A [X â€™â€™aixVaW c â€™rAnVÃÃâ€™V(x) = lrA1>A1' â€™rÃ1),rÃÃâ€™rM.(x) = Ati,rA)P(lrAM.â€™rAr\(x> = v â€™vSiW = F 7T (x) , (X |X by the definition of FA, F^ and the equality = 7r^. The fact that (7T-^F(x) , tta , A) forms an inverse system follows from Lemma 4.4. (ii) For each A e A and any x e X^, , we have x e tta tta(x) and hence, 7T.F(x) C u\F irt^Ax) = (tt,F ir~Â±)vy{x) = F,ttAx) -1> Thus, lim (tt^F(x), tt^, A)C^ (Fatta(x), tt^, A). To prove the other inclusion, we show Xoo - lim (ttaF(x), ttA[i, A) C Xm- lim (F^r^(x) , ir^, A). Let y be in Xa - lim (ttaF(x), , /\ ) then, by Lemma 4.4 44 there exists a p e A such that ir^(y) ^ tt^FÃx) . Let U| M- M- and V be two disjoint open sets in X such that P M- so that Vy) 6 um. and V(x) c V F It follows then from Lemma 4.3 (i) and the continuity of F that there exists a 6 Â£ A and an open set in X^ such that x a tt^U^ , and <*) F(uÂ¡\) c ir'\ â€¢ Since A is directed, there is a Aq e A such that Aq > p and Aq > 6, we shall use this Aq throughout the proof of this lemma. If we write U.. = ir-T^eU* and use Aq AqÃ“ 0 the equality ttT1 = tt-.1 irr,1,- , then ( Hr) may be rewritten 0 aq Aq0 as F<< V c r-\ , -1, and hence F, U, = 7T, F 7T,1(U, ) ( 7f, 7T 1V = 7T, (iT, TT, ) 1V Aq Aq Ao A0 Aq V Aq P P A0 AqP Aq p - â€¢ In particular, F, TT, (x) ( IT,1 V . Ao Ao ^ Ao^ ^ Similarly, one obtains ^o(y) 6 T^un â€™ since ^(y) e D(i and ^ = v ^ The fact that U are disjoint implies TTA0pVpn^AopUp â€”.UM = â–¡ and and 45 consequently 7r, (y) Â¿ F, tt, (x) . From this we conclude Ao * Ao Ao y ^ lim (F^tt^(x), tt^ , A ) as desired. (iii) This follows immediately from (ii) and Lemma 4.4. Lemma 4.8. Let F : X^-* X,*, be a continuous multiÂ¬ function, let F^ : X, -> X^ be defined as in Lemma 4.6 Let Ex = Ãex l ex e xx and ex â‚¬ Vex>} fchen forms an inverse system. Proof. It suffices to prove Â£ E^ whenever A > p. Let e^ Â£ F-^(e^), then V^x) 6 TxnFx = WaK1*^ = V'xn^xh- Thus, ir^E^ C E|j_ as is to be proven, Proof of Theorem 4.5Â» Since each X^ has the F. p. p. and by Lemma 4.6 each F^ : X^ -Â» X-^ is continuÂ¬ ous, each E^ is closed and non-void. By Lemma 4.8, (E^, , A ) is an inverse system of compact spaces, so it has a non-void inverse limit space lim (E-^, tt^, A ) . We now conclude the proof by showing that each x in iim (Ea, ttA/U, A ) is a fixed point under F; i.e., xÂ£F(x). 46 If x is in lim (E^, tt^, A) then tta(x) for-all A G A ; i.e., tta(x) e E^7r7v(x) fÂ°r &H A G A . ConseÂ¬ quently, by Lemmas 4.4 and 4.7, we have x = lim (irA(x), tt^, A) g lim (F^tt^x) , vA) = F(x). In fact, with the assumption of Theorem 4*5 and the notation of Lemma 4.8 together with the notation E = | x j x G F(x) J. , we can make the following sharper assertion. Theorem 4.5' . E = lim (E^, tt^, A ) . Proof. Prom the proof of Theorem 4.5, we have E } lim (Ea, ttA[X, A) . It remains to prove that E C lim (Ea, tt^, A ) â€¢ Let x be in E, then x G F(x) and therefore, for all A e A , ^(x) g tt^F(x) C = Fa(tta(x)). That is, tta(x) g E^ for all A; consequently, by Lemma 4.4 E C lim (Ea, ir^ , A ) â€¢ A chain (U^, U^, â€¢â€¢â€¢, U ) is a finite sequence of sets U. such that U. r\ U. Â± â–¡ if and only if J | i - j [ <_ 1. A Hausdorff space X is said to be chainable 47 .if to each open cover \J* of X there is a finite open cover /[X = (U^, Ug, . .., Un) such that (i) U refines V"> (ii) u= ( Uf, Ug, . .., Un) forms a chain. It follows that a chainable space is a continuum. It is implicit in a paper by Isbell [9] that each metrizable chainable continuum is the inverse limit space of a sequence of (real) arcs. This together with a theorem of Strother [13] that a bounded closed interval of the real numbers has the F. p. p. imply the following result of Ward [23] as a consequence of Theorem 4.5- Corollary 4.9. Each chainable metric continuum has the-F. p. p. Examples of inverse limit spaces of inverse systems of real arcs exist which are not metrizable; for instance, the "long line" is one such. Thus, Theorem 4.5 is a proper generalization of that of Ward's [25]* BIBLIOGRAPHY 1. Anderson, L. W. and Ward, L. E., Jr. One-dimensional topological semilattices. Ill. J. Math., 5 (1961), 182-186. , 2. Borsuk, K. lÃber die Abbildungen Raume auf die Kreislinie. der metrischen kompakten Fund. Math., 20 (1933), 224-231. 3. Sur un continu acyclique qui se laisse transÂ¬ former topologiquement en lui meme sans points invariants. Fund. Math., 24 (1935), 51-58. 4. Capel, C. E. Inverse limit spaces. Duke Math. J., 21 (1954), 233-245. 5. Cech, E. Sur les continus Peaniens unicoherents. Fund. Math., 20 (1933), 232-243- 6. Cohen, H. A cohomological definition of dimension for locally compact Hausdorff. spaces . Duke Math. J. , 21 (1954), 209-224. 7* Eilenberg, S. and Steenrod, N. Foundations of algebraic topology. Princeton University Press, Princeton 1952. 8. Hamilton, 0. H. A fixed point theorem for pseudo-arcs and certain other metric continua. Proc. Amer. Math. Soc., 2 (l95l), 173-174. 9. 10. 11. / 12. 14. Isbell, J. R. Embeddings of inverse limits 4 Ann. of Math., 70 (1959), 73-84. Lefschetz, S, Algebraic topology. Amer. Math. Soc. Colloq. Publ., no. 27, New York 1942. Mardesic, S. On inverse limits of compact spaces. Glasnik Mat. Fiz. Astr., 13 (1958), 249-255- Strother, W. L. Continuity for multi-valued functions and some applications to topology. Doctoral Dissertation, Tulane University 1952. On an open question concerning fixed points. Proc. Amer. Math. Soc., 4 (1953), 988-993- Wallace, A. D. A fixed point theorem. Bull. Amer. Math Soc., 51 (1945), 613-616. 15- Cohomology, dimension and mobs. Summa Brasil. Math., 3 (1953), 43-55- l6. Struct ideals. Proc. Amer. Math. Soc., 6 (1955), 634-638. 17â€¢ Acyclicity of compact connected semigroups. Fund. Math., 50 (l96l), 99-105- 18. Relations on topological spaces. Proc. Symp on General Topology and its Relations to Modern Analysis and Algebra. Prague 1961, 356-360. 19. Relation-theory, Lecture Notes. University of Florida, 1963-1964. 20. Topological semigroups, Lecture Notes. University of Florida, 1964-1965- 50 21. Wallace, A. D. Algebraic topology, Lecture Notes. University of Florida, 1964-65Â» 22. Ward, L. E. Jr. Partially ordered topological spaces. Proc. Amer. Math. Soc., 5 (1954), l44-l6l. 23- A. note on dendrites and trees. Proc. Amer. Math. Soc., 5 (1954), 992-994. 24. Mobs, trees, fixed points. Proc. Amer. Math. Soc., 8 (1957), 798-804. 25- A_ fixed point theorem. Amer. Math. Monthly, 65 (1958), 271-272. 26. Whyburn, G. T. Analytic topology. Amer. Math. Soc., 1942. BIOGRAPHICAL SKETCH Shwu-yeng Tzeng Lin was born May 11, 193^ at Tainan, Formosa. In June, 1953, she was graduated from Tainan Girl's High School. In June, 1958> she received the degree of Bachelor of Sciences from Taiwan Normal University. She worked as a Research Assistant at the Mathematics Institute of Academia Sinica from 1958 to I960. In the fall of I960 she enrolled, and worked as a teaching assistant, in the Department of Mathematics, Graduate School of the Tulane University, New Orleans, Louisiana, and received the degree of Master of Sciences in May, 1963- From September, 1963, until the present time she has pursued her work toward the degree of Doctor of Philosophy at the University of Florida. Shwu-yeng Tzeng Lin is married to You-Feng Lin and is the mother of one child. 51 This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been approved by all members of that committee. It was submitted to the Dean of the College of Arts and Sciences and to the Graduate Council, and was approved as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August, 1965 Dean, College of Arts and Sciences Supervisory Committee: Ãº j\ Chairman SLL. k-w/. c/ (-j/c-i Ã³Ã¼. 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