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RELATIONS ON SPACES By SHWUYENG 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. TABLE OF CONTENTS Page ACKIOWLEDGMENTS ..................................... ii Chapter I. PRELIMINARIES............................. 1 II. A CHARACTERIZATION OF THE CUTPOINT 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 ((A x X) (\ R) and RA = p1 ((X x A) {\ R) where p1, p2 are respectively the first projections. For x in X we shall write i x R. and the second simply xR for Let R(I) = { (y, x) I (x, y) E R ) , AR[~1] = { x xR C A ., and R1A { y I 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 lower semicontinuous (abbreviated l.s.c.) said to be upper semicontinuous (abbreviated u.s.c.) 2 VR(l) at x in X if and only if x in { implies x in (VR(l)) o w ] ,, S( O for all V = V C X. R is said to be (VR[1])0 l.s.c. l.s.c. { } on X if and only if R is lsc at u.s.c. u.s.c. 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.) f and only if xR Slower semiclosed (ls.cl.)Rx is closed for any x in X. Definition 1.3. A relation R on X is said to be a partial orderI if and only if the following conditions are satisfied: (a) (reflexivity): (x, x) E R for all x in X, (b) (antisymmetry): (x, y) E R and (y, x) E R imply x = y, (c) transitivityy): (x, y) E R and (y, z) E R imply (x, z) e R. 1. A relation R is said to be a quasiorder if (a) and (c) are satisfied. 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 RI1 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 C (RA) for all A C X. (c) A* R C (AR)* for all A C X. (d) RV is open for all open V C X. (e) p, I R is open, where p, I 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(l) is closed for all closed A C X. (d) A R[1] ( (AR[l])o 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, 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 Rchain if and only if C x C C R U R(1). As a consequence of the wellknown Hausdorff maximality principle, there is a maximal Rchain 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. quasiorder, then every maximal Rchain is closed. Definition 1.5. An element a in X is Rminimal (Rmaximal), 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 nonvoid closed subset of X, then A contains an Rmaximal (Rminimal) element a of A. The AlexanderKolmogoroff cohomology groups will be 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 ( A C X, then H(X,A) 4 HO(X,B) H(A,B)6     HI1XA) j H(XB) (A,IB)  6 H n( XA)   Hn(X,A) i 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) ? H(A) 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 a 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 i:(X, A) C (X, B). Theorem 1.9. If f:(X, A, continuous, and if u: (X, A) (X', v: (X, B) (X', w: (A, B) > (A', B) (X' A', B') is A'), B'), B') are defined by u(x) = v(x) = w(x) = f(x), then the ladder SHP(X, A) HP(X, B) HP(A, B) _ u* J V* W* ,HP(X, A) HP(X, B) i HP(A, B)  is analytic, that is, each rectangle of the ladder is analytic. We state here special case of The MayerVietoris Sequence which will be sufficient in what follows for our purpose. Theorem 1.10. If X is a compact Hausdorff space, and if X = X1 U X2 where X1 and X2 are closed subsets of X, then there exists an exact sequence I )Hql(x 1, X2) H q(X) JT Hq(x1) X Hq(X2)  Hq(X, 1 xn ) A where (j i 6) and (j 6) are the homo morphisms in the exact sequences for the triples (X, Xa, j ) and (X X1 r X2, 0 ) respectively for a = 1i, 2; k: (X2, X1 r X2) C (X, X1) ; J*= j*l x j*2 I* i* i k 1 = i 1 ; and A = J k 62 As a consequence of the foregoing theorem we have the following Theorem 1.11. With the hypotheses of Theorem 1.10 and if X1 \ X2 is connected, the homomorphism A in H(X1 r X2) H H (X(x ) X H (X2) is 0. Proof. Observe the following ladder H(X)  Hl(X, X1) 1 HI1() 1) 62 H <>  T H(x x2) 2 X, XI X) Hx) where u: X1 r\ X2 C X and where (j*, i, 6) (J 2' 2) and k are as in Theorem 1.10. It follows from Theorem 1.8 that if h is in H(X r\ X2) then there is an h' in H(X) such that u*(h') = h. Thus by Theorem 1.9., we have 1 k* 62(h) = k k 6(h which contains 61(h'). And A = j k* 1 62 being a welldefined homomorphism, we see that for any a and P in k*' 32(h), J*(a) = J(P). Therefore, A(h) =j k 1 2(h) = J61(h) = 0. The following notation is convenient. If P is a subset of Q and if h e HP(Q) then h I 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 X is a compact Hausdorff space, if A is closed, and if h e HP(X) such that h I A = 0, then there is an open set U containing 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; 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 f* : HP(Y, B) HP(X, A). 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 ht : (X, A) > (X x T, A x T) is defined by ht(x, t) = (x, t), then K*r = ?h* for r and s any elements of T. Definition 1.7. A space X is unicoherent if and only if X is connected and X = A V B with A and B closed and connected implies A \ B is connected. X is hereditarily unicoherent if every subcontinuum2 of X is unicoherent. Theorem 1.15. If X is a continuum and if H1(X) = 0 for G t 0, then X is unicoherent. Definition 1.8. If X is a space, if A C X and if h is a nonzero member of HP(A), then a closed set F C A is a floor for h if and only if h I F 6 0 while h I F' = 0 for any closed proper subset F' of F. 2. A continuum is a compact connected Hausdorff space. Theorem 1.16. (Floor Theorem). If (X, A) is a compact pair and if h is a nonzero 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) I x E X } then HP(X, A) HP(X/R) for all p> 1. CHAPTER II A CHARACTERIZATION OF THE CUTPOINT 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 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 turn s 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 cutpointorder [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 Relationtheoretic and Cohomological viewpoints. 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 semilocallyconnected (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 codimensional compact connected and locally connected topological semigroup with unit and zero is a tree. L. W. Anderson and L. E. Ward, Jr. in 1961 [1] 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]. 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 Relationtheoretic analogy. Lemma 2.2. If R is a relation on a compact Hausdorff space X with RX a onecodimensional semi locally connected subspace, such that (i) the relation R is closed, i.e., R = R , (ii) H1(Rx) = 0 for every x in X, (iii) the collection { RxJ 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 nonvoid 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 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 nonmetric case was implicit in a paper by Wallace [171 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 I 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 H1(RX) = 0. If there were a nonzero h E Hl(RX), then there would be a maximal (non void) tower 3 of closed subsets A of X such that h IRA # 0. Let A= { A I A Then hi RAo :j 0, for if h I RAo = 0, then by the Reduction Theorem (Chapter 1, Theorem 1.12.) there woulcdi'be an open V ) RAo such that hi V* = 0. It would then follow from Theorem 1.1. that R[1] V would be an open set containing A If R11]V is designated by U then R U ( V so that there is an A in 7 with A ( U and RA C RU ( V*; therefore h I RA = 0, a contradiction. Case 1. Card Ao = 1, i.e., Ao = x By (ii) H1(RA ) = 0, a contradiction. Case 2. Card Ao > 1. Write A = A1J A2 where both A1 and A2 are proper closed subsets of A . We consider the following part of the MayerVietoris exact sequence (Chapter I, Theorem 1.10.), j* HO(RA 1 n RA2) 1(RA) H1(RA1) X H1(RA2). Since by (iii) and (iv) RA1 r\ RA2 = J { RA ^ Rb I (a, b) e A, x A21 is connected, then A = 0 (Theorem 1.11.), and h RA E Ker J = ImA = 0, 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 Hl(K) = 0 for every closed subset K of RX [21], and thus every subcontinuum of RX is unicoherent. 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 J Q would be a subcontinuum of RX which would not be unicoherent, since P ri 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 1codimension and s.l.c., (ii) P is a closed partial order, (iii) P is left monotone, i.e., Px is connected, and Hl(px) = 0 for every x in X, (iv) {Px J 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 2 tree, and that P be a cutpointorder. Proof: We first prove the sufficiency. Conditions (ii), (iv) and the first half of (iii) imply that Pa n Pb = J { Pxl 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 E X} has the f.i.p., then {I Pxx X } 0 Indeed, it is a single point, the unique Pminimal element of X. Let us denote by 0 the set Px I x X } We prove that P = Q(0). If (a, b) E 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(O) C P. Conversely, if (a, b) is in P, then since a is in aP r\ Pb, and since both aP and Pb are continue, then aP vk Pb is a subcontinuum of the tree X, and therefore 2. Condition (v) is not necessary for X to be a tree. 3. Only reflexivity and transitivity of P are used. by Lemma 2.1. it is unicoherent. Thus aP r, Pb is also a continuum. Now, by virtue of the Hausdorffi.4Lxima1lity Principle, aP n Pb has a maximal Pchain, C, and (a) C is closed (Theorem 1.5.), and (b) C is connected. For if C were not connected there would be two nonvoid disjoint closed sets A and B such that C = A u B and b E B. The set A contains a maximal element m. Define A' and B' by the equations A' = Pm r\ C and B' = C Pm. B C mP, and since A C A', then B' C B. Now A' \ B' C Pm n (mP n B) = (Pm n mP) \ B = , therefore C = A' V B' is a separation. If b0 designates the minimal element in B', then by the maximality of C mP r" Pb0 = { m, bo ) which contradicts the connectedness of mP n Pbo Therefore, any maximal Pchain in aP r Pb is connected. (c) aP rN Pb has a unique maximal Pchain, which we denote by C p(a, b). If C and C' were two distinct maximal Pchains in aP r\ 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 E C C' C n C' = (Px rA C r\ C') v (xP R C C') = C A (Px u xP) n C' is obviously a separation, a contradiction. Since (0, b) E Q(O) C P and since X is a tree, there is a unique connected Qchain [22], [23] CQ(0, b) C Pb which contains both 0 and b. Pb must also have a connected Pchain containing both 0 and b and this Pchain must be unique. We denote by Cp (0, b) the unique connected Pchain in Pb containir; 0 and b. Since a Qchain is also a Pchain, then Cp(O, b) = CQ(O, b). Similarly, there is a unique connected Pchain Cp(O, a) in Pa containing both 0 and a. It is clear that Cp(O, a) Q Cp(a, b) = Cp(O, b) = CQ(0, b). As a consequence, a E CQ(0, b) and hence (a, b) E 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 HI(x) = 0. We now show H1(A) = 0 for every A = A* C X and thus X is of one codimension, unless X '.is degenerate. Suppose on the contrary that H1(A) 4 0 for some closed subset A of X. If h is a nonzero member of HI(A), then by the Floor Theorem (Theorem 1.16.) there is a floor F Cj A for h, which is connected. The set F beirn a subcontinuum of a tree is itself a tree and hence is acyclic. Therefore, H1(F) = 0 which contradicts the fact that F is a floor, and thus H1(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 cutpoint order P is order dense [23] and since by (ii) P = P* we have that every maximal Pchain in Px is connected [22], thus Px is connected. Indeed, Px itself is a tree and therefore as has been proved in (i),' H1(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. (1) There is a component A 0of A such that M does not separate A and B in X. For if otherwise, to each component A? of A there is a pair of disjoint open sets Gh and Th such that X M = G\ U T and G? ) A?, T B. Since A is either compact or consisting of finitely many,components, there is a finite subfamily {G1, G2, ..., GmJ of G'S such that A C, { Gi. i = 1, 2, ..., m } Whence m m G = Gi and T = (\ Ti are two disjoint open sets i=1= such that X M = G T and G ) A, T ) B so that the hypothesis (i) is contradicted. Throughout the rest of the proof, let f[S] be the number of components of the space S. Let I = VI ? A EA be the collection of all open sets Vh containing M but missing A such that #[X VQ] is finite. For each V E let R1 be the component of X V, that contains the eCWijpoiKclLt Ao of A. Delignate R U ( RX N e A } (2) R is open. Let y E R, and let V E  be such that Rh contains y. By the normality of X there is a V e ( with VC V V ( Since X V has only a finite number of components and since (X V )o ) X V ) V, we have Rh ( R C. (R. Therefore R is open. (3) For each V E there is an R, containing R such that # [R4 n (X Va)] = [ [R (X Va)] for all R containing R Let C1, C2, ..., Cn be the n components of X Va, so that X Va = C1 J C2 C ... n Cn where C, = R., then for any Vg contained in Va we have X Vo containing X Vo, so that each Ci must be either totally contained in R., the component of X VP contain ing Ao, or disjoint from Rp. Thus #[RI t (X V,)] # (X V,) = n. But, if the for # [R,6 n (X v)] < #[R? n (X Va)] R. > R6 ) R Hence there is an R, ) Ra such that number #[R, n (X Va)] is the maximal so that #[R (x Va)] = #[R n (x Va)] all R containing R . (4) For each Va E Z there is an Rh R L (X V,) = R n (X V,) for all R containing R,. By virtue of (3) al is an Rh containing R suchthat #[RI (X V,)] = #i[R; (X V( for all R containing R,. We note, further, 1 R\ (X V) = RN n (X V,) for all R containing R For if as in (3) we express such that above, there ,) ] bhat above, X Va = C1 U C2 j ... Cn where, without loss of generality, Ci is contained in Rh for 1 1, 2, ..., k (k < n) and Ci n R = 0 for i = k + 1, ..., n, then it follows from R containingg Rh and #[R \ (X V,)] = [R; n (X V,)] that Ci r R4 = for i = k + 1, ..., n. And therefore, R / (X Va)= C1 U C2 U ... Q Ck = Rh ) (X V,) (5) 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 t missing y. By the normality of X there is a Vp in t, such that Vg ( V* C V By (4) there is an Rh containing Rp such that R ^ (x vf) = R? n (x v ) for all R containing Rh. Furthermore, if U is an open set about y, then U f R O* .0 For, if we designate the set U r\ (X VP*) by W then y is in both R M and W so that W R O0 and there is an R 6 such that W n R6 0 Without loss of generality we may assume that R, contains Rh. Then SP W n R 6 W r\ R6 C (X VP) w n R, r\ (x v) Sn R (x Thus, U n Rh ) W n R* L so that y is in R = R,. Therefore, R is closed in X M. 24 It is to be noted that R meets B and hence RT r\ B 4 0 for some T ; because otherwise.' X M= R U (X M R) would be a separation.of X M between Ao and B, so that (1) would be contradicted. We now conclude the result by taking N = R T . CHAPTER III FIXED POINTS AND :IIi'IriAL ELELEii;TS In the Symposium of General Topology and its Relations to Modern Analysis and Algebra (Prague 1961), 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 < 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 (PA)* for each A < X, then the set K of Pminimal 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 3.1. We will also show that under certain conditions the state ments (i) If z separates Pa and Pb in X then 25 Pz = z. (ii) The set of Pminimal elements *is connected. are equivalent (Theorem 3.4). Furthermore, if P is a partial order on the continuum X and K is the Pminimal 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 nonnegative dimensions for certain classes of X with suitable choice of P. The following lemma will be used in the proofs of foregoing theorems. Lemma 3.3. [cf. 16]. If X is a compact Haus dorff space and if P is a. lower semiclosed partial order on X such that PA ( (PA)* for each A < X, then the set K of Pminimal 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 Pminimal element in X, and thus Pxr\ K Q Let y be any element in PxrK. Obviously x 4 y, so that there is an open set U containing x whose closure excludes y. If V = U ( K, then PV = V, and x is in V*. Since PVC (Pvf = v", then PV*< U and hence y E Px C PV* C U*. This contra dicts the fact that y k UY. 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 nonvoid closed sets A and B such that K = A V B. Since P is closed, then AP and BP are closed and X = AP v BP. Furthermore, the connectedness of X yields AP r\ BP 4 Thus AP n BP has a minimal element, say t. Since Pt = (Pt r\ AP) V (Pt r\ BP), then (Pt r AP) ^ (Pt BP) = Pt n (AP r BP) = t. If we designate C = (Pt (\ AP) t and D = (Pt/\ BP) t, then both C and D are open in Pt. Furthermore, C and D are both nonvoid. For if C is void, then Pt n\ 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 CM= Pt \ AP and D = Pt n 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, for each x in C we see that Px C C, and hence PC C C, and it follows that C* (PC)* 4 PC = P(C U t) : Pt ) D which is a contradiction. Therefore K is connected. Proof of Theorem 3.1. Let X z = A V B where A and B are nonvoid separated sets and aP C A and bP ( B. Let K be the set of Pminimal elements. Now, if Pzp z, then K C X z and K = (K r\ A) V (K r B) is a separation for K which contradicts the connectedness of K. Theorem 3.4. If X is a continuum and P is a closed left monotone partial order on X such that the set of Pminimal 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 nonvoid closed sets A and B. Since K is the set of Pminimal elements then X = KP = (A V B)P = AP U BP. And the hypotheses that X is connected and P is closed yield AP r BP t Q Let z be a Pminimal element of AP BP. Then there are elements a e A and bEB such that {a, b} ( Pz z. The qualities Pz = (Pz r AP) V (Pz rBP), and Pz, (AP ^ BP) = z show that Pz z = (Pz r AP z) v (Pz r 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 3.5. Example. Let X = { (a, b) \ a2 + b2 = 1 U {(0, b) 1 1 < b < 2} for reals a and b. Let X be endowed with the Euclidean topology. Let P = { ((0, bl), (0, b2)) 1 1 <_ b <_ 2} U A where A = { (x,x) I x X} Then P = P* is a both left and right monotone partial order on X such that PAXC (PA)* for each A ( X and K = { (a, b) I a2 + b2 = 1} = S1 (1sphere) is the set of Pminimal elements. It is to be noted in this example that HI(Px) = 0 for each x X while H1(K) H1(S) + 0 for any non trivial coefficient group. However the following equality HP(X) 2 HP(K) holds for all nonnegative dimensions. It is interesting to seek conditions on X and a relation P which imply the equality HP(X) s HP(K) for all nonnegative 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 < Pu where K = { x I Px = x and (iii) the quotient space X/K x K u A modulo K X K V A is a topological semilattice under the natural partial order induced by P, where A = 1 (x, x) I x E X , then, HP(X) n! HP(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) HP(X, K) M 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 takes X K topologically onto Y z. Thus'by the I.;ap 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 ft : (Y,z) > (Y,z) by ft(y) = t A y for all t E Y, where A is the semilattice operation on Y. If for each t E Y define t: (Y, z) > (YX Y, z x Y) by At(y) = (y, t), then ft = A ? \t. Since by the Homotopy Lemma (Theorem 1.14) A*= ? "*, thus Since f is the identity map, so is f. Let u z i : (z,z) ( (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 fz = i h, the following diagram f = f U Z HP(y, z) HP(Y, z) i* HP(z, z) is analytic, for all nonnegative integers p, that is f= f*= h*o i*. Since HP(z, z) = 0 for all integers u z p > 0 and HP(Y, z) = fu(HP(Y, z))( h*(HP(z, z)) for all integers p > 0, then HP(Y, z) = 0 for all integers p > 0 as desired. (c) HP(X) t H(K) for all integers p > 0. By combining the above results with the exact sequence for the triple (X, K, 0 ) (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 nonnegative dimensions. We now center our attention on the case in which X is of codimension one. Theorem 3.6. If P is a closed partial order on a continuum X and if T : X X/KXKUA designates the natural map, such that (i) PA*( (PA)* for all A C X, (ii) P is left monotone and Hl(v(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 KUA and P= i (T(x), 7(y))  (x, y) E P then since X is compact and wT continuous, P is closed and left monotone; indeed we have Pw(x) = vr(Px). By virtue of Theorem 1.17, we have HP(X, K) S HP(") for all p > 1. Since X is of codimension one we have' HP(X, K) = 0 for p >2 [21]. We show H1(X, K) = 0 by proving H (x) = 0 Using the same argument employed in the proof of Lemma 2.2 one sees that Hl(PS) = 0 for all closed sets S in X; in particular we have HI () = 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) c HP(K) for all p L 1. The equality HO(x) = HO(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) c HP(K) for all p > 0. Proof. Let X, w and P be defined as in Theorem 3.6. Since P is closed and 7rPx is a connected chain P is closed and P7r(x) is a connected chain. It follows then that each PTr(x) is a generalized arc and hence HP(P7r(x)) = 0 for all p > 1. As it has been noted in the proof of Theorem 3.6, HI(PS) = 0 for All 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) # 0 for some closed set A in X, then n would have to be greater than 1, and A + D Let h be a non zero member of Hn(PA), then h I 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 0% of all open subsets U of A such that h PU*= 0 forms an open cover of A. Also 0 is closed under finite union, for if U1 and U2 are in , denote h h IPU1 U PU". Since P(U1 U UV P(U1 v U2) * 1 2 1 2= = 1PU V PU to show G0 is closed under finite union it suffices to show h = 0. In the following part of Mayer Vietoris exact sequence : HnI(puI U,) A n JU n K, H PU1 PU2) Hn(1PU VPU) 2 H (PU) X Hn(PU2) since J(ho) = (h U, h i PU) = (0, 0), h is in the image of A. But Hn (UIp PUP) = Hn (PS) where S = PUI PU is a closed subset of X, and hence IHn(PS) = 0 by the minimality of n. This proves h = 0 so that a is closed under finite union. Since A is compact, A is a union of some finitely many elements of Thus A must be in and so h = 0, a contradiction. This 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, C) [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 replacing the hypothesis (ii) by (ii') each v(Px) is acyclic. We conclude this chapter by exhibiting an e:.;;iple 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) L (x, y) E S X S, xS V Sx C ySw Sy then R is a closed quasiorder on S and the set of Rminimal 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 ar'c 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 U I be the subset of Euclidean 3space such that X = { (x,0,0) j 1 < x < 1] V {(0,y,0) I 1 y < 1} I= { (o,o,t) 0 o< t l ; let S be endowed with the (0,0,I) Euclidean topology and let (0,l,0) (l,0,i) the multiplication o on S be defined below (the usual multiplication of reals is denoted by (i,0,0) (0,1,0) juxtaposition) : i. Nop. = N for all \ E X and for all p E S, ii. (I, o) is the usual semigroup of the real unit interval; i.e., (0,O,tl)o(O;O,t2) = (O,O,tlt2), iii. For each (O,O,t) E I and for any (0,y,0)EX, (0,0,t)o(x,0,0) = (tx,0,O) and (0,O,t)o(O,y,O) = (O,ty,O). Then, (1) (S, o) is associative. (2) o : S S > S is continuous. (3) (S, o) is a clan with (0,0,1) 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 onetoone correspond ence t x+ y J7_ (x,y,t) It I O 1 3 37 where 1 < x, y < 1, 0 < t < 1 and (tx)2 + (:xy)2 + (yt)2 = 0. The correspondence is indeed an isomorphism. From this the assertions (1), (2) and (3) in the example are selfevident. CHAPTER IV FIXED POINT PROPERTIES AND INVERSE LIMIT SPACES A space X is said to have the f. p. p. (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 continue have the f. p. p. A space X is said to have the F. p. p. (fixed point property for multifunctions) if every continuous "multifunction1 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 continue X and Y both have the F. p. p. but their Cartesian product X Y fails to have the F. p. p. Borsuk [3] has constructed a decreasing sequence of threecells 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 (Xh, hT, t\ ) 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 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 nonmetric chainable continue as well. Definition 4.1. The collection (X,, w A ) is as inverse system of spaces if: (i) A is a directed set, (ii) ? in A implies that X, is a Hausdorff space, (iii) whenever A > j there is a continuous function v : X> > X, , (iv) if A > and 4 > v, then n = IV7T The function vn is called a bonding map. If A is in A let Sh be the subset of the Cartesian product P { Xx A e A } defined by S = { x if ?A > j then wr\x(?) = x(u)} , where x(?) denotes the 7\th coordinate of x. Definition 4.2. The inverselimit space X, of the inverse system of spaces (X,, w, A ) is defined to be X= r{ S 1h E A endowed with the relative topology inherited from the product topology for P( X  A E A} ; in notation X = lim (XA, 7 T,' A). We write p, : P { X E A) X for the Ath projection of P { X E 1 A A} i.e., p,(x) = x(A) for all x in P{ X  AI EA); the restriction phI X, will be de noted by 7rA which will be called a projection map. It is readily seen from the definition that an element x of P{ X I A E A} is in X, if and only if "hrM,(x) = 7T(x) whenever ?\ > t. A more detailed account of inverse limit space may be found in Lefschetz [10], Eilenberg and Steenrod [71, Capel [4] and Mardesic [11]. The following known results (see, e.g., [4], [101) will be used. Lemma 4.3. (i) The collection {(rIJ(UN)I ) E A and UT is an open subset of XN J forms a basis for the topology of Xe. (ii) The inverse limit space X. is Hausdorff; if N EA SN is a closed subset of P i X3 N E A} so that X0 is closed in Pi X I A E A} (iii) If XA is compact for each h in A then X, is compact; if, in addition, each Xh is nonvoid then X, is nonvoid. (iv) If Xh is a continuum for each NA EA then the inverse limit space is a continuum. Lemma 4.4. If A is a compact subs et of X, and if T =w k I w(A), then (I(A), 7( A) is an inverse system of spaces such that A = lim (7ir(A), 7 A) and each bonding map T is onto. In the sequel, since we are only interested in compact spaces, each projection map T x will be assumed to be onto; for if otherwise, by virtue of Lemma 4.4, each Xh may be replaced by 7rx(X.) without disturbing the resulting inverse limit space. We are now ready to state our main result. Theorem 4.5. Let (XV, T,' /A) be an inverse system of compact spaces such that each Xh has the F. 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 (XX, 7wgT, A ) of compact spaces. Lemma 4.6. If F : X > X_ is a continuous I multifunction, define Fh : Xh X> by Fh = 7hF xT7 for each X. Then Fh is a continuous multifunction. 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, Pl1(t) is a closed subset of XV. Thus, Fh(t) is closed for every t in X (ii) If CA is a closed subset of X., then F lC.) is closed. It is readily seen that F1 Vl(cC) is closed in X~, and hence compact; therefore 'AF"'A(CA) = FA (CC) is compact and hence closed. (iii) If UA is open in X,, then. F~I(U) is open. 7r and F being continuous, F1 V N(U1 ) is an open subset of X, It follows then, by virtue of Lemma 4.3 (i), that VnFIvF 1 (lu) = F1(U) is open. Thus, by (i), (ii) and (iii) above, FA: X jXN is continuous. Lemma 4.7. Let F : X~ X, be a continuous multifunction, let F, : Xh XN be defined as in Lemma 4.6. Then, for each x in Xo,, (i) (Fzw,(x), wA, A )2 and (7TF(x), 7T\, A ) are inverse systems of compact spaces, (ii) lim (F 7,h(x), 7', A) = lim (7TwF(x) v,' /A), (iii) F(x) = lim (FV?(x) 7hw, A). Proof. Since each F, is continuous (Lemma 4.6) and each Xh is compact, so is Fhr,(x) for all ?\ E A . 2. For simplicity in symbolism, henceforth if AClim (X,, A r, A) then (rrA, 7r4, A) will mean (vA, V A ~?h' A )' To show that (Frn,(x), h*,, A ) forms an inverse system, it suffices to show 7h4 Fh,(x) ( FT w(x) whenever \ > p. To this end we first observe 7r^() C (V 7r )V,(X) = V1 since vhLp = "r. From this we have V CF() 7%^p.FQ7T ,i"(x) V p.(7vrF 7w 1)V>r1(x) F= ( )F( p:w?) 1(x) = vw pl7r j(x) = F r (x), by the definition of F,, Fp. and the equality n = T .P, The fact that (w7F(x), p., 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 iThl1 (x) and hence, 7rF(x) C 7NF rr (x) = (wF 7rl)7(x) = Fw(x). Thus, rim (7hF(x), 7TT?, A) im (F7j(x), wP, A). To prove the other inclusion, we show X.. lir (wrF(x), rp, A) C X, lim (Fh7T(x), Tp), A). Let y be in X lm (rF(x), ) then, by Lemma 4.4 Let y be in X, lir (7r F(x), 7 then, by Lemma 4.4 L y e X,  there exists a L e A such that P(y) 4 V PF(x). Let UP and VL be two disjoint open sets in X such that (y) E U and 7rF(x) C V so that F(x)( C7r.1V P It follows then from Lemma 4.3 (i) and the continuity of F that there exists a 6 E A and an open set U6 in X8 1 such that x e 76 U6 and (*) F(61U6) C v 7VP Since A is directed, there is a E A such that 7 > p. and 0 > 6, we shall use this 0W throughout the proof of this lemma. If we write Uo = 6U6 and use the equality r = then (*) may be rewritten as Fr U ) and hence F oUo = r F h(U ( 1.o)'V = ro(o7 ~ l In particular, FVoo(x) C ^o 1 Similarly, one obtains .ro (y) E tf U since vP,(y) E UP and wr, = AOi,.lo The fact that UP and V are disjoint implies l oV4 (% V 1 U = 0 and P. 0.V ?\t 71)0p.U consequently o, (y) J F ow o(x). From this we conclude y lim (FaX,,(x), vh;\, A) as desired. (iii) This follows immediately from (ii) and Lemma 4.4. Lemma 4.8. Let F : Xw X, be a continuous multi function, let Fh : X > X be defined as in Lemma 4.6 Let E = {e I e X E X and eA E F,(eh)} then (E', wr,' A ) forms an inverse system. Proof. It suffices to prove 7rhL E,( EL whenever X > L. Let eX F,(e9), then V (e.) E 7xF(e.) = 2Fr(e,) l(eF) i vi : F l~e ) C 7rF(v17T )7 A (e?) F.m7v_'(e\) F= ( \A)1 (e,) = Thus, "v E4 ( EL as is to be proven. Proof of Theorem 4.5. Since each Xh has the F. p. p. and by Lemma 4.6 each F, : X > Xh is continu ous, each E is closed and nonvoid. By Lemma 4.8, (Eh, iTv, A ) is an inverse system of compact spaces, so it has a nonvoid inverse limit space lim (E,\, FT7, A). We now conclude the proof by showing that each x in lim (E., Ay,' A) is a fixed point under F; i.e., xeF(x). If x is in lim (Eh, 7r" A) then i,(x) e'Eh for all A E A ; i.e., w,(x) E F.7r,(x) for all ? e A Conse quently, by Lemmas 4.4 and 4.7, we have x = lim (wr?(x) ., A) E lim (Fh,(x) 7r,,, A) = 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 E F(x) we can make the following sharper assertion. Theorem 4.5'. E = lim (E,, wI, A). Proof. From the proof of Theorem 4.5, we have E ) lim (E,' 7r,, A). It remains to prove that E Cl1im (E?, IT^ 7,) A Let x be in E, then x E F(x) and therefore, for all E A , TTh(x) E v F(x) ( vF(Tl, ) (x) = F?(7r(x)). That is, 7,r(x) E Eh for all X; consequently, by Lemma 4.4 E ( lim (E?, 7r,,, A). A chain (UI, U2, ..., Un) is a finite sequence of sets Ui such that Ui r U, f D if and only if Ii J I 1. A Hausdorff space X is said to be chainable if to each open cover LJ of X there is a finite open cover U = (UI, U2, ..., U n) such that (i) U refines 1; (ii) ;J= (U1, U2, ..., 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 [131 that a bounded closed interval of the real numbers has the F. p. p. imply the following result of Ward [251 as a consequence of Theorem 4.5. Corollary 4.9. Each chainable metric continuum has theF. 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 [251. BIBLIOGRAPHY 1. Anderson, L. W. and Ward, L. E., Jr. Onedimensional topological semilattices. Ill. J. Math., 5 (1961), 182186. 2. Borsuk, K. Uber die Abbildungen der metrischen kompakten Raume auf die Kreislinie. Fund. Math., 20 (1933), 224231. 3. Sur un continue acyclique qui se laisse trans former topologiquement en lui meme sans points invariants. Fund. Math., 24 (1935), 5158. 4. Capel, C. E. Inverse limit spaces. Duke Math. J., 21 (1954), 233245. 5. Cech, E. Sur les continues Peaniens unicoherents. Fund. 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BIOGRAPHICAL SKETCH Shwuyeng Tzeng Lin was born May 11, 1934 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 1960. In the fall of 1960 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. Shwuyeng Tzeng Lin is married to YouFeng Lin and is the mother of one child. 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 Dean, Graduate School Su pervisory Comittee: (i i ^'A Cha1 rn rn  .. ,  ," {Z ~ ^ ,. i7 '/. .^/ L U y '. r.a.r~c~tu UNIVERSITY OF FLORIDA 3 1262 08556 7369 *I ,*  \\ '` '.  v' * ,. '. .. * 