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1 Topology Proceedings Web: Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA ISSN: COPYRIGHT c by Topology Proceedings. All rights reserved.

2 TOPOLOGY PROCEEDINGS Volume 29, No. 1, 2005 Pages WHEN ARE LOCAL CONNECTIVITY FUNCTIONS CONNECTIVITY? FRANCIS JORDAN Abstract. Suppose X is a continuum, then X is Peano if and only if every real valued local connectivity function defined on X is a connectivity function if and only if every real valued local connectivity function defined on X is a Darboux function. This gives a partial answer to a question of J. Stallings. We also characterize the continua for which every local connectivity function has connected graph, answering a question of S. B. Nadler, Jr. 1. Introduction In his 1959 paper, J. Stallings asked for conditions under which a local connectivity function f : X Y was a connectivity function [6, #5, p. 262]. We give an answer to this question by characterizing continua for which every real valued local connectivity function is connectivity. Theorem 1. Let X be a continuum. The following are equivalent (a1) Every real valued local connectivity function defined on X is connectivity. (b1) Every real valued local connectivity function defined on X is Darboux. (c1) X is Peano Mathematics Subject Classification. Primary 54. Key words and phrases. connectivity functions, local connectivity functions, Peano continuum. The results presented here are a direct result of a visit I made to West Virginia University in the summer of I would like to thank Georgia Southern University and West Virginia University for their support. I would also like thank Bruce McLean and Sam Nadler for stimulating discussions. 185

3 186 F. JORDAN Until now, it was known only that (a1) is true for peripherally connected locally unicoherent spaces ([1] and [6]) and hereditarily arcwise connected spaces [5]. We also complete work begun by Sam B. Nadler, Jr. [5] on characterizing continua with the property that every real valued local connectivity function has connected graph. We prove the following characterization of these continua: Theorem 2. Let X be a continuum. The following conditions are equivalent. (a2) Every real valued local connectivity function defined on X has connected range. (b2) X is continuum chainable. (c2) Every real valued local connectivity function defined on X has connected graph. For the interested reader, the continua for which every real valued function with connected graph is connectivity have been characterized by Nadler [4]. 2. Preliminaries A continuum is a nonempty compact connected metric space. A space is a Peano continuum provided that it is a locally connected continuum. Suppose X is a metric space with metric d. Given nonempty sets A, B X we define d(a, B) = inf({d(x, y): x A & y B}). The diameter of a nonempty set A X is defined by diam(a) = sup{d(x, y): x, y A}. For a set A X we write cl(a), int(a), bd(a) for the topological closure, interior, and boundary of A in X, respectively. Given δ > 0, we say a continuum is a δ-continuum provided that its diameter is less than δ. Given a space X, a point p X, and δ > 0, we define the δ-component of x X to be the set of all points x in X such that there is a finite collection of δ-continua C such that x, p C and C is connected. We say an indexed collection of sets C = {C 1,..., C n } is a δ-chain provided that C i C j if and only if i j < 2. If x C 1 and y C n, we say C is a chain from x to y. If C C, we say C is a link of C. We define the length of a chain to be the number of links it contains, i.e., its cardinality.

4 WHEN ARE LOCAL CONNECTIVITY FUNCTIONS CONNECTIVITY? 187 If C is a chain such that diam(c) < δ for every C C, we say C is a δ-chain. A continuum X such that for every δ > 0 and x, y X there is a δ-chain {C 1,... C n } of continua such that x C 1 and y C n is said to be continuum chainable [2]. Let X and Y be topological spaces. We say f : X Y is Darboux provided that f[c] is connected for every connected C X. We say f : X Y is connectivity provided that the graph of f C is connected for every connected C X. If there is an open cover U of X such that f U is connectivity for every U U, then we say that f is a local connectivity function. In general, we will identity a function with its graph and use the same symbol to describe both objects. We denote the real line by R. 3. Proof of Theorem 2 Lemma 3. If x, y X are in the same δ-component of a space X, then there is a δ-chain of continua from x to y. Proof: Since x and y are in the same δ-component, there is a finite collection A of δ-continua such that x, y A and A is connected. By 8.13 of [3], we may index A to form a weak chain {A 1,..., A n } from x to y, i.e., x A 1, y A n, and A i A i+1 for each 1 i n 1. Let j 1 = max{i: x A i }. If y A j1, then {A j1 } is the desired δ-chain. Otherwise, begin an inductive construction. Assume we have defined {j 1,..., j k } so that j 1 < j 2 <... < j k, y / A jk, and j l = max{i > j l 1 : A i A jl 1 } for all 1 < l k. To see the inductive step, let j k+1 = max{i > j k : A i A jk }. Notice that the set we are taking the maximum over is nonempty since j k < n 1 and A jk A jk +1. Clearly, j k < j k+1. If y A jk+1 the construction stops. Since A is finite, we know there is an L such that y A jl It is immediate from the construction that {A j1,..., A jl } is a δ-chain of continua from x to y. Proof of Theorem 2: We show that (a2) implies (b2). Suppose X is not continuum chainable. Let x, y X be such that there is no δ-chain from x to y. By Lemma 3, x and y are in different

5 188 F. JORDAN δ-components of X. Let A be the δ-component of x. The characteristic function of A is easily checked to be local connectivity (just cover X with open δ/2 balls), but the range of f is not connected since A is a proper subset of X. We show that (b2) implies (c2). Suppose X is continuum chainable and that f : X R is local connectivity. Let U be a cover of X by open sets such that f U is connectivity for every U U. Let δ > 0 be a Lebesgue number for U. Fix p X. For every x X there is a δ-chain C x of continua from p to x. Since f C is connected for each C C, f C is connected. Finally, f = x X C x is connected since p C x for all x X. That (c2) implies (a2) is obvious. 4. Proof of Theorem 1 We first note that (a1) implies (b1) is obvious from the definitions. Lemma 4. Suppose X is a continuum such that every real valued local connectivity function is Darboux. For every connected set C X, x cl(c), and δ > 0 there is a connected set E of diameter less than δ such that E C and x E. Proof: Suppose there is a connected set C, an x cl(c), and a δ > 0 such that C E = for every connected set E of diameter less than δ containing x. Let U be an open set of diameter less than δ with x U. Let W be an open set such that x W cl(w ) U. Let T be the component of x in U. Define a function f on X so that f(x) = 1, f[bd(w ) T ] = {0}, f (cl(w ) T ) is continuous, and f[x \ (cl(w ) T )] = {0}. Since the restriction of f to any component of U is continuous and f (X \ cl(w )) is constant, f is a local connectivity function. Since diam(t ) < δ and x T and T is connected, T C =. So, C {x} is connected and f[c {x}] = {0, 1}. Thus, f is a local connectivity function which is not Darboux. Proof that (b1) implies (c1): By way of contradiction, assume that X is not Peano and condition (b1) holds. Let p be a point where X is not connected im kleinen. Let U be an open set such that p U and no neighborhood of p contained in U is connected. Let K be the component of p in U. Let W U be a compact

6 WHEN ARE LOCAL CONNECTIVITY FUNCTIONS CONNECTIVITY? 189 neighborhood of p. Since X is not connected im kleinen at p, there is a sequence of points {p n } n ω in int(w ) \ K converging to p. Let L n denote the component of W that contains p n. Since p n / K for every n, we may assume that L n L k if and only if n = k. Clearly, L n K = for all n. Since each L n must bump the boundary of W, we may assume there is a ɛ > 0 such that diam(l n ) > ɛ for all n. Moreover, we may assume that the sequence {L n } n ω converges to some continuum L. Clearly, L W, p L, and diam(l) ɛ. Let δ < ɛ/8. For each n ω we may find, by Theorem 2, a δ-chain C n of continua from p n to p. Moreover, we assume C n has minimal length among all such chains. For A ω we let S A = n A ( C n ). Notice that S A is connected. We claim that L cl(s A ) for all infinite A ω. Let x L and γ > 0. Clearly, x cl(s A ( n A L n)), and S A ( n A L n) is connected. By Lemma 4, there is a connected set T of diameter strictly less than min{γ, d(x, X \ W )} such that x T and T (S A ( n A L n)). Notice that T n A L n = ; otherwise, p n and p would be in the same component of W for some n. Thus, T S A. So, d(x, S A ) < γ. Since γ was arbitrary, x cl(s A ). We claim that q S A for every infinite A ω and q L such that d(p, q) > ɛ/2. By the previous claim, we have q cl(s A ). Let {γ n } n ω be a sequence of real numbers such that lim γ n = 0 and 0 < γ n < ɛ/8. By Lemma 4, there is for every γ n a connected set of T n of diameter γ n such that q T n and T n S A. Let T = n ω T n. Let B A be the set of all k A such that T C k. Assume B is infinite. For each k B let Ck be the shortest subchain of C k from p k to any point in T. Since d(p, T ) > 3ɛ/8 and the last link of Ck has diameter less than ɛ/8, C k is at least two links shorter than C k. Let Q = T ( ( k B C k )). Notice Q is connected. Since p cl(q), there is a connected set R such that p R, R Q, and diam(r) < ɛ/8. Since d(p, T ) > 3ɛ/8, it follows that for some k B, R ( Ck ). Now p k, p (Ck {R}) and (Ck {R}) is connected. Arguing as in Lemma 3, C k {R} contains a subchain D from p k to p. However, the length of D must be at least one less than the length of C k, contradicting minimality. We may now assume B is finite. Since T n S A for all n, there is a k B such that T n C k for infinitely many n ω.

7 190 F. JORDAN Since lim diam(t n ) = 0, q cl( C k ). However, C k is closed. So, q C k S A, establishing the claim. Let q L and d(p, q) > ɛ/2. By the preceding claim, we may find an infinite A ω such that q C k for every k A. For each k A let Ck be the shortest subchain of C k from p k to q. Since d(p, q) > ɛ/2 and the last link of Ck has diameter less than ɛ/8, C k is at least two links shorter than C k. Let Q = k A C k. Since p cl(q) and Q is connected, there is a connected set R such that p R, R Q, and diam(r) < ɛ/8. It follows that for some k A, p, p k (Ck {R}) and (Ck {R}) is connected. By Lemma 3, Ck {R} contains a subchain D from p k to p. However, the length of D would be at least one less than the length of C k, contradicting minimality. Proof that (c1) implies (a1): Let X be Peano and f : X R be local connectivity. By way of contradiction, assume C X is connected and f C is not connected. Let W, V X R be open sets such that f C V W, V f C, W f C, and V W f C =. By way of contradiction, assume that for every x C there is an open O x X such that f (O x C) V or f (O x C) W. Now {x: f (O x C) W } and {x: f (O x C) V } is a partition of C into two nonempty disjoint open sets contradicting that C is connected. So there is a p C such that for every neighborhood N of p, we have f (N C) V and f (N C) W. Assume that (p, f(p)) W. Let U 1 be an open neighborhood of p such that cl(u 1 ) is a Peano continuum and f cl(u 1 ) is connectivity. Since cl(u 1 ) is Peano, cl(u 1 ) is semi-locally-connected by [3, 8.44(d)]. So, we can find an open U 2 such that p U 2 cl(u 2 ) U 1 such that cl(u 1 ) \ U 2 has finitely many components E 1,..., E n. Let L = (C U 2 ) (cl(u 1 )\U 2 ). Suppose L x is a quasicomponent of x in L. By way of contradiction, assume that L x (cl(u 1 )\U 2 ) =. For every w cl(u 1 )\U 2 there exist L-open sets S w and T w such that L = S w T w, S w T w =, and w S w and x T x. Since cl(u 1 ) \ U 2 is compact, there are w 1,... w k cl(u 1 ) \ U 2 such that cl(u 1 ) \ U 2 k, x k, ( k ) ( k ) =, and ( k ) ( k ) = L. Now (C \ cl(u 1 )) (C k ) and C k form a partition of C into two disjoint C-open

8 WHEN ARE LOCAL CONNECTIVITY FUNCTIONS CONNECTIVITY? 191 sets, contradicting that C is connected. Thus, L x (cl(u 1 )\U 2 ) for every x L. Since each quasicomponent of L has nonempty intersection with cl(u 1 ) \ U 2 and cl(u 1 ) \ U 2 has finitely many components, we conclude that L has only finitely many quasicomponents. It follows that L has finitely many components. Let D be the component of p in L. Notice that D \ U 2 has finitely many components and is compact. Let M = (D \ U 2 ) (D π X [f V ]). Suppose M x is a quasicomponent of x in M. By way of contradiction, assume that M x (D \ U 2 ) =. For every w D \ U 2 there exist M-open sets S w and T w such that M = S w T w, S w T w =, and w S w and x T w. Since D \ U 2 is compact, there are w 1,... w k D \ U 2 such ) =, that D \ U 2 k, x k, ( k ) ( k and ( k ) ( k ) = M. Now (D \ U 2 ) ( k ) and k form a partition of M into two disjoint M-open sets. Let A = k. Notice that cl D (A) U 2 since D \ U 2 k. In particular, cl D (A) C. We get a contradiction by showing that f A is closed and open in f D. We first show that f A is closed in f D. Let {(x n, f(x n ))} n ω be a sequence of points in f A and (x, f(x)) f D be such that lim(x n, f(x n )) = (x, f(x)). Since A U 2, A π X [f V ]. So, (x n, f(x n )) V for every n ω. Since cl D (A) C and f V is closed in f C, we have (x, f(x)) V. Now x cl D (A) π X [f V ] π X [f V ] D M. So, x cl M (A). Since A is M-closed, we have x A. So, (x, f(x)) f A. Thus, f A is closed. We now show that f A open in f D. Suppose x A and {z n } n ω is a sequence of points in f D such that lim(z n, f(z n )) = (x, f(x)). Since A U 2, we may assume that z n C for all n ω and that x π X [f V ]. Since x π X [f V ] and f V is open in f C, (z n, f(z n )) V for almost all n ω. Now all but finitely many z n π X [f V ] U 2 D M. Since A is open in M, we must have z n A for almost all n ω. Thus, f A is open in f D. Thus, M x (D \ U 2 ) for every x D. Since each quasicomponent of M has nonempty intersection with D \ U 2 and D \ U 2 has finitely many components, we conclude that M has only finitely many quasicomponents. It follows that M has finitely many components.

9 192 F. JORDAN Since every neighborhood of p has nonempty intersection with π X [f V ], there is a component E of M such that p cl(e). Now E {p} is connected, f (E U 2 ) V, and (p, f(p)) W. Thus, f (E {p}) is not connected. However, E U 1 so f (E {p}) is connected, a contradiction. References [1] Melvin R. Hagan, Equivalence of connectivity maps and peripherally continuous transformations, Proc. Amer. Math. Soc. 17 (1966), [2] Charles L. Hagopian and Leland E. Rogers, Arcwise Connectivity and Continuum Chainability, Houston J. Math. 7 (1981), no. 2, [3] Sam B. Nadler, Jr., Continuum Theory: An Introduction. Monographs and Textbooks in Pure and Applied Mathematics, 158. New York: Marcel Dekker, Inc., [4], Continua on which all real-valued connected functions are connectivity functions. Preprint [5], Locally connectivity functions on arcwise connected spaces and certain continua. Preprint [6] J. Stallings, Fixed point theorems for connectivity maps, Fund. Math. 47 (1959), Department of Mathematics and Computer Science; Georgia Southern University; Statesboro, GA address: fjord@georgiasouthern.edu