On Jones' set function T and the property of Kelley for Hausdorff continua

Topology and its Applications 226 (2017) 51–65

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Topology and its Applications www.elsevier.com/locate/topol

On Jones’ set function T and the property of Kelley for Hausdorff continua Sergio Macías Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, México D.F., C.P. 04510, Mexico

a r t i c l e

i n f o

Article history: Received 30 August 2016 Received in revised form 20 April 2017 Accepted 24 April 2017 Available online 3 May 2017 In memoriam Professor Sam B. Nadler Jr. MSC: 54B20 54C60

a b s t r a c t We extend to Hausdorff continua the result of Camargo and Uzcátegui that says that if X is a metric continuum, Jones’ set function T is continuous on singletons and T is idempotent on closed sets, then G = {T ({x}) | x ∈ X} is a decomposition of X. We also present important implications of this result, a couple of them answer questions of the celebrated Houston Problem Book. We study Hausdorff continua with the property of Kelley and with the property of Kelley weakly. We establish a Hausdorff version of Jones’ Aposyndetic Decomposition Theorem. To this end, we introduce the uniform property of Effros. © 2017 Elsevier B.V. All rights reserved.

Keywords: Atomic map Continuous decomposition Effros continuum Hausdorff continuum Homogeneous continuum Idempotency on closed sets Lower semicontinuous decomposition Metric continuum Property of Kelley Property of Kelley weakly Set function T Uniformity Uniform property of Effros Upper semicontinuous decomposition

1. Introduction F. Burton Jones defined the set function T to study aposyndetic properties of metric continua [20]. The first study of the set function T for Hausdorff continua is in [13]. Since then, several authors have used T E-mail address: [email protected]. http://dx.doi.org/10.1016/j.topol.2017.04.025 0166-8641/© 2017 Elsevier B.V. All rights reserved.

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to study Hausdorff continua; e.g., [16,12,1,4,2,3,17,25–27]. We use this opportunity to continue with such studies. The paper consists of six sections. After this Introduction and Definitions, in section 3, we extend to Hausdorff continua the result of Camargo and Uzcátegui that says that if X is a metric continuum, T is continuous on singletons and T is idempotent on closed sets, then G = {T ({x}) | x ∈ X} is a decomposition of X (Theorem 3.4). We also present important implications of this result, a couple of them answer questions of the celebrated Houston Problem Book [11] (Theorems 3.5 and 3.6). We mention that results about the continuity of T on subcontinua of metric continua have their natural extension to Hausdorff continua. In section 4, we use the Hausdorff definition of the property of Kelley given independently in [9] and [32], and prove that for Hausdorff continua with the property of Kelley, the set function T is idempotent on closed sets (Theorem 4.2). We also show that if X is a Hausdorff continuum with the property of Kelley and G = {T ({x}) | x ∈ X} is a decomposition of X, then the elements of G are terminal subcontinua of X (Theorem 4.6). In section 5, we study homogeneous Hausdorff continua. We obtain a weaker version of F. Burton Jones’ Aposyndetic Decomposition Theorem [21, Theorem 1] for homogeneous Hausdorff continua with the property of Kelley (Theorem 5.6). In order to obtain the full version of Jones’ theorem (Theorem 5.13), we introduce the uniform property of Effros, we show that each homogeneous Effros continuum has the uniform property of Effros (Theorem 5.9). We also prove that if a homogeneous Hausdorff continuum has the uniform property of Effros, then it has the property of Kelley (Theorem 5.11). In section 6, we study the property of Kelley weakly. The property of Kelley weakly is introduced in [31] to study points of connectedness im kleinen of the n-fold hyperspaces of metric continua. We extend this property to Hausdorff continua. We show that if U is an open subset of the n-fold hyperspace of a Hausdorff continuum X with the property  of Kelley weakly, then IntX ( U) = ∅ (Theorem 6.2); and if X is an indecomposable Hausdorff continuum with the property of Kelley weakly, then X is the only point of the n-fold hyperspace of X at which it is locally connected (Theorem 6.5). Many of our results are known for the metric case, at the appropriate places, we put a reference where the metric version of the result can be found. 2. Definitions If Z is a topological space, then given a subset A of Z, the interior of A is denoted by Int(A) and the closure of A by Cl(A). We write IntZ (A) and ClZ (A), respectively, if there is a possibility of confusion. A map is a continuous function. A surjective map f : X  Y between topological spaces is monotone provided that f −1 (C) is connected for every connected subset C of Y . The surjective map f is open (closed, respectively) if f (U ) is open (closed, respectively) in Y for each open (closed, respectively) set U of X. If f : X  Y is a map and Z is a nonempty subset of X, then f |Z : Z → Y denotes the restriction of f to Z. Given a space X, 1X denotes the identity map on X. Given a topological space Z, a decomposition of Z is a family G of nonempty and mutually disjoint subsets  of Z such that G = Z. A decomposition G of a topological space Z is said to be upper semicontinuous if for each element G of G and each open subset U of Z such that G ⊂ U , there exists an open subset V of Z such that G ⊂ V and for every G ∈ G such that G ∩ V = ∅, we have that G ⊂ U . The decomposition G of Z is lower semicontinuous provided that for each G ∈ G any two points z1 and z2 of G and any open subset U of Z containing z1 , there exists an open subset V of Z such that z2 ∈ V and for each G ∈ G such that G ∩ V = ∅, we obtain that G ∩ U = ∅. The decomposition G is continuous if it is both upper and lower semicontinuous. A continuum is a compact connected Hausdorff space. A subcontinuum of a space Z is a continuum contained in Z. A continuum is decomposable if it is the union of two of its proper subcontinua. A continuum is indecomposable if it is not decomposable. A continuum X is homogeneous provided that for each pair of points x1 and x2 of X, there exists a homeomorphism h : X  X such that h(x1 ) = x2 .

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An arc is a continuum with exactly two points such that the complement of each of them is connected. These points are called the end points of the arc. A continuum X is arcwise connected provided that for each two points x0 and x1 of X, there exists an arc whose end points are x0 and x1 . Let X be continuum and let x0 be a point of X. Then X is almost connected im kleinen at x0 , provided that for each open subset U of X containing x0 , there exists a subcontinuum W of X such that Int(W ) = ∅ and W ⊂ U . X is almost connected im kleinen if X is almost connected im kleinen at each of its points. Also, X is connected im kleinen at x0 if for each open subset U of X containing x0 , there exists a subcontinuum W of X such that x0 ∈ Int(W ) ⊂ W ⊂ U . It is clear that if a continuum is connected im kleinen at a point, then it is almost connected im kleinen at that point. The converse implication is not true even in the metric case [23, 1.7.5]. Given a continuum X, we define the set function T as follows: if A is a subset of X, then T (A) = X \ {x ∈ X | there exists a subcontinuum W of X such that x ∈ Int(W ) ⊂ W ⊂ X \ A}. We write TX if there is a possibility of confusion. Let us observe that for any subset A of X, T (A) is a closed subset of X and A ⊂ T (A). A continuum X is aposyndetic provided that T ({p}) = {p} for every p ∈ X. A continuum X is T -additive provided that T (A ∪ B) = T (A) ∪ T (B) for each pair of nonempty closed subsets A and B of X. We say that X is point T -symmetric if for any two points p and q of X, p ∈ T ({q}) if and only if q ∈ T ({p}). The set function T is idempotent on X (idempotent on closed sets of X, on subcontinua of X, respectively) provided that T 2 (A) = T (A) for each subset A of X (for every nonempty closed subset A of X, for each subcontinuum A of X, respectively), where T 2 = T ◦ T . The set function T is idempotent on singletons if T 2 ({x}) = T ({x}) for all x ∈ X. We say that T is continuous for a continuum X provided that T : 2X → 2X is continuous, where 2X is the hyperspace of nonempty closed subsets of X, topologized with the Vietoris topology (or the Hausdorff metric if X is metric) [37]. A base for the Vietoris topology for 2X is given by the sets of the form

U1 , . . . , Um  =

⎧ ⎨ ⎩

A ∈ 2X

⎫ m  ⎬   Uj and A ∩ Uj = ∅ for all j , A⊂ ⎭ j=1

where U1 , . . . , Um are open subsets of X. If f : X → Y is a map, then 2f : 2X → 2Y given by 2f (A) = f (A) is a map [37, (1.168)]. Observe that there exists a copy of X inside 2X , namely, F1 (X) = {{x} | x ∈ X}. The map ξ : X  F1 (X) given by ξ(x) = {x} is an embedding. We say that T is continuous on singletons provided that T |F1 (X) : F1 (X) → 2X is continuous. Let X be a continuum. Then C1 (X) denotes the hyperspace of subcontinua of X; i.e., C1 (X) = {A ∈ 2X | A is connected}. If U1 , . . . , Um are open subsets of X, then U1 , . . . , Um  ∩ C1 (X) is denoted by U1 , . . . , Um 1 . Note that if K is a subcontinuum of X, then T (K) is a subcontinuum of X (the proof given in [23, 3.1.21] does not use the fact that the continuum is metric). We say that T is continuous on subcontinua provided that T |C1 (X) : C1 (X) → C1 (X) is continuous. 3. Jones’ set function T We extend to continua the main results about metric continua that appear in [8]. The proofs are modifications of the original ones, we include the details for the convenience of the reader. The answers for the metric case to the questions of the Houston Problem Book [11] are given in [8].

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The following two results remove the metric assumption in [8, Proposition 3.1] and [8, Theorem 3.2], respectively. Lemma 3.1. Let X be a continuum. Let A and L be two elements of 2X and let {Aλ }λ∈Λ be a net of elements of 2X converging to A. If the net {T (Aλ )}λ∈Λ converges to L, then L ⊂ T (A) ⊂ T (L). Proof. We prove that L ⊂ T (A). Let l ∈ L and let W be a subcontinuum of X such that l ∈ Int(W ). Since {T (Aλ )}λ∈Λ converges to L, there exists λ0 ∈ Λ such that T (Aλ ) ∩ Int(W ) = ∅ for all λ ≥ λ0 . This implies that Aλ ∩ W = ∅ for every λ ≥ λ0 . By [15, 3.1.23 and 1.6.1], there exist Λ ⊂ Λ and zλ ∈ Aλ ∩ W , for each λ ∈ Λ , and a point z in X such that {zλ }λ∈Λ converges to z [36, Theorem 4]. Note that z ∈ A ∩ W . Since W is an arbitrary subcontinuum of X containing l in its interior, l ∈ T (A). Hence, L ⊂ T (A). Since Aλ ⊂ T (Aλ ), {Aλ }λ∈Λ converges to A and {T (Aλ )}λ∈Λ converges to L, we have that A ⊂ L. Thus, T (A) ⊂ T (L). 2 Theorem 3.2. Let X be a continuum. Then T is continuous if and only if T is idempotent on closed sets and T (2X ) is closed in 2X . Proof. Suppose T is continuous. Then, by [1, Lemma 3], T is idempotent on all X and, clearly, T (2X ) is closed in 2X . Suppose T is idempotent on closed sets and T (2X ) is closed in 2X . Let {Aλ }λ∈Λ be a net of elements of X 2 converging to A ∈ 2X . By [15, 3.1.23 and 1.6.1], without loss of generality, we assume that {T (Aλ )}λ∈Λ converges to an element L ∈ 2X . By Lemma 3.1, L ⊂ T (A) ⊂ T (L). Since T (2X ) is closed in 2X , there exists K ∈ 2X such that T (K) = L. Hence, T (L) = T 2 (K) = T (K) = L (the second equality holds because T is idempotent on closed sets). Therefore, T (A) = L and T is continuous [33, 3.38]. 2 Let X be a continuum and let x be a point of X. Then T ({x}) has property BL provided that for each z ∈ T ({x}), T ({x}) ⊂ T ({z}). Remark 3.3. Let X be a continuum such that T is idempotent on singletons and let x be a point of X. If T ({x}) has property BL, then for each z ∈ T ({x}), T ({x}) = T ({z}). The next theorem removes the metric assumption from [8, Theorem 3.4]. Note that we only require that T is idempotent on singletons instead of on closed sets. Theorem 3.4. Let X be a continuum. If T is continuous on singletons and it is idempotent on singletons, then G = {T ({x}) | x ∈ X} is a decomposition of X. Proof. Let M = {x ∈ X | T ({x}) has property BL} (see Remark 3.3). Note that for every m ∈ M , T ({m}) ⊂ M . By [25, Theorem 3.7], M = ∅, and M ∩ T ({x}) = ∅ for all x ∈ X. We show that M = X. First, we prove that M is closed in X. Let x ∈ Cl(M ). Then there exists a net {mλ }λ∈Λ of elements of M converging to x [15, 1.6.3]. Let z ∈ T ({x}). Since T is continuous on singletons, the net {T ({mλ })}λ∈Λ converges to T ({x}). Hence, for each λ ∈ Λ, there exists zλ ∈ T ({mλ }) such that {zλ }λ∈Λ converges to z [36, Theorem 4]. Since T is continuous on singletons, the net {T ({zλ })}λ∈Λ converges to T ({z}). Since T ({zλ }) = T ({mλ }) for all λ ∈ Λ, we have that T ({z}) = T ({x}). Therefore, M is closed. Now, we see that M = X. Suppose this is not true. Assume first that M is not connected. Then there exist two nonempty closed subsets M1 and M2 of X such that M = M1 ∪ M2 . Since X is normal, there exist two disjoint open subsets U1 and U2 of X such that M1 ⊂ U1 and M2 ⊂ U2 . Let j ∈ {1, 2}. Observe that for each mj ∈ Mj , T ({mj }) ⊂ Mj ⊂ Uj . Since T is upper semicontinuous [1, Lemma 1], for every mj ∈ Mj , there exists an open subset Vmj of X such that mj ∈ Vmj and T ({z}) ⊂ Uj for each z ∈ Vmj .

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 Let Vj = {Vmj | mj ∈ Mj }. Clearly, Vj is an open subset of X, Vj ⊂ Uj , Mj ⊂ Vj and T ({z}) ⊂ Uj for every z ∈ Vj . Let Rj = {x ∈ X | T ({x}) ⊂ X \ Mj } = ξ −1 ((T |F1 (X) )−1 ( X \ Mj 1 )); and let R = {x ∈ X | T ({x}) ∩ M1 = ∅ and T ({x}) ∩ M2 = ∅} = ξ −1 ((T |F1 (X) )−1 ( X, M1 1 ∩ X, M2 1 )). Note that Rj is open in X and R is closed. Also, Vk ⊂ Rj , j, k ∈ {1, 2} and k = j, and (V1 ∪ V2 ) ∩ R = ∅. Thus, Rj = ∅, j ∈ {1, 2}. By [25, Theorem 3.7], if x ∈ X, then T ({x}) ∩ M = ∅. Hence, X ⊂ R ∪ R1 ∪ R2 , and X = R ∪ R1 ∪ R2 . Observe that if x ∈ R1 ∩ R2 , then T ({x}) ⊂ X \ (M1 ∪ M2 ), a contradiction to [25, Theorem 3.7]. This implies that R1 ∩ R2 = ∅. As a consequence of this, since X is connected, R = ∅. Let x ∈ R ∩ (Cl(R1 ) ∪ Cl(R2 )). Without loss of generality, we assume that x ∈ Cl(R2 ). Then, by [15, 1.6.3], there exists a net {rλ }λ∈Λ of elements of R2 converging to x. Since x ∈ R, T ({x}) ∩ M2 = ∅. Also, since M2 ⊂ V2 , we have that T ({x}) ∈ X, V2 . Since T is continuous on singletons, there exists λ0 ∈ Λ such that T ({rλ0 }) ∈ X, V2 . Let z ∈ T ({rλ0 }) ∩ V2 . By the construction of V2 , T ({z}) ⊂ U2 . By [25, Theorem 3.7], T ({z}) ∩ M = ∅. Since M1 ∩ U2 = ∅, we have that T ({z}) ∩ M2 = ∅. Since T is idempotent on singletons, T ({z}) ⊂ T 2 ({rλ0 }) = T ({rλ0 }), and T ({rλ0 }) ∩ M2 = ∅, a contradiction to the fact that rλ0 ∈ R2 . Thus, M cannot be disconnected. Hence, M is connected. Let x ∈ X \ M . By [25, Theorem 3.7], there exists m0 ∈ M such that T ({m0 }) ⊂ T ({x}). Since X is normal, there exist two disjoint open subsets U and V of X such that M ⊂ U and x ∈ V . Since T is upper semicontinuous [1, Lemma 1], there exists an open subset W of X such that m0 ∈ W and  T ({z}) ⊂ U , for all z ∈ W . Let K = Cl ( {T ({z}) | z ∈ W }) ∪ M . Since M is a subcontinuum of X and T ({z}) ∩M = ∅ for each z ∈ W , we have that K is a subcontinuum of X. Note that K ⊂ Cl(U ) ⊂ X \V and m0 ∈ W ⊂ K. Thus, K ∩{x} = ∅ and m0 ∈ X \T ({x}), a contradiction to the fact that T ({m0 }) ⊂ T ({x}). Therefore, M = X and G = {T ({x}) | x ∈ X} is a decomposition of X. 2 We present several consequences of Theorem 3.4. We begin with the following result that answers Problem 161 of the Houston Problem Book [11, p. 389]. Theorem 3.5. If X is a decomposable continuum for which T is continuous, then X is T -additive. Proof. By Theorem 3.4, X is point T -symmetric. Hence, by [1, Lemma 9], X is T -additive. 2 The following theorem gives a positive answer to two questions of David Bellamy, see Problems 162 and 163 of the Houston Problem Book [11, p. 390]. Theorem 3.6. Let X be a decomposable continuum for which TX is continuous. Then G = {TX ({x}) | x ∈ X} is a continuous decomposition of X such that the quotient space X/G is a locally connected continuum and TX (2X ) is homeomorphic to 2X/G . (In particular, if X is metric, then TX (2X ) is homeomorphic to the Hilbert cube.) Moreover, all the elements of G are nowhere dense in X; if X is metric, there exists a dense Gδ subset W of X/G such that if q(z) ∈ W, then TX ({z}) is an indecomposable continuum, where q : X  X/G is the quotient map. Proof. The fact that G is a decomposition of X follows from Theorem 3.4. The rest of the proof is contained in [25, Theorem 3.4]. 2

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A weaker version of Theorem 3.6, in which we only assume that T is continuous on singletons, is: Theorem 3.7. Let X be a decomposable continuum for which TX is idempotent on singletons and TX is continuous on singletons. Then G = {TX ({x}) | x ∈ X} is a continuous decomposition of X such that the quotient space X/G is an aposyndetic continuum. Moreover, all the elements of G are nowhere dense in X; and if X is metric, then there exists a dense Gδ subset W of X/G such that if q(z) ∈ W, then TX ({z}) is an indecomposable continuum, where q : X  X/G is the quotient map. Proof. The fact that G is a decomposition of X follows from Theorem 3.4. For the rest see [27, Theorem 6.5]. 2 Corollary 3.8. If X is a metric continuum for which T is idempotent on singletons and it is continuous on singletons, then X is not hereditarily decomposable. Note that [8, Theorem 3.9] may be strengthened as follows: Theorem 3.9. Let X be a continuum. Then T is idempotent on subcontinua of X and T (C1 (X)) = C1 (X) if and only if X is locally connected. Proof. Suppose that T is idempotent on subcontinua of X and T (C1 (X)) = C1 (X). Let M be a subcontinuum of X. Since T (C1 (X)) = C1 (X), there exists a subcontinuum L of X such that T (L) = M . Thus, since T is idempotent on continua, T (M ) = T 2 (L) = T (L) = M . Therefore, by [2, Theorem 3], X is locally connected. If X is locally connected, then T (A) = A for all A ∈ 2X [1, (2), p. 582]. 2 Remark 3.10. Observe that the proof of Theorem 3.9 does not use the fact that T is continuous. The proofs of the following three results are the same as the ones given in [28]. For Theorem 3.11, we have to use [34, 5.10.2] instead of [23, 1.8.24] to conclude that the map ζ = f −1 is continuous. Theorem 3.11. [28, Theorem 4.1] Let X be a continuum for which TX is continuous on continua and let Z be a continuum. If f : X  Z is a surjective, monotone, and open map, then TZ is continuous on continua. Corollary 3.12. [28, Corollary 4.2] Let X and Y be continua. If TX×Y is continuous on continua for X × Y , then TX and TY are continuous on continua for X and Y , respectively. Theorem 3.13. [28, Theorem 4.4] Let X be a decomposable continuum for which TX is idempotent on singletons and TX is continuous on continua. Then TX/G is continuous on continua for X/G. 4. Continua with the property of Kelley The property of Kelley for Hausdorff continua is defined independently in [9] and [32]. In [9], a homogeneous continuum without the property of Kelley is given, showing that the fact that homogeneous metric continua have the property of Kelley [38, (2.5)] is not true in this more general setting. It is also mentioned that hereditarily indecomposable continua have the property of Kelley [9, Proposition 2.7]. Note that the

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homogeneous continuum presented in [9] is not an Effros continuum, by Corollary 5.12. In [32], the property of Kelley is used to study points of various types of local connectedness in the hyperspace of subcontinua of a continuum. A continuum X has the property of Kelley at a point x0 if for every subcontinuum K of X containing x0 and each open subset U of C1 (X) containing K, there exists an open subset V of X such that x0 ∈ V and if x ∈ V , then there exists a subcontinuum L of X such that x ∈ L and L ∈ U. The open set V is called a Kelley set for U and x0 . The continuum X has the property of Kelley if X has the property of Kelley at each of its points. The following two results remove the metric assumption in [5, Lemma 4.7 and Theorem 4.8], respectively. Theorem 4.1. Let X be a continuum with the property of Kelley. Let A be a nonempty closed subset of X and let W be a subcontinuum of X such that Int(W ) = ∅ and W ∩ A = ∅. If w ∈ W , then there exists a subcontinuum K of X such that w ∈ Int(K) and K ∩ A = ∅. Proof. Let w ∈ W . Since X is normal, there exists an open subset U of X such that W ⊂ U and Cl(U ) ∩A = ∅. Let U = U, Int(W )1 . Note that W ∈ U. Also observe that if L ∈ U, then L ⊂ U and L ∩W = ∅. Since X has the property of Kelley at w, there exists a Kelley set V for U and w. Let v ∈ V \ {w}. Then there exists  a subcontinuum Lv of X such that v ∈ Lv and Lv ∈ U. Let Lw = W , and let K = Cl ( {Lv | v ∈ V }). Then K is a subcontinuum of X, w ∈ V ⊂ K and K ⊂ Cl(U ) ⊂ X \ A. 2 Theorem 4.2. If X is a continuum with the property of Kelley, then T is idempotent on closed sets. Proof. Let A be a nonempty closed set of X. We know that T (A) ⊂ T 2 (A). Let x ∈ X \ T (A). Then there exists a subcontinuum W of X such that x ∈ Int(W ) ⊂ W ⊂ X \ A. Since X is normal, there exists an open subset U of X such that W ⊂ U and Cl(U ) ∩ A = ∅. Let U = U, Int(W )1 . Since X has the property of Kelley, for each w ∈ W , there exists a Kelley set Vw for U and w. Since W is compact, there exists n w1 . . . , wn in W such that W ⊂ j=1 Vwj . Let j ∈ {1, . . . , n}. By the proof of Theorem 4.1, there exists a n subcontinuum Mj of X such that Vwj ⊂ Mj and Mj ∩A = ∅. Let M = j=1 Mj . Then M is a subcontinuum of X, W ⊂ Int(M ) and M ∩ A = ∅. Since Int(M ) ⊂ X \ T (A), x ∈ Int(W ) ⊂ W ⊂ X \ T (A). Hence, x ∈ X \ T 2 (A). Therefore, T is idempotent on closed sets. 2 Remark 4.3. Theorem 4.2 cannot be improved to obtain that the function T is idempotent, even for metric continua. Let X be the product of a solenoid and a simple closed curve. Then X is a homogeneous metric continuum. By [23, 4.2.32], TX is idempotent on closed sets and, by [27, Theorem 3.3], TX is not idempotent. We continue with the following results about decompositions [7, Theorem 5.1]: Theorem 4.4. Let X be a continuum. If G = {TX ({x}) | x ∈ X} is a decomposition of X, then G is upper semicontinuous. Hence, X/G is a continuum. Proof. By [1, Lemma 1], T is an upper semicontinuous function. Hence, G is an upper semicontinuous decomposition. By [18, Theorem 3-33], X/G is a Hausdorff space. Therefore, X/G is a continuum. 2 A version for metric continua of the following corollary is in [7, Corollary 5.2]. Corollary 4.5. Let X be a continuum such that TX is idempotent on singletons. If G = {TX ({x}) | x ∈ X} is a decomposition of X, then X/G is an aposyndetic continuum. Proof. By Theorem 4.4, G is an upper semicontinuous decomposition and X/G is a continuum. Let q : X  X/G be the quotient map. Then q is a monotone map. Let χ ∈ X/G and let x ∈ X be such that TX ({x}) =

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q −1 (χ). Since q is a monotone map, by [3, Theorem 1, c)], TX/G ({χ}) = qTX q −1 (χ) = qTX (TX ({x})) = qTX ({x}) = qq −1 (χ) = {χ}, the third equality is true because TX is idempotent on singletons. Therefore, X/G is an aposyndetic continuum [3, d), p. 5]. 2 Let X be a continuum and let K be a subcontinuum of X. Then K is a terminal subcontinuum of X if for each subcontinuum L of X such that L ∩ K = ∅, then either L ⊂ K or K ⊂ L. In the following theorem, the metric assumption in [7, Theorem 5.3] is removed. Theorem 4.6. Let X be a continuum with the property of Kelley. If G = {TX ({x}) | x ∈ X} is a decomposition of X, then T ({x}) is a terminal subcontinuum of X for all x ∈ X. Proof. Suppose there exists a point x0 in X such that T ({x0 }) is not a terminal subcontinuum of X. Then there exists a subcontinuum L of X such that L ∩ T ({x0 }) = ∅, L \ T ({x0 }) = ∅ and T ({x0 }) \ L = ∅. Let p ∈ L \ T ({x0 }) and  ∈ L ∩ T ({x0 }). Since G is a decomposition, without loss of generality, we assume that x0 ∈ T ({x0 }) \ L. Since p ∈ L \ T ({x0 }), there exists a subcontinuum W of X such that p ∈ Int(W ) ⊂ W ⊂ X \ {x0 }. Let K = L ∪ W . Then K is a subcontinuum of X, p ∈ Int(K) and K ⊂ X\{x0 }. By Theorem 4.1, there exists a subcontinuum M of X such that  ∈ Int(M ) and M ⊂ X\{x0 }, a contradiction to the choice of . Therefore, T ({x}) is a terminal subcontinuum of X for all x ∈ X. 2 The proof of the following lemma is essentially the same as in the metric case [29, Lemma 3.1], we only need the Hausdorff version of the Boundary Bumping Theorem [18, Theorem 2-16]. Lemma 4.7. Let X be a continuum. If W is a proper terminal subcontinuum of X, then Int(W ) = ∅. The following theorem is proved originally for homogeneous metric continua [24, Theorem 3.5] and it is extended to metric continua with the property of Kelley [7, Theorem 5.4]. Theorem 4.8. Let X be a decomposable continuum with the property of Kelley. If TX is continuous on singletons and G = {TX ({x}) | x ∈ X}, then TX (Z) = q −1 TX/G q(Z) for each nonempty closed subset Z of X, where q : X  X/G is the quotient map. Proof. By Theorem 4.2, TX is idempotent on closed sets, in particular, TX is idempotent on singletons. Hence, by Theorems 3.4 and 4.6, G is a terminal decomposition of X. The rest of the proof, using Lemma 4.7, is essentially the same as the one given in [7, Theorem 5.4]. 2 5. Homogeneous continua We prove a weak version of F. Burton Jones’ Aposyndetic Decomposition Theorem [21, Theorem 1] for homogeneous continua with the property of Kelley (Theorem 5.6). In order to obtain a full version of the mentioned theorem (Theorem 5.13), we need to introduce the uniform property of Effros. We use uniformities to do this. To know more about uniformities see [15, Chapter 8]. We present the basic notation. Let X be a Hausdorff space. If V and W are subsets of X × X, then −V = {(x , x) ∈ X × X | (x, x ) ∈ V } and

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V + W = {(x, x ) | there exists x ∈ X such that (x, x ) ∈ V and (x , x ) ∈ W }. We write 1V = V and for each positive integer n, (n + 1)V = nV + 1V . The diagonal of X is the set ΔX = {(x, x) | x ∈ X}. An entourage of ΔX is a subset V of X × X such that ΔX ⊂ V and V = −V . The family of entourages of the diagonal of X is denoted by DX . If V ∈ DX and x ∈ X, then B(x, V ) = {x ∈ X | (x, x ) ∈ V }. By [15, 8.1.3], Int(B(x, V )) is an open neighborhood  of x. If A is a subset of X and V ∈ DX , then B(A, V ) = {B(a, V ) | a ∈ A}. If V ∈ DX and (x, x ) ∈ V , then we write ρ(x, x ) < V . If (x, x ) ∈ V , then we write ρ(x, x ) ≥ V . We have that if x, x and x are points of X, and V and W belong to DX then the following hold [15, p. 426]: (i) ρ(x, x) < V . (ii) ρ(x, x ) < V if and only if ρ(x , x) < V . (iii) If ρ(x, x ) < V and ρ(x , x ) < W , then ρ(x, x ) < V + W . Let X be a Tychonoff space. A uniformity on X is a subfamily U of DX such that: (1) (2) (3) (4)

If V ∈ U, W ∈ DX and V ⊂ W , then W ∈ U. If V and W belong to U, then V ∩ W ∈ U. For every V ∈ U, there exists W ∈ U such that 2W ⊂ V .  {V | V ∈ U} = ΔX .

Notation 5.1. Let X be a Tychonoff space and let U a uniformity of X that induces its topology. If V ∈ U, then we define the cover C(V ) = {B(x, V ) | x ∈ X}. Remark 5.2. Note that by [15, 8.3.13], for every compact Hausdorff space X, there exists a unique uniformity UX on X that induces the original topology of X. Notation 5.3. Let X be a compact Hausdorff space and let UX be the unique uniformity of X that induces its topology. If Y is a subspace of X and U ∈ UX , then UY = U ∩ (Y × Y ). We need the following result [15, 8.3.G]: Theorem 5.4. If X is a compact Hausdorff space, then for every open cover U of X, there exists V ∈ UX such that C(V ) refines U. It is known that if X is a compact Hausdorff space and H(X) is the group of homeomorphisms of X, then H(X) is a topological group with the compact-open topology [6, p. 594]. the entourage of the diagonal Let X be a compact Hausdorff space. For every U ∈ UX (Remark 5.2), let U ΔH(X) ⊂ H(X) × H(X) defined by: = {(g1 , g2 ) ∈ H(X) × H(X) | ρ(g1 (x), g2 (x)) < U for all x ∈ X}. U | U ∈ UX } is a base for a uniformity on H(X); the uniformity generated It is known that the family U = {U by this family is called the uniformity of uniform convergence induced by UX and is denoted by U X. We also need a special case of [15, 8.2.7]: Theorem 5.5. Let X be a compact Hausdorff space. Then the topology on H(X) by the uniformity U X of uniform convergence coincides with the compact-open topology on H(X), and depends only on the topology induced on X by the uniformity UX (Remark 5.2).

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Let X and Z be continua and let f : X  Z be a map. Then f is an atomic map if for each subcontinuum K of X such that f (K) is not degenerate, then K = f −1 (f (K)). Theorem 5.6. Let X be a homogeneous continuum with the property of Kelley. If G = {T ({x}) | x ∈ X}, then G is an upper semicontinuous terminal decomposition of X, X/G is an aposyndetic homogeneous continuum, and the quotient map q : X  X/G is an atomic map. Proof. Let X be a point of X. We show that if z ∈ T ({x}), then T ({z}) = T ({x}). Since X has the property of Kelley, by Theorem 4.2, T is idempotent on closed sets. Hence, T ({z}) ⊂ T ({x}). Also, by [25, Theorem 3.7], there exists x0 ∈ T ({x}) such that T ({x0 }) has property BL. Since X is homogeneous, there exists a homeomorphism h : X  X such that h(x0 ) = x. Since z ∈ T ({x}), h−1 (z) ∈ T ({x0 }). Hence, by [25, Lemma 3.5], T ({h−1 (z)}) = T ({x0 }). Thus, by [23, 5.1.1], T ({x}) = h(T ({x0 })) = h(T ({h−1 (z)})) = T ({z}). Therefore, G is a decomposition of X. By Theorem 4.4, G is upper semicontinuous. The fact that each element of G is a terminal subcontinuum of X follows from Theorem 4.6. Also, by Corollary 4.5, X/G is an aposyndetic continuum. We show that X/G is homogeneous. Let q : X  X/G be the quotient map. Let χ1 and χ2 be two elements of X/G. We define ζ : X/G  X/G as follows: Let x1 and x2 be elements of X such that q(x1 ) = χ1 and q(x2 ) = χ2 . Since X is homogeneous, there exists a homeomorphism h : X  X such that h(x1 ) = x2 . Note that, by [23, 5.1.1], h(T ({x1 })) = T ({x2 }); i.e., h(q −1 (χ1 )) = q −1 (χ2 ). Define ζ(χ) = q ◦ h(q −1 (χ)). Then ζ is well defined and ζ(χ1 ) = χ2 . To see that ζ is continuous, let U be an open subset X/G. Then q −1 (U) is a saturated open subset of X. Since h is a homeomorphism, h−1 (q −1 (U)) is a saturated open subset of X. Hence, q(h−1 (q −1 (U))) = ζ −1 (U) is an open subset of X/G. Thus, ζ is continuous. Similarly, ζ −1 is defined and is continuous. Therefore, X/G is homogeneous. By [14, Theorem 3], the quotient map is atomic. 2 The proof of the following corollary is the same as the one given in [23, 5.1.20], this result uses [23, 5.1.11 and 5.1.13], whose proofs do not use the fact that the space is metric. Corollary 5.7. Let X be a homogeneous continuum with the property of Kelley. If x is a point of X, then T ({x}) is the maximal terminal proper subcontinuum of X containing x. As a consequence of Corollary 5.7, we have the following result, compare with [23, 5.1.22]. Corollary 5.8. If X is an arcwise connected homogeneous continuum with the property of Kelley, then X is aposyndetic. Let X be a homogeneous continuum, let H(X) be its group of homeomorphisms. Let x be a point of X and let γx : H(X) → X be given by γx (h) = h(x). The homogeneous continuum X is said to be Effros if γx is an open map for all x ∈ X. In [6] a homogeneous continuum that is not Effros is presented. Since this continuum is locally connected, it has the property of Kelley [9, Proposition 2.6]. A homogeneous continuum X has the uniform property of Effros provided that for each U ∈ UX , there exists V ∈ UX such that if x1 and x2 are two points of X with ρ(x1 , x2 ) < V , there exists a homeomorphism h : X  X such that h(x1 ) = x2 and ρ(z, h(z)) < U for all z ∈ X. The entourage V is called an Effros entourage for U . A homeomorphism h : X  X satisfying ρ(z, h(z)) < U , for all z ∈ X, is called a U -homeomorphism. Theorem 5.9. If X is a homogeneous Effros continuum, then X has the uniform property of Effros.

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Proof. By Theorem 5.5, the topology induced by U X on H(X) coincides with the compact-open topology.  Also, U is a base for the uniformity U X . Let x0 be a point of X and let U ∈ UX (Remark 5.2). Let U ∈ UZ be 

 ))) is such that 2U ⊂ U . Since X is an Effros continuum, the map γx0 is an open map. Then γx0 (Int(B(1X , U

 ))). an open neighborhood of x0 in X. Then there exists Vx0 ∈ UX such that B(x0 , Vx0 ) ⊂ γx0 (Int(B(1X , U 

)) such that h(x0 ) = x and ρ(z, h(z)) < U  Thus, if x ∈ B(x0 , Vx0 ), then there exists h ∈ Int(B(1X , U for every z ∈ X. Observe that {Int(B(x, Vx )) | x ∈ X} is an open cover of X. Since, X is compact, by Theorem 5.4, there exists V ∈ UX such that C(V ) refines {Int(B(x, Vx )) | x ∈ X}. Let x1 and x2 be points of X such that ρ(x1 , x2 ) < V . Then there exists x3 in X such that x1 and x2 both belong to Int(B(x3 , Vx3 )).

 )) be such that h1 (x3 ) = x1 and h2 (x3 ) = x2 . Let h = h2 ◦ h−1 . Then h is Let h1 , h2 ∈ Int(B(1X , U 1 a homeomorphism of X and h(x1 ) = x2 . Let z be a point of X. Since ρ(z, h(z)) = ρ(z, h2 ◦ h−1 1 (z)), −1 −1 −1 −1    ρ(z, h−1 (z)) = ρ(h ◦ h (z), h (z)) < U , ρ(h (z), h ◦ h (z)) < U and 2U ⊂ U , we obtain that 1 2 1 1 1 1 1 ρ(z, h(z)) < U . Therefore, X has the uniform property of Effros. 2 As a first consequence of the uniform property of Effros, we extend [30, Theorem 3.5] to continua: Theorem 5.10. Let X be a homogeneous continuum with the uniform property of Effros. Then the following are equivalent: (1) (2) (3) (4) (5)

X X X X X

is is is is is

locally connected; locally connected at some point; connected im kleinen at some point; almost connected im kleinen at every point of X; almost connected im kleinen at some point of X.

Proof. We only prove that (5) implies (1), the other implications are clear. Suppose X is almost connected im kleinen at x0 . Let A be an open subset of X such that x0 ∈ A. Let U ∈ UX be such that B(x0 , U ) ⊂ A. Let U  ∈ UX be such that 2U  ⊂ U . Since X has the uniform property of Effros, there exists an Effros entourage V for U  . Without loss of generality, we assume that V ⊂ U  . Since X is almost connected im kleinen at x0 , there exists a subcontinuum W of X such that W ⊂ Int(B(x0 , V )) and Int(W ) = ∅. Let w0 ∈ Int(W ). Then ρ(x0 , w0 ) < V . Hence, there exists a U  -homeomorphism h : X  X such that h(w0 ) = x0 . Thus, x0 ∈ Int(h(W )). Since for each w ∈ W , ρ(x0 , w) < V , ρ(w, h(w)) < U  and V ⊂ U  , we obtain that for every w ∈ W , ρ(x0 , h(w)) < 2U  . Hence, since 2U  ⊂ U , we have that h(W ) ⊂ B(x0 , U ) ⊂ A. Thus, X is connected im kleinen at x0 . Since X is homogeneous, X is connected im kleinen at each of its points. Therefore, X is locally connected [23, 1.7.12]. 2 Next, we show that homogeneous continua with the uniform property of Effros have the property of Kelley. To this end, we need some terminology. Let X be a continuum, we define a uniformity on C1 (X) as follows: If U ∈ UX (Remark 5.2), then let C1 (U ) = {(A, A ) ∈ C1 (X) × C1 (X) | A ⊂ B(A , U ) and A ⊂ B(A, U )}. Let BX = {C1 (U ) | U ∈ UX }. Then BX is a base for a uniformity, denoted by C1 (UX ) [15, 8.5.16]. Observe that the topology generated by C1 (UZ ) coincides with the Vietoris topology [34, 3.3]. Hence, C1 (X) is compact and Hausdorff [34, 4.9]. Thus, C1 (UX ) is unique (Remark 5.2), and C1 (UX ) = {W ⊂ C1 (X) × C1 (X) | there exists U ∈ UX such that C1 (U ) ⊂ W}. Theorem 5.11. If X is a continuum with the uniform property of Effros, then X has the property of Kelley. Proof. Let x0 be a point of X, let K be a subcontinuum of X with x0 ∈ K and let U be an open subset of C1 (X) containing K. Then there exists U ∈ UX (Remark 5.2) such that B(K, C1 (U )) ⊂ U. Since X has the uniform property of Effros, there exists an Effros entourage V ∈ UX for U . Let x ∈ Int(B(x0 , V )). Then

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there exists a U -homeomorphism h : X  X such that h(x0 ) = x. Then h(K) is a subcontinuum of X, x ∈ h(K) and h(K) ∈ B(K, C1 (U )) ⊂ U. Therefore, X has the property of Kelley. 2 As a consequence of Theorems 5.9 and 5.11, we obtain Corollary 5.12. If X is an Effros continuum, then X has the property of Kelley. Now, we are ready to prove a Hausdorff version of F. Burton Jones’ Aposyndetic Decomposition Theorem [21, Theorem 1]: Theorem 5.13. Let X be a homogeneous continuum with the uniform property of Effros. If G = {T ({x}) | x ∈ X}, then G is a continuous terminal decomposition of X, X/G is an aposyndetic homogeneous continuum, and the quotient map q : X  X/G is an atomic map. Proof. By Theorem 5.13, X has the property of Kelley. Thus, by Theorem 5.6, G is an upper semicontinuous terminal decomposition of X, X/G is an aposyndetic homogeneous continuum, and the quotient map q is an atomic map. We need to prove that G is lower semicontinuous. Let x be an element of X, let z1 and z2 be two elements of T ({x}) and let A be an open subset of X containing z1 . Then there exists U ∈ UX (Remark 5.2) such that B(z1 , U ) ⊂ A. Since X has the uniform property of Effros, there exists V ∈ UX such that if x1 and x2 be two points of X with ρ(x1 , x2 ) < V , then there exists a U -homeomorphism h : X  X such that h(x1 ) = x2 . Note that Int(B(z2 , V )) is an open subset of X containing z2 . Let z3 be an element of Int(B(z2 , V )). Then there exists a U -homeomorphism h : X  X such that h(z2 ) = z3 . Note that, since T ({z2 }) = T ({x}) = T ({z1 }), h(T ({z1 })) = T ({z3 }) [23, 5.1.1] and ρ(z1 , h(z1 )) < U , we have that T ({z3 }) ∩ B(z1 , U ) = ∅. Thus, T ({z3 }) ∩ A = ∅. Therefore, G is lower semicontinuous. 2 6. Property of Kelley weakly The property of Kelley weakly is introduced in [31] to study points of connectedness im kleinen of the n-fold hyperspaces of metric continua. We extend this property to continua. A continuum X has the property of Kelley weakly, if there exists a dense subset A of C1 (X) such that X has the property of Kelley at some point of each element A ∈ A. Let X be a continuum and let n be a positive integer. Then Cn (X) denotes the n-fold hyperspace of X; i.e., Cn (X) = {A ∈ 2X | A has at most n components}. With the Vietoris topology. If U1 , . . . , Um are open subsets of X, then U1 , . . . , Um  ∩ Cn (X) is denoted by

U1 , . . . , Um n . A version for metric continua of the following lemma is in [31, (∗), p. 45]. Lemma 6.1. Let X be a continuum with the property of Kelley weakly and let n be a positive integer. Then for each open subset U of Cn (X), there exists A ∈ U such that X has the property of Kelley at some point a of A. Proof. Let U be an open subset of Cn (X), let K ∈ U and let V1 , . . . , Vt be open subsets of X such that K ∈

V1 , . . . , Vt n ⊂ U. Let L be a component of K and let Vj1 , . . . , Vjl denote all the elements of {V1 , . . . , Vt } that L intersects. Then L ∈ Vj1 , . . . , Vjl 1 . Since X has the property of Kelley weakly, there exists M ∈ C1 (X) such that M ∈ Vj1 , . . . , Vjl 1 and X has the property of Kelley at some point a of M . Let A = (K \ L) ∪ M . Then A ∈ V1 , . . . , Vt n ⊂ U and X has the property of Kelley at a ∈ M ⊂ A. 2

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Part (2) of the following theorem is known for the hyperspace of subcontinua of a continuum [32, Theorem 11], it is extended to the n-fold hyperspaces of a metric continuum in [31, Lemma 4.1]. Theorem 6.2. Let X be a continuum and let U be an open subset of Cn (X).  (1) If X has the property of Kelley weakly, then Int( U) = ∅.  (2) If X has the property of Kelley, then U is an open subset of X. Proof. We prove (1). By Lemma 6.1, there exists A ∈ U such that X has the property of Kelley at some point a of A. Let V1 , . . . , Vt be open subsets of X such that A ∈ V1 , . . . , Vt n ⊂ U. Let A1 be the component of A containing a. Let Vj1 , . . . , Vjl denote all the elements of {V1 , . . . , Vt } that A1 intersects. Since X has  the property of Kelley at a, there exists a Kelley set V for a and Vj1 , . . . , Vjl 1 . We show that V ⊂ U. Let z ∈ V . Then there exists D ∈ C1 (X) such that z ∈ D and D ∈ Vj1 , . . . , Vjl 1 . Let B = (A \ A1 ) ∪ D.    Then B ∈ Cn (X) and B ∈ V1 , . . . , Vt n ⊂ U. Hence, B ⊂ U. Since z ∈ D ⊂ B, z ∈ U, and V ⊂ U.  Therefore, Int( U) = ∅.  To prove (2), let x ∈ U. Then there exists C ∈ U such that x ∈ C. Let C1 , . . . , Ct be the components of C. Without loss of generality, we assume that x ∈ C1 . Let V1 , . . . , Vt be open subsets of X such that C ∈ V1 , . . . , Vt n ⊂ U. Let Vj1 , . . . , Vjl denote all the elements of {V1 , . . . , Vt } that C1 intersects. Since X  has the property of Kelley at x, there exists a Kelley set V for x and Vj1 , . . . , Vjl 1 . We show that V ⊂ U. Let z ∈ V . Then there exists L ∈ C1 (X) such that z ∈ L and L ∈ Vj1 , . . . , Vjl 1 . Let R = (C \ C1 ) ∪ L.   Then R ∈ Cn (X) and R ∈ V1 , . . . , Vt n ⊂ U. Since z ∈ L ⊂ R, z ∈ U and V ⊂ U. Thus, x is an interior   point of U. Therefore, U is open in X. 2 The following lemma extends [19, Lemma 3.1] to continua. Lemma 6.3. Let X be a continuum. If K is a subcontinuum of 2X and K ∈ K, then each component of intersects K.



K

  Proof. First, note that K ∈ 2X [34, 2.5.2]. Suppose that there exists a component C of K such that   K ∩ C = ∅. Hence, by [10, (4.A.11)], there exist two closed subsets K1 and K2 of K such that K = K1 ∪ K2 , K ⊂ K1 and C ⊂ K2 . Let K1 = {L ∈ K | L ⊂ K1 } and K2 = {L ∈ K | L ∩ K2 = ∅}. Then K1 and K2 are disjoint closed subsets of K. Also, K = K1 ∪ K2 , a contradiction to the fact that K is connected.  Therefore, each component of K intersects K. 2 As a consequence of Theorem 6.2 part (1) and Lemma 6.3, we have the following corollary, it extends [22, Lemma 3.6]: Corollary 6.4. Let X be a continuum that has the property of Kelley weakly, and let n ≥ 1. If W is a   subcontinuum of Cn (X) such that IntCn (X) (W) = ∅, then W ∈ Cn (X) and IntX ( W) = ∅. The following result strengthens [22, Theorem 3.7]. Theorem 6.5. Let X be a continuum that has the property of Kelley weakly, and let n ≥ 1. If X is indecomposable, then Cn (X) is locally connected only at X. Proof. Let U1 , . . . , Um n be a basic open subset of Cn (X) containing X. Then, by [35, Theorem 2.4 and (1.4)], we have that U1 , . . . , Um n is arcwise connected. Therefore, Cn (X) is locally connected at X. Assume that Cn (X) is locally connected at A and A = X. Then there exists a continuum neighborhood K    of A such that K = X. By part (1) of Theorem 6.2, IntX ( K) = ∅; also, by Lemma 6.3, K is compact

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and has at most n components. Hence, by the Baire Theorem [18, Theorem 2-82], some component C of  K has nonempty interior in X. Therefore, since C = X, X is decomposable by [18, Theorem 3-41]. 2 Corollary 6.6. Let n be a positive integer. If X is a hereditarily indecomposable continuum, then X is the only point at which Cn (X) is locally connected. Proof. By [9, Proposition 2.7], hereditarily indecomposable continua have the property of Kelley. The corollary now follows from Theorem 6.5. 2 Our final result extends [31, Lemma 5.2] to continua. Theorem 6.7. Let X be a continuum that has the property of Kelley, and let A be a subcontinuum of X such that IntX (A) = ∅. If U is an open neighborhood of A and K is the component of U containing A, then A ⊂ IntX (K). Proof. Let U be an open neighborhood of A and let K be the component of U containing A. Suppose A ⊂ IntX (K). Let a ∈ A \ IntX (K). Since X has the property of Kelley at a, there exists a Kelley set V for a and U, IntX (A)1 Since a ∈ A \ IntX (K), V \ K = ∅. Let b ∈ V \ K. Then there exists a subcontinuum B of X such that b ∈ B and B ∈ U, IntX (A)1 . Note that B ⊂ U , B ∩ K = ∅ and B \ K = ∅, which is impossible. Therefore, A ⊂ IntX (K). 2 Acknowledgement The author thanks the referee for the valuable suggestions made that improved the paper. References [1] D.P. Bellamy, Continua for which the set function T is continuous, Trans. Am. Math. Soc. 151 (1970) 581–587. [2] D.P. Bellamy, Set functions and continuous maps, in: L.F. McAuley, M.M. Rao (Eds.), General Topology and Modern Analysis, Academic Press, 1981, pp. 31–38. [3] D.P. Bellamy, Some topics in modern continua theory, in: R.H. Bing, W.T. Eaton, M.P. Starbird (Eds.), Continua Decompositions Manifolds, University of Texas Press, 1983, pp. 1–26. [4] D.P. Bellamy, H.S. Davis, Continuum neighborhoods and filterbases, Proc. Am. Math. Soc. 27 (1971) 371–374. [5] D.P. Bellamy, L. Fernández, S. Macías, On T -closed sets, Topol. Appl. 195 (2015) 209–225. [6] D.P. Bellamy, K.F. Porter, A homogeneous continuum that is non-Effros, Proc. Am. Math. Soc. 113 (1991) 593–598. [7] J. Camargo, S. Macías, C. Uzcátegui, On the images of Jones’ set function T , Colloq. Math., forthcoming. [8] J. Camargo, C. Uzcátegui, Continuity of the Jones’ set function T , Proc. Am. Math. Soc. 145 (2017) 893–899. [9] W.J. Charatonik, A homogeneous continuum without the property of Kelley, Topol. Appl. 96 (1999) 209–216. [10] C. Christenson, W. Voxman, Aspects of Topology, Monographs and Textbooks in Pure and Applied Mathematics, vol. 39, Marcel Dekker, New York, Basel, 1977. [11] H. Cook, W.T. Ingram, A. Lelek, A list of problems known as Houston Problem Book, in: H. Cook, W.T. Ingram, K.T. Kuperberg, A. Lelek, P. Minc (Eds.), Continua with The Houston Problem Book, in: Lecture Notes in Pure and Applied Mathematics, vol. 170, Marcel Dekker, New York, Basel, Hong Kong, 1995, pp. 365–398. [12] H.S. Davis, A note on connectedness im kleinen, Proc. Am. Math. Soc. 19 (1968) 1237–1241. [13] H.S. Davis, D.P. Stadtlander, P.M. Swingle, Properties of the set functions T n , Port. Math. 21 (1962) 113–133. [14] A. Emeryk, Z. Horbanowicz, On atomic mappings, Colloq. Math. 27 (1973) 49–55. [15] R. Engelking, General Topology, Sigma Series in Pure Mathematics, vol. 6, Heldermann, Berlin, 1989. [16] R.W. FitzGerald, P.M. Swingle, Core decompositions of continua, Fundam. Math. 61 (1967) 31–50. [17] S. Gorka, Several Set Functions and Continuous Maps, Ph.D. dissertation, University of Delaware, 1997. [18] J. Hocking, G. Young, Topology, Dover, 1988. [19] H. Hosokawa, Induced mappings on hyperspaces, Tsukuba J. Math. 21 (1997) 239–250. [20] F.B. Jones, Concerning non-aposyndetic continua, Am. J. Math. 70 (1948) 403–413. [21] F.B. Jones, On a certain type of homogeneous plane continuum, Proc. Am. Math. Soc. 6 (1955) 735–740. [22] S. Macías, On the hyperspaces Cn (X) of a continuum X, II, Topol. Proc. 25 (2000) 255–276. [23] S. Macías, Topics on Continua, Pure and Applied Mathematics, vol. 275, Chapman & Hall/CRC, Taylor & Francis Group, Boca Raton, London, New York, Singapore, 2005. [24] S. Macías, Homogeneous continua for which the set function T is continuous, Topol. Appl. 153 (2006) 3397–3401.

S. Macías / Topology and its Applications 226 (2017) 51–65

65

[25] S. Macías, A decomposition theorem for a class of continua for which the set function T is continuous, Colloq. Math. 109 (2007) 163–170. [26] S. Macías, On the continuity of the set function K, Topol. Proc. 34 (2009) 167–173. [27] S. Macías, On the idempotency of the set function T , Houst. J. Math. 37 (2011) 1297–1305. [28] S. Macías, A note on the set functions T and K, Topol. Proc. 46 (2015) 55–65. [29] S. Macías, Atomic maps and T -closed sets, Topol. Proc. 50 (2017) 97–100. [30] S. Macías, S.B. Nadler Jr., On hereditarily decomposable homogeneous continua, Topol. Proc. 34 (2009) 131–145. [31] S. Macías, S.B. Nadler Jr., Various types of local connectedness in n-fold hyperspaces, Topol. Appl. 154 (2007) 39–53. [32] W. Makuchowski, On local connectedness in hyperspaces, Bull. Pol. Acad. Sci. 47 (1999) 119–126. [33] A. McCluskey, B. McMaster, Undergraduate Topology, Oxford University Press, 2014. [34] E. Michael, Topologies on spaces of subsets, Trans. Am. Math. Soc. 71 (1951) 152–182. [35] A.K. Misra, C-supersets, piecewise order-arcs and local arcwise connectedness in hyperspaces, Quest. Answ. Gen. Topol. 8 (1990) 467–485. [36] S. Mrówka, On the convergence of nets of sets, Fundam. Math. 45 (1958) 237–246. [37] S.B. Nadler Jr., Hyperspaces of Sets, Monographs and Textbooks in Pure and Applied Mathematics, vol. 49, Marcel Dekker, New York, Basel, 1978. Reprinted in Aportaciones Matemáticas de la Sociedad Matemática Mexicana, Textos, vol. 33, 2006. [38] R.W. Wardle, On a property of J.L. Kelley, Houst. J. Math. 3 (1977) 291–299.