On the property of Kelley for Hausdorff continua

Abstract. We introduce the concepts Hausdorff maximal limit continuum and Hausdorff strong maximal limit continuum, for Hausdorff continua; these definitions extend the concepts of maximal limit continuum and strong maximal limit continuum, respectiveley, introduced by J. J. Charatonik and W. J. Charatonik in 1998 for metric continua [1, Definitions 2.2 and 2.3]. We show that in metric continua, being a maximal limit continuum is equivalent to being a Hausdorff maximal limit continuum. We also show that in metric continua, being a strong maximal limit continuum implies being a Hausdorff strong maximal limit continuum. Finally, we show an equivalence of having the property of Kelley, in terms of these new definitions, whose analog version for metric continua was given by J. J. Charatonik and W. J. Charatonik.


Introduction
A continuum is a compact, connected Hausdorff space with more than one point. A metric continuum is a continuum with a metric d that generates its topology.
The property of Kelley for metric continua was introduced by J. L. Kelley as property 3.2 in [4, p. 26]; he used it to study the contractibility of hyperspaces (see [13,Chapter XVI] and [3, pp. 167-172]). In 1999, W. J. Charatonik [2, Definition 2.1] and W. Makuchowski [9, p. 124] extended the property of Kelley for continua; in particular, Charatonik shows an example of a homogeneous continuum that does not have the property of Kelley, and Makuchowski uses the property of Kelley to show that several definitions of local connectivity are equivalent in the hyperspace C(X) of a continuum X having the property of Kelley. Concerning the generalization of some properties of metric continua to continua, S. Macías studied the property of Kelley for continua and introduces the uniform Effros property [6, p. 60]. In [7] and [8], the author proved that several properties of Jones' set function T , valid for metric continua, hold for continua as well.
In 1998, J. J. Charatonik and W. J. Charatonik defined the concepts of maximal limit continuum and strong maximal limit continuum, for metric continua [1, Definitions 3.2 and 3.3]; the authors used those definitions to show several properties of continua having the property of Kelley, and to show that some properties are equivalent to the property of Kelley. In this paper we extend the metric concepts of maximal limit continuum and strong maximal limit continuum, to continua, which we call Hausdorff maximal limit continuum and Hausdorff strong maximal limit continuum, respectively. We show that in metric continua, the definition of maximal limit continuum is equivalent to the definition of Hausdorff maximal limit continuum (Theorem 4.8), and the definition of strong maximal limit continuum is stronger that the definition of Hausdorff strong maximal limit continuum (Proposition 4.13). To end the paper, we show that the equivalences of [1,Theorem 3.11] still hold under these new extensions (Theorem 4.19).

Preliminaries
Given a continuum X, we consider the collection of all nonempty closed subsets of X, which is denoted by 2 X ; in other words, 2 X = {A ⊂ X : A is a nonempty closed subset of X}, topologized with the Vietoris topology, which can be described as follows: for each n ∈ N and each finite collection U 1 , . . . , U n of open subsets of X, we define The collection of all sets of the form U 1 , . . . , U n , is a basis for a topology for 2 X , which is called the Vietoris topology [11,Definition 1.7]. The set 2 X , endowed with the Vietoris topology, is called hyperspace of closed subsets of X. Also, we consider the collection of all subcontinua of X, denoted and defined by [Revista Integración, temas de matemáticas as a subspace of 2 X . The collection C(X) is called the hyperspace of subcontinua of X. It is known that if X is a continuum, then 2 X and C(X) are also continua [11,Theorems 4.9 and 4.10]. In this way, given a continuum X, we have that 2 C(X) = {A ⊂ C(X) : A is a nonempty closed subset of C(X)} and C(C(X)) = {A ∈ 2 C(X) : A is connected} are also continua.
If X is a topological space, given a subset A of X, the interior of A is denoted by int(A) and the closure of A by cl(A). Given A, B ∈ C(X) with A ⊂ B, we consider the collection Let X be a continuum and A ⊂ 2 X ; we denote A = {x ∈ X : there exists A ∈ A such that x ∈ A}. Now, if X is a metric continuum, the hyperspace 2 X is also considered with the Hausdorff metric, which we denote by H. It is known that the Hausdorff metric of 2 X generates the Vietoris topology [13, (0.13) Theorem]. Given r > 0, x ∈ X and A ∈ 2 X , let B(r, x) be the open ball in X with center x and radius r and let B H (r, A) be the open ball in 2 X with center A and radius r; also, let N d (r, A) = {B(r, x) : x ∈ A}.

Limit superior of nets in 2 X
A directed set is a pair (D, ≤), where D is a nonempty set and ≤ is a partial order in D, Given a continuum X, if {A d } d∈D is a net in 2 X , we define the limit superior of {A d } d∈D as follows: The limit superior of a net in 2 X was first considered by Mrówka [12, p. 237]. We present two properties of the limit superior that will be used in this paper. Definition 4.1. Let X be a metric continuum with metric d. We say that X has the property of Kelley if for each ε > 0 there exists δ > 0 such that for any p, q ∈ X, if d(p, q) < δ, then for each K ∈ C({p}, X), there exists L ∈ C({q}, X) such that H(K, L) < ε.

Main results
Let us recall that H is the Hausdorff metric of the hyperspaces 2 X and C(X); for A ∈ 2 X and r > 0, B H (r, A) is the open ball in 2 X of radius r and center A. In the following theorem we summarize two characterizations of continua having the property of Kelley. The equivalence of (1) and (3)  (1) X has the property of Kelley.
(2) For each open subset U of C(X), it holds that U is an open subset of X.
Proof. In this proof, since we consider the hyperspaces 2 X and 2 C(X) , we will use the notation · 2 C(X) to denote the basic open sets of the Vietoris topology in 2 C(X) .
We prove that (2) Let W = U 1 ∪ · · · ∪ U n and define V ′ = {v ∈ X : C({v}, X) ⊂ W}. We will show that V ′ is a neighborhood of p in X. In order to prove this assertion, first we prove that U is an open subset of X, p / ∈ U and p ∈ int(X − U )}. (1) Notice that C({p}, X) ⊂ W. If A ∈ C(X) and A / ∈ C({p}, X), then we choose two disjoint nonempty open subsets U and V of X such that A ⊂ U and p ∈ V . Then A ∈ U ∩ C(X) and p ∈ V ⊂ int(X − U ), and thus (1) follows.
On the other hand, by compactness of C(X), there exist k ∈ N and U 1 , . . . , U k open subsets of X such that p ∈ int(X − U i ) for each i ∈ {1, . . . , k} and We have that V is a neighborhood of p. We consider a point v ∈ V , and we will prove that f (v) ∈ U 1 , . . . , is an open subset of X, and this ends the implication (2)⇒(3).
We see that (3)⇒(1). Let p ∈ X, let K ∈ C({p}, X) and let U be an open subset of we have that U is an open subset of X and p ∈ U . Let q ∈ U , then f (q) ∈ U, C(X) 2 C(X) , hence f (q) = C({q}, X) and C({q}, X) ∩ U = ∅. Let L ∈ C({q}, X) ∩ U, then q ∈ L and L ∈ U. Therefore, X has the property of Kelley. For U ⊂ C(X), we define the collection We extend the concept maximal limit continuum for continua as follows:  Proof. Let L be a subcontinuum of X such that M L ⊂ K. By hypothesis, there is an open subset U of C(X) such that L ∈ U and the collection F (U) is not a neighborhood of M . Now, let V ⊂ C(X) be a neighborhood of M . Since F (U) is not neighborhood of M , we obtain that V F (U). Hence, there exists B ∈ V such that C(B, X) ∩ U = ∅.
The following lemma follows from the definition. Lemma 4.7. Let K be a subcontinuum of a continuum X. Then K is a Hausdorff maximal limit continuum in K.

[Revista Integración, temas de matemáticas
We are going to prove that in metric continua Definitions 4.4 and 4.5 are equivalent: Theorem 4.8. Let X be a metric continuum, K ∈ C(X) and M ∈ C(K). Then the following statements are equivalent: (1) M is a maximal limit continuum in K.
(2) M is a Hausdorff maximal limit continuum in K.
Proof. We prove that (1)⇒ (2). Assume that X is a metric continuum that satisfies (1) but does not satisfy (2). Let {M n } n∈N be the sequence converging to M , given by (1). Since X does not satisfy (2), there exists L ∈ C(X) such that M L ⊂ K and for each k ∈ N, we have that the set F (B H ( 1 k , L)) is a neighborhood of M . Since {M n } n∈N converges to M , for each k ∈ N there exists n k ∈ N such that for every n ≥ n k , we have that M n ∈ F (B H ( 1 k , L)). We can assume that the sequence {n k } k∈N is strictly increasing. Let n ∈ N with n ≥ n 1 , let k ∈ N such that n k ≤ n < n k+1 ; thus we obtain that . Then the sequence {M ′ n } ∞ n=n1 converges to L ⊂ K and L = M , which is a contradiction.
We prove that (2) n , for each n ∈ N, and satisfying the following properties: , and so on inductively. We consider a sequence {M ′ n } n∈N in C(X) convergent to some M ′ ∈ C(K) and such that M n ⊂ M ′ n , for each n ∈ N; hence, M ⊂ M ′ . Assume that M = M ′ , then there exists k ∈ N such that M ′ ∈ U k , and choose a subsequence {M ′ nj } j∈N of the sequence {M ′ n } n∈N such that M nj / ∈ F (U k ), for each j ∈ N. So, for each j ∈ N, we have that C(M nj , X) ∩ U k = ∅, then M ′ nj ∈ C(X) − U k . Since C(X) − U k is a closed subset of C(X), we have that M ′ ∈ C(X) − U k , which is a contradiction. We conclude that M = M ′ .
In 1998, J. J. Charatonik and W. J. Charatonik introduced the following definition for metric continua [1, Definition 3.3.]: Definition 4.9. Let K be a subcontinuum of a metric continuum X. A continuum M ⊂ K is called a strong maximal limit continuum in K provided that there is a sequence {M n } n∈N of subcontinua of X converging to M , such that for each subsequence {M n k } k∈N of {M n } n∈N and for each sequence For M, U ⊂ C(X), we define the collection Concerning the definition of strong maximal limit continuum, we propose the following concept:  We show that a strong maximal limit continuum is also a Hausdorff strong maximal limit continuum.
Proposition 4.13. Let X be a metric continuum, K ∈ C(X) and M ∈ C(K). If M is a strong maximal limit continuum in K, then M is a Hausdorff strong maximal limit continuum in K.
Proof. Let d be a metric on X, and let H be the Hausdorff metric on C(X). Let {M n } n∈N be a sequence that witnesses that M is a strong maximal limit continuum in K.
We consider an open subset M of C(X) such that M ∈ M. Suppose that for each open subset U of C(X) such that C(M, K) ⊂ U, we have that the collection Taking a subsequence if necessary, we may assume that {M ′ k } k∈N converges to an element M ′ ∈ C(X). For each k ∈ N, notice that M ′ k ∈ U k and M ′ k ∈ C(X)−M, then M ′ ∈ C(M, K) and M ′ ∈ C(X)−M, therefore M ′ = M , which contradicts the choice of the sequence {M n } n∈N .
[Revista Integración, temas de matemáticas Question 4.14. Let X be a metric continuum, K ∈ C(X) and M ∈ C(K), such that M is a Hausdorff strong maximal limit continuum in K. Is it true that M is a strong maximal limit continuum in K? Proposition 4.15. Let X be a continuum, let K be a subcontinuum of X, let {A d } d∈D be a net in C(X) converging to A ∈ C(K) and let M be a maximal element in C(K)∩lim sup{C(A d , X)} d∈D (with respect to inclusion). Then M is a Hausdorff strong maximal limit continuum in K.
Proof. Assume that M is not a Hausdorff strong maximal limit continuum in K. We will prove that Theorem 3.11 of [1] is still valid under these new definitions. We start with two lemmas. that C(B, X) ∩ U = ∅. Let D ∈ V, choose a point d ∈ D ⊂ U , then there exists J ∈ U 1 , . . . , U n ∩C(X) such that d ∈ J. Since D∪J ⊂ U 1 ∪· · ·∪U n and (D∪J)∩U i = ∅, for each i ∈ {1, . . . , n}. It follows that D ∪ J ∈ U 1 , . . . , U n ∩ C(X) ⊂ U. Notice that D ∪ J ∈ C(D, X). Therefore, C(D, X) ∩ U = ∅, which is a contradiction.
Lemma 4.17. Let K be a subcontinuum of a continuum X and let M be a subcontinuum of K. If M is a Hausdorff strong maximal limit continuum in K, then M is a Hausdorff maximal limit continuum in K. Let U = U ′ ∩ M 2 . We will show that the set F (U) is not a neighborhood of M , first we prove that: Indeed, let B ∈ F (U); then B ∈ C(X) and C(B, X) We have proved that B ∈ G(M 1 , U ′ ).
Thus (2) follows. Since G(M 1 , U ′ ) is not a neighborhood of M , then F (U) is not a neighborhood of M .
Corollary 4.18. Let X be a continuum with the property of Kelley, K ∈ C(X) and M ∈ C(K). If M = K, then M is not a Hausdorff strong maximal limit continuum in K.
To conclude this paper, we prove that Theorem 3.11 of [1] is still valid under these new definitions.
Theorem 4.19. Let X be a continuum; then, the following statements are equivalent: (1) X has the property of Kelley.
(2) For each subcontinuum K of X, the only Hausdorff maximal limit continuum in K is K itself.
(3) For each subcontinuum K of X, the only Hausdorff strong maximal limit continuum in K is K itself.
We prove ( [Revista Integración, temas de matemáticas be a directed set, where V 1 ≤ V 2 if V 2 ⊂ V 1 , for each V 1 , V 2 ∈ D. For each V ∈ D, choose p V ∈ V − U. Notice that the net {p V } V ∈D converges p in X and p V / ∈ U, for each V ∈ D. Let K ∈ U such that p ∈ K. Since the net {{p V }} V ∈D converges to {p} in C(X), by Lemma 3.2 we have that {p} ∈ lim sup{C({p V }, X)} V ∈D . Therefore C(K) ∩ lim sup{C({p V }, X)} V ∈D = ∅. By the Kuratowski-Zorn Lemma [5, p. 33], there exists M ∈ C(K) ∩ lim sup{C({p V }, X)} V ∈D maximal with respect to inclusion. By Proposition 4.15, we have that M is a Hausdorff strong maximal limit continuum in K. By (3), we have that M = K. It follows that K ∈ lim sup{C({p V }, X)} V ∈D ; since U is an open subset of C(X) and K ∈ U, then for each R ∈ D, there exists S ∈ D such that S ≥ R and U ∩ C({p S }, X) = ∅. Choose B ∈ U ∩ C({p S }, X), then p S ∈ B ∈ U. Hence p S ∈ U, which is a contradiction. We have proved that U is an open subset of X. This ends the proof of the theorem.