A NOTE ON CONNECTEDNESS IM KLEINEN IN C(X)

Abstract

Abstract. In this paper, we investigate the relationships between the space X and the hyperspace C(X) concerning admissibility and connectedness im kleinen. The following results are obtained: Let X be a Hausdorff continuum, and let A ∈ C(X). (1) If for each open set U containing A there is a continuum K and a neighborhood V of a point of A such that V ⊂ IntK ⊂ K ⊂ U, then C(X) is connected im kleinen. at A. (2) If IntA ≠ ø, then for each open set U containing A there is a continuum K and a neighborhood V of a point of A such that V ⊂ IntK ⊂ K ⊂ U. (3) If X is connected im kleinen. at A, then A is admissible. (4) If A is admissible, then for any open subset U of C(X) containing A, there is an open subset V of X such that A ⊂ V ⊂ ∪U. (5) If for any open subset U of C(X) containing A, there is a subcontinuum K of X such that A ∈ IntK ⊂ K ⊂ U and there is an open subset V of X such that A ⊂ V ⊂ ∪ IntK, then A is admissible.

0. INTRODUCTION

Let X be a Hausdorff continuum, and let 2X (C(X), K(X), CK (X)) the hyperspace of nonempty closed subsets(connected closed subsets, compact subsets, continua) of X with the Vietoris topology. Throughout by a continuum we mean a compact connected Hausdorff space. For a continuum X, C(X) is endowed with the Vietoris topology and, since X is a continuum, the hyperspace C(X) is also a continuum [8].

Wojdyslawsk [13] established the conditions of local connectedness between a space X and its hyperspace 2X (C(X)). Goodykoontz [3, 4, 5] investigated local connectedness as a pointwise property in the hyperspace 2X (C(X)) of metric continua. And Goodykoontz and Rhee [6] investigated the relationships between the space X and the hyperspaces concerning the properties of local compactness and local connectedness. They proved that a Hausdorff space X is connected im kleinen at x ∈ X if and only if 2X (K(X), CK (X)) is connected im kleinen at {x} and a locally compact Hausdorff space X is connected im kleinen at x ∈ X if and only if 2X (C(X), K(X), CK (X)) is connected im kleinen at {x}. In 2003, Makuchowski [9, 10] investigated with respect to local connectedness at a subcontinuum of continua.

The purpose of this paper is to investigate the relationships between the space X and the hyperspace C(X) concerning admissibility and connectedness im kleinen.

For notational purposes, small letters will denote elements of X, capital letters will denote subsets of X and elements of 2X , and script letters are reserved for subsets of 2X . If B ⊂ 2X , ∪B = {A : A ∈ B}. If A ⊂ X, the symbol will denote the interior(closure, boundary) of the set A.

 

1. PRELIMINARIES

Let X be a topological space. Let 2X = {E ⊂ X : E is nonempty and closed}, K(X) = {E ∈ 2X : E is compact}, C(X) = {E ∈ 2X : E is connected}, and CK (X) = K(X) ∩ C(X), and endow each with the Vietoris topology. A basis for 2X consists of all elements of the form

< U1, U2, ⋯ , Un >= {A ∈ 2X : A ∩ Ui ≠ ø

for each i and where U1, U2, ⋯ , Un are open sets in X.

Let T (x) = {A ∈ C(X) : x ∈ A}. An element A ∈ T (x) is said to be admissible at x in X if, for each basic open set < U1, U2, ..., Un > ∩C(X) containing A, there is a neighborhood Vx of x in X such that whenever y ∈ Vx there is an element B ∈ T (y) such that B ∈< U1, U2, ..., Un > ∩C(X) [11].

The space X is said to be locally connected at x in X, if for each neighborhood U of x there is a connected neighborhood V of x such that V ⊂ U [7]. The space X is said to be connected im kleinen at x, if for each neighborhood U of x there is a component of U which contains x in its interior [7, 9]. The space X is said to be locally connected provided that X is locally connected at each of its points. If a space X is connected im kleinen at each of its points, then X is locally connected. The space X is said to be locally arcwise connected at x, if for each neighborhood U of x there is an arcwise connected neighborhood V of x such that V ⊂ U . The space X is said to be locally arcwise connected, if X is locally arcwise connected at each of its points. The space X is said to be arcwise connected im kleinen at x, if for each neighborhood U of x there is an arcwise connected, component of U which contains x in its interior. If a space X is arcwise connected im kleinen at each of its points, then X is locally arcwise connected.

A continuum X is said to be connected im kleinen at a subcontinuum A, if for each open subset U of X containing A, there is a subcontinuum K such that A ⊂ IntK ⊂ K ⊂ U [10]. A continuum X is said to be locally connected at a subcontinuum A, if for each open subset U of X containing A, there is an open connected subset V such that A ⊂ V ⊂ U [1]. Obviously, if a subcontinuum is degenerate, then the notion of connectedness im kleinen(local connectedness) at a subcontinuum is the same as the notion of connectedness im kleinen(local connectedness) at a point. Note that if X is connected im kleinen(locally connected) at each point of A, then X is connected im kleinen(locally connected) at a subcontinuum A, but not conversely,

Result 1.1 ([12]). (Boundary Bumpping Theorem) Let X be a Hausdorff continuum, and let A ∈ C(X). Then for each open set U in X containing A, the component CA of containing A intersects Bd(U ).

 

2. CONNECTEDNESS IM KLEINEN AND ADMISSIBILITY

Theorem 2.1. Let X be a Hausdorff continuum, and let A ∈ C(X). If for each open set U containing A there is a continuum K and a neighborhood V of a point of A such that V ⊂ IntK ⊂ K ⊂ U, then C(X) is connected im kleinen at A.

Proof. Let U =< U1, ⋯ , Un > ∩C(X) be an open subset of C(X) containing A. Then is an open subset of X containing A. And, there is a continuum K and a neighborhood V of a point x of A such that

V ⊂ IntK ⊂ K ⊂ U.

And

Let L1, L2 ∈< U1, ⋯ , Un, V > ∩C(X). Then L1 ∩ K ≠ ø and L2 ∩ K ≠ ø. It follows that L1 ∪ L2 ∪ K ∈< U1, ⋯ , Un, V > ∩C(X). Hence there is order arcs L1 and L2 in < U1, ⋯ , Un, K > ∩C(X) from L1 to L1 ∪ L2 ∪ K and from L2 to L1 ∪ L2 ∪ K. It follows that there is an arc in L1 ∪ L2 from L1 to L2, and it is clear that L1 ∪ L2 ⊂< U1, ⋯ , Un, V > ∩C(X). Therefore C(X) is locally arcwise connected at A. ☐

Theorem 2.2. Let A ∈ C(X). If IntA ≠ ø, then for each open set U containing A there is a continuum K and a neighborhood V of a point of A such that V ⊂ IntK ⊂ K ⊂ U .

Proof. Let U be an open set containing A. Let x ∈ IntA. Then there is an open set V such that x ∈ V ⊂ IntA, and hence x ∈ IntA ⊂ A ⊂ U. In this case A is a continuum which satisfies the condition of the continuum K in this theorem. ☐

We get the below Corollary from Theorem 2.1 and Theorem 2.2.

Corollary 2.3 ([Theorem 3 of [4]]). Let A ∈ C(X). If IntA ≠ ø, then C(X) is locally arcwise connected at A.

Proof. Let A ∈ C(X) and let < U1, ⋯ , Un < ∩C(X) be a basic open set containing A. Let x ∈ IntA and let V be an open set such that x ∈ V ⊂ IntA and such that Then A ∈< U1, ⋯ , Un, V >⊂< U1, ⋯ , Un >. Let L1, L2 ∈< U1, ⋯ , Un, V > ∩C(X). Then L1 ∩V ≠ ø and L2 ∩V ≠ ø, so L1 ∩A ≠ ø and L2 ∩ A ≠ ø. It follows that L1 ∪ L2 ∪ A ∈< U1, ⋯ , Un, V > ∩C(X). Hence there is order arcs L1 and L2 in < U1, ⋯ , Un, V > ∩C(X) from L1 to L1 ∪ L2 ∪ A and from L2 to L1 ∪ L2 ∪ A. It follows that there is an arc in L1 ∪ L2 from L1 to L2, and it is clear that L1 ∪ L2 ⊂< U1, ⋯ , Un, V > ∩C(X). ☐

Theorem 2.4. Let X be a Hausdorff continuum, and let A ∈ C(X). If X is connected im kleinen at A, then A is admissible.

Proof. Let x ∈ A ∈ C(X) and X is connected im kleinen at A. Let < U1, ⋯ , Un > ∩C(X) be a basic open set containing A, and let . Then A ⊂ U and there is a continuum K such that A ⊂ IntK ⊂ K ⊂ U . Set Vx = IntK. Then for every y ∈ Vx, y is an element of K. And since A ⊂ K and K ⊂ U , K ∈< U1, ⋯ , Un > ∩C(X). Thus A is admissible. ☐

Theorem 2.5. Let X be a Hausdorff continuum, and let A ∈ C(X). If A is admissible, then for any open subset U of C(X) containing A, there is an open subset V of X such that .

Proof. Let U be an open set containing A in C(X), and let x ∈ A. Then by the definition of admissibility there is an open set Vx containing x in X such that for every y ∈ Vx there is a continuum B in C(X) such that y ∈ B ∈ U . Set . Then .

Theorem 2.6. Let X be a Hausdorff continuum, and let A ∈ C(X). If for any open subset U of C(X) containing A, there is a subcontinuum K of X such that A ∈ IntK ⊂ K ⊂ U and there is an open subset V of X such that then A is admissible.

Proof. Let U =< U1, ⋯ , Un > ∩C(X) be a basic open subset of C(X) containing A, let K a continuum in C(X) contains A in its interior, let V an open subset of X such that . Then for any element y of V , is a continuum in U containing y.