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A NOTE ON CONNECTEDNESS IM KLEINEN IN C(X)
A NOTE ON CONNECTEDNESS IM KLEINEN IN C(X)
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2015. May, 22(2): 139-144
Copyright © 2015, Korean Society of Mathematical Education
  • Received : November 18, 2014
  • Accepted : February 12, 2015
  • Published : May 31, 2015
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About the Authors
BONG SHIN BAIK
DEPARTMENT OF MATHEMATICS EDUCATION, WOOSUK UNIVERSITY, JEONBUK, 565-701, KOREAEmail address:baik@ws.ac.kr
CHOON JAI RHEE
DEPARTMENT OF MATHEMATICS, WAYNE STATE UNIVERSITY, DETROIT, MI, 48202, USAEmail address:rhee@math.wayne.edu

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.
Keywords
0. INTRODUCTION
Let X be a Hausdorff continuum, and let 2 X ( 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 2 X ( C ( X )). Goodykoontz [3 , 4 , 5] investigated local connectedness as a pointwise property in the hyperspace 2 X ( 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 2 X ( 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 2 X ( 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 2 X , and script letters are reserved for subsets of 2 X . If B ⊂ 2 X , ∪ B = { A : A B }. If A X , the symbol
PPT Slide
Lager Image
will denote the interior(closure, boundary) of the set A .
1. PRELIMINARIES
Let X be a topological space. Let 2 X = { E X : E is nonempty and closed}, K ( X ) = { E ∈ 2 X : E is compact}, C ( X ) = { E ∈ 2 X : E is connected}, and CK ( X ) = K ( X ) ∩ C ( X ), and endow each with the Vietoris topology. A basis for 2 X consists of all elements of the form
< U 1 , U 2 , ⋯ , Un >= { A ∈ 2 X : A Ui ≠ ø
for each i and
PPT Slide
Lager Image
where U 1 , U 2 , ⋯ , 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 < U 1 , U 2 , ..., 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 ∈< U 1 , U 2 , ..., 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
PPT Slide
Lager Image
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 =< U 1 , ⋯ , Un > ∩ C ( X ) be an open subset of C ( X ) containing A . Then
PPT Slide
Lager Image
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
PPT Slide
Lager Image
Let L 1 , L 2 ∈< U 1 , ⋯ , Un , V > ∩ C ( X ). Then L 1 K ≠ ø and L 2 K ≠ ø. It follows that L 1 L 2 K ∈< U 1 , ⋯ , Un , V > ∩ C ( X ). Hence there is order arcs L 1 and L 2 in < U 1 , ⋯ , Un , K > ∩ C ( X ) from L 1 to L 1 L 2 K and from L 2 to L 1 L 2 K . It follows that there is an arc in L 1 L 2 from L 1 to L 2 , and it is clear that L 1 L 2 ⊂< U 1 , ⋯ , 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 < U 1 , ⋯ , 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
PPT Slide
Lager Image
Then A ∈< U 1 , ⋯ , Un , V >⊂< U 1 , ⋯ , Un >. Let L 1 , L 2 ∈< U 1 , ⋯ , Un , V > ∩ C ( X ). Then L 1 V ≠ ø and L 2 V ≠ ø, so L 1 A ≠ ø and L 2 A ≠ ø. It follows that L 1 L 2 A ∈< U 1 , ⋯ , Un , V > ∩ C ( X ). Hence there is order arcs L 1 and L 2 in < U 1 , ⋯ , Un , V > ∩ C ( X ) from L 1 to L 1 L 2 A and from L 2 to L 1 L 2 A . It follows that there is an arc in L 1 L 2 from L 1 to L 2 , and it is clear that L 1 L 2 ⊂< U 1 , ⋯ , 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 < U 1 , ⋯ , Un > ∩ C ( X ) be a basic open set containing A , and let
PPT Slide
Lager Image
. 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 ∈< U 1 , ⋯ , 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
PPT Slide
Lager Image
.
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
PPT Slide
Lager Image
. Then
PPT Slide
Lager Image
.
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
PPT Slide
Lager Image
then A is admissible.
Proof. Let U =< U 1 , ⋯ , 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
PPT Slide
Lager Image
. Then for any element y of V ,
PPT Slide
Lager Image
is a continuum in U containing y .
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