ADMISSIBILITY AND CONNECTEDNESS IM KLEINEN IN HYPERSPACES

• Journal title : Honam Mathematical Journal
• Volume 36, Issue 4,  2014, pp.913-919
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2014.36.4.913
Title & Authors
ADMISSIBILITY AND CONNECTEDNESS IM KLEINEN IN HYPERSPACES
Baik, Bong Shin; Rhee, Choon Jai;

Abstract
We investigate the relationships between the space X and the hyperspaces concerning admissibility and connectedness im kleinen. The following results are obtained: Let X be a Hausdorff continuum, and let A, $\small{B{\in}C(X)}$ with $\small{A{\subset}B}$. (1) If X is c.i.k. at A, then X is c.i.k. at B if and only if B is admissible. (2) If A is admissible and C(X) is c.i.k. at A, then for each open set U containing A there is a continuum K and a neighborhood V of A such that $\small{V{\subset}IntK{\subset}K{\subset}U}$. (3) If for each open subset U of X containing A, there is a continuum B in C(X) such that $\small{A{\subset}B{\subset}U}$ and X is c.i.k. at B, then X is c.i.k. at A. (4) If X is not c.i.k. at a point x of X, then there is an open set U containing x and there is a sequence $\small{\{S_i\}^{\infty}_{i=1}}$ of components of $\small{\bar{U}}$ such that $\small{S_i{\longrightarrow}S}$ where S is a nondegenerate continuum containing the point x and $\small{S_i{\cap}S={\emptyset}}$ for each i = 1, 2, $\small{{\cdots}}$.
Keywords
Language
English
Cited by
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