ADMISSIBILITY AND CONNECTEDNESS IM KLEINEN IN HYPERSPACES

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, <TEX>$B{\in}C(X)$</TEX> with <TEX>$A{\subset}B$</TEX>. (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 <TEX>$V{\subset}IntK{\subset}K{\subset}U$</TEX>. (3) If for each open subset U of X containing A, there is a continuum B in C(X) such that <TEX>$A{\subset}B{\subset}U$</TEX> 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 <TEX>$\{S_i\}^{\infty}_{i=1}$</TEX> of components of <TEX>$\bar{U}$</TEX> such that <TEX>$S_i{\longrightarrow}S$</TEX> where S is a nondegenerate continuum containing the point x and <TEX>$S_i{\cap}S={\emptyset}$</TEX> for each i = 1, 2, <TEX>${\cdots}$</TEX>.