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A NOTE ON CONNECTEDNESS IM KLEINEN IN C(X)
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  • Journal title : The Pure and Applied Mathematics
  • Volume 22, Issue 2,  2015, pp.139-144
  • Publisher : Korea Society of Mathematical Education
  • DOI : 10.7468/jksmeb.2015.22.2.139
 Title & Authors
A NOTE ON CONNECTEDNESS IM KLEINEN IN C(X)
BAIK, BONG SHIN; RHEE, CHOON JAI;
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 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
hyperspace;connected im kleinen;admissible;
 Language
English
 Cited by
 References
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