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  • P-ISSN1226-0657
  • E-ISSN2287-6081
  • KCI

ISSN : 1226-0657

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Vol.22 No.2

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A NOTE ON CONNECTEDNESS IM KLEINEN IN C(X)

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics / Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, (P)1226-0657; (E)2287-6081
2015, v.22 no.2, pp.139-144
https://doi.org/10.7468/jksmeb.2015.22.2.139
BAIK, BONG SHIN
RHEE, CHOON JAI
BAIK,, B. S. , & RHEE,, C. J. (2015). A NOTE ON CONNECTEDNESS IM KLEINEN IN C(X). Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, 22(2), 139-144, https://doi.org/10.7468/jksmeb.2015.22.2.139

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

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Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics

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