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On the property of kelley in the hyperspace and Whitney continua

Authors
Journal
Topology and its Applications
0166-8641
Publisher
Elsevier
Publication Date
Volume
30
Issue
2
Identifiers
DOI: 10.1016/0166-8641(88)90015-6

Abstract

Abstract In this paper, we introduce the notion of property [ K] ∗ which implies property [ K], and we show the following: Let X be a continuum and let ω be any Whitney map for C( X). Then the following are equivalent. (1) X has property [ K] ∗. (2) C( X) has property [ K] ∗. (3) The Whitney continuum ω −1( t) (0⩽ t< ω( X)) has property [ K] ∗. As a corollary, we obtain that if a continuum X has property [ K] ∗, then C( X) has property [ K] and each Whitney continuum in C( X) has property [ K]. These are partial answers to Nadler's question and Wardle's question ([10, (16.37)] and [11, p. 295]). Also, we show that if each continuum X n ( n=1,2,3,…) has property [ K] ∗, then the product ∏ X n has property [ K] ∗, hence C(∏ X n ) and each Whitney continuum have property [ K] ∗. It is known that there exists a curve X such that X has property [ K], but X× X does not have property [ K] (see [11]).

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