On the property of Kelley in the hyperspace and Whitney continua
type: Item
instance of: scholarly article
title: On the property of Kelley in the hyperspace and Whitney continua
zbMATH Open document ID: 0663.54005
DOI: 10.1016/0166-8641(88)90015-6
author: Hisao Kato
published in: Topology and its Applications
publication date: 1988-01-01T00:00:00Z
review text: A (metric) continuum X is said to have property K if for each subcontinuum A of X, each point a in A and for each positive \(\epsilon\) there exists a positive \(\delta\) such that if b is a point of X at the distance \(<\delta\) from a then there exists a subcontinuum B of X containing b and being at the Hausdorff distance \(<\epsilon\) from A. This property was discovered by \textit{J. L. Kelley} [Trans. Am. Math. Soc. 52, 22-36 (1942; Zbl 0061.401)] who proved that the hyperspaces C(X) and \(2^ X\) are contractible if X has this property. Later this property was studied as interesting in itself by \textit{R. W. Wardle} [Houston J. Math. 3, 291-299 (1977; Zbl 0355.54025); see also book by \textit{B. S. Nadler jun.}, ``Hyperspaces of sets'' (1978; Zbl 0432.54007)]. The author introduces a much more restrictive property \(K^*\) so that a two-spiral curve from the Wardle paper, having obviously the property K, has no longer the property \(K^*\). However, locally connected continua, homogeneous continua and \(\sin(1/x)\) curve, which are known as having the property K, have the property \(K^*\). The author proves that if X has property \(K^*\) then C(X) has property \(K^*\), and therefore the property K, too. This is the main result of the paper (it is a still open question whether C(X) should have property K if X has property K). Another result: the property \(K^*\) is preserved under countable products. It should be mentioned that the property K, as was shown by Wardle, is not necessarily preserved even under operation of squaring the space.
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54F15
zbMATH DE Number: 4084533
zbMATH Keywords: Whitney map
zbMATH Keywords: Kelley property
zbMATH Keywords: Whitney continuum
zbMATH Keywords: eqi-\(LC^ 0\)
zbMATH Keywords: property \(K^*\)
zbMATH Keywords: countable products
reviewed by: Jerzy Mioduszewski
MaRDI profile type: Publication
full work available at URL: 0166-8641(88)90015-6
OpenAlex ID: W2083377874
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description: scientific article; zbMATH DE number 4084533