Rays and the fixed point property in noncompact spaces
type: Item
instance of: scholarly article
title: Rays and the fixed point property in noncompact spaces
zbMATH Open document ID: 0885.54026
author: Tadeusz Dobrowolski
author: Witold Marciszewski
published in: Tsukuba Journal of Mathematics
publication date: 1998-01-12T00:00:00Z
review text: A problem is discussed in the paper of whether a noncompact space with a nice local structure contains a ray, i.e., a closed homeomorphic copy of \([0,1)\). After the introduction, \(LC^0\)-spaces containing a ray are considered in the second section. It is shown that if a metrizable \(LC^0\)-space either is noncompact, connected and locally compact, or is not completely metrizable, then it contains a ray. The third section is devoted to convex sets which lack the fixed point property. Answering an old question of \textit{V. L. Klee} jun. [Trans. Am. Math. Soc. 78, 30-45 (1955; Zbl 0064.10505)] the authors show that each noncompact convex subset of a metric linear space contains a ray, and therefore fails the fixed point property. The main result of this section says that if a convex subset of a topological vector space is not totally bounded, then it contains a ray and does not have the fixed point property. In the last section two examples of noncompact convex set with the fixed point property are constructed, showing that the previous result cannot be extended to all totally bounded noncompact convex subsets of topological vector spaces.
Mathematics Subject Classification ID: 54F35
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 54C55
zbMATH DE Number: 1102912
zbMATH Keywords: absolute retract
zbMATH Keywords: \(LC^ n\)-space
zbMATH Keywords: locally contractible space
zbMATH Keywords: ray
zbMATH Keywords: fixed point
zbMATH Keywords: convex set
reviewed by: Janusz J. Charatonik
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496163163
OpenAlex ID: W1644746493
DOI: 10.21099/TKBJM/1496163163
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description: scientific article; zbMATH DE number 1102912