\(\beta X\) and fixed-point free maps
type: Item
instance of: scholarly article
title: \(\beta X\) and fixed-point free maps
zbMATH Open document ID: 0792.54037
DOI: 10.1016/0166-8641(93)90152-4
author: Eric K. van Douwen
published in: Topology and its Applications
publication date: 1993-11-15T00:00:00Z
review text: It is proved that whenever \(f\) is a fixed-point free map of an \(n\)- dimensional paracompact space \(X\) into itself and there exists an integer \(k\) such that the size of the preimage of every point is not greater than \(k\), then there exists an open cover \(\mathcal P\) of \(X\) such that the cardinality of \(P\) is not greater than \((n+1)\cdot (k+1)\) and \(f(U) \cap U = \emptyset\) for every \(U \in {\mathcal P}\). In particular it follows that if \(f\) is a fixed-point free homeomorphism of a finite dimensional paracompact space onto itself, then the extension of \(f\) over the Čech-Stone compactification is also fixed-point free. In the 0- dimensional case the result was obtained independently by the reviewer and \textit{Dok Yong Kim} [Commentat. Math. Univ. Carol. 29, No. 4, 657-663 (1988; Zbl 0687.54011)].
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 446219
zbMATH Keywords: paracompact space
zbMATH Keywords: open cover
zbMATH Keywords: Čech-Stone compactification
reviewed by: Aleksander Błaszczyk
MaRDI profile type: Publication
full work available at URL: 0166-8641(93)90152-4
OpenAlex ID: W2011967035
cites work: Q3994935
cites work: Fixed Points Arising Only in the Growth of First Countable Spaces
description: scientific article; zbMATH DE number 446219