Fixed point sets of Tychonov cubes
type: Item
instance of: scholarly article
title: Fixed point sets of Tychonov cubes
zbMATH Open document ID: 0832.54036
DOI: 10.1016/0166-8641(95)00104-O
author: John R. Martin
author: Alex Chigogidze
published in: Topology and its Applications
publication date: 1996-03-05T00:00:00Z
review text: The authors, applying advanced methods of infinite-dimensional topology, give a spectral characterization of compact subsets of a Tikhonov cube \(Q^\tau\), \(\tau \geq \omega_1\), which are the sets of fixed points of continuous maps of the cube \(Q^\tau\) into itself. One of the main results says that \(X\) is the set of fixed points of a continuous map \(f : Q^\tau \to Q^\tau\) if and only if \(X\) can be represented as the limit of an inverse system \(\{X_\alpha, p_\alpha^{\alpha + 1}, \alpha < \tau\}\) whose short projections \(p_\alpha^{\alpha + 1} : X_{\alpha + 1} \to X_\alpha\) are stable and whose first element \(X_0\) is metrizable. The paper is interesting in view of the following known facts: (1) [\textit{J. R. Martin}, Proc. Am. Math. Soc. 103, No. 4, 1293-1298 (1988; Zbl 0652.54029)] For each closed subset \(X\) of the Hilbert cube \(Q^\omega\) there exists a continuous map \(f:Q^\omega \to Q^\omega\) such that \(X = \{x : f(x) = x\}\). (2) [\textit{P. Koszmider}, Can. Math. Bull. 35, No. 2, 221-229 (1992; Zbl 0796.54056)] There exists a zero-dimensional closed subset of \(Q^\tau\), \(\tau \geq \omega_1\), which does not coincide with the set of fixed points of any continuous map \(f : Q^\tau \to Q^\tau\).
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 54B35
zbMATH DE Number: 792535
zbMATH Keywords: limits of inverse systems over well-ordered sets
reviewed by: Władysław Kulpa
MaRDI profile type: Publication
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cites work: On the Complete Invariance Property in Some Uncountable Products
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cites work: Fixed Point Sets of Metric and Nonmetric Spaces
cites work: Q3914041
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description: scientific article; zbMATH DE number 792535