Cell structures and topologically complete spaces
type: Item
instance of: scholarly article
title: Cell structures and topologically complete spaces
zbMATH Open document ID: 1390.54014
author: Wojciech Dębski
author: E. D. Tymchatyn
published in: Topology and its Applications
publication date: 2018-03-27T00:00:00Z
review text: The overarching theme encompassing this paper is that of approximations in topology: given a topological space, we would like to approximate it by simpler spaces. The idea goes back to Alexandroff who approximated compact metric spaces by polyhedra. Freudenthal proved that if \(X\) is a compact metric space then it is homeomorphic to the inverse limit of an inverse sequence of polyhedra whose dimension is bounded by the dimension of \(X\). These ideas developed into Shape Theory and later influenced similar approaches in the asymptotic setting (Anti-Čech approximations by Dranishnikov and the combinatorial approach by Dydak et al.). While approximations by complexes are in a way most natural, it is in principle possible to construct approximations by other spaces. For example, Kopperman et al. used finite \(T_0\) spaces in such approximations. In their paper [\textit{W. Dębski} and \textit{E. D. Tymchatyn}, Colloq. Math. 147, No. 2, 181--194 (2017; Zbl 1382.54018)], the authors introduced cell structures, which are approximations by combinatorial graphs, the latter being considered as discrete spaces. They showed that a complete metric space can be obtained from a cell structure by taking the appropriate quotient of the inverse limit. Furthermore, they showed that continuous maps between such spaces are determined by their approximations via a cell structure. In this paper the authors generalize these results to topologically complete spaces (and continuous maps between them). In contrary to approximations by complexes, a topologically complete space is only a quotient of the obtained inverse limit of a cell structure. On the other hand, cell structures turn out to be technically much more manageable than approximations by complexes.
Mathematics Subject Classification ID: 54C05
Mathematics Subject Classification ID: 54B35
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54E15
zbMATH DE Number: 6853367
zbMATH Keywords: cell structures
zbMATH Keywords: inverse systems of graphs
zbMATH Keywords: discrete approximation of spaces
zbMATH Keywords: discrete approximation of mappings
zbMATH Keywords: topological completeness
reviewed by: Ziga Virk
MaRDI profile type: Publication
full work available at URL: j.topol.2018.02.020
OpenAlex ID: W2793169977
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Wikidata QID: Q130155067
DOI: 10.1016/J.TOPOL.2018.02.020
description: scientific article; zbMATH DE number 6853367