Universal elements for families of separable metrizable spaces
type: Item
instance of: scholarly article
title: Universal elements for families of separable metrizable spaces
zbMATH Open document ID: 0913.54024
DOI: 10.1007/BF02434916
published in: Journal of Mathematical Sciences (New York)
publication date: 1998-05-12T00:00:00Z
review text: The authors describe a very general way of constructing universal elements in families of separable metric spaces. The rough idea is to represent each space by a pair \((S,D)\), where \(S\) is a subset of the Cantor set \(C\) and \(D\) is an upper-semicontinuous decomposition of \(S\) with compact fibers. The number of these pairs is reduced to `at most \(\mathfrak c\)' by amalgamating equivalent pairs into one, where the equivalence of \((S,D)\) and \((S',D')\) seems to mean that \(S\) and \(S'\) have the same closure and that \(D\cup D'\) is upper-semicontinuous on \(S\cup S'\). The set of \(\mathcal E\) of these amalgamations is topologized so as to be separable metric. Their decompositions induce an upper-semicontinuous decomposition of the subspace \[ \bigcup\Bigl\{\bigl\{(S,D)\bigr\}\times S:(S,D)\in{\mathcal E}\Bigr\} \] of the product \(C\times\mathcal E\); the corresponding quotient space \(Q\) contains a copy of every element of the family. To make \(Q\) an element of the family, for example if the family is defined by some dimensional condition, the authors have to develop a huge amount of notation for bookkeeping.
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54B15
Mathematics Subject Classification ID: 54D65
Mathematics Subject Classification ID: 54F45
zbMATH DE Number: 1103438
zbMATH Keywords: separable metrizable space
zbMATH Keywords: universal space
author: Dimitris Georgiou
author: S. D. Iliadis
reviewed by: K. P. Hart
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02434916
OpenAlex ID: W2020637085
description: scientific article; zbMATH DE number 1103438