Triquotient and inductively perfect maps
type: Item
instance of: scholarly article
title: Triquotient and inductively perfect maps
zbMATH Open document ID: 0591.54010
DOI: 10.1016/0166-8641(86)90014-3
published in: Topology and its Applications
publication date: 1986-01-01T00:00:00Z
review text: A map \(f: X\to Y\) of topological spaces is said to be triquotient if for each \(y\in Y\) there is a family \(\eta_ y\) of open subsets of X such that \(X\in \eta_ y\), and the following conditions are satisfied: (i) if \(U\in \eta_ y\) and \(\gamma\) is an open cover of \(U\cap f^{-1}(y)\) by open subsets of X, then there exist a finite number of elements \(U_ i\in \gamma\), \(1\leq i\leq k\), such that \(\cup^{k}_{i=1}U_ i\in \eta_ y\); (ii) if \(U\in \eta_ y\), then \(y\in int f(u)\); (iii) if \(U\in \eta_ y\), then there is a neighborhood 0(y) such that \(U\in \eta_{\xi}\) for every \(\xi\in 0(y)\). Triquotient maps were introduced by \textit{E. Michael} [Ill. J. Math. 21, 716-733 (1977; Zbl 0386.54007)]. A mapping \(f: X\to Y\) of topological spaces is said to be inductively perfect if there is a closed set \(F\subset X\) such that \(f(F)=Y\) and f/F is perfect. The main results of this paper are as follows: Theorem 1. Let f:X\(\to Y\) be a triquotient map onto a paracompact space Y. If there is a perfect extension \(f^*: X^*\to Y\) of f such that X is a \(G_{\delta}\) set in \(X^*\), then f is inductively perfect. Theorem 2. If f:X\(\to Y\) is a triquotient map of a metric space (X,d) onto a metric space Y such that every fiber \(f^{-1}(y)\) is a complete metric space with respect to d and for every open \(U\subset X\), f(U) is an \(\epsilon_{\delta}\)-set in Y, then f is inductively perfect.
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 3949272
zbMATH Keywords: s-covering maps
zbMATH Keywords: completions
zbMATH Keywords: inductively perfect maps
zbMATH Keywords: separable
zbMATH Keywords: metric spaces
zbMATH Keywords: Čech complete space
zbMATH Keywords: compact covering maps
zbMATH Keywords: triquotient map
zbMATH Keywords: perfect extension
author: Alexey Ostrovsky
reviewed by: Edwin Duda
MaRDI profile type: Publication
full work available at URL: 0166-8641(86)90014-3
OpenAlex ID: W1998821751
cites work: Q5553887
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cites work: Necessary and sufficient conditions for the measurability of certain sets of closed subsets
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cites work: Inductively Perfect Maps and Tri-Quotient Maps
cites work: Q3854315
cites work: A theorem on semi-continuous set-valued functions
cites work: Q5664551
cites work: Q4104546
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cites work: Q4082013
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description: scientific article; zbMATH DE number 3949272