On zero-dimensional Milutin maps and Michael selection theorems
type: Item
instance of: scholarly article
title: On zero-dimensional Milutin maps and Michael selection theorems
zbMATH Open document ID: 0794.54023
DOI: 10.1016/0166-8641(93)90052-F
published in: Topology and its Applications
publication date: 1994-03-27T00:00:00Z
review text: The authors study the relationship among the following selection theorems of E. Michael: (A) Let \(X\) be a paracompact space with \(\dim X=0\), \(M\) a complete metric space and \(F:X \to M\) a lower-semicontinuous closed- valued map. Then \(F\) admits a single-valued continuous selection. (B) Let \(X\) be a paracompact space, \(B\) a Banach space and \(F:X \to B\) a lower semicontinuous map with closed convex values. Then \(F\) admits a continuous single-valued selection. (C) Let \(X\) be a paracompact space, \(M\) a complete metric space and \(F:X \to M\) a lower semicontinuous map with closed values. Then \(F\) has an upper semicontinuous compact-valued selection, which admits a lower semicontinuous compact-valued selection. It is shown that theorem (A) implies both theorem (B) and (C). This follows from the main result in the paper that every paracompact space is the image of a zero-dimensional paracompact space under a Milutin map. Remark: An alternative proof that theorem (A) implies theorem (C) was given also by \textit{M. Choban} and \textit{E. Michael} in their recent paper [Topology Appl. 49, No. 3, 217-220 (1993; see the review above)].
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 60B05
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 28B20
zbMATH DE Number: 503636
zbMATH Keywords: selection theorem
author: Dušan Repovš
author: Pavel V. Semenov
author: Evgeny V. Shchepin
reviewed by: V. Valov
MaRDI profile type: Publication
cites work: Application of a Selection Theorem to Hyperspace Contractibility
cites work: Averaging Operators in C(S) and Lower Semicontinuous Sections of Continuous Maps
cites work: Continuous selections. I
cites work: Continuous selections. II
cites work: Selected Selection Theorems
cites work: A theorem on semi-continuous set-valued functions
cites work: Convex Structures and Continuous Selections
cites work: Q4287537
cites work: Q5667074
cites work: Q5550062
cites work: Q3777052
cites work: Q3852404
full work available at URL: 0166-8641(93)90052-f
OpenAlex ID: W2070590979
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description: scientific article; zbMATH DE number 503636