Equivariant generalization of Dugundji's theorem
type: Item
instance of: scholarly article
title: Equivariant generalization of Dugundji's theorem
zbMATH Open document ID: 0604.57026
DOI: 10.1007/BF01158413
published in: Mathematical Notes
publication date: 1985-01-01T00:00:00Z
review text: The author provides an equivariant generalization of Dugundji's extension of Tietze's theorem. The main result of this paper is the following: Theorem. Let G be a compact Lie group, A an invariant closed subset of the metrizable G-space X, and V a convex invariant subset of some locally convex G-space Z. Then each equivariant map \(f: A\to V\) admits an equivariant extension \(F: L\to V\) to some invariant neighbourhood L of the set A in X. The author shows by an example that F cannot be defined on all of X, in general. If, though, the set \(\{\) \(v\in V:\) \(gv=v\) for all \(g\in G\}\) is nonempty, then \(L=X\). Also, if V is a complete set with respect to the natural group of uniformities of the linear topological space Z, then \(L=X\).
Mathematics Subject Classification ID: 57S10
Mathematics Subject Classification ID: 54C55
zbMATH DE Number: 3975904
zbMATH Keywords: Dugundji's theorem
zbMATH Keywords: invariant subset of locally convex G-space
zbMATH Keywords: compact Lie group
zbMATH Keywords: equivariant map
zbMATH Keywords: equivariant extension
author: Sergey A. Antonyan
reviewed by: Joachim Grispolakis
MaRDI profile type: Publication
cites work: An extension of Tietze's theorem
cites work: Q3935836
cites work: The classification of 𝐺-spaces
cites work: Q3943674
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description: scientific article; zbMATH DE number 3975904