Tychonov's theorem for G-spaces
type: Item
instance of: scholarly article
title: Tychonov's theorem for G-spaces
zbMATH Open document ID: 0632.54010
DOI: 10.1007/BF01903940
published in: Acta Mathematica Hungarica
publication date: 1987-01-01T00:00:00Z
review text: This note presents an English version of most of the results of the first author's paper in Dokl., Akad. Nauk Arm. SSR 71, 212-216 (1980; Zbl 0468.54008). It concerns a version for G-spaces of the well-known result that every Tikhonov space of weight \(\tau\) can be topologically embedded in \(I^{\tau}\), the product of \(\tau\) copies of the unit interval I. Theorem. Let G be locally compact and sigma-compact. Then for every G- space \(<X,\pi >\) with X a compact Hausdorff space of weight \(w(X)=:\tau\) there exists an action \({\tilde \pi}\) of G on \({\mathbb{R}}^{\tau}\) such that (i) the cube \(I^{\tau}\) is an invariant subset of \({\mathbb{R}}^{\tau}\) under this action; (ii) X can equivariantly be embedded in \(I^{\tau}\). Moreover, there exists a linear structure on \({\mathbb{R}}^{\tau}\) making \({\mathbb{R}}^{\tau}\) (with its ordinary product topology) a locally convex topological vector space such that (iii) \(I^{\tau}\) is a convex subset of \({\mathbb{R}}^{\tau}\); (iv) the action \({\tilde \pi}\) is linear (i.e. \({\tilde \pi}^ t: {\mathbb{R}}^{\tau}\to {\mathbb{R}}^{\tau}\) is linear for every \(t\in G)\).
Mathematics Subject Classification ID: 54C25
Mathematics Subject Classification ID: 54H15
Mathematics Subject Classification ID: 47D03
zbMATH DE Number: 4028379
zbMATH Keywords: G-spaces
zbMATH Keywords: locally convex topological vector space
author: Sergey A. Antonyan
author: Jan de Vries
MaRDI profile type: Publication
cites work: Some Topological Properties of Convex Sets
cites work: Q4076463
cites work: Extending certain transformation group actions in separable, infinite-dimensional Frechet spaces and the Hilbert cube
full work available at URL: https://doi.org/10.1007/bf01903940
OpenAlex ID: W2006585004
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description: scientific article; zbMATH DE number 4028379