Tychonov's theorem for G-spaces
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 \(t\in G)\).
scientific article; zbMATH DE number 4028379