On Embeddings of Topological Groups

Abstract

The problem of the existence of universal elements in the class of all topological groups of weight ≤τ ≠ ω remains open. In this paper, it is proved that for many classes of topological groups there are so-called continuously containing spaces. Let 𝕊 be a saturated class of completely regular spaces of weight ≤τ and 𝔾 be the subclass of elements of 𝕊 that are topological groups. Then there exists an element T ∈ 𝕊 having the following property: for every G ∈ 𝕋, there exists a homeomorphism \( {h}_{\mathrm{T}}^G \) of G into T such that if the points x, y of T belong to the set \( {h}_{\mathrm{T}}^H \) (H) for some H ∈ 𝔾, then for every open neighborhood U of xy in T there are open neighborhoods V and W of x and y in T, respectively, such that for every G ∈ 𝔾 we have