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

$$ \left(V\cap {h}_{\mathrm{T}}^G(G)\right){\left(W\cap {h}_{\mathrm{T}}^G(G)\right)}^{-1}\subset U\cap {h}_{\mathrm{T}}^G(G). $$

In this case, we say that T is a continuously containing space for the class 𝔾. We recall that as the class 𝕊 we can consider, for example, the following classes of completely regular spaces: n-dimensional spaces, countable-dimensional spaces, strongly countable-dimensional spaces, locally finite-dimensional spaces. Therefore, in all these classes there are elements that are continuously containing spaces for the corresponding subclasses consisting of topological groups. In this paper, some open problems are considered.