On absolutely C-embedded topological groups

Abstract

A topological group H is called absolutely C-embedded if H is C-embedded into every topological group containing H as a subgroup. It is shown that a left uniformly continuous real-valued function on a subgroup of a topological group G can extend to a continuous function on G. Therefore, fine groups are absolutely C-embedded. We also prove that a CM-factorizable groups are absolutely C-embedded. Additionally, we explore the factorizability of absolutely C-embedded groups and establish that an absolutely C-embedded group G is R-factorizable if and only if G is an ω-narrow group. We also show that a non-metrizable absolutely C-embedded group G is M-factorizable if and only if G is τ-precompact, where τ=Fi(G).