On [formula omitted]-factorizable topological groups

Abstract

A topological group G is called M-factorizable if for every continuous real-valued function f:G→R on G, there exist a continuous homomorphism φ of G onto a first-countable topological group H and a continuous real-valued function g on H such that f=g∘φ.It is shown that an R-factorizable topological group is exactly an M-factorizable ω-narrow topological group. It is also proved that a subgroup H of an M-factorizable group G is M-factorizable if and only if H is z-embedded in G.We show that for a P-group G, G is M-factorizable if and only if every continuous real-valued function on G is uniformly continuous and G is ω-balanced. We also show that for a locally compact group G, G is M-factorizable if and only if G is either metrizable or σ-compact.