Simply sm-factorizable (para)topological groups and their quotients

Abstract

We say that a (para)topological group G is strongly submetrizable if it admits a coarser separable metrizable (para)topological group topology and is projectively strongly submetrizable if for each open neighborhood U of the identity in G, there is a closed invariant subgroup N contained in U such that the quotient (para)topological group G/N is strongly submetrizable. We show that a quotient group of a simply sm-factorizable ω-narrow topological abelian group can fail to be simply sm-factorizable. This answers a question posed by Arhangel'skii and the first listed author in 2018. If, however, the kernel of a quotient homomorphism is a bounded subgroup, then the homomorphism preserves simple sm-factorizability in the classes of topological and paratopological groups.We also prove that a regular (para)topological group G is simply sm-factorizable if and only if G is projectively strongly submetrizable and every continuous real-valued function on G is uniformly continuous on Gω, the P-modification of G. Making use of this fact we show that all weakly Lindelöf projectively strongly submetrizable paratopological groups and all weakly Lindelöf paratopological abelian groups are simply sm-factorizable. It is also established that every precompact paratopological group is simply sm-factorizable.