M-factorizability of products and τ-fine topological groups

摘要

Our main objective is a further study of M-factorizability in topological groups as defined in Zhang, Peng, He, Tkachenko (2020) [15]. We focus on topological-algebraic implications of M-factorizability such as tau-precompactness, pseudo-tau-compactness and tau-fineness. We also study products of topological groups and present necessary and sufficient conditions on the factors guaranteeing the M-factorizability of products. Our main technical tool for this study is the new notion of tau-fine topological group, where tau omega is a cardinal. We prove the following dichotomy theorem: Every M-factorizable topological group is either R-factorizable or omega(1)-fine.Another dichotomy is established for the product of two groups. We prove that if the product G x H of topological groups is M-factorizable, then for every cardinal tau omega, either G is tau-fine or H is pseudo-tau-compact. We also show that the product G x H is M-factorizable provided G is a metrizable topological group with omega(G) = tau and H is a tau-fine topological group with hl(H) = tau.It is also proved that the product G x H is M-factorizable (R-factorizable) whenever G is an arbitrary M-factorizable (R-factorizable) topological group and H is a locally compact separable metrizable topological group. (C) 2021 Elsevier B.V. All rights reserved.