On [formula omitted]-factorizable P-groups

Abstract

A topological group G is M-factorizable if for every continuous real-valued function f on G, one can find a continuous homomorphism π:G→H onto a metrizable topological group H and a continuous function h on H such that f=h∘π. We continue the study of M-factorizability in topological groups started in Zhang et al. (2020) [14], with a special emphasis on P-groups, i.e., the groups in which all Gδ-sets are open. It turns out that a P-group G is M-factorizable iff for every clopen subset U of G, there exists an open invariant subgroup N of G satisfying U=UN.Let p(G) be the least cardinality of a family γ of open sets in G such that the set ⋂γ fails to be open. We find an alternative characterization of M-factorizability in P-groups in purely topological terms as follows: A P-group G is M-factorizable iff G is pseudo-p(G)-compact. The latter characterization is applied to prove the validity of the conjecture of Comfort and Hager formulated in 1975: A topological group G is fine iff it is pseudo-p(G)-compact. In particular, a P-group is M-factorizable iff it is fine.We also study products of M-factorizable P-groups and their powers. In particular, it is proved that all finite and countable powers of an M-factorizable P-group are M-factorizable.