The Baire set fixing property and uniform continuity in topological groups

Abstract

Every topological group that satisfies theM-Baire set fixing property and is left separable isdense in a retract of a product of metrizable groups. Since compact groups satisfy MBSFP, it is a corollary that compact groups are dyadic spaces (Kuz'minov). IfG satisfies MBSFP, then the left uniformity is separable if and only ifG satisfies the countable chain condition. Dense subgroups of left separable MBSFP groups satisfy MBSFP, and any product of left separable MBSFP groups satisfies MBSFP. A left separable group satisfies MBSFP if and only if the left uniformity is weak generated by natural maps to metrizable quotient groups. If for each α,Aα is a denseC-embedded set in the left separable MBSFP groupGα, then the real-compactification νΠAα = ΠνAα. It is a corollary that the product of pseudocompact groups is pseudocompact (Comfort and Ross). A locally compact group satisfies MBSFP if and only if it is metrizable or sigma compact (Ross and Stromberg, forG Abelian).