Projectively first-countable semitopological groups with certain D-properties

Abstract

In this article, we give an internal characterization of subgroups of products of semitopological groups which satisfy certain properties that imply D-property. For example, we give an internal characterization of subgroups of products of regular semitopological groups which satisfy open (G), give an internal characterization of subgroups of products of regular first-countable semitopological groups which satisfy property (σ-A) (property (σ-B)). Every first-countable semitopological group with Collins-Roscoe property satisfies pre-(G) and property (pre-σ-B). We finally show that if G is a Hausdorff countably compact semitopological group with Hs(G)≤ω and G satisfies property (pre-σ-B) (pre-(G)), then G is a topological group.