On reflections and three space properties of semi(para)topological groups

Abstract

We show that many topological properties are invariant and/or inverse invariant under taking T2-reflections in semitopological groups. We also extend some three space properties in topological groups (paratopological groups) to semitopological groups. The following result is established: Let G be a regular semitopological group and let H be a closed subgroup of G such that all compact (resp., countably compact, sequentially compact) subsets of the semitopological group H are first-countable. If the quotient space G/H has the following property, then so does the semitopological group G.(*) all compact (resp., countably compact, sequentially compact) subsets are Hausdorff and strongly Fréchet (strictly Fréchet).