Some three-space properties of semitopological and paratopological groups

Abstract

In this paper, some three-space properties of semitopological and paratopological groups are considered. We show that: (1) if the closed invariant subgroup H of semitopological group G and the natural quotient group G/H have Gδ-diagonals, then G has a Gδ-diagonal; (2) if H is a compact invariant subgroup of paratopological group G, then the canonical quotient mapping from G to G/H is closed. An example is given to show that the result (2) can not be generalized to quasitopological groups.