Every regular countably sieve-complete semitopological group is a topological group

Abstract

    In this note, we show that if $G$ is a paratopological group such
that $G$ is a $G_{\delta}$-set in a regular countably
sieve-complete space then $G$ is a topological group. We finally
show that every regular countably sieve-complete semitopological
group is a topological group. Thus every locally countably
compact regular semitopological group is a topological group. If
$G$ is regular countably sieve-complete semitopological group,
then $G$ is a $D$-space if and only if $G$ is paracompact. We
point out that some conditions in Theorem 2.14 and Corollary 2.15
in [11]  are not essential.