Cellularity in subgroups of paratopological groups (Q491812)

A paratopological (semitopological) group is a group with a topology such that multiplication on the group is jointly (separately) continuous. In this paper, it is proved that every subgroup of a \(\sigma\)-compact \(T_1\) paratopological group has countable cellularity (but this conclusion fails for subgroups of \(\sigma\)-compact \(T_0\) paratopological groups) and that if \(S\) is a non-empty subsemigroup of a topologically periodic semitopological group \(G\), then the closure of \(S\) is a subgroup of \(G\). This implies, in particular, that the closure of every non-empty subsemigroup of a precompact topological group \(G\) is a subgroup of \(G\) and that every subsemigroup of \(G\) has countable cellularity.