Suitable sets for paratopological groups

Abstract

A paratopological group $G$ has a {\it suitable set} $S$. The latter means that $S$ is a discrete subspace of $G$, $S\cup \{e\}$ is closed, and the subgroup $\langle S\rangle$ of $G$ generated by $S$ is dense in $G$. Suitable sets in topological groups were studied by many authors. The aim of the present paper is to provide a start-up for a general investigation of suitable sets for paratopological groups, looking to what extent we can (by proving propositions) or cannot (by constructing examples) generalize to paratopological groups results which hold for topological groups, and to pose a few challenging questions for possible future research. We shall discuss when paratopological groups of different classes have suitable sets. Namely, we consider paratopological groups (in particular, countable) satisfying different separation axioms, paratopological groups which are compact-like spaces, and saturated (in particular, precompact) paratopological groups. Also we consider the permanence of a property of a group to have a suitable set with respect to (open or dense) subgroups, products and extensions.