The extensions of paratopological groups (Q484277)

A \textit{paratopological} group is a pair \((G,\mathcal T)\) consisting of a group \(G\) and a topology \(\mathcal T\) on it such that the multiplication is continuous (in other words, \((G,\mathcal T)\) is a topological semigroup). The authors study the following general problem: ``Let \(\mathcal P\) be a (topological, algebraic, or a mixed nature) property. Let \(N\) be a closed invariant subgoup of a paratopological group \(G\) such that both \(N\) and the quotient paratopological group \(G/N\) have \(\mathcal P\). When can we conclude that \(G\) has \(\mathcal P\)?''. A typical result: The following properties are preserved under extensions in the class of regular paratopological groups: (a) local compactness; (b) compactness; (c) connectedness; (d) every compact set is first-countable; (e) every countably compact set is first countable; (f) every sequentially compact set is first countable; (g) every countably compact set is compact; (h) every sequentially compact set is compact. (Theorem 2.3). Further results are obtained for paratopological Abelian groups and \(k\)-gentle paratopological groups. It is proved, for instance, that the property ``first-countable'' is preserved under extensions in the class of paratopological Abelian groups.