Three-space properties in paratopological groups
A (semitopological) paratopological group is a group \(G\) with a topology such that just the the multiplication in \(G\) is (separately) continuous and the inversion in \(G\) is not necessarily continuous. A topological-algebraic property \(P\) is a three-space property in the class of semitopological (paratopological) groups provided that for every semitopological (paratopological) group \(G\) and a closed invariant subgroup \(N\) of \(G\), the fact that both \(N\) and \(G/N\) have \(P\) implies that \(G\) also has \(P\). In the paper the authors show that neither first-countable nor second-countable are three-space properties in the class of paratopological groups. They present a countable regular paratopological abelian group \(H\) which contains a closed discrete subgroup \(F\) such that \(H/F\) is topologically isomorphic to the rational numbers with the Sorgenfrey topology and \(H\) is not first-countable and prove that there exists a countable regular paratopological abelian group \(H\) which contains a closed discrete subgroup \(F\) such that \(H/F\) is topologically isomorphic to the rational numbers with the Sorgenfrey topology. However, \(H\) is not first-countable.
scientific article; zbMATH DE number 6668173
scientific article