Abstract:In this paper, we discuss certain networks on paratopological (or topological) groups and give positive or negative answers to the questions in \cite{Lin2013}. We also prove that a non-locally compact, $k$-gentle paratopological group is metrizable if its remainder (in the Hausdorff compactification) is a Fr\'echet-Urysohn space with a point-countable cs*-network, which improves some theorems in [Liu C., {\it Metrizability of paratopological $($semitopological$)$ groups\/}, Topology Appl. {\bf 159} (2012), 1415--1420], [Liu C., Lin S., {\it Generalized metric spaces with algebraic structures\/}, Topology Appl. {\bf 157} (2010), 1966--1974].
Keywords: paratopological groups; topological groups; sequential neighborhood; networks; metrizable; compactifications; remainders
AMS Subject Classification: 54E20 54E35 54H11