A note on paratopological groups with countable networks of sequential neighborhoods (Q2850616)

A \textit{paratopological group} is a pair \((G, \tau)\), where~\(G\) is a group and \(\tau\) is a topology on~\(G\) such that the mapping \((x, y)\mapsto xy\) of \(G\times G\) into~\(G\) is continuous. If in addition, the mapping \(x\mapsto x^{-1}\) of~\(G\) into \(G\) is continuous, then \((G, \tau)\) is a \textit{topological group}.NEWLINENEWLINELet \(X\) be a topological space. Let \(\mathcal{P}=\bigcup_{x\in X}\mathcal{P}_x\) be a cover of \(X\), where for each \(x\in X\), \(\mathcal{P}_x\) is a family of subsets of \(X\) containing \(x\) and such that (a) if \(U, V\in \mathcal{P}_x\), then \(W\subseteq U\cap V\) for some \(W\in \mathcal{P}_x\); (b) the family \(\mathcal{P}_x\) is a netwotk of \(x\) in \(X\) (if \(x\in U\), where \(U\) is open in \(X\), then \(P\subseteq U\) for some \(P\in \mathcal{P}_x\)).NEWLINENEWLINEWe call \(\mathcal{P}\) an \textit{\(sn\)-network (sequential-neighborhood network) for \(X\)} if each element of \(\mathcal{P}_x\) is a sequential neighborhood of \(x\) in \(X\) for each \(x\in X\). We call \(\mathcal{P}\) an \(so\)-network (sequentially-open network) for \(X\) if each element of \(\mathcal{P}_x\) is a sequentially open neighborhood of \(x\) in \(X\) for each \(x\in X\).NEWLINENEWLINEA regular space \(X\) is called \textit{\(sn\)-metrizable} if \(X\) has a \(\sigma\)-locally finite \(sn\)-network and a regular space \(X\) is called \textit{\(so\)-metrizable} if \(X\) has a \(\sigma\)-locally finite \(so\)-network.NEWLINENEWLINEIn this paper, the author discusses generalized metric properties of paratopological groups and he proves that a paratopological group is \(sn\)-merizable if and only if it is \(so\)-metrizable. Moreover, the author poses some questions concerning generalized metric properties of paratopological groups.