Submetrizability in semitopological groups (Q2452038)

Recall that a \textit{semitopological group} is a group with a topology such that the multiplication in the group is separately continuous, and if the multiplication is jointly continuous, then the group is called a \textit{paratopological group}. Thus, every topological group is a paratopological group and every paratopological group is a semitopological group. It is known that every Hausdorff topological group of countable pseudocharacter is submetrizable. On the other hand, \textit{F. Lin} and \textit{C. Liu} [Topology Appl. 159, No. 10--11, 2764--2773 (2012; Zbl 1276.54027)] gave an example of a Hausdorff paratopological group of countable pseudocharacter, which is not submetrizable. Recently, \textit{L.-H. Xie} and \textit{S. Lin} [Topol. Proc. 44, 139--149 (2014; Zbl 1300.54040)] discussed submetrizability of paratopological groups. In this paper, the authors discuss submetrizability of semitopological groups, in particular, they discuss the following questions that were posed by Tkachenko: 1. Does every Hausdorff (or regular) semitopological group \(G\) with \(l(G)\cdot\psi(G)\leq\omega\) satisfy \(|G|\leq2^{\omega}\)? 2. Does a Hausdorff (or regular) paratopological group \(G\) with \(l(G)\cdot\psi(G)\leq\omega\) admit a continuous bijection onto a Hausdorff space with a countable base? The authors give a partial answer to Question 1 and a positive answer to Question 2. For example, they prove the following result: Every Hausdorff semitopological group \(G\) with \(H_s(G)\cdot\text{\textit{In}}_r(G) \cdot\psi(G)\leq\omega\) admits a continuous bijection onto a Hausdorff space with a countable base, in particular, \(|G|\leq2^{\omega}\); in addition, the diagonal of \(G\) is a \(G_{\delta}\)-set in \(G\times G\), where \(H_s(G)\) means the Hausdorff number of \(G\) and \textit{In}\(_r(G)\) means the right index of narrowness of \(G\).