Some properties of semicompletions of paratopological groups (Q2042102)

A topological group \(G\) such that every Cauchy filter on \(G\) converges is called Raĭkov complete. It is well-known that every Čech-complete group is Raĭkov complete and a feathered group is Čech-complete if and only if it is Raĭkov complete. \textit{T. Banakh} and \textit{A. Ravsky} [Topology Appl. 284, Article ID 107363, 36 p. (2020; Zbl 1468.22003)] extended the notion of Raĭkov complete topological groups to the class of \(T_{0}\) paratopological groups and proved that for every \(T_{0}\) paratopological group \(G\), there exists a \(T_{0}\) paratopological group \(\check{G}\) such that \(\check{G}\) is Raĭkov complete and \(G\) is dense in \(\check{G}\). In this paper, it is proved that if \(G\) is a Raĭkov complete paratopological group, then \(G_{\omega}\) is a Raĭkov complete paratopological group. Moreover, if \(N\) is a closed invariant topological subgroup of a paratopological group \(G\), and both of \(N\) and \(G/N\) are Raĭkov complete paratopological groups, then \(G\) is also Raĭkov complete. Then, the \(\mathcal{C}\)-semicompletion of paratopological groups is investigated. It is claimed that if \(G\) is a \(T_{3}\) (Tychonoff) paratopological group, then every \(\mathcal{C}\)-semicompletion of \(G\) is \(T_{3}\) (Tychonoff). If \(G\) is first-countable (second-countable), then every \(\mathcal{C}\)-semicompletion of \(G\) is first-countable (second-countable).