On reflections and three space properties of semi(para)topological groups (Q501602)

Given a semitopological group \(G\), a \(T_2\)-reflection \(T_2(G)\) of \(G\) is a pair \((H,\varphi_{G,2})\), where \(H\) is a Hausdorff semitopological group and \(\varphi_{G,2}:G\to H\) a continuous surjective homomorphism such that, for every continuous map \(f:G\to X\) with \(X\) a Hausdorff topological space, there exists a continuous map \(h:H\to X\) such that \(f=h\circ \varphi_{G,2}\) (see [\textit{M. Tkachenko}, Topology Appl. 161, 364--376 (2014; Zbl 1287.54047)]). The first part of the paper is devoted to the study of topological properties of semitopological groups that are invariant and/or inversely invariant with respect to \(T_2\)-reflections. In particular, a description is given of \(T_2(G)\) as an appropriate quotient of the semitopological group \(G\). Moreover, among several other results, we mention the following: the connected component of \(G\) coincides with the connected component of \(T_2(G)\); \(G\) is connected if and only if \(T_2(G)\) is connected; \(G\) is weakly Lindelöf if and only if \(T_2(G)\) is weakly Lindelöf. Finally, an explicit proof is furnished of the topological isomorphism given in [\textit{M. Tkachenko}, Topology Appl. 192, 176--187 (2015; Zbl 1419.54052)] between \(T_2(\prod_{i\in I} G_i)\) and \(\prod_{i\in I}T_2(G_i)\), for a family of semitopological groups \(G_i\). The second part of the paper concerns three space properties for semitopological groups (see [\textit{M. Bruguera} and \textit{M. Tkachenko}, ibid. 153, No. 13, 2278--2302 (2006; Zbl 1102.54039)] in the case of topological groups). The main result, which generalizes theorems from [\textit{Li-Hong Xie} and \textit{Shou Lin}, ibid. 180, 91--99 (2015; Zbl 1316.54015)] and [\textit{Shou Lin, Fucai Lin} and \textit{Li-Hong Xie}, ibid. 180, 167--180 (2015; Zbl 1312.54016)], is the following. Given a regular semitopological group \(G\), and a closed subgroup \(H\) of \(G\) such that all compact (respectively, countably compact, sequentially compact) subsets of \(H\) are first countable, if all compact (respectively, countably compact, sequentially compact) subsets of \(G/H\) are Hausdorff and strongly Fréchet (respectively, strictly Fréchet), then \(G\) has the same property.