Compactness-like properties on Hattori spaces (Q6181140)

Let \(G\) be a group and \(\tau\) a topology on \(G\) which makes the multiplication mapping continuous (that is, \((G,\tau)\) is a paratopological group). We say that \((G,\tau)\) is an almost topological group if there is a Hausdorff group topology \(\sigma\) on \(G\) and a local base \(\beta\) of the identity \(e\) on \((G,\tau)\) such that \(\sigma\le \tau\) and \(U\setminus\{e\}\in \sigma\) for every \(U\in \beta.\) If \((G,\tau)\) has an almost topological group structure witnessed by \(\sigma\) and \(\beta\) as above, then \(\sigma\) is neccessarily \(\tau_*,\) the strongest Hausdorff group topology weaker than \(\tau.\) Almost topological groups were defined by \textit{M. Fernández} [Topol. Proc. 40, 63--72 (2012; Zbl 1271.54068)]. By generalizing a construction initially given for the Sorgenfrey line by \textit{Y. Hattori} [Mem. Fac. Sci. Eng., Shimane Univ., Ser. B, Math. Sci. 43, 13--26 (2010; Zbl 1196.54048)], the same authors defined in [\textit{A. Calderón-Villalobos} and \textit{I. Sánchez}, Topology Appl. 326, Article ID 108411, 15 p. (2023; Zbl 1518.54025)] a family of topologies \(\{\tau(A):A\subseteq G\}\) on an arbitrary almost topological group \((G,\tau)\) with \(\tau_*=\tau(G)\le\tau(A)\le\tau(\emptyset)=\tau\) for every \(A\subseteq G.\) In this article the authors continue the study of the Hattori spaces \((G,\tau(A))\) for an almost topological group \((G,\tau)\), and their relations with the topologies \(\tau\) and \(\tau_*.\) Among other results, they give sufficient conditions for the closure of a subgroup in a Hattori space to be also a subgroup, and they show that \((G,\tau(A))\) is compact if and only if \((G,\tau_*)\) is compact and \(A=G.\) Necessary and/or sufficient conditions are also given for Hattori spaces to be locally compact, \(\sigma\)-compact, Polish, (hereditarily) Lindelöf and other topological properties.