The Baire property in the remainders of semitopological groups (Q2865134)

All spaces are assumed to be Tychonoff. A remainder of a space \(X\) is the subspace \(bX\setminus X\) of \(bX\), where \(bX\) is a Hausdorff compactification of \(X\). The study of remainders of topological groups was begun by Arhangel'skiĭ in 2005 and many topologists have obtained various results.NEWLINENEWLINEIn this paper, the authors improve a result of \textit{A. Arhangel'skiĭ} in [Commentat. Math. Univ. Carol. 50, No. 2, 273--279 (2009; Zbl 1212.54098)] and give a positive answer to a question posed by \textit{F. Lin} and \textit{S. Lin} [About remainders in compactifications of paratopological groups; \url{arXiv:1106.3836}]. A group \(G\) is called a \textit{semitopological group} if \(G\) has a topology such that the multiplication map from \(G\times G\) to \(G\) is separately continuous. A space \(G\) is called a \textit{rectifiable space} provided that there exist a homeomorphism \(\varphi:G\times G\to G\times G\) and an element \(e\in G\) such that \(\pi_1\circ\varphi=\pi_1\) and for every \(x\in G\) \(\varphi(x,x)=(x,e)\), where \(\pi_1:G\times G\to G\) is the projection to the first coordinate. Every topological group is a semitopological group and a rectifiable space, and both of them are homogeneous. The following result is a key theorem of this paper: If some remainder of a homogeneous space \(G\) is not Baire, then \(G\) has a dense Čech-complete subspace and \(G\) has a compact subset which has a countable base of open neighborhoods in \(G\). Applying the result, the authors show the following: Let \(G\) be a semitopological group; then either every remainder of \(G\) has the Baire property, or every remainder of \(G\) is \(\sigma\)-compact. A remainder of a nonlocally compact semitopological group cannot be a \(\sigma\)-compact Baire space. Let \(G\) be a rectifiable space; then every remainder of \(G\) is either Baire, or meagre and Lindelöf.