Notes on remainders of paratopological groups (Q464219)

The remainder of a Tychonoff topological space \(X\) is the subspace \(bX\setminus X\) of some compactification \(bX\) of \(X\). The authors study remainders of paratopological and of semitopological groups \(G\), in particular they relate properties of \(G\) with properties of the remainder \(bG\setminus G\).NEWLINENEWLINELet \(G\) be a non-locally compact paratopological group and \(bG\) a compactification of \(G\).NEWLINENEWLINEThe main result of this paper extends a theorem by Arhangel'skii from topological groups to paratopological groups. It states that \(bG\setminus G\) is a \(p\)-space if and only if either \(G\) is a Lindelöf \(p\)-space or \(G\) is \(\sigma\)-compact. Several consequences of the main theorem are considered. For example, if \(bG\setminus G\) is a paracompact \(p\)-space, then \(G\) is a Lindelöf space.NEWLINENEWLINEMoreover, the authors generalize another theorem by Arhangel'skii to the case of paratopological groups, restricting appropriately the hypothesis: if \(bG\setminus G\) is locally metrizable, then \(G\) and \(bG\) are separable and metrizable.NEWLINENEWLINEIn the final part of the paper, remainders of semitopological groups are treated. It is proved that if \(G\) is a non-locally compact separable semitopological group and \(bG\setminus G\) has countable \(\pi\)-character, then either \(bG\setminus G\) is countably compact or \(G\) has a countable \(\pi\)-base. Finally, if the remainder \(bG\setminus G\) of a pseudocompact non-compact semitopological group \(G\) has countable \(\pi\)-character, then it is countably compact.