Notes on remainders of topological spaces in some compactifications (Q290615)

In this paper, the authors improve some of Arhangelskii's results on remainders of compactifications of topological spaces. Here, a remainder of a topological space \(X\) is a space of the form \(bX\setminus X\) for some compactification \(bX\) of \(X\). It is shown for example, that if \(X\) is a first-countable space with a \(G_\delta\)-diagonal and \(bX\) is a compactification of \(X\) such that the remainder \(bX\setminus X\) has countable tightness, then \(bX\) has countable fan-tightness. The authors show moreover that if a non-locally compact paratopological group \(G\) has a compactification \(bG\) such that the remainder \( bG \setminus G\) is the union of a finite family of metrizable subspaces, then \(G\) is locally separable and locally metrizable. They ask whether the Sorgenfrey line has a compactification whose remainder is \(\sigma\)-metrizable. Several other interesting observations are made.