The compactificability classes: The behavior at infinity (Q884171)

Let \(X\) and \(Y\) be topological spaces. The author says that \(X\) is compactifyable by \(Y\) iff there exists a compact topology on the disjoint union \(Z\) of \(X\) and \(Y\) such that \(X\) and \(Y\) become subspaces of \(Z\) and any two points \(x\in X\) and \(y\in Y\) have disjoint neighbourhoods in \(Z\). His main result states that if a locally compact, non-compact, second countable Hausdorff space \(X\) has the property that each neighbourhood of the point of infinity in the Alexandroff compactification of \(X\) contains uncountably many elements of \(X\), then \(X\) and \(\mathbb{R}\) are compactifyable by the same spaces \(Y\).