Maximal metrizable remainders of locally compact spaces (Q388029)

All spaces \(X\) are assumed to be locally compact, noncompact, separable, and metrizable. A remainder of a space \(X\) is a topological space homeomorphic to \(\alpha X \setminus X\) for some Hausdorff compactification \(\alpha X\) of \(X\). Inverse limits are used to construct a zero-dimensional metrizable remainder \(Z\) of \(X\) which is maximal in the sense that any other zero-dimensional metrizable remainder \(Y\) can be embedded in \(Z\), or equivalently, any such \(Y\) is a continuous image of \(Z\). If \(\mathcal{R}(X)\) is the set of metrizable remainders of \(X\) (or equivalently, remainders of metrizable compactifications of \(X\)), let \(\mathcal{R}_{con}\) be the collection of all \(\mathcal{R}(X)\) for all connected spaces \(X\). Ordered by inclusion, \(\mathcal{R}_{con}\) is shown to be isomorphic to \(\omega_1 + 1\).