On metrizable remainders of locally compact separable metrizable spaces (Q2862664)

Let \(\mathcal C\) denote the class of all compact metrizable non-empty spaces, \(\mathcal M\) the class of all locally compact, non-compact separate metric spaces, and \(\mathcal M_C\) the class of \(X\in \mathcal M\) such that \(X\) is connected. If \(Y\) is a compact Hausdorff space such that \(X\) is dense in \(Y,\) then \(Y\setminus X\) is called the remainder of \(X\) in \(Y\). For \(X\in \mathcal M\), let \(\mathcal R(X)\) denote the class of all metrizable remainders of \(X\). A result of \textit{K. D. Magill jun.} [Trans. Am. Math. Soc. 160, 411--417 (1971; Zbl 0224.54029)], shows that for all \(X\in\mathcal M\) a compactification \(Y\) of \(X\) is metrizable if and only if the remainder \(Y\setminus X\) is metrizable. The authors prove two main results: (1) A characterization of those \(X\in\mathcal M\) such that \(\mathcal R(X)=\mathcal C\), and (2) the theorem that for any two \(X,X^\prime\in \mathcal M_C\), either \(R(X)\subset \mathcal R(X^\prime)\) or \(R(X^\prime)\subset \mathcal R(X)\).