A remainder for some non-locally compact spaces (Q1126337)

All spaces considered are completely regular and Hausdorff. A subset \(F\) of a space \(X\) has countable character if there is a sequence of open sets \(V_1\supseteq V_2\supseteq\dots \supseteq F\) such that if \(V\) is open in \(X\) and \(F\subseteq V\) then there is an \(n\) for which \(V_n\subseteq V\). The residue \(R(X)\) of \(X\) is the set of points in \(X\) which do not possess compact neighbourhoods. It was shown by \textit{R. E. Chandler} and \textit{F.-C. Tzung} [Proc. Am. Math. Soc. 70, 196-202 (1978; Zbl 0419.54011)] that if \(X\) is real compact, \(R(X)= \{p\}\) and \(p\) is contained in a compact set of countable character, then \([0,1)\) is a remainder of \(X\). In the current paper, the authors provide a partial generalization of this result. To that end, \(T_n\) is defined as the set of points in \(\mathbb{R}^n\) on the positive coordinate axes whose distance to the origin is less than or equal to 1, \(\{e_1,\dots, e_n\}\) as the endpoints of \(T_n\) and \(T_n^0= T_n-\{e_1, e_2,\dots, e_n\}\). The major result is Theorem 7: Let \(X\) be realcompact and \(R(X)= \{p_1,\dots,p_n\}\). If \(R(X)\) has countable character then \(T_n^0\) is a remainder of \(X\).