Extremally disconnected remainders of nowhere locally compact spaces (Q2105033)

All topological spaces considered in this paper are Tychonoff. A space is \textit{extremally disconnected} if the closure of each open set is open. A \textit{compactification} of a space \(X\) is any compact space \(bX\) such that \(X\) is a dense subspace of \(bX\). A \textit{remainder} \(Y\) of \(X\) is the subspace \(Y = bX \setminus X\) of a compactification \(bX\) of \(X\). A space is \textit{nowhere locally compact} if the closure of any nonempty open subset of the space is not compact. The following theorem is the main result in this paper: Theorem: Let \(Z\) be an extremally disconnected space, and let \(bX\) be a compactification of a nowhere locally compact nonpseudocompact space \(X\) such that \(Z = bX \setminus X\). Then: \begin{itemize} \item[1)] \(X\) contains a nonempty open and closed in \(X\) extremally disconnected subspace, and \item[2)] \(bX\) contains a nonempty open and closed in \(bX\) extremally disconnected subspace. \end{itemize} Many corollaries, variations, or generalizations of this theorem related to products, topological groups, paratopological groups, or \(P\)-spaces are also presented.