Discrete reflexivity in squares (Q2822583)

\textit{V. V. Tkachuk} [Trans. Mosc. Math. Soc. 1988, 139--156 (1988); translation from Tr. Mosk. Mat. O.-va 50, 138--155 (1987; Zbl 0662.54007)] proved that a Hausdorff space \(X\) is compact if and only if \(\overline{D}\) is compact for every discrete subspace \(D \subset X\). Tkachuk and other authors have proved variations of this result. There are two main topics in this paper. One topic is to generalize results of \textit{D. Burke} and \textit{V. V. Tkachuk} [Commentat. Math. Univ. Carol. 54, No. 1, 69--82 (2013; Zbl 1274.54101)] involving countably compact, completely regular, Hausdorff spaces.NEWLINENEWLINEFor example the authors prove that a regular, Hausdorff, initially \(\kappa\)-compact space \(X\) has character \(\leq\kappa\) if and only if for all discrete subspaces \(D \subset X\), \(D\) has character \(\leq\kappa\) (or even pseudo-character \(\leq\kappa\)). This generalizes a result of Burke-Tkachuk who prove the same result for initial \(\aleph_0\)-compactness (i.e., countable compactness). They extend the result of Burke-Tkachuk by proving for \(X\) a feebly compact regular Hausdorff space, \(X\) is first countable if and only if \(\overline{D}\) is first countable (or even if \(\overline{D}\) has countable pseudocharacter) for all discrete subspaces \(D \subset X\).NEWLINENEWLINEThe second topic in this paper concerns discretely reflexive properties in squares, i.e., in \(X \times X\) with the product topology. For instance they prove for a closed-hereditary property \(P\), if \(X\) is a space with countable \(\pi\)-weight, and \(\overline{D}\) has \(P\) for every discrete subspace \(D\subset X\times X\), then \(X\) has \(P\). Another result answers a problem of Burke-Tkachuk: If \(X\) is a regular space with a countable network and \(\overline{D}\) is zero-dimensional for any discrete \(D\subset X\times X\), then \(X\) is zero-dimensional. Two examples show some limitations for this kind of results. Other reflexive properties and other hypotheses on a space \(X\) are considered, and the paper ends with a list of thirteen open problems.