On discrete reflexivity of Lindelöf degree and pseudocharacter (Q2049882)

Given a topological property \(\mathscr{P}\), a space \(X\) is called \textit{discretely} \(\mathscr{P}\) if the closure of every discrete subspace of \(X\) has \(\mathscr{P}\). A property \(\mathscr{Q}\) is \textit{discretely reflexive} in the class \(\mathscr{A}\) if a space \(X\) from the class \(\mathscr{A}\) has \(\mathscr{Q}\) if and only if the closure of every discrete subspace of \(X\) has \(\mathscr{Q}\). In this well-written paper the authors establish several results concerning discretely \(\kappa\)-Lindelöf spaces, tightness, pseudocharacter, and calibers. The main results include: \begin{itemize} \item[1.] If \(\kappa\) is an infinite cardinal and \(X\) is a Hausdorff discretely \(\kappa\)-Lindelöf space such that \(t(X)\leq\kappa\), then \(X\) is \(\kappa\)-Lindelöf. (This was previously proved in [\textit{A. V. Arhangel'skii} and \textit{R. Z. Buzyakova}, Proc. Am. Math. Soc. 127, No. 8, 2449--2458 (1999; Zbl 0930.54003)] for Tychonoff spaces). \item[2.] Let \(\kappa\) be an infinite cardinal and \(X\) a Hausdorff space such that \(L(X)\leq\kappa\) and \(t(X)\leq\kappa\). If \(\psi(\overline{D})\leq\kappa\) for any discrete set \(D\subseteq X\), then \(\psi(X)\leq\kappa\). (However, it is an open question whether countable pseudocharacter must be discretely reflexive in Lindelöf spaces). \item[3.] In Lindelöf \(\Sigma\)-spaces countable pseudocharacter is discretely reflexive under the hypothesis \(\mathfrak{c}<\omega_\omega\). \item[4.] If \(X\) is a Tychonoff space and \(\psi(\overline{D})\leq\kappa\) for any discrete set \(D\subseteq C_p(X)\), then \(\kappa^+\) is a caliber of \(X\). \item[5.] The property of having pseudocharacter \(\leq\kappa\) is discretely reflexive in \(C_p(X)\) whenever \(X\) is a compact space with \(t(X)\leq\kappa\). \end{itemize} A systematic study of discrete reflexivity can be found in [\textit{O. T. Alas} et al., Topol. Proc. 25(Spring), 27--44 (2000; Zbl 1002.54021)] where it was established that initial \(\kappa\)-compactness, hereditary Lindelöf number, and sequential compactness are discretely reflexive for all spaces. It was also proved in [\textit{D. Burke} and \textit{V. V. Tkachuk}, Acta Math. Hung. 139, No. 1--2, 120--133 (2013; Zbl 1299.54070)] that first countability, countable tightness, and the Fréchet-Urysohn property are discretely reflexive in countably compact spaces of weight \(\leq\omega_1\). Tkachuk and Wilson showed in [\textit{V. V. Tkachuk} and \textit{R. G. Wilson}, Glas. Mat., III. Ser. 49, No. 2, 433--446 (2014; Zbl 1310.54020)] that paracompactness and the Lindelöf property are both discretely reflexive in GO spaces. The paper concludes with a series of 11 open questions. The authors state that the most intriguing open question is whether countable pseudocharacter is discretely reflexive in the class of all Tychonoff spaces.