On linearly H-closed spaces (Q1738934)

In this work the authors expand the theory of linearly $H$-closed spaces, a class of spaces introduced in [\textit{M. Baillif}, Topol. Proc. 54, 109--124 (2019; Zbl 1421.54015)] and studied further in [\textit{A. Bella}, Topology Appl. 228, 355--362 (2017; Zbl 1375.54002)]. A space is defined to be linearly $H$-closed if in every open chain cover of the space there is a dense element. It was shown in [Baillif, loc. cit.] that an $H$-closed space is linearly $H$-closed and that a linearly $H$-closed space is feebly compact (equivalent to pseudocompact in Tychonoff spaces). It is shown in the present paper that a space $X$ is linearly $H$-closed if and only if there is no free open filter on $X$ with a nested base. A corollary is Corollary 2.4 of [loc. cit.]: A ccc feebly compact space is linearly $H$-closed. Several other characterizations of linearly $H$-closed are given. It is also shown that a monotonically normal, linearly $H$-closed space is compact. This is analogous to Theorem 3.17 in [\textit{I. Juhász} et al., Stud. Sci. Math. Hung. 54, No. 4, 523--535 (2017; Zbl 1413.54082)] that a monotonically normal weakly linearly Lindelöf space is Lindelöf. Spaces in which all closed subspaces are linearly $H$-closed, equivalent to the authors' definition of weakly discretely complete, are studied, as well as their relationship to the property of discretely complete, defined in [\textit{O. T. Alas} and \textit{R. G. Wilson}, Appl. Gen. Topol. 8, No. 2, 273--281 (2007; Zbl 1202.54005)]. It is shown that if $X$ is a regular sequential space in which all closed subspaces are linearly $H$-closed and $\chi(X)\leq\mathfrak{c}$, then $|X|\leq\mathfrak{c}$. It is asked whether every weakly discretely complete space is discretely complete. Maximal and $T$-maximal linearly $H$-closed spaces are considered and it is shown, for example, that a LOTS is $T$-maximal linearly $H$-closed if and only if it is compact. Products are also considered, and the following are proved: a) A product of an $H$-closed space with a linearly $H$-closed space is linearly $H$-closed, and b) if $X$ is a countable product of spaces, feebly compact, and every finite subproduct is linearly $H$-closed, then $X$ is linearly $H$-closed. Finally, an example is given of a pseudocompact product of two linearly $H$-closed spaces that is not linearly $H$-closed.