countably compact hereditarily normal topological group
On countably compact spaces satisfying \(wD\) hereditarily
A topological space \(X\) is said to satisfy property \(wD\) if for every infinite closed discrete subset \(C\) there exists a countably infinite discrete (hence pairwise disjoint) family \({\mathcal U}\) of open subsets of \(X\) such that for all \(U\in{\mathcal U}\), \(U\) intersects \(C\) in exactly one point. A space is said to satisfy property \(wD\) hereditarily if every subspace of \(X\) satisfies property \(wD\). Among spaces that statisfy \(wD\) hereditarily are the hereditarily normal, and countably compact first countable spaces [\textit{J. E. Vaughan}, Topology Proceedings Vol. 3, No. 1, Proc. Topol. Conf. Univ. Okla. 1978, 237-265 (1979; Zbl 0407.54013)]. The author studies the relation between tightness and hereditary \(\pi\)-character in the class of spaces that satisfy property \(wD\) hereditarily. The main result states that if \(X\) is a countably compact \(T_3\)-space that satisfies \(wD\) hereditarily, and \(A\) is a countable subset of \(X\) with \(x\in\overline A\smallsetminus A\), and \(x\) has countable \(\pi\)-character in the subspace \(\overline A\smallsetminus A\), then \(x\) has a countable \(\pi\)-base \({\mathcal B}\) in the space \(X\) such that every member of \({\mathcal B}\) meets \(A\). Another theorem states that if \(X\) is a \(T_3\)-space and \(\omega\)-bounded (i.e., every countable set is contained in a compact set) then \(X\) has countable tightness if and only if the hereditary \(\pi\)-character of \(X\) is countable. Some results are proved under the assumption of PFA: the proper forcing axiom [\textit{J. E. Baumgartner}, Handbook of set-theoretic topology, 913-959 (1984; Zbl 0556.03040)]. Assuming PFA, if \(X\) is countably compact and hereditarily normal then \(X\) has countable tightness if and only if the hereditary \(\pi\)-character of \(X\) is countable. An example of \textit{A. Hajnal} and \textit{I. Juhász} [Fundam. Math. 81, 147-158 (1974; Zbl 0274.54002)] shows that this statement is not a theorem of ZFC. The author also proves that assuming PFA every countably compact hereditarily normal topological group is metrizable.
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