A note on pseudocompactifications
Recall that a space \(X\): is feebly compact if every locally finite family of open subsets of \(X\) is finite; has the discrete finite chain condition (is DFCC) if every discrete collection of nonempty open sets of \(X\) is finite; and is pseudocompact if every continuous real-valued function defined on \(X\) is bounded. It is known that in a regular space the first two of these properties are equivalent, and in a completely regular space all of them are equivalent. In this article the author shows that there exists a countable space \(V_\omega\) having a single nonisolated point which has no regular feebly compact extension (and therefore no Tikhonov pseudocompactification) with countable tightness. He notes that \(V_\omega\) is from [\textit{A. V. Arkhangel'skij}, Usp. Mat. Nauk 33, No. 6(204), 29-84 (1978; Zbl 0414.54002)] and belongs to the class of spaces studied in [\textit{M. G. Bell}, Topology Appl. 21, 159-166 (1985; Zbl 0568.54019)]. The author then asks if every Tikhonov space with countable tightness can be embedded (as a closed subspace) in a pseudocompact Tikhonov space with countable tightness. He concludes by giving a proof of the following: For every Hausdorff space \(X\) there exists a Hausdorff feebly compact extension \(Y\) of \(X\) such that \(X\) is open in \(Y\), \(|Y |\leq |X |+ w(X)^\omega\), and \(\chi (x,Y) = \omega\) for every \(x \in Y \smallsetminus X\) (and therefore \(\chi (Y) = \chi (X)\) and \(t(Y) = t(X))\). For additional related results in this area, see [\textit{I. J. Tree}, ibid. 52, No. 3, 267-278 (1993; Zbl 0793.54018)] and its review.
scientific article; zbMATH DE number 792935