Variations on \(\omega\)-boundedness
Let \({\mathcal P}\) be a class of topological spaces. The authors call a Tychonoff space \(X\) \({\mathcal P}\)-bounded if every subspace of \(X\) belonging to \({\mathcal P}\) has a compact closure. For \({\mathcal P}=\) \{countable spaces\}, \({\mathcal P}\)-boundedness coincides with \(\omega\)-boundedness as introduced by \textit{S. L. Gulden}, \textit{W. F. Fleischman} and \textit{J. H. Weston} [Proc. Am. Math. Sec. 24, 197--203 (1970; Zbl 0203.55104)]. For \({\mathcal P}=\) \{\(\sigma\)-compact spaces\}, \({\mathcal P}\)-boundedness is identical with strong \(\omega\)-boundedness as introduced by P. Nyikos. The authors prefer to call these spaces \(\sigma\)C-bounded. Similarly, if \({\mathcal P}=\) \{weakly Lindelöf-spaces\} (respectively, = \{Lindelöf-spaces\}), then the \({\mathcal P}\)-bounded spaces are called wL-bounded (respectively, L-bounded). Obviously, every wL-bounded space is L-bounded, every L-bounded space is \(\sigma\)C-bounded, and every \(\sigma\)C-bounded space is \(\omega\)-bounded. In response to a question of \textit{P. Nyikos}, \textit{L. F. Aurichi} has constructed a locally compact and first countable \(\omega\)-bounded space that is not \(\sigma\)C-bounded in [Topology Appl. 156, No. 4, 775--782 (2009, Zbl 1162.54017)]. In this paper, the authors provide an example of a space that is \(\sigma\)C-bounded but not L-bounded. Whether there exists an L-bounded space that is not wL-bounded remains an open question. However, it is shown that, for locally compact spaces, wL-boundedness, L-boundedness, and \(\sigma\)C-boundedness are equivalent. For \({\mathcal P}=\) \{spaces satisfying the countable chain condition\}, \({\mathcal P}\)-bounded spaces are called ccc-bounded, and for \({\mathcal P}=\) \{hereditarily Lindelöf spaces\}, \({\mathcal P}\)-bounded spaces are called HL-bounded. Obviously, every wL-bounded space is ccc-bounded, every ccc-bounded space is HL-bounded, and every HL-bounded space is \(\omega\)-bounded. It is shown that there exists an \(\omega\)-bounded space that is not HL-bounded. Assuming CH, there even exists a normal, locally compact, and first countable \(\omega\)-bounded space that is not HL-bounded. On the other hand, \(\text{MA}_{\aleph_1}\) implies that HL-boundedness and w-boundedness coincide for first countable spaces. And \(\text{MA}_{\aleph_1}\) also implies that ccc-boundedness, HL-boundedness, and \(\omega\)-boundedness are equivalent for spaces that are locally compact and first countable. In ZFC alone, a locally compact HL-bounded space is constructed that is not ccc-bounded. However, CH implies that ccc-boundedness and HL-boundedness are identical for spaces of character at most \(\omega_1\). Additionally, a ZFC example is given of a zero-dimensional, normal, locally compact, and first countable ccc-bounded space that is not \(\sigma\)C-bounded, hence not wL-bounded. Almost every proof in this interesting paper uses nontrivial results from various areas of set-theoretic topology.
scientific article; zbMATH DE number 6176740
scientific article