A note on quasi-Menger and similar spaces
Let \(\mathcal{A}\) and \(\mathcal{B}\) be families of sets, with \(\emptyset \notin \mathcal{A}\). The \textit{selection principles} \(S_1(\mathcal{A},\mathcal{B})\) and \(S_{\mathrm{fin}}(\mathcal{A},\mathcal{B})\) are stated as follows:NEWLINENEWLINE\(S_1(\mathcal{A},\mathcal{B})\) declares that for every sequence \(\langle \mathcal{A}_n: n < \omega \rangle\) of elements of \(\mathcal{A}\) there is a sequence \(\langle B_n: n < \omega\rangle\) such that \(B_n \in \mathcal{A}_n\) for all \(n < \omega\) and \(\{B_n: n < \omega\} \in \mathcal{B}\).NEWLINENEWLINE\(S_{\mathrm{fin}}(\mathcal{A},\mathcal{B})\) declares that for every sequence \(\langle \mathcal{A}_n: n < \omega \rangle\) of elements of \(\mathcal{A}\) there is a sequence \(\langle \mathcal{B}_n: n < \omega\rangle\) such that \(\mathcal{B}_n\) is a finite subset of \(\mathcal{A}_n\) for all \(n < \omega\) and \(\bigcup\{\mathcal{B}_n: n < \omega\} \in \mathcal{B}\).NEWLINENEWLINEThese selection principles are related to classical covering properties of Rothberger, Menger and Hurewicz. Let \(\mathcal{O}_X\) denote the family of all open covers of a topological space \(X\). \(X\) is said to be a \textit{Rothberger space} if \(S_1(\mathcal{O}_X,\mathcal{O}_X)\) holds, and it is said to be a \textit{Menger space} if \(S_{\mathrm{fin}}(\mathcal{O}_X,\mathcal{O}_X)\) holds.NEWLINENEWLINEIn the paper under review, the authors introduce a number of weaker forms of these classical covering properties.NEWLINENEWLINELet \(\mathcal{\overline{O}}\) be the family of all collections of open sets \(\mathcal{U}\) such that \(\{\overline{U}: U \in \mathcal{U}\}\) covers \(X\), and let \(\mathcal{O}_D\) be the family of all collections of open sets \(\mathcal{U}\) such that \(\bigcup \mathcal{U}\) is a dense subset of \(X\).NEWLINENEWLINE\(X\) is \textit{almost Rothberger (almost Menger)} if \(S_1(\mathcal{O}_X, \mathcal{\overline{O}})\) (respectively, \(S_{\mathrm{fin}}(\mathcal{O}_X, \mathcal{\overline{O}})\)) holds. \(X\) is \textit{weakly Rothberger (weakly Menger)} if \(S_1(\mathcal{O}_X, \mathcal{O}_D)\) (respectively, \(S_{\mathrm{fin}}(\mathcal{O}_X, \mathcal{O}_D)\)) holds.NEWLINENEWLINE\(X\) is \textit{weakly Hurewicz} if for every sequence \(\langle \mathcal{U}_n: n < \omega\rangle\) of open covers of \(X\) there is a sequence \(\langle \mathcal{V}_n: n < \omega\rangle\) such that \(\mathcal{V}_n\) is a finite subset of \(\mathcal{U}_n\) for all \(n < \omega\) and every point of \(X\) belongs to \(\overline{\bigcup \mathcal{V}_n}\) for all but finitely many of \(n < \omega\).NEWLINENEWLINE\(X\) is \textit{quasi-Rothberger} if for every closed set \(F \subseteq X\) and for every sequence \(\langle \mathcal{U}_n: n < \omega\rangle\) of covers of \(F\) by open sets in \(X\) there is a sequence \(\langle U_n: n < \omega\rangle\) such that \(U_n \in \mathcal{U}_n\) for all \(n < \omega\) and \(F \subseteq \overline{\bigcup\limits_{n < \omega} U_n}\).NEWLINENEWLINE\(X\) is \textit{quasi-Menger} if for every closed set \(F \subseteq X\) and for every sequence \(\langle \mathcal{U}_n: n < \omega\rangle\) of covers of \(F\) by open sets in \(X\) there is a sequence \(\langle \mathcal{V}_n: n < \omega\rangle\) such that \(\mathcal{V}_n\) is a finite subset of \(\mathcal{U}_n\) for all \(n < \omega\) and \(F \subseteq \overline{\bigcup\limits_{n < \omega} \bigcup \mathcal{V}_n}\).NEWLINENEWLINEThe main results of the paper are the following:NEWLINENEWLINE\noindent (i) Every hereditarily separable space is quasi-Rothberger.NEWLINENEWLINE\noindent (ii) If a Lindelöf space is the union of less than \(\mathfrak{d}\) \(H\)-closed subspaces, then \(X\) is almost Menger.NEWLINENEWLINE\noindent (iii) If a Lindelöf space is the union of less than \(\mathfrak{b}\) \(H\)-closed subspaces, then \(X\) is weakly Hurewicz.NEWLINENEWLINERecall that a topological space \(X\) is said to be \textit{\(H\)-closed} if every open cover of \(X\) contains a finite subfamily whose union is dense in \(X\); restricted to the class of Hausdorff spaces, \(H\)-spaces are precisely those which are closed in every Hausdorff space containing them.NEWLINENEWLINE\noindent \((iv)\) A number of examples are presented (in order to show that these weaker forms are different from each other), e.g.\,: there are quasi-Rothberger spaces which are not Rothberger; there is a weakly Menger space wich is neither almost Menger nor quasi-Menger; there is an almost Rothberger space which is not quasi-Rothberger.
scientific article; zbMATH DE number 6361147
scientific article