Set-Menger and related properties
Let \(\mathcal P\) be a collection of subsets of a space \(X\) and say that \(X\) is \(\mathcal P\)-Menger (resp. weakly \(\mathcal P\)-Menger, almost \(\mathcal P\)-Menger) if for each \(A\in\mathcal P\) and each sequence \(\langle\mathcal U_n\rangle\) of families of open subsets such that \(\overline{A}\subset\cup\mathcal U_n\) for each \(n\) there is a sequence \(\langle\mathcal V_n\rangle\) such that each \(\mathcal V_n\) is a finite subset of \(\mathcal U_n\) and \(A\subset\cup_{n\in\omega}\cup\mathcal V_n\) (resp. \(A\subset\overline{\cup_{n\in\omega}\cup\mathcal V_n}\), \(A\subset\cup_{n\in\omega}\overline{\cup\mathcal V_n}\)); if each \(\mathcal V_n\) can be taken as a singleton then the word ``Menger'' is replaced by ``Rothberger.'' When \(\mathcal P=\{X\}\) then \(\mathcal P\) may be dropped and when \(\mathcal P\) is the power set of \(X\) then \(X\) is said to be set-Menger, etc. Hurewicz and Gerlits-Nagy versions are also given but the emphasis is on set-Menger and set-Rothberger, especially their behaviour under subspaces and (pre-)images. Topological games are described and it is shown that if \(X\) is Lindelöf then \(X\) is (weakly, almost) set-Menger if and only if the first player does not have a winning strategy in the respective game.
scientific article; zbMATH DE number 7187643