A note on the Menger (Rothberger) property and topological games
Making use of several topological games, a number of conditions either equivalent to or implying either the Menger or Rothberger property are presented. As an example it is shown that a \(T_1\) space \(X\) is Menger (Rothberger) if and only if for each sequence \(\langle\phi_n\rangle\) of neighbourhood assignments for \(X\) and each \(n\) there is a finite (singleton) set \(D_n\subset X\) so that \(X=\bigcup\{\phi_n(d) \mid d\in D_n, n\in\omega\}\) and \(\bigcup_{n\in\omega} D_n\) is discrete and closed in \(X\). \(\mathbf K\)-like spaces are also considered for certain classes \(\mathbf K\). For example if \(\mathbf K\) is the class of all spaces with the Menger (Rothberger) property then a \(\mathbf K\)-like space is Menger (Rothberger).
scientific article; zbMATH DE number 6534253
scientific article