On Hurewicz properties.
\textit{W. Hurewicz} in 1925 introduced several properties of a topological space \(X\) concerning sequences of real functions and open covers, e.g., Property \(H^{**}\): For any sequence \(\{f_n\}^\infty_{n= 0}\) of continuous function from \(X\) into \(\mathbb{R}\) the family of sequences in \(\mathbb{R}\) \(\{\{t_n(x)\}_{n=0}^\infty |x\in X\}\) is bounded. Property \(E^{**}\): for every sequence \(\{U_n\}^\infty_{n =0}\) of open covers of \(X\) there exist finite susets \(V_n\subseteq U_n\) such that \(\{\bigcup V_n|n\in\omega\}\) is a \(\gamma\)-cover of \(X\) is a finite cover. The authors of the present paper investigate relations between these properties and several generalizations. Their main result is the following: Consider the property of being an \(mQN\)-space, introduced by \textit{L. Bukovský}, \textit{I. Reclaw} and \textit{M. Repický} [Topology Appl. 41, 25--40 (1991; Zbl 0768.54025) and ibid. 112, 13--40 (2001; Zbl 1067.54028)]: \(X\) is called an \(mQN\)-space if every sequence \(f_n\to 0\) of contiuous real-valued functions converges quasi-normally. The authors prove: Theorem 6: Every \(mQN\)-space possesses property \(H^{**}\).
scientific article; zbMATH DE number 1961617