Monotone weak Lindelöfness
The authors call a topological space \(X\) a monotonically weakly Lindelöf space (mwL-space) if one can assign to every open cover \({\mathcal U}\) of \(X\) a collection \(r({\mathcal U})\) of open subsets of \(X\) in such a way that (i) \(\bigcup r({\mathcal U})\) is dense in \(X\), (ii) \(r({\mathcal U})\) refines \({\mathcal U}\), and (iii) \(r({\mathcal U})\) refines \(r({\mathcal V})\) whenever \({\mathcal U}\) refines \({\mathcal V}\). They prove some elementary facts concerning subspaces, products, and mappings of mwL-spaces. In particular, they show that the class of mwL-spaces is much broader than the class of monotonically Lindelöf spaces, which has been introduced by M. Matveev and studied, e.g., by \textit{H. Bennett}, \textit{D. Lutzer} and \textit{M. Matveev} in [Topology Appl. 151, No. 1--3, 180--186 (2005; Zbl 1069.54021)] and by \textit{R. Levy} and \textit{M. Matveev} in [ibid. 154, No. 11, 2333--2343 (2007; Zbl 1134.54006)]. In fact, every topological space with a countable \(\pi\)-base is mwL, hence every separable first countable space is mwL. If \(X\) is second countable, then the Alexandroff duplicate \(A(X)\) of \(X\) is mwL. All \(\Psi\)-like spaces are mwL. If \(\kappa<{\mathfrak c}\), then the one-point compactification of the discrete space of cardinality \(\kappa\) is mwL. Every finite or countably infinite power of the Sorgenfrey line is mwL. \(C_p(X)\) is mwL if and only if \(X\) is countable. Every Tychonoff space of weight \(\leq{\mathfrak c}\) can be embedded in a Tychonoff mwL-space as a closed subspace. However, if \(\kappa>\omega\), then \(2^\kappa\) is not mwL.
scientific article; zbMATH DE number 5997904
scientific article