A measurable cardinal associated with an epireflective subcategory of Hausdorff spaces
The authors associate with every non-trivial epireflective subcategory \(\mathbf{A}\) of the category \textbf{Haus} of Hausdorff spaces and continuous maps the cardinal \(\sigma (\mathbf{A}) = \sup \{ | X| \;\mid X\) is a discrete space belonging to \(\mathbf{A}\}\), if the latter set is bounded, and \(\sigma (\mathbf{A})= \infty\) otherwise. Generalizing known results about \(k\)-compact and topologically complete spaces, they prove in particular that (1) \(\sigma (\mathbf{A})\) is non-measurable or equals \(\infty\). (2) If \(k\) is non-measurable or equals \(\infty\) and \(\mathbf{A}\) is the epireflective hull of all discrete spaces \(X\) with \(| X| < k\), then \(\sigma (\mathbf{A}) = k\). (3) A discrete space \(X\) belongs to \(\mathbf{A}\) iff \(| X| < \sigma (\mathbf{A})\). (4) \(\mathbf{A}\) is closed under the formation of topological sums ( = coproducts with at most \(k\) summands) iff \(k < \sigma (\mathbf{A})\).
Epireflective subcategory of Hausdorff spaces
scientific article; zbMATH DE number 2142288
scientific article