Separation properties in algebraic categories of topologies spaces
Concrete categories C over Set are called quasivarietal or algebraic if they are concretely equivalent to a quasivariety of universal algebras with not necessarily finitary operations [cf. \textit{H. Herrlich} and \textit{G. E. Strecker}, Category theory, 2nd ed. (1979; Zbl 0437.18001)]. A well-known example is the category \(C=Comp_ 2\) of compact Hausdorff spaces which is even varietal. Moreover, \(Comp_ 2\) can be characterized as the only nontrivial, full, isomorphism-closed, productive, and closed hereditary subcategory of all Hausdorff spaces which is varietal [cf. \textit{H. Herrlich} and \textit{G. E. Strecker}, Gen. Topology Appl. 1, 283- 287 (1971; Zbl 0231.18007) and the book cited above], or as the greatest algebraic one [the author, Lect. Notes Math. 962, 263-271 (1982; Zbl 0508.54005)]. The Hausdorff axiom is essential for the proofs. But is it really independent from the others? We state that Hausdorffness can be obtained from some of the other axioms in the varietal case. But even in the weaker algebraic case there are pleasant separation properties available, for instance \(T_ 1\) for all spaces in C and \(T_ 2\) for first countable spaces. It remains as an open question whether \(Comp_ 2\) is maximal algebraic with respect to all spaces.
scientific article; zbMATH DE number 3919312