On product bases
Using new nomenclature the author first summarizes results he established earlier concerning category bases (i.e., \({\mathfrak K}\)-families) (X,\({\mathcal C})\). A subfamily \({\mathcal B}\) of \({\mathcal C}\) is called a quasibase if each abundant (i.e., \({\mathcal C}_{II})\) set is abundant everywhere in at least one member of \({\mathcal B}\). The product (X\(\times Y,{\mathcal C}\times {\mathcal D})\) of two category bases need not satisfy the axioms for a category base, but when it does it is called a product base, and then if (Y,\({\mathcal D})\) has a countable quasibase the conclusion of the Kuratowski-Ulam theorem holds. Even when specialized to topological spaces this result is more general than previously known versions, since a topological space may have a countable quasibase but no countable pseudobase. Related theorems concerning local separability, product sets, Baire spaces and the Baire property are similarly generalized to product category bases. Numerous examples are discussed.
scientific article; zbMATH DE number 3919304