On product bases (Q1064559)

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.