On product bases
type: Item
instance of: scholarly article
title: On product bases
zbMATH Open document ID: 0575.54002
published in: Pacific Journal of Mathematics
publication date: 1982-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 54A05
Mathematics Subject Classification ID: 54E52
zbMATH DE Number: 3919304
zbMATH Keywords: product base
zbMATH Keywords: countable quasibase
zbMATH Keywords: local separability
zbMATH Keywords: Baire spaces
zbMATH Keywords: Baire property
zbMATH Keywords: product category bases
author: John C. II. Morgan
MaRDI profile type: Publication
full work available at URL: pjm.1982.99.105
OpenAlex ID: W1963597222
DOI: 10.2140/PJM.1982.99.105
description: scientific article; zbMATH DE number 3919304