The generalised Banach Category theorem
by
Julian Dontchev
University of Helsinki
Coauthors: Maximilian Ganster (Graz University of Technology, Austria)

A nonempty family I of subsets on a topological space (X, \tau) is called an ideal on X if it satisfies the following two conditions: (1) If A in I and B subset or equal A, then B in I (heredity) and (2) If A in I and B in I, then A \cup B in I (finite additivity). A \sigma-ideal on a topological space (X, \tau) is an ideal which satisfies: (3) If { A_i\colon i = 1, 2, 3, ... } subset or equal I, then \cup { A_i \colon i = 1, 2, 3, ... } in I (countable additivity). By an ideal topological space, we mean a topological space (X, \tau) with an ideal I on X and we denote it by (X, \tau, I).

The topology \tau of an ideal topological space (X, \tau, I) is compatible with the ideal I, denoted \tau ~ I, if the following condition holds for every subset A of X: if for every x in A there exists a U in \tau(x) such that U \cap A in I, then A in I. If this condition holds, then the ideal I is also sometimes said to be local relative to the topology or to have the strong Banach's localization property. For example, the \sigma-ideal of meager sets is always local and every topology is compatible with the ideal of meager subsets-this result is known as the Banach category theorem and was first proven by Banach for metric spaces and extended to general topological spaces by Kuratowski. We give a generalised version of the Banach Category theorem.

Date received: July 9, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadg-07.