The ideal generated by σ-nowhere dense sets

Abstract

In this paper, we consider the ideal I σ generated by all σ-nowhere dense sets in a topological space. Properties of this ideal and its relations with the Volterra property are explored. We show that I σ is compatible with the topology for any given topological space, an analogue to the Banach category theorem. Some applications of this result and the Banach category theorem are also given.