Topologies making a given ideal nowhere dense or meager

Abstract

Let X be a set and let I be an ideal on X. In this paper we show how to find a topology τ on X such that τ-nowhere dense (or τ-meager) sets are exactly the sets in I We try to find the “best” possible topology with such property. In Section 1 we discuss the ideals {∅} and P ( X). We also show that for every ideal I ≠ P ( X) there is a topology T 0 making it nowhere dense and that this topology is T 1 if ∪ I = X. Section 2 concerns principal ideals P ( S) for S ⊂ X. It contains characterization of cardinal pairs ( κ, λ) = (| S|, | X⧹ S|) for which P ( S) can be made nowhere dense or meager by compact Hausdorff, metric, and complete metric topologies. Section 3 deals with the ideals containing all singletons. We prove there that it is consistent with ZFC + CH that for every σ-ideal I on R containing all singletons and such that every element of I is either null or meager, there exists a Hausdorff zero-dimensional topology making I nowhere dense. Section 4 contains the discussion of the above theorem. In particular, it is noticed there that the theorem follows from CH for the ideals with the cofinality ⩽ ω 1.