Topologies with prescribed nowhere dense sets

by

Krzysztof Ciesielski & Jakub Jasinski

Topology Appl. 63 (1995), 277-298

Let X be a set and let J be an ideal on X. In this paper we show how to find a topology \tau on X such that \tau-nowhere dense (or \tau-meager) sets are exactly the sets in J. We try to find the best'' possible topology with such property. In Section 1 we discuss the ideals {\emptyset} and P(X). We also show that for every ideal J not equal to P(X) there is a topology T0 making it nowhere dense and that this topology is T1 if union of J is equal to X. Section 2 concerns principal ideals P(S) for subset S of X. It contains characterization of cardinal pairs (\kappa,\lambda)=(|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 \sigma-ideal J on R containing all singletons and such that every element of J is either null or meager, there exists a Hausdorff zero dimensional topology making J 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 at most \omega1.

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