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
T_{0} making
it nowhere dense and that this topology is T_{1}
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 \omega_{1}.

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**Last modified April 24, 1999.**