  This paper studies the combinatorics of ideals which recently appeared in
ergodicity results for analytic equivalence relations. The ideals have the
following topological representation. There is a separable metrizable space
$X$, a $\sigma$-ideal $I$ on $X$ and a dense countable subset $D$ of $X$ such
that the ideal consists of those subsets of $D$ whose closure belongs to $I$.
It turns out that this definition is indepedent of the choice of $D$. We show
that an ideal is of this form if and only if it is dense and countably
separated. The latter is a variation of a notion introduced by Todor\vcevi\'c
for gaps. As a corollary, we get that this class is invariant under the
Rudin--Blass equivalence. This also implies that the space $X$ can be always
chosen to be compact so that $I$ is a $\sigma$-ideal of compact sets. We
compute the possible descriptive complexities of such ideals and conclude that
all analytic equivalence relations induced by such ideals are
$\mathbf{\Pi}^0_3$. We also prove that a coanalytic ideal is an intersection of
ideals of this form if and only if it is weakly selective.
