Topological representations

Abstract

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\vcević 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 ${\mathbf{\Pi }}_3^0$. We also prove that a coanalytic ideal is an intersection of ideals of this form if and only if it is weakly selective.