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 are a separable metrizable space X, a σ-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 independent 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 Todorčević 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 σ-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 Π30. We also prove that a coanalytic ideal is an intersection of ideals of this form if and only if it is weakly selective.