Topological representations

TL;DR

This paper characterizes a class of ideals with topological representations, showing their invariance under certain equivalences, and analyzes their descriptive complexities and related properties in descriptive set theory.

Contribution

It provides a complete characterization of ideals with topological representations, including invariance under Rudin–Blass equivalence and complexity classification.

Findings

Ideals are dense and countably separated, equivalent to having a topological representation.

The class is invariant under Rudin–Blass equivalence.

All such ideals have descriptive complexity at most Pi^0_3.

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\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.