
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 , a -ideal on and a dense countable subset of such that the ideal consists of those subsets of whose closure belongs to . It turns out that this definition is indepedent of the choice of . 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 can be always chosen to be compact so that is a -ideal of compact sets. We compute the possible descriptive complexities of such ideals and…
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