拓扑表示

摘要：

本文研究了最近出现在解析等价关系遍历性结果中的理想组合学。理想具有以下拓扑表示。存在可分可度量化空间$X$，$X$上的$σ$-理想$I$和$X$的稠密可数子集$D$，使得理想由$D$的那些子集组成，这些子集的闭包属于$I$。事实证明，该定义与$D$的选择无关。我们证明了理想是这种形式的当且仅当它是稠密的和可数分离的。后者是Todor \vcevi ć引入的间隙概念的变体。作为推论，我们得到这个类在Rudin-Blass等价下是不变的.这也意味着空间$X$总是可以被选择为紧的，使得$I$是紧集的$σ$-理想。我们计算了这种理想的可能的描述复杂性，并得出结论，所有的解析等价关系诱导的理想是$\mathbfΠ^0_3$。我们还证明了一个共解析理想是这种形式的理想的交当且仅当它是弱选择的。

英文摘要：

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 $σ$-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 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 $σ$-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 $\mathbfΠ^0_3$. We also prove that a coanalytic ideal is an intersection of ideals of this form if and only if it is weakly selective.

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