The descriptive complexity of the set of all closed zero-dimensional subsets of a Polish space

TL;DR

This paper analyzes the descriptive complexity of the set of all closed zero-dimensional subsets of a Polish space, establishing upper bounds and existence results within the projective hierarchy.

Contribution

It determines the maximum complexity class of these sets for analytic spaces and provides bounds for Borel spaces, also constructing a specific example with high complexity.

Findings

The maximum complexity class for analytic spaces is a.

For Borel spaces, the complexity exceeds certain projective pointclasses.

Existence of a Polish subspace with a set of zero-dimensional closed subsets outside standard pointclasses.

Abstract

Given a space $X$ we investigate the descriptive complexity class $\G_X$ of the set $\FF_0(X)$ of all its closed zero-dimensional subsets, viewed as a subset of the hyperspace $\FF(X)$ of all closed subsets of $X$. We prove that $\max \{ \G_X; \ X \text{ analytic } \}=\pca $ and $\sup \{ \G_X; \ X \text{ Borel } \borm \xi\} \supseteq \Game \bora \xi$ for any countable ordinal $\xi\geq1$. In particular we prove that there exists a one-dimensional Polish subpace of $2^\wo\times \R^2$ for which $\FF_0(X)$ is not in the smallest non trivial pointclass closed under complementation and the Souslin operation $\mathcal A\,$.