Classical and Effective Descriptive Complexities of omega-Powers

TL;DR

This paper establishes the existence of omega-powers with complex topological classifications across various levels of the arithmetical hierarchy, extending previous results with effective versions and closure properties.

Contribution

It introduces effective versions of topological complexity results for omega-powers and proves the existence of recursive languages with omega-powers of specified complexity.

Findings

Existence of Sigma^0_alpha-complete omega-powers for each non null countable ordinal alpha.

Existence of Pi^0_alpha-complete omega-powers for each non null countable ordinal alpha.

Effective versions of topological complexity results for omega-powers and closure properties in the hyperarithmetical hierarchy.

Abstract

We prove that, for each non null countable ordinal alpha, there exist some Sigma^0_alpha-complete omega-powers, and some Pi^0_alpha-complete omega-powers, extending previous works on the topological complexity of omega-powers. We prove effective versions of these results. In particular, for each non null recursive ordinal alpha, there exists a recursive finitary language A such that A^omega is Sigma^0_alpha-complete (respectively, Pi^0_alpha-complete). To do this, we prove effective versions of a result by Kuratowski, describing a Borel set as the range of a closed subset of the Baire space by a continuous bijection. This leads us to prove closure properties for the classes Effective-Pi^0_alpha and Effective-Sigma^0_alpha of the hyperarithmetical hierarchy in arbitrary recursively presented Polish spaces. We apply our existence results to get better computations of the topological complexity of some sets of dictionaries considered by the second author in [Omega-Powers and Descriptive Set Theory, Journal of Symbolic Logic, Volume 70 (4), 2005, p. 1210-1232].