# Classes of Polish spaces under effective Borel isomorphism

## TL;DR

This paper explores the complex structure of effective Borel isomorphism classes among complete separable metric spaces with recursive presentations, revealing intricate hierarchies and classifications unlike the non-effective case.

## Contribution

It introduces a new framework for classifying Polish spaces under effective Borel isomorphism, including the construction of infinite hierarchies and the analysis of specific space categories.

## Key findings

- Existence of infinite increasing and decreasing sequences of classes
- Construction of infinite antichains under $ ext{Δ}^1_1$-reduction
- Characterization of Baire space up to $ ext{Δ}^1_1$-isomorphism

## Abstract

We study the equivalence classes under $\Delta^1_1$ isomorphism, otherwise effective-Borel isomorphism, between complete separable metric spaces which admit a recursive presentation and we show the existence of strictly increasing and strictly decreasing sequences as well as of infinite antichains under the natural notion of $\Delta^1_1$-reduction, as opposed to the non-effective case, where only two such classes exist, the one of the Baire space and the one of the naturals. A key tool for our study is a mapping $T \mapsto \mathcal{N}^T$ from the space of all trees on the naturals to the class of Polish spaces, for which every recursively presented space is $\Delta^1_1$-isomorphic to some $\mathcal{N}^T$ for a recursive $T$, so that the preceding spaces are representatives for the classes of $\Delta^1_1$-isomorphism. We isolate two large categories of spaces of the type $\mathcal{N}^T$, the Kleene spaces and the Spector-Gandy spaces and we study them extensively. Moreover we give results about hyperdegrees in the latter spaces and characterizations of the Baire space up to $\Delta^1_1$-isomorphism.