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# memo_has_moved_text();Classes of Polish spaces under effective Borel isomorphism

Publication: Memoirs of the American Mathematical Society
Publication Year: 2016; Volume 240, Number 1135
ISBNs: 978-1-4704-1563-1 (print); 978-1-4704-2822-8 (online)
DOI: http://dx.doi.org/10.1090/memo/1135
Published electronically: October 9, 2015
Keywords:Recursively presented metric space, effective Borel isomorphism, $Δ^{1}_1$ isomorphism, $Δ^{1}_1$ injection, Kleene space, Spector-Gandy space.

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Chapters

• Preface
• Chapter 1. Introduction
• Chapter 2. The spaces $\mathcal {N}^T$
• Chapter 3. Kleene spaces
• Chapter 4. Characterizations of $\mathcal {N}$ up to $\Delta ^1_1$ isomorphism
• Chapter 5. Spector-Gandy spaces
• Chapter 6. Questions and related results

### Abstract

We study the equivalence classes under $Δ^{1}_{1}$ 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 $Δ^{1}_{1}$-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 $Δ^{1}_{1}$-isomorphic to some $\mathcal N^{T}$ for a recursive $T$, so that the preceding spaces are representatives for the classes of $Δ^{1}_{1}$ 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 $Δ^{1}_{1}$ isomorphism.