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## Results in descriptive set theory on some represented spaces

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1
Mathieu Hoyrup

#### Abstract

Descriptive set theory was originally developed on Polish spaces. It was later extended to ω-continuous domains [Selivanov 2004] and recently to quasi-Polish spaces [de Brecht 2013]. All these spaces are countably-based. Extending descriptive set theory and its effective counterpart to general represented spaces, including non-countably-based spaces has been started in [Pauly, de Brecht 2015]. We study the spaces $O(N^N)$, $C(N^N, 2)$ and the Kleene-Kreisel spaces $N\langle α\rangle$. We show that there is a $Σ^0_2$-subset of $O(N^N)$ which is not Borel. We show that the open subsets of $N^{N^N}$ cannot be continuously indexed by elements of $N^N$ or even $N^{N^N}$, and more generally that the open subsets of $N\langle α\rangle$ cannot be continuously indexed by elements of $N\langle α\rangle$. We also derive effective versions of these results. These results give answers to recent open questions on the classification of spaces in terms of their base-complexity, introduced in [de Brecht, Schröder, Selivanov 2016]. In order to obtain these results, we develop general techniques which are refinements of Cantor's diagonal argument involving multi-valued fixed-point free functions and that are interesting on their own right.
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### Dates and versions

hal-01657883 , version 1 (07-12-2017)

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• HAL Id : hal-01657883 , version 1

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Mathieu Hoyrup. Results in descriptive set theory on some represented spaces. 2018. ⟨hal-01657883⟩

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