Results in descriptive set theory on some represented spaces
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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〈α〉. 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〈α〉 cannot be continuously indexed by elements of N〈α〉. 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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