Results in descriptive set theory on some represented spaces

Abstract

Descriptive set theory was originally developed on Polish spaces. It was later extended to $\omega$-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 $\mathcal{O}(\mathbb{N}^\mathbb{N})$, $\mathcal{C}(\mathbb{N}^\mathbb{N},2)$ and the Kleene-Kreisel spaces $\mathbb{N}\langle\alpha\rangle$. We show that there is a $\Sigma^0_2$-subset of $\mathcal{O}(\mathbb{N}^\mathbb{N})$ which is not Borel. We show that the open subsets of $\mathbb{N}^{\mathbb{N}^\mathbb{N}}$ cannot be continuously indexed by elements of $\mathbb{N}^\mathbb{N}$ or even $\mathbb{N}^{\mathbb{N}^\mathbb{N}}$, and more generally that the open subsets of $\mathbb{N}\langle\alpha\rangle$ cannot be continuously indexed by elements of $\mathbb{N}\langle\alpha\rangle$. 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, Schr\"oder, 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.