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Descriptive set theory was originally developed on Polish spaces. It waslater extended to $\omega$-continuous domains [Selivanov 2004] and recently toquasi-Polish spaces [de Brecht 2013]. All these spaces are countably-based.Extending descriptive set theory and its effective counterpart to generalrepresented 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 $\Sigma^0_2$-subsetof $\mathcal{O}(\mathbb{N}^\mathbb{N})$ which is not Borel. We show that theopen subsets of $\mathbb{N}^{\mathbb{N}^\mathbb{N}}$ cannot be continuouslyindexed by elements of $\mathbb{N}^\mathbb{N}$ or even$\mathbb{N}^{\mathbb{N}^\mathbb{N}}$, and more generally that the open subsetsof $\mathbb{N}\langle\alpha\rangle$ cannot be continuously indexed by elementsof $\mathbb{N}\langle\alpha\rangle$. We also derive effective versions of theseresults.  These results give answers to recent open questions on the classification ofspaces in terms of their base-complexity, introduced in [de Brecht, Schr\"oder,Selivanov 2016]. In order to obtain these results, we develop generaltechniques which are refinements of Cantor's diagonal argument involvingmulti-valued fixed-point free functions and that are interesting on their ownright.

Results in descriptive set theory on some represented spaces