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.