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 $mathbb{N}langlealpharangle$. 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 $mathbb{N}langlealpharangle$ cannot be continuously indexed by elements of $mathbb{N}langlealpharangle$. We also derive effecti
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