Results in descriptive set theory on some represented spaces

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

Topological analysis of representations

International audienceComputable analysis is the theoretical study of the abilities of algorithms to process infinite objects. The algorithms abilities depend on the way these objects are presented to them. We survey recent results on the problem of identifying the properties of objects that are decidable or semidecidable, for several concrete classes of objects and representations of them. Topology is at the core of this study, as the decidable and semidecidable properties are closely related to the open sets induced by the representation

Overt choice

We introduce and study the notion of overt choice for countably-based spaces and for CoPolish spaces. Overt choice is the task of producing a point in a closed set specified by what open sets intersect it. We show that the question of whether overt choice is continuous for a given space is related to topological completeness notions such as the Choquet-property; and to whether variants of Michael’s selection theorem hold for that space. For spaces where overt choice is discontinuous it is interesting to explore the resulting Weihrauch degrees, which in turn are related to whether or not the space is Fréchet–Urysohn

Game characterizations and lower cones in the Weihrauch degrees

We introduce a parametrized version of the Wadge game for functions and show that each lower cone in the Weihrauch degrees is characterized by such a game. These parametrized Wadge games subsume the original Wadge game, the eraser and backtrack games as well as Semmes's tree games. In particular, we propose that the lower cones in the Weihrauch degrees are the answer to Andretta's question on which classes of functions admit game characterizations. We then discuss some applications of such parametrized Wadge games. Using machinery from Weihrauch reducibility theory, we introduce games characterizing every (transfinite) level of the Baire hierarchy via an iteration of a pruning derivative on countably branching trees