Computability and continuity are closely linked – in fact, continuity can be seen as computability relative to an arbitrary oracle. As such, concepts from topology and descriptive set theory feature heavily in the foundations of computable analysis. Conversely, techniques developed in computability theory can be fruitfully employed in topology and descriptive set theory, even if the desired results mention no computability at all.

In this Dagstuhl Seminar, we bring together researchers from computable analysis, from classical computability theory, from descriptive set theory, formal topology, and other relevant areas. Our goals are to identify key open questions related to this interplay, to exploit synergies between the areas and to intensify collaboration between the relevant communities.

Particular topics to be discussed include:

Quasi-Polish spaces as a common generalization of omega-continuous domains and Polish spaces. Many key results from descriptive set theory have already been extended to quasi-Polish spaces. Recently, reasonable candidates for computable quasi-Polish spaces were identified. We plan to continue the investigation of computability-aspects of quasi-Polish spaces.

The nascent synthetic descriptive set theory, where ideas from synthetic topology and category theory are linked to classical and effective descriptive set theory. The focus here is on properties of the spaces as a category rather than on their internal structures.

The connection between sigma-homeomorphism typesof topological spaces and recursion-theoretic degree structures, in particular substructures of the enumeration degrees. In recent years, this connection has enabled Kihara and Pauly to solve a long-open problem by Jayne from topological dimension theory and has sparked significant new interest in the enumeration degrees.

CoPolish spaces as introduced by Schröder are a natural dual of the quasi-Polish spaces. They commonly arise in analysis (the space of real polynomials or of analytic functions are typical examples). They are also characterized as the class of spaces in computable analysis where complexity theory takes a familiar form.

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Mathieu Hoyrup, Arno Pauly, Victor Selivanov, and Mariya I. Soskova