Overt choice

Issue title: Dagstuhl Seminar on Measuring the Complexity of Computational Content 2018

Guest editors: Vasco Brattka, Damir D. Dzhafarov, Alberto Marcone and Arno Pauly

Article type: Research Article

Authors: de Brecht, Matthew^{a; *} | Pauly, Arno^b | Schröder, Matthias^c

Affiliations: [a] Graduate School of Human and Environmental Studies, Kyoto University, Japan. matthew@i.h.kyoto-u.ac.jp | [b] Department of Computer Science, Swansea University, UK. Arno.M.Pauly@gmail.com | [c] Department of Mathematics, Technical University Darmstadt, Germany. Matthias.Schroeder@cca-net.de

Correspondence: [*] Corresponding author. matthew@i.h.kyoto-u.ac.jp.

Abstract: 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.

Keywords: Weihrauch reducibility, computable topology, quasi-Polish space

DOI: 10.3233/COM-190253

Journal: Computability, vol. 9, no. 3-4, pp. 169-191, 2020