Overt choice

TL;DR

This paper investigates the concept of overt choice in certain topological spaces, linking its continuity to properties like the Choquet property and Michael's selection theorem, and exploring its computational complexity.

Contribution

It introduces overt choice for countably-based and CoPolish spaces, establishing connections with topological completeness and selection theorems, and analyzing Weihrauch degrees.

Findings

Overt choice's continuity relates to the Choquet property.

Discontinuity of overt choice correlates with Weihrauch degrees.

The study links overt choice properties to Frechet-Urysohn spaces.

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 Frechet-Urysohn.