Notes on overt choice

Abstract

Overt choice was recently introduced and thoroughly studied by de Brecht, Pauly and Schröder. They give estimates on the Weihrauch degree of overt choice on various spaces, and relate it to the topological properties of the space. In this article, we pursue this line of research, answering some of the questions that were left open. We show that overt choice on the rationals is not limit-computable. We identify the Weihrauch degree of overt choice on the space of natural numbers with the co-finite topology. We prove that the quasi-Polish spaces are the countably-based  T 0 -spaces on which a variant of overt choice, called  Π ~ 2 0 overt choice, is continuous. It extends a previous result that holds in the class of  T 1 -spaces. We also prove an effective version of this equivalence.