Recursive Polish spaces

Abstract

This paper is concerned with the proper way to effectivize the notion of a Polish space. A theorem is proved that shows the recursive Polish space structure is not found in the effectively open subsets of a space \({\mathcal {X}}\), and we explore strong evidence that the effective structure is instead captured by the effectively open subsets of the product space \(\mathbb {N}\times {\mathcal {X}}\).