Representations of totally bounded metric spaces and their categorical formulation

Abstract

In the framework of Type-2 Theory of Effectivity, representations of continuous spaces affect computability and computational complexity of problems drastically. We propose ``quantitative admissibility'' as a criterion for sensible representations. Quantitative admissibility is a refinement of classical admissibility notion by Kreitz and Weihrauch, 1985. Classical setting of second-countable $T_0$ spaces is concretized to totally bounded metric spaces. We show that there is a close correspondence between modulus of continuity of a function and that of its realizer when the representations are quantitatively admissible. We formulate the represented spaces as categories and show that they have all finite products.