Relativized topological size of sets of partial recursive functions

Abstract

In [1], a recursive topology on the set of unary partial recursive functions was introduced and recursive variants of Baire topological notions of nowhere dense and meagre sets were defined. These tools were used to measure the size of some classes of partial recursive (p.r.) functions. Thus, for example, it was proved that measured sets or complexity classes are recursively meagre in contrast with the sets of all p.r. functions or recursive functions, which are sets of recursively second Baire category. In this paper we measure the size of sets of p.r. functions using the above Baire notions relativized to the topological spaces induced by these sets. In this way we strengthen, in a uniform way, most results of [4, 5, 6, 3, 2], and we also obtain new results. For many sets of p.r. functions, strong differences between “local” and “global” topological size are established.