Relativized topological size of sets of partial recursive functions

AbstractIn [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

If not empty, NP — P is topologically large

AbstractIn the classical Cantor topology or in the superset topology, NP and, consequently, classes included in NP are meagre. However, in a natural combination of the two topologies, we prove that NP — P, if not empty, is a second category class, while NP-complete sets form a first category class. These results are extended to different levels in the polynomial hierarchy and to the low and high hierarchies. P-immune sets in NP, NP-simple sets, P-bi-immune sets and NP-effectively simple sets are all second category (if not empty). It is shown that if C is any of the above second category classes, then for all B∈NP there exists an A∈C such that A is arbitrarily close to B infinitely often