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