If not Empty, NP - P is Topologically Large

In the classical Cantor topology or in the superset topology. NP and. consequently, classes included in NP are mccr~~rr. However, in a natural combination of the two topologies. we prove that NP-P, if not empty. is a swontl coregor~a class, while NP-complete sets form a first cutegory 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 arc all second category (if not empty). It is shown that if C is any of the above second category classes. then for all BENP there exists an AEC such that A is arbitrarily close to B infinitely often.