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 hiera...

Denote by C the class of oracles relative to which P = NP (collapsing oracles), and by S the class of oracles relative to which P 6= NP (separating oracles). We present structural results on C and S. Using a diagonalization argument, we show that neither C nor S is closed under disjoint union, also known as join. We show that this implies that neither C nor S is closed under union, intersection...