Structural properties of oracle classes

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≠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, or symmetric difference. Consequently EL1, the first level of the extended low hierarchy, is not closed under join.