I know both are closed under finite intersection and union but what about inf versions?
The recursively enumerable and recursive languages are closed under finite union and intersection but not the infinite versions--does the same thing happen with these classes?
For any $n$ the problem of deciding whether a given Turing machine halts in $n$ steps is in both $\sf P$ and $\sf NP$. But the union of such problems is that of deciding whether a Turing machine halts, which isn't even computable.
The complementary problems give a counterexample for intersections.
