Many important (non-deterministic) complexity classes like NP are believed not to be closed under complement. But have any of them been proven not to be?
I'm sure one could construct some contrived example, but is there any "natural" complexity class that has this property, by which I mean one that has been studied by theoretical computer scientist for reasons other than its lack of closure under complement?
I know that RE has been proven not to be closed under complement, but I don't know of any examples from complexity theory rather than computability theory - i.e. any classes defined in terms of some kind of resource-constrained Turing machine.