I'm trying to prove that the cone of copositive matrices is closed and in S. Boyd, L. Vandenberghe's Convex Optimization it says that:
K has nonempty interior, because it includes the cone of positive semidefinite matrices, which has nonempty interior.
I can't make sense of this. Since the definition of a copositive matrix is
$$ x^TAx \geq0,\,\forall x \geq 0, $$
but for a positive semidefinite matrix all $x$ would be considered. It seems to me that positive semidefnite matrices are a more "general" concept and a less strict constraint than copositivity.
Am I misunderstanding the relationship between cones and sets?