I'm trying to prove that the cone of copositive matrices is closed and in Boyd & 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
$$ x^T A x \geq 0,\quad \forall x \geq 0 $$
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?