First, in this discussion, I am only considering real matrices. Second, I have a few questions I am ruminating on related to symmetric matrices. Some of these questions I need someone to say my logic is correct while others I need you to help provide the logic.
I am curious as to what the closure of the set of positive definite matrices is. I know that this set is an open cone. Just thinking about it seems to point to the closure being the positive semi-definite matrices. However, I cannot think of a way to prove this. Also, is the closure in the set of symmetric matrices, the same as the closure in the set of all matrices. I believe this to be the same as any sequence of symmetric matrices should be a symmetric matrix.
Also, is every symmetric matrix, either positive definite, negative definite or in the boundary of both? Again this seems somewhat logical but I don't know why.
Finally, if the closure of the positive definite matrices is the positive semi-definite ones, do all the matrices in the boundary have determinant zero? This seems true as I believe these matrices have at least one eigenvalue as $0$ implying $0=det(A-\lambda I)= det (A-0I) =det (A)$. If I was wrong about the closure is there any way to characterize the matrices in the boundary?