Is the set of diagonal matrices with positive entries open in the set of positive definite symmetric matrices? I suspect that it's not, but would like to know a proof for why the set of diagonal matrices with positive entries is or isn't open in the set of positive definite symmetric matrices. 
I am familiar with what it means to being open as in for any point in the subset there exists a small ball around that point also in that subset, but I don't know how this translates to sets of matrices.

Let $A$ be diagonal with positive diagonal entries, choose as $B$ any positive definite matrix that is not diagonal and consider $A_n := A + \frac B n$. Then each of the $A_n$ is positive definite and none of them is diagonal. So, your set is definitely not open (except for the case where the space is one-dimensional).
