I'm not sure if the set of symmetric positive-definite matrices is open in the set of symmetric matrices. I'm almost certain that it is not open in the larger set of n by n matrices. So far most of the proofs I've seen don't use symmetry, so I'm not sure if they are correct. Any hints? I've been trying to use that the set of positive-definite matrices is path connected, but I can't come up with a solution. Thanks in advance.