Title basically says it all: Is the space of positive (semi- or not) definite correlation matrices Polish?
As an aside, I'm interested in general comments/references about the space(s).
Edit: For the sake of completeness, a $p\times p$ positive definite correlation matrix $C$ is a real symmetric matrix with ones on the diagonal and x'Cx>0 for any nonzero $x\in \mathbb{R}^p$. I updated the question to include positive semidefinite correlation matrices as well (ie relaxing to $x'Cx\geq0$) because I suppose it's interesting too :)