Is the space of positive (semi- or not) definite correlation matrices Polish? 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 :)

Let me do it this way:

* The symmetric matrices are closed in all $p \times p$-matrices (obvious), hence they are Polish.
* The positive definite matrices are an open cone in the symmetric matrices, hence they are Polish as well (open subsets of a Polish space are Polish -- this is a bit easier to prove than the fact on $G_{\delta}$'s).
* Finally, since the functions $f_{i}:A \mapsto a_{ii}$ are continuous, their simultaneous pre-image of $1$ is closed in the positive definite matrices, hence the positive definite correlation matrices are Polish.



The positive semi-definite case is even simpler, as the conditions are all closed.

Finally, I can only recommend working through the first few sections of Kechris, as most of these arguments become rather simple once one gets used to them.