Between any two uncountable Polish spaces, there is a Borel isomorphism; that is, a bijection that preserves the Borel structure. (Polish space, Wikipedia)
So since the Borel-$\sigma$-Algebra on $\mathbb R$ is countably generated, it should follow that the same is true for all Polish spaces, am I right here?
(Another argument for that would be, that Polish spaces are completely separable, so they contain a dense countable subset)
Additional question: Can you point me to a book stating this statement, so I can cite it?