# Is the Borel-sigma-Algebra of a Polish space always countably generated?

Wikipedia says:

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?

To see that a Polish space is second-countable, simply fix a countable dense subset $D$, and a compatible metric, and consider $\{B(x,\frac1n)\mid x\in D,n\in\Bbb N\}$, which is of course a basis for the topology.
This is a far simpler argument than relying on the Borel isomorphism. Although you are correct, a generating set is moved to a generating set. So if $\Bbb R$ has a countably generated Borel algebra, then every space which is Borel isomorphic to it also has a countably generated Borel algebra.