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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?

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This is nearly trivial when you note that Polish spaces are always second-countable.

Simply note that the Borel sets are generated by the open sets, but if every open set is the countable union of basic open sets (as in the case of second-countable spaces), then a countable basis also generates the Borel sets.

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.

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