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In some books, like Srivastava (1998) A Course on Borel Sets, p. 96, a standard Borel space is defined as a measurable space which isomorphic to a Borel subset of a Polish space (i.e. there exists a bimeasurable bijection between the two).

Elsewhere, a standard Borel space is defined as a measurable space, say $(X,\mathcal{X})$, for which a Polish topology $\mathcal{T}$ exists which generates the $\sigma$-algebra $\mathcal{X}$.

Are the two definitions equivalent?

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As I understood your terms, the answer is affirmative, by the following theorem from “Classical Descriptive Set Theory” (Springer, 1995) by Alexander Kechris.

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