While studying Bayesian statistics, I came up with de Finetti's rappresentation theorem, presented in a general setting:
$$ Let: \mathbb X \ \text{ a separable and complete metric space,} \ \mathcal{X}=\mathcal{B}(\mathbb X) \\ (X_n)_{n \ge 1} \ \text{a sequence of random elements on} \ (\Omega, \mathcal F, \mathbb P) \ \text{taking values in} \ (\mathbb X^{\infty}, \mathcal X^{\infty}) \\ \mathbb P_{\mathbb X}= \ \text{the set of all probability measures on} \\ (\mathbb X, \mathcal X) \ \text{endowed with the} \ \underline{topology \ of \ the \ weak \ convergence} \\ \mathcal P_\mathbb X = \mathcal B(\mathbb P_{\mathbb X}) $$
My question is: what is exactly the topology of the weak convergence? When is it equivalent to the weak topology, when it's not and why?