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Does the topology of Wasserstein space $(\mathcal P_p (E), W_p)$ coincide with the initial topology induced by $\mathcal C_b(E) \cup \{g_p\}$?

A topology generated by countably many point-separating real functions is metrizable. To apply this here, it suffices to show that there is a countable family $\mathcal{G}$ of bounded real functions ...
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Optimal transport: how $\varphi^c$ can be written as $\varphi^c = \lim _{\ell \rightarrow \infty} \psi_{\ell}$?

Are you sure there aren’t additional conditions on $\varphi$? Because otherwise taking $X = \mathbb R$ and $Y$ to be a one point space, the following gives a counterexample: $c(x, y) = 0$ if $x = 0$; $...
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