In the book Computational Optimal Transport, by Peyré and Cuturi, the following assertion is made.
Let $X$ and $Y$ be two compact probability spaces, with $c$ continuous and $$ \mathcal U(\mu, \nu):= \left\{ \gamma \in \mathcal P(X \times Y) \quad : \quad (\text{Proj}_X)_\# \gamma = \mu, (\text{Proj}_Y)_\# \gamma = \nu \right\} $$
The author then claims that $\mathcal U(\mu, \nu)$ is compact for the weak topology of measures.
How does one proves this assertion?
Are X and Y considered to be Polish?
If yes: every probability measure on a Polish space is tight. Using this you can show that $\mathcal{U}(\mu,\nu)$ is also tight. By Prokhorov's Theorem its closure is compact. Using how $\mathcal{U}(\mu,\nu)$ is defined you can show that it is closed w.r.t. the weak topology and therefore it is compact.