1-Wasserstein spaces over unbounded Polish metric spaces are not Gromov-compactifiable

In this paper, we prove that the 1-Wasserstein space P 1 ( X ) \mathcal {P}_{1}(\mathcal {X}) is non-Gromov-compactifiable when X \mathcal {X} is an unbounded Polish metric space (separable and complete metric space). This shows that the statement “ P 1 ( X ) \mathcal {P}_1(\mathcal {X}) is Gromov-compactifiable when X \mathcal {X} is Gromov-compactifiable” is false, although the converse statement holds [Trans. Amer. Math. Soc. Ser. B 12 (2025), 1130–1155].