What can say about $2^X= \{A\subseteq X: \text{ A is closed set} \}$, when $(X, \mathcal{U})$ is a compact uniform space?

It is known that if $ (X, d)$ is a compact metric space, then hyperspace $ 2^X= \{A\subseteq X: \text{ A is closed set} \}$ is a compact space with Hausdorff metric

What can say about $ 2^X= \{A\subseteq X: \text{ A is closed set} \}$ , when $ (X, \mathcal{U})$ is a compact uniform space?