Is there an equal definition of $d(x, A)= \inf \{d(x, a): a\in A\}$ in the case of uniform space?

Question

If $(X, d)$ is a metric space and $A\subseteq X$, $x\in X$, then $$d(x, A)= \inf \{d(x, a): a\in A\}.$$
Question. Is there an equal definition of $$d(x, A)= \inf \{d(x, a): a\in A\}$$ in the case of uniform space $(X, \mathcal{U})$?

A uniformity for a set $X$ is a non-empty $\mathcal{U}$ of subsets of $X\times X$ such that:

1. If $U\in\mathcal{U}$, then $U^{-1}\in\mathcal{U}$;
2. Each member of $\mathcal{U}$ contains diagonal of $X$;
3. If $U\in \mathcal{U}$, then $V\circ V\subseteq U$ for some $V\in\mathcal{U}$;
4. If $U$ and $V$ are elements of $\mathcal{U}$, then $U\cap V\in\mathcal{U}$;
5. If $U\in\mathcal{U}$ and $U\subseteq V\subseteq X\times X$, then $V\in \mathcal{U}$.

The pair $(X, \mathcal{U})$ is a uniform space.