@MikeMiller Would you happen to know the definition of a uniformly continuous function, with respect to the definition of a uniform space with covers?
@Jake1234 Maybe Proposition 8.1.22 from Engelking is what you are looking for.
Proposition 8.1.22. Let $(X,\mathcal U)$ and $(Y,\mathcal V)$ be uniform spaces and $f$ a mapping of $X$ to $Y$. The following conditions are equivalent:
(i) The mapping $f$ is uniformly continuous with respect to $\mathcal U$ and $\mathcal V$.
(iii) For every cover $\mathcal A$ of the set $Y$ which is uniform with respect to $\mathcal V$ the cover $\{f^{-1}(A): A\in\mathcal A\}$ of the set $X$ is uniform with respect to $\mathcal U$.
I also found a direct definition in J.R.Isbell - Uniform spaces, the definition there is, that for every $V \in \mathcal V$ (where $\mathcal V$ is a family of coverings, that satisfies certain claims) there exists a $U \in \mathcal U$, such that for every $A \in U$, $f[A] \subset B$ for some $B \in V$, that is, the image of $U$ is a refinement of $V$.
