The closure operation in the uniform topology is $$\overline{A} = \bigcap_{E \in \mathcal{U}} E \cdot A$$ where $E \cdot A$ is the set of all $E$-left-relatives of $A$. I cannot seem to prove that $$\overline{A \cup B} = \overline{A} \cup \overline{B}$$ It is fairly straightforward to show that $\bar{A} \cup \bar{B} \subseteq \overline{A \cup B}$, but the reverse direction eludes me because I can't see how to get from $$(p \in E \cdot A \text{ or } p \in E \cdot B) \text{ for all } E \in \mathcal{U}$$ to $$(p \in E \cdot A \text{ for all } E \in \mathcal{U}) \text{ or } (p \in E \cdot B \text{ for all } E \in \mathcal{U})$$
Any tips?