Uniform structure induces proximity structure

Statement
Suppose $$(X,\mathcal{U})$$ is a fact about::uniform space: there is an underlying set $$X$$ and a uniform structure $$\mathcal{U}$$ on $$X$$. The induced proximity structure on $$X$$ is defined as the following proximity structure $$\delta$$:

$$A \delta B \iff \ \forall \ U \in \mathcal{U}, (A \times B) \cap U \ne \varnothing$$.

Related facts

 * Induced proximity structure from uniform structure is functorial

Other induced structures

 * Metric induces topology
 * Metric induces uniform structure
 * Metric induces proximal structure
 * Uniform structure induces topology
 * Proximity structure induces topology