Uniform structure induces proximity structure

Statement

Suppose $(X,{\mathcal {U}})$ is a 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 $\delta$:

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

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