Let be a uniform structure on a set . We define a subset to be open if and only if for each there exists an entourage such that whenever , then .

Let us verify that this defines a topology on .

Clearly, the subsets and are open. If and are two open sets, then for each , there exist an entourage such that, whenever , then , and an entourage such that, whenever , then . Consider the entourage : whenever , then , hence is open.

Suppose is an arbitrary family of open subsets. For each , there exists such that . Let be the entourage whose existence is granted by the definition of open set. We have that whenever , then ; hence , which concludes the proof.