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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.
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