topology induced by uniform structure
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.
|Title||topology induced by uniform structure|
|Date of creation||2013-03-22 12:46:44|
|Last modified on||2013-03-22 12:46:44|
|Last modified by||Mathprof (13753)|