Let be a uniform space with uniformity . For each and , define the following items
We show that all five defining conditions of a neighborhood system on a set are met:
For each , , since every entourage contains the diagonal relation.
Every and every entourage , with
iff iff . This implies that if , then .
Suppose . There is such that . We show that is what we want. Clearly, . For any , and any , we have , or . So for any . In order to show that , we must find such that . By the third step above, since , there is with . Thus .
Definition. For each in a uniform space with uniformity , a uniform neighborhood of is a set for some entourage . In general, for any , the set
is called a uniform neighborhood of .
Two immediate properties that we have already seen in the proof above are: (1). for each , ; and (2). . More generally, .
Remark. If we define , then is a topology induced by the uniform structure (http://planetmath.org/TopologyInducedByAUniformStructure) . Under this topology, uniform neighborhoods are synonymous with neighborhoods.
|Date of creation||2013-03-22 16:42:31|
|Last modified on||2013-03-22 16:42:31|
|Last modified by||CWoo (3771)|