complete uniform space
We say that is complete if every Cauchy filter is a convergent filter in the topology induced (http://planetmath.org/TopologyInducedByUniformStructure) by . in this case is called a complete uniformity.
A Cauchy sequence in a uniform space is a sequence in whose section filter is a Cauchy filter. A Cauchy sequence is said to be convergent if its section filter is convergent. is said to be sequentially complete if every Cauchy sequence converges (every section filter of it converges).
Remark. This is a generalization of the concept of completeness in a metric space, as a metric space is a uniform space. As we see above, in the course of this generalization, two notions of completeness emerge: that of completeness and sequentially completeness. Clearly, completeness always imply sequentially completeness. In the context of a metric space, or a metrizable uniform space, the two notions are indistinguishable: sequentially completeness also implies completeness.
|Title||complete uniform space|
|Date of creation||2013-03-22 16:41:40|
|Last modified on||2013-03-22 16:41:40|
|Last modified by||CWoo (3771)|