Cauchy completeness
E825636
Cauchy completeness is a property of a metric or uniform space ensuring that every Cauchy sequence in the space converges to a limit within the space.
Statements (40)
| Predicate | Object |
|---|---|
| instanceOf |
property of metric spaces
ⓘ
property of uniform spaces ⓘ topological property ⓘ |
| characterizedBy | convergence of all Cauchy sequences ⓘ |
| contrastsWith | sequential completeness in general topological spaces ⓘ |
| definedOn |
metric space
ⓘ
uniform space ⓘ |
| ensures |
every Cauchy sequence converges in the space
ⓘ
limits of Cauchy sequences lie in the space ⓘ no proper metric completion is needed ⓘ |
| equivalentCondition | every Cauchy filter converges (in uniform spaces) ⓘ |
| equivalentTo | metric completeness ⓘ |
| failsFor | rational numbers with the usual metric ⓘ |
| formalizedBy | Cauchy’s criterion for convergence ⓘ |
| generalizedBy | Cauchy completeness for uniform spaces NERFINISHED ⓘ |
| historicallyNamedAfter | Augustin-Louis Cauchy NERFINISHED ⓘ |
| holdsFor |
complex numbers with the usual metric
ⓘ
real numbers with the usual metric ⓘ |
| implies |
Cauchy sequences are bounded in normed spaces
ⓘ
every Cauchy net converges in a complete uniform space ⓘ every absolutely convergent series converges in Banach spaces ⓘ space has no Cauchy sequence without a limit in the space ⓘ |
| relatedTo |
Banach space
ⓘ
Cauchy completion NERFINISHED ⓘ Cauchy sequence ⓘ Hilbert space NERFINISHED ⓘ complete metric space ⓘ complete uniform space ⓘ metric completion ⓘ |
| requires |
notion of Cauchy sequence
ⓘ
notion of distance or uniform structure ⓘ |
| studiedIn |
general topology
ⓘ
uniform space theory ⓘ |
| usedIn |
analysis
ⓘ
construction of real numbers from rationals ⓘ functional analysis ⓘ metric geometry ⓘ proofs of existence theorems in analysis ⓘ topology ⓘ |
| usedToDefine | completion of a metric space ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.