Cauchy net
E825424
concept in analysis
concept in topology
concept in uniform spaces
generalization of Cauchy sequence
topological concept
A Cauchy net is a generalization of a Cauchy sequence to arbitrary topological or uniform spaces, capturing the idea that the elements of the net eventually become arbitrarily close to each other.
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
concept in analysis
ⓘ
concept in topology ⓘ concept in uniform spaces ⓘ generalization of Cauchy sequence ⓘ topological concept ⓘ |
| appearsIn |
textbooks on functional analysis
ⓘ
textbooks on general topology ⓘ |
| associatedWith | Nets and Filters in topology NERFINISHED ⓘ |
| assumesStructure | uniform structure or compatible uniformity on the space ⓘ |
| capturesIdeaOf | Cauchy convergence in general spaces ⓘ |
| characterizes |
completeness of metric spaces (via sequences as special case)
ⓘ
completeness of uniform spaces ⓘ |
| definedBy |
for every entourage U there exists i_0 such that for all i,j ≥ i_0, (x_i,x_j) ∈ U in a uniform space
ⓘ
for every neighborhood V of the diagonal there exists i_0 such that for all i,j ≥ i_0, (x_i,x_j) ∈ V in a topological space with a compatible uniformity ⓘ |
| definedIn |
topological spaces
ⓘ
uniform spaces ⓘ |
| ensures | eventual pairwise closeness of terms ⓘ |
| formalizedAs | net (x_i) indexed by a directed set I ⓘ |
| generalizes | Cauchy sequence ⓘ |
| hasIndexSet | directed set ⓘ |
| hasMotivation | sequences are insufficient to capture convergence in general topological spaces ⓘ |
| hasProperty | elements eventually become arbitrarily close to each other ⓘ |
| implies | Cauchy sequence when the directed set is the natural numbers with usual order ⓘ |
| isAlternativeTo | Cauchy filter in describing completeness ⓘ |
| isSpecialCaseOf | Cauchy filter when considering tails of the net ⓘ |
| isToolFor | extending sequence-based arguments to non-metrizable spaces ⓘ |
| relatedTo |
Cauchy filter
ⓘ
Cauchy sequence ⓘ filter ⓘ net ⓘ |
| requires | directed index set for definition ⓘ |
| usedIn |
completion of spaces
ⓘ
convergence theory ⓘ functional analysis ⓘ general topology ⓘ study of completeness ⓘ uniform space theory ⓘ |
| usedToDefine |
completion of topological vector spaces
ⓘ
completion of uniform spaces ⓘ |
| usedToProve | existence of limits in complete spaces ⓘ |
| usedToStudy |
convergence in function spaces
ⓘ
convergence in product spaces ⓘ non-first-countable spaces ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.