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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Cauchy completeness canonical | 1 |
| Cauchy completeness for uniform spaces | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9843667 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Cauchy completeness Context triple: [Cauchy convergence criterion, relatedTo, Cauchy completeness]
-
A.
Cauchy completion
Cauchy completion is a construction in metric space theory that embeds a given space into a complete metric space by formally adding limits of all its Cauchy sequences.
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B.
Cauchy convergence criterion
The Cauchy convergence criterion is a fundamental concept in mathematical analysis that characterizes convergence of sequences (and series) by requiring that their terms become arbitrarily close to each other beyond some index.
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C.
Cauchy sequence
A Cauchy sequence is a sequence whose terms become arbitrarily close to each other as the sequence progresses, providing a fundamental criterion for convergence in metric and normed spaces.
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D.
Cauchy net
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.
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E.
Bolzano–Weierstrass theorem
The Bolzano–Weierstrass theorem is a fundamental result in real analysis stating that every bounded infinite sequence in ℝⁿ has a convergent subsequence.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Cauchy completeness Target entity description: 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.
-
A.
Cauchy completion
Cauchy completion is a construction in metric space theory that embeds a given space into a complete metric space by formally adding limits of all its Cauchy sequences.
-
B.
Cauchy convergence criterion
The Cauchy convergence criterion is a fundamental concept in mathematical analysis that characterizes convergence of sequences (and series) by requiring that their terms become arbitrarily close to each other beyond some index.
-
C.
Cauchy sequence
A Cauchy sequence is a sequence whose terms become arbitrarily close to each other as the sequence progresses, providing a fundamental criterion for convergence in metric and normed spaces.
-
D.
Cauchy net
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.
-
E.
Bolzano–Weierstrass theorem
The Bolzano–Weierstrass theorem is a fundamental result in real analysis stating that every bounded infinite sequence in ℝⁿ has a convergent subsequence.
- F. None of above. chosen
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
ⓘ
linked to:
Cauchy completeness
|
| historicallyNamedAfter | Augustin-Louis Cauchy ⓘ |
| 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 ⓘ Cauchy sequence ⓘ Hilbert space ⓘ
linked to:
Hilbert spaces
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Cauchy completeness Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Cauchy completeness