Lindelöf space
E518483
A Lindelöf space is a topological space in which every open cover has a countable subcover, generalizing a key compactness property.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Lindelöf space canonical | 1 |
| Lindelöf space is Lindelöf | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5425860 — 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: Lindelöf space Context triple: [Ernst Lindelöf, hasConceptNamedAfter, Lindelöf space]
-
A.
Alexandrov–Hausdorff theorem
The Alexandrov–Hausdorff theorem is a result in descriptive set theory that characterizes analytic sets as continuous images of Baire space, playing a key role in the study of definable sets in Polish spaces.
-
B.
Hausdorff
Hausdorff is a topological separation property requiring that any two distinct points in a space can be enclosed in disjoint open sets.
-
C.
Noetherian space
A Noetherian space is a topological space in which every descending chain of closed subsets stabilizes, mirroring the finiteness conditions of Noetherian rings in algebra.
-
D.
Tychonoff theorem for products of compact spaces
The Tychonoff theorem for products of compact spaces is a fundamental result in topology stating that any product of compact topological spaces is compact, a statement that is equivalent in strength to the axiom of choice.
-
E.
Alexandrov compactification
The Alexandrov compactification is a topological construction that adds a single “point at infinity” to a non-compact space to make it compact.
- 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: Lindelöf space Target entity description: A Lindelöf space is a topological space in which every open cover has a countable subcover, generalizing a key compactness property.
-
A.
Alexandrov–Hausdorff theorem
The Alexandrov–Hausdorff theorem is a result in descriptive set theory that characterizes analytic sets as continuous images of Baire space, playing a key role in the study of definable sets in Polish spaces.
-
B.
Hausdorff
Hausdorff is a topological separation property requiring that any two distinct points in a space can be enclosed in disjoint open sets.
-
C.
Noetherian space
A Noetherian space is a topological space in which every descending chain of closed subsets stabilizes, mirroring the finiteness conditions of Noetherian rings in algebra.
-
D.
Tychonoff theorem for products of compact spaces
The Tychonoff theorem for products of compact spaces is a fundamental result in topology stating that any product of compact topological spaces is compact, a statement that is equivalent in strength to the axiom of choice.
-
E.
Alexandrov compactification
The Alexandrov compactification is a topological construction that adds a single “point at infinity” to a non-compact space to make it compact.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
topological property ⓘ |
| characterizedInMetricSpacesBy | every open cover has a countable subcover if and only if the space is separable and second countable ⓘ |
| closedSubspaceOf | Lindelöf space is Lindelöf ⓘ |
| continuousImageOf |
Lindelöf space is Lindelöf
ⓘ
linked to:
Lindelöf space
|
| countableUnionOf | Lindelöf subspaces need not be Lindelöf ⓘ |
| definition | a topological space in which every open cover has a countable subcover ⓘ |
| doesNotImply |
compactness
ⓘ
paracompactness in general ⓘ σ-compactness ⓘ |
| field | topology ⓘ |
| generalizes | compact space ⓘ |
| hasCardinalityRestriction | no restriction on underlying set cardinality ⓘ |
| hasExample |
every compact space
ⓘ
every second countable space ⓘ the real line with the usual topology ⓘ |
| hasNonExample |
an uncountable discrete space
ⓘ
the product of uncountably many copies of the unit interval with the product topology ⓘ |
| hasOpenCoverCondition | every open cover has a countable subcover ⓘ |
| hasProperty | every open cover admits a countable subcover ⓘ |
| implies |
Lindelöf property
ⓘ
every family of pairwise disjoint nonempty open sets is countable ⓘ every open cover has a countable dense subfamily ⓘ |
| impliesInRegularSpaces | normality under additional set-theoretic assumptions ⓘ |
| isEquivalentInMetricSpacesTo |
Lindelöfness is equivalent to separability and second countability
ⓘ
separability plus completeness does not characterize Lindelöfness ⓘ |
| isHereditaryWithRespectTo | closed subspaces ⓘ |
| isImpliedBy |
compact space
ⓘ
second countable space ⓘ σ-compact space ⓘ |
| isNotPreservedBy |
arbitrary products
ⓘ
uncountable disjoint unions ⓘ |
| isPreservedBy |
closed subspaces
ⓘ
quotient maps ⓘ |
| isStrongerThan | second countability in general ⓘ |
| isWeakerThan | compactness ⓘ |
| namedAfter |
Ernst Leonard Lindelöf
ⓘ
linked to:
Ernst Lindelöf
|
| productWith |
Lindelöf space need not be Lindelöf in general
ⓘ
compact space is Lindelöf ⓘ |
| relatedConcept |
compact space
ⓘ
countably compact space ⓘ paracompact space ⓘ second countable space ⓘ separable space ⓘ σ-compact space ⓘ |
| usedIn |
general topology
ⓘ
set-theoretic topology ⓘ |
| usedToStudy | covering properties of topological spaces ⓘ |
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: Lindelöf space Description of subject: A Lindelöf space is a topological space in which every open cover has a countable subcover, generalizing a key compactness property.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Lindelöf space