Hausdorff
E259764
Hausdorff is a topological separation property requiring that any two distinct points in a space can be enclosed in disjoint open sets.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Hausdorff canonical | 8 |
| Hausdorff space | 2 |
| Hausdorff property | 1 |
| Hausdorff separation axiom | 1 |
| Hausdorff space concept | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2364456 — 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: Hausdorff Context triple: [Riemann surface, hasProperty, Hausdorff]
-
A.
Pringsheim
Pringsheim is a German-Jewish family name historically associated with a prominent bourgeois and intellectual family in 19th- and early 20th-century Germany.
-
B.
Menger
Menger is a surname most notably associated with Austrian mathematician Karl Menger, known for his work in topology, dimension theory, and the foundations of geometry.
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C.
Baire
Baire is a town in eastern Cuba historically notable as the site where the Cuban War of Independence was ignited by the Grito de Baire uprising in 1895.
-
D.
Arciszewski
Arciszewski is a Polish surname most notably borne by Tomasz Arciszewski, a socialist politician and Prime Minister of the Polish government-in-exile during World War II.
-
E.
Hungerford
Hungerford is a historic market town and civil parish in West Berkshire, England, situated on the River Kennet and the Kennet and Avon Canal.
- 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: Hausdorff Target entity description: Hausdorff is a topological separation property requiring that any two distinct points in a space can be enclosed in disjoint open sets.
-
A.
Pringsheim
Pringsheim is a German-Jewish family name historically associated with a prominent bourgeois and intellectual family in 19th- and early 20th-century Germany.
-
B.
Menger
Menger is a surname most notably associated with Austrian mathematician Karl Menger, known for his work in topology, dimension theory, and the foundations of geometry.
-
C.
Baire
Baire is a town in eastern Cuba historically notable as the site where the Cuban War of Independence was ignited by the Grito de Baire uprising in 1895.
-
D.
Arciszewski
Arciszewski is a Polish surname most notably borne by Tomasz Arciszewski, a socialist politician and Prime Minister of the Polish government-in-exile during World War II.
-
E.
Hungerford
Hungerford is a historic market town and civil parish in West Berkshire, England, situated on the River Kennet and the Kennet and Avon Canal.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
separation axiom
ⓘ
topological space property ⓘ |
| alsoKnownAs |
Hausdorff separation axiom
ⓘ
linked to:
Hausdorff
T2 space ⓘ |
| appliesTo | topological space ⓘ |
| characterization |
limits of nets, if they exist, are unique in Hausdorff spaces
ⓘ
limits of sequences, if they exist, are unique in Hausdorff spaces ⓘ |
| closureProperty |
arbitrary products of Hausdorff spaces are Hausdorff
ⓘ
closed subspaces of Hausdorff spaces are Hausdorff ⓘ finite products of Hausdorff spaces are Hausdorff ⓘ subspaces of Hausdorff spaces are Hausdorff ⓘ |
| definition | a topological space in which any two distinct points have disjoint open neighbourhoods ⓘ |
| equivalentCondition | X is Hausdorff iff the diagonal Δ = {(x,x) : x ∈ X} is closed in X × X ⓘ |
| example |
any metric space is Hausdorff
ⓘ
the real line with the standard topology is Hausdorff ⓘ |
| field | topology ⓘ |
| historicalNote | introduced in the early 20th century in the development of axiomatic topology ⓘ |
| implies | T1 space ⓘ |
| isStrongerThan | T1 separation axiom ⓘ |
| isWeakerThan |
Tychonoff space
ⓘ
metric space ⓘ normal Hausdorff space ⓘ regular Hausdorff space ⓘ |
| namedAfter | Felix Hausdorff ⓘ |
| nonClosureProperty | quotients of Hausdorff spaces need not be Hausdorff ⓘ |
| nonExample |
the Sierpiński space is not Hausdorff
ⓘ
the trivial topology on a set with more than one point is not Hausdorff ⓘ |
| property |
singletons are closed in any Hausdorff space
ⓘ
the diagonal is closed in the product space X × X for a Hausdorff space X ⓘ |
| relatedConcept |
Kolmogorov space (T0 space)
ⓘ
Tychonoff space ⓘ metric space ⓘ normal space ⓘ regular space ⓘ |
| requirementIn |
definition of compact Hausdorff space
ⓘ
definition of topological manifold ⓘ |
| requires | for any two distinct points x and y there exist open sets U and V with x in U, y in V, and U ∩ V = ∅ ⓘ |
| symbolicNotation | often denoted by T2 in the separation axiom hierarchy ⓘ |
| usedIn |
algebraic topology
ⓘ
functional analysis ⓘ general topology ⓘ topological groups ⓘ topological manifolds ⓘ |
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: Hausdorff Description of subject: Hausdorff is a topological separation property requiring that any two distinct points in a space can be enclosed in disjoint open sets.
Referenced by (13)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Riemann surfaces
linked to: Hausdorff
linked to: Hausdorff
subject linked to:
Hausdorff
linked to: Hausdorff
subject linked to:
Paul Mongré
linked to: Hausdorff
linked to: Hausdorff
subject linked to:
CW complexes