T1 separation axiom
E898480
The T1 separation axiom is a topological property requiring that for any two distinct points, each has an open set containing it but not the other, ensuring all singletons are closed.
All labels observed (2)
| Label | Occurrences |
|---|---|
| T1 axiom | 1 |
| T1 separation axiom canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991576 — 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: T1 separation axiom Context triple: [Hausdorff space, isStrongerThan, T1 separation axiom]
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A.
T1
T1 is one of the tram routes of the Trambaix light rail network serving the Barcelona metropolitan area.
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B.
T1
T1 is a tram line serving the Lyon metropolitan area in France, connecting key districts including Villeurbanne.
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C.
T1
T1 is one of the main tram lines in Casablanca’s urban light rail network, providing mass transit service across key districts of the city.
-
D.
the T
The T is the public transit system serving the Greater Boston area, operated by the Massachusetts Bay Transportation Authority.
-
E.
T2-SE-A1
T2-SE-A1 is a World War II–era American oil tanker design, built in large numbers for the U.S. Maritime Commission as a standard T2-class ship used to transport fuel across the oceans.
- 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: T1 separation axiom Target entity description: The T1 separation axiom is a topological property requiring that for any two distinct points, each has an open set containing it but not the other, ensuring all singletons are closed.
-
A.
T1
T1 is one of the tram routes of the Trambaix light rail network serving the Barcelona metropolitan area.
-
B.
T1
T1 is a tram line serving the Lyon metropolitan area in France, connecting key districts including Villeurbanne.
-
C.
T1
T1 is one of the main tram lines in Casablanca’s urban light rail network, providing mass transit service across key districts of the city.
-
D.
the T
The T is the public transit system serving the Greater Boston area, operated by the Massachusetts Bay Transportation Authority.
-
E.
T2-SE-A1
T2-SE-A1 is a World War II–era American oil tanker design, built in large numbers for the U.S. Maritime Commission as a standard T2-class ship used to transport fuel across the oceans.
- F. None of above. chosen
Statements (41)
| Predicate | Object |
|---|---|
| instanceOf |
separation axiom
ⓘ
topological property ⓘ |
| alsoKnownAs |
Fréchet axiom
ⓘ
T1 axiom ⓘ
linked to:
T1 separation axiom
|
| appliesTo | topological space ⓘ |
| category | mathematics ⓘ |
| characterizes | spaces in which convergence of sequences is determined by limits being unique when the space is also Hausdorff ⓘ |
| closureProperty | The closure of a singleton {x} is {x} itself. ⓘ |
| consequence | No point is isolated by being contained in every nonempty open set unless the space is trivial ⓘ |
| definition | For any two distinct points x and y in a topological space, there exists an open set containing x but not y, and an open set containing y but not x. ⓘ |
| ensures | points are topologically distinguishable by closed sets ⓘ |
| equivalentCondition |
Every singleton set {x} is closed in the space.
ⓘ
Finite subsets of the space are closed. ⓘ For each point x and each point y ≠ x, there exists an open set containing x but not y. ⓘ |
| failsIn | indiscrete topology on a set with more than one point ⓘ |
| field | topology ⓘ |
| formalizes | the idea that points can be separated by open sets in at least one direction ⓘ |
| historicalNote | Named after Maurice Fréchet in some literature ⓘ |
| holdsIn |
discrete topology on any set
ⓘ
standard topology on the real numbers ⓘ |
| implies | T0 separation axiom ⓘ |
| interactionWithCompactness | In a T1 space, compact subsets are closed ⓘ |
| interactionWithConnectedness | In a T1 space, components are intersections of clopen sets ⓘ |
| interactionWithConvergence | In a T1 space, a sequence can converge to a point only if every neighborhood of the point contains all but finitely many terms of the sequence ⓘ |
| isImpliedBy |
Hausdorff property
ⓘ
linked to:
Hausdorff
T2 separation axiom ⓘ normal Hausdorff property ⓘ regular Hausdorff property ⓘ |
| isStrongerThan |
T0 separation axiom
ⓘ
linked to:
Kolmogorov space (T0 space)
|
| isWeakerThan |
Hausdorff property
ⓘ
T2 separation axiom ⓘ |
| logicalStrength | strictly stronger than T0 and strictly weaker than T2 in general ⓘ |
| notPreservedUnder | arbitrary quotient maps ⓘ |
| preservedUnder |
arbitrary products
ⓘ
continuous open images ⓘ finite products ⓘ taking subspaces ⓘ |
| topologicalInvariance | invariant under homeomorphisms ⓘ |
| usedIn |
general topology
ⓘ
topological separation theory ⓘ |
| usedToDefine | T1 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: T1 separation axiom Description of subject: The T1 separation axiom is a topological property requiring that for any two distinct points, each has an open set containing it but not the other, ensuring all singletons are closed.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Hausdorff
linked to: T1 separation axiom