Kolmogorov space (T0 space)
E898482
A Kolmogorov (T0) space is a topological space in which any two distinct points are topologically distinguishable, meaning at least one has an open neighborhood not containing the other.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Kolmogorov space (T0 space) canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991598 — 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: Kolmogorov space (T0 space) Context triple: [Hausdorff space, relatedConcept, Kolmogorov space (T0 space)]
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A.
Lindelöf space
A Lindelöf space is a topological space in which every open cover has a countable subcover, generalizing a key compactness property.
-
B.
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.
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C.
Freudenthal compactification
The Freudenthal compactification is a topological construction that extends a non-compact, locally compact space by adding a boundary of “ends” to obtain a compact space that more finely captures its asymptotic structure than the one-point (Alexandrov) compactification.
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D.
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.
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E.
Baire space
Baire space is a fundamental topological space—typically the set of all infinite sequences of natural numbers with the product topology—that serves as a central object in descriptive set theory and general topology.
- 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: Kolmogorov space (T0 space) Target entity description: A Kolmogorov (T0) space is a topological space in which any two distinct points are topologically distinguishable, meaning at least one has an open neighborhood not containing the other.
-
A.
Lindelöf space
A Lindelöf space is a topological space in which every open cover has a countable subcover, generalizing a key compactness property.
-
B.
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.
-
C.
Freudenthal compactification
The Freudenthal compactification is a topological construction that extends a non-compact, locally compact space by adding a boundary of “ends” to obtain a compact space that more finely captures its asymptotic structure than the one-point (Alexandrov) compactification.
-
D.
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.
-
E.
Baire space
Baire space is a fundamental topological space—typically the set of all infinite sequences of natural numbers with the product topology—that serves as a central object in descriptive set theory and general topology.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
T0 space
ⓘ
separation axiom ⓘ separation axiom ⓘ topological space property ⓘ topological space property ⓘ |
| alsoKnownAs |
Kolmogorov T0 space
NERFINISHED
ⓘ
T0 space ⓘ |
| appearsIn | classification of topological spaces by separation axioms ⓘ |
| characterizedBy | specialization preorder is antisymmetric ⓘ |
| closureProperty |
arbitrary products of T0 spaces are T0
ⓘ
finite products of T0 spaces are T0 ⓘ quotients of T0 spaces need not be T0 ⓘ subspaces of T0 spaces are T0 ⓘ |
| context |
mathematical logic and semantics via specialization order
ⓘ
point-set topology ⓘ |
| definedOn | topological space ⓘ |
| definition | a topological space in which any two distinct points are topologically distinguishable ⓘ |
| ensures |
no two distinct points have exactly the same open neighborhoods
ⓘ
points are determined by their neighborhoods ⓘ |
| equivalentCondition | specialization preorder is a partial order ⓘ |
| hasAbbreviation | T0 ⓘ |
| hasCondition | for any two distinct points, there exists an open set containing one but not the other ⓘ |
| hasExample |
Sierpinski space
NERFINISHED
ⓘ
any T1 space ⓘ discrete space ⓘ |
| hasNonExample | indiscrete space with more than one point ⓘ |
| hasProperty | any two distinct points are topologically distinguishable ⓘ |
| implies |
Kolmogorov quotient is T0
ⓘ
Kolmogorov space NERFINISHED ⓘ Kolmogorov space ⓘ Kolmogorov space NERFINISHED ⓘ |
| isFirstInHierarchy | T0–T4 separation axioms ⓘ |
| isWeakerThan |
Hausdorff space
ⓘ
T1 space ⓘ T2 space ⓘ normal space ⓘ regular space ⓘ |
| logicalForm | for all distinct x,y there exists an open set containing x and not y or containing y and not x ⓘ |
| namedAfter | Andrey Kolmogorov NERFINISHED ⓘ |
| nonExampleCondition | if every nonempty open set contains all points, the space is not T0 ⓘ |
| relatedConcept |
Alexandrov topology
NERFINISHED
ⓘ
Kolmogorov quotient NERFINISHED ⓘ specialization preorder ⓘ |
| usedIn |
domain theory
ⓘ
general topology ⓘ order theory ⓘ theoretical computer science ⓘ |
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: Kolmogorov space (T0 space) Description of subject: A Kolmogorov (T0) space is a topological space in which any two distinct points are topologically distinguishable, meaning at least one has an open neighborhood not containing the other.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Hausdorff space