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.
Observed surface forms (5)
| Surface form | Occurrences |
|---|---|
| Kolmogorov space | 0 |
| T0 space | 0 |
| Hausdorff space | 0 |
| T1 space | 0 |
| T2 space | 0 |
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Hausdorff space