Kolmogorov space: Difference between revisions
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Latest revision as of 21:57, 24 January 2012
Definition
A topological space is termed a space or Kolmogorov space if it satisfies the following equivalent conditions:
- For any two distinct points , there is either an open subset containing but not , or an open subset containing but not .
- For any two distinct points , there is either a closed subset containing but not , or a closed subset containing but not .
- For any two distinct points such that , i.e., the closures of any two distinct one-point subsets must be distinct.
This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces
In the T family (properties of topological spaces related to separation axioms), this is called: T0
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| T1 space | singleton subsets are closed | Kolmogorov not implies T1 | ||
| Hausdorff space | KC-space, Locally Hausdorff space, Sober T0 space|FULL LIST, MORE INFO | |||
| locally Hausdorff space | |FULL LIST, MORE INFO | |||
| totally disconnected space | |FULL LIST, MORE INFO | |||
| regular space | |FULL LIST, MORE INFO | |||
| normal space | |FULL LIST, MORE INFO | |||
| metrizable space | Functionally Hausdorff space, Hausdorff space, Normal Hausdorff space, Regular Hausdorff space|FULL LIST, MORE INFO |