Hausdorff space
A topological space is said to be (or said to satisfy the axiom) if givendistinct , there exist disjointopen sets (that is, )such that and .
A space is also known as a Hausdorff space.A Hausdorff topology for a set is a topology such that is a Hausdorff space.
Properties
The following properties are equivalent:
- 1.
is a Hausdorff space.
- 2.
The set
is closed in the product topology of .
- 3.
For all , we have
Important examples of Hausdorff spaces are metric spaces, manifolds,and topological vector spaces.
Title | Hausdorff space |
Canonical name | HausdorffSpace |
Date of creation | 2013-03-22 12:18:18 |
Last modified on | 2013-03-22 12:18:18 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 23 |
Author | yark (2760) |
Entry type | Definition |
Classification | msc 54D10 |
Synonym | Hausdorff topological space |
Synonym | T2 space |
Related topic | SeparationAxioms |
Related topic | T1Space |
Related topic | T0Space |
Related topic | T3Space |
Related topic | RegularSpace |
Related topic | MetricSpace |
Related topic | NormalTopologicalSpace |
Related topic | ASpaceMathnormalXIsHausdorffIfAndOnlyIfDeltaXIsClosed |
Related topic | SierpinskiSpace |
Related topic | HausdorffSpaceNotCompletelyHausdorff |
Related topic | Tychonoff |
Related topic | PropertyThatCompactSetsInASpaceAreClosedLies |
Defines | Hausdorff |
Defines | Hausdorff topology |
Defines | T2 |
Defines | T2 topology |
Defines | T2 axiom |