Hausdorff space: Difference between revisions
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* Given any two points in the topological space, there are disjoint open sets containing the two points respectively. | * Given any two points in the topological space, there are disjoint open sets containing the two points respectively. | ||
* Every ultrafilter of subsets converges to at most one point | * Every ultrafilter of subsets converges to at most one point | ||
===Definition with symbols=== | |||
A [[topological space]] <math>X</math> is said to be '''Hausdorff''' if given any two points <math>x \ne y \in X</math>, there exist disjoint [[open subset]]s <math>U \ni x</math> and <math>V \ni y</math>. | |||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 21:16, 25 January 2008
This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces
In the T family (properties of topological spaces related to separation axioms), this is called: T2
This article is about a basic definition in topology.
VIEW: Definitions built on this | Facts about this | Survey articles about this
View a complete list of basic definitions in topology
For survey articles related to this, refer: Category:Survey articles related to Hausdorffness
Please also read the Topospaces Convention page: Convention:Hausdorffness assumption
Definition
Symbol-free definition
A topological space is said to be Hausdorff if it satisfies the following equivalent conditions:
- Given any two points in the topological space, there are disjoint open sets containing the two points respectively.
- Every ultrafilter of subsets converges to at most one point
Definition with symbols
A topological space is said to be Hausdorff if given any two points , there exist disjoint open subsets and .
Relation with other properties
This property is a pivotal (important) member of its property space. Its variations, opposites, and other properties related to it and defined using it are often studied
Stronger properties
Weaker properties
- Locally Hausdorff space
- KC-space: This is based on the fact that any compact subset of a Hausdorff space is closed. For full proof, refer: Hausdorff implies KC
- US-space: This is based on the fact that any convergent sequence in a Hausdorff space has a unique limit. For full proof, refer: Hausdorff implies US
- Sober space
- T1 space
- T0 space
Metaproperties
Products
This property of topological spaces is closed under taking arbitrary products
View all properties of topological spaces closed under products
An arbitrary (finite or infinite) product of Hausdorff spaces is Hausdorff. For full proof, refer: Hausdorffness is product-closed
Hereditariness
This property of topological spaces is hereditary, or subspace-closed. In other words, any subspace (subset with the subspace topology) of a topological space with this property also has this property.
View other subspace-hereditary properties of topological spaces
Any subspace of a Hausdorff space is Hausdorff. For full proof, refer: Hausdorffness is hereditary
Refining
This property of topological spaces is preserved under refining, viz, if a set with a given topology has the property, the same set with a finer topology also has the property
View all refining-preserved properties of topological spaces OR View all coarsening-preserved properties of topological spaces
Moving to a finer topology increases the number of possible open sets to choose from, and hence, preserves the property of Hausdorffness.