Hausdorff space: Difference between revisions
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{{pivotal topospace property}} | {{pivotal topospace property}} | ||
{{T family|T2}} | |||
==Definition== | ==Definition== | ||
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==Relation with other properties== | ==Relation with other properties== | ||
{{pivotalproperty}} | |||
* [[:Category:Variations of Hausdorffness]] | |||
===Stronger properties=== | ===Stronger properties=== | ||
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* [[T1 space]] | * [[T1 space]] | ||
* [[T0 space]] | * [[T0 space]] | ||
==Metaproperties== | |||
{{DP-closed}} | |||
An arbitrary (finite or infinite) direct product of Hausdorff spaces is Hausdorff. {{proofat|[[Hausdorffness is direct product-closed]]}} | |||
{{subspace-closed}} | |||
Any subspace of a Hausdorff space is Hausdorff. | |||
Revision as of 05:54, 18 August 2007
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
Definition
A topological space is said to be Hausdorff if given any two points in the topological space, there are disjoint open sets containing the two points respectively.
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
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) direct product of Hausdorff spaces is Hausdorff. For full proof, refer: Hausdorffness is direct 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.