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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. The article text may, however, contain more than just the basic definition. Rate its utility as a basic definition article on the talk page
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 X 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

• Category:Variations of Hausdorffness

Stronger properties

• Normal space
• Completely regular space
• Regular space
• Metrizable space
• CW-space
• Manifold

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 a complete list of properties hereditary to subspaces

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

Moving to a finer topology increases the number of possible open sets to choose from, and hence, preserves the property of Hausdorffness.

References

Textbook references

• Topology (2nd edition) by James R. Munkres^{More info}, Page 98 (formal definition)
• Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. Thorpe^{More info}, Page 26 (formal definition)

External links

Definition links

• Wikipedia page