Hausdorff space

A Hausdorff space (or space) is a type of topological space in which points can be "cleanly separated" by neighborhoods. Specifically, for any two distinct points, there exist disjoint open sets containing each point. Consequently, Hausdorff property ensures that limits of sequences are unique when they exist.

Definitions

A topological space is Hausdorff, if for any two points , there exists two disjoint open sets , , such that and .

Equivalent Definitions

Any convergent sequence in has at most one limit.

Properties

Property.

Subspaces of Hausdorff spaces are Hausdorff.

Proof

Let with Hausdorff. For , there exist disjoint open sets with and . Then and are disjoint open sets in containing and .

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Property.

Finite products of Hausdorff spaces are Hausdorff.

Proof

Let be Hausdorff. Consider points . There exists an index with .

Since is Hausdorff, choose disjoint open sets containing and . Then and are disjoint open sets containing the two points.

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Property.

Compact subsets of Hausdorff spaces are closed.

Proof

Let be compact and Hausdorff. For any , for each choose disjoint open sets and .

The collection covers . By compactness, finitely many cover . Then

is open, contains , and is disjoint from . Hence is closed.

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Examples

Every metric space is a Hausdorff space.

Proof

Let be a metric space, take two distinct points such that .

Consider open balls and where . The open balls are both open with and .

For contradiction, assume there exists , then and . By triangular inequality,

there exists a contradiction.

Thus, ; therefore is Hausdorff.

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See also

• Topological space
• Convergence

Terminology

en	fr	de	zh		ja
Hausdorff space	espace de Hausdorff (espace séparé)	hausdorff-Raum (hausdorffscher Raum)	Hausdorff 空间	Hausdorff 空間	ハウスドルフ空間