Hausdorff space

A Hausdorff space (or $\text{[math]}$ 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

Definition.

A topological space $\text{[math]}$ is Hausdorff, if for any two points $\text{[math]}$ , there exists two disjoint open sets $\text{[math]}$ , $\text{[math]}$ , such that $\text{[math]}$ and $\text{[math]}$ .

Equivalent Definitions

Any convergent sequence in $\text{[math]}$ has at most one limit.

Properties

Property.

Subspaces of Hausdorff spaces are Hausdorff.

Proof.

Let $\text{[math]}$ with $\text{[math]}$ Hausdorff. For $\text{[math]}$ , there exist disjoint open sets $\text{[math]}$ with $\text{[math]}$ and $\text{[math]}$ . Then $\text{[math]}$ and $\text{[math]}$ are disjoint open sets in $\text{[math]}$ containing $\text{[math]}$ and $\text{[math]}$ . □

Property.

Finite products of Hausdorff spaces are Hausdorff.

Proof.

Let $\text{[math]}$ be Hausdorff. Consider points $\text{[math]}$ . There exists an index $\text{[math]}$ with $\text{[math]}$ .

Since $\text{[math]}$ is Hausdorff, choose disjoint open sets $\text{[math]}$ containing $\text{[math]}$ and $\text{[math]}$ . Then $\text{[math]}$ and $\text{[math]}$ are disjoint open sets containing the two points. □

Property.

Compact subsets of Hausdorff spaces are closed.

Proof.

Let $\text{[math]}$ be compact and $\text{[math]}$ Hausdorff. For any $\text{[math]}$ , for each $\text{[math]}$ choose disjoint open sets $\text{[math]}$ and $\text{[math]}$ .

The collection $\text{[math]}$ covers $\text{[math]}$ . By compactness, finitely many $\text{[math]}$ cover $\text{[math]}$ . Then

$\text{[math]}$

is open, contains $\text{[math]}$ , and is disjoint from $\text{[math]}$ . Hence $\text{[math]}$ is closed. □

Examples

Example.

Every metric space is a Hausdorff space.

Terminology

The following table lists the principal terms used in this article.

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Hausdorff space	espace de Hausdorff	hausdorff-Raum	ハウスドルフ空間	Hausdorff 空间 · Hausdorff 空間	5 languages
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Definitions

Equivalent Definitions

Properties

Examples

See also

Terminology