Definition:Hausdorff Space

Definition

Let $T = \left({X, \vartheta}\right)$ be a topological space.

$\left({X, \vartheta}\right)$ is a Hausdorff space or $T_2$ space iff:

$\forall x, y \in X, x \ne y: \exists U, V \in \vartheta: x \in U, y \in V: U \cap V = \varnothing$

That is, for any two distinct points $x, y \in X$ there exist disjoint open sets $U, V \in \vartheta$ containing $x$ and $y$ respectively.

That is:

$\left({X, \vartheta}\right)$ is a $T_2$ space iff every two points in $X$ are separated by neighborhoods.

This condition is known as the Hausdorff condition.

For short, we can say $T$ is Hausdorff, using the name as an adjective.

Conveniently, a topological space is Hausdorff if any two distinct points can be housed off from one another in separate disjoint open sets.

Equivalent Definitions

$\left({X, \vartheta}\right)$ is a Hausdorff space or $T_2$ space iff each point is the intersection of all its closed neighborhoods.

This is proved in Equivalent Definitions for $T_2$ Space.

Source of Name

This entry was named for Felix Hausdorff.

See Also

• Basic Properties of a Hausdorff Space

• Results about $T_2$ (Hausdorff) spaces can be found here.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 2$