# Definition:Hausdorff Space/Definition 3

## Definition

Let $T = \struct {S, \tau}$ be a topological space.

$\struct {S, \tau}$ is a **Hausdorff space** or **$T_2$ space** if and only if:

- $\forall x, y \in S, x \ne y: \exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \subseteq U \subseteq N_x, y \subseteq V \subseteq N_y: N_x \cap N_y = \O$

That is:

- for any two distinct elements $x, y \in S$ there exist disjoint neighborhoods $N_x, N_y \subseteq S$ containing $x$ and $y$ respectively.

That is:

- $\struct {S, \tau}$ is a
**$T_2$ space**if and only if every two elements in $S$ are separated by neighborhoods.

## Also known as

This condition is known as **the Hausdorff condition**.

For short, **$T$ is Hausdorff** is used to mean **$T$ is a Hausdorff space**.

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

Some sources use the term **separated space** for **Hausdorff space** but this is discouraged as there already exists considerable confusion and ambiguity around the definition of the word **separated** in the context of topology.

Some authors **require** a space to be **Hausdorff** before allowing it to be classed as a topological space, but this approach is unnecessarily limiting.

## Also see

## Source of Name

This entry was named for Felix Hausdorff.

## Sources

- 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $3$: Topological Spaces: $\S 3$: Neighborhoods and Neighborhood Spaces: Definition $3.3$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Hausdorff space** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Hausdorff space**