# Definition:Hausdorff Space/Definition 1

## 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:

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

That is:

- for any two distinct elements $x, y \in S$ there exist disjoint open sets $U, V \in \tau$ 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 open sets.

## 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 note that a **Hausdorff space** is the same thing as **$T_2$ space**.

## Also see

## Source of Name

This entry was named for Felix Hausdorff.

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $4$: The Hausdorff condition: $4.2$: Separation axioms: Definition $4.2.1$ - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms - 1991: Walter Rudin:
*Functional Analysis*(2nd ed.) ... (previous) ... (next): $1.5$: Topological spaces