# Definition:Hausdorff Space

## Contents

## Definition

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

### Definition 1

$\left({S, \tau}\right)$ is a **Hausdorff space** or **$T_2$ space** if and only if:

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

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.

### Definition 2

$\left({S, \tau}\right)$ is a **Hausdorff space** or **$T_2$ space** if and only if each point is the intersection of all its closed neighborhoods.

### Definition 3

$\left({S, \tau}\right)$ is a **Hausdorff space** or **$T_2$ space** if and only if:

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

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.

## 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.

## Equivalence of Definitions

See Equivalence of Definitions of $T_2$ Space for a proof that these definitions are equivalent.

Note that while some sources give Definition 1 and others Definition 3, it is rarely indicated specifically that the definitions are equivalent.

In particular, there may exist places on the internet where a page using Definition 3 may directly link to another page which uses Definition 1 without comment that the definitions are linguistically different.

## Also see

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

## Source of Name

This entry was named for Felix Hausdorff.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**T-axioms**or**Tychonoff conditions**:**2a.**(**$T_2$-space**) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Hausdorff space**