---
tag: 0MXB4CKHFW
kind: definition
name: Hausdorff space
msc: [54D10]
defines: 0S1FJ8VEX9
uses: [0CNQXVBHCX, 01ZQQDCERP]
sources:
  - url: https://en.wikipedia.org/wiki/Hausdorff_space
    locator: Definitions
---

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is Hausdorff when for all points $x \neq y$ of $X$ there are [[open sets|01ZQQDCERP]] $U$ and $V$ with $x \in U$, $y \in V$ and $U \cap V = \varnothing$.

## Remarks

Each [[metric space is Hausdorff|0P9DCDES73]]. Subspaces and products of Hausdorff spaces are Hausdorff.
In a Hausdorff space, [[limits of sequences are unique|0D8T984XEE]] and [[compact subsets are closed|0MYFJVMN7M]].
Each Hausdorff space is [[T1|05GQJXBP3F]]. An infinite set with the [[cofinite topology|0B906ZBTMZ]] is T1 but not Hausdorff.
The Hausdorff condition is also called T2.

## Explanations

### L1 human

In a Hausdorff space, you can put any two distinct points into two open sets that do not meet.
Most spaces of analysis, such as $\mathbb{R}^n$, are Hausdorff.