## Hausdorff space

### Set

definiendum | $\langle X,\mathcal{T}_X\rangle \in\mathrm{it} $ |

inclusion | $\langle X,\mathcal{T}_X\rangle$ … topological space |

for all | $x,y\in X, x\ne y$ |

exists | $U_x\in$ Neighbourhood$(\mathcal{T}_X,x)$, $V_y\in$ Neighbourhood$(\mathcal{T}_X,y)$ |

postulate | $U_x\cap V_y=\emptyset$ |

### Discussion

#### Idea

A Hausdorff space $\langle X,\mathcal{T}_X\rangle$ is one where the topology $\mathcal{T}_X$ is fine enough so that separate points also have seperate neighbourhoods.

Also, boobs.

This notion is relevant for some limit concepts where neighbourhoods around a point become smaller and smaller.

#### Examples

Any metric space.

#### Non-examples

An ordered set like $\mathbb R$ and the right-ordered topology, i.e. the “infinite to the right” sets ${x\|\ x>a}$. Here a neighbourhood of $3$ can not be small enough to not contain the number $7$.

#### Reference

Wikipedia: Neighbourhood