===== Hausdorff space =====
==== Set ====
| @#FFBB00: definiendum | @#FFBB00: $\langle X,\mathcal{T}_X\rangle \in\mathrm{it} $ |
| @#AAFFAA: inclusion   | @#AAFFAA: $\langle X,\mathcal{T}_X\rangle$ ... topological space |
| @#FFFDDD: for all     | @#FFFDDD: $x,y\in X, x\ne y$ |
| @#FFFDDD: exists      | @#FFFDDD: $U_x\in$ Neighbourhood$(\mathcal{T}_X,x)$, $V_y\in$ Neighbourhood$(\mathcal{T}_X,y)$ |
| @#55EE55: postulate   | @#55EE55: $U_x\cap V_y=\emptyset$  |

==== 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.

{{hausdorff.png?X350}}

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: 
[[https://en.wikipedia.org/wiki/Neighbourhood_%28mathematics%29|Neighbourhood]]

==== Parents ====
=== Subset of ===
[[Topological space]]
=== Requirements ===
[[Neighbourhood]]