---
tag: MF3BWZ9YSW
kind: concept
name: Urysohn space
msc: [54D10]
specializes: [0S1FJ8VEX9]
sources:
  - url: https://en.wikipedia.org/wiki/Urysohn_and_completely_Hausdorff_spaces
---

## Summary

An Urysohn space is a topological space in which any two distinct points have disjoint closed neighbourhoods.
This separation axiom, also written $T_{2\frac12}$, lies between [[Hausdorff|0S1FJ8VEX9]] and regular Hausdorff.
Every Urysohn space is Hausdorff, and every regular Hausdorff space is Urysohn.
Some authors use the name for the stronger notion of a completely Hausdorff space.

## Explanations

### L1 human

In a Hausdorff space, two points can be put in two open sets that do not meet.
An Urysohn space asks a little more: the closures of these open sets must not meet either.
So there is a real gap between the two points, not just a thin border.
Every metric space is an Urysohn space.
For points $x \ne y$ at distance $r$, the closed balls of radius $r / 3$ around them are disjoint neighbourhoods.