---
tag: R1H00F1668
kind: object
name: Urysohn universal space
msc: ["54E35", "54E50"]
sources:
  - url: https://en.wikipedia.org/wiki/Urysohn_universal_space
same_as:
  wikidata: Q7901611
  wikipedia: Urysohn universal space
---

## Description

The *Urysohn universal space* $\mathbb{U}$ is a [[complete|0BY52JW6NK]] [[separable|0Z2P41SVDP]] [[metric space|0VKD1TP7F3]] with two properties.

1. Universality: every separable metric space is [[isometric|02HPP5VCZH]] to a subset of $\mathbb{U}$.
2. Ultrahomogeneity: every isometry between two finite subsets of $\mathbb{U}$ extends to an isometry of $\mathbb{U}$ onto itself.

Up to isometry, there is exactly one complete separable metric space with both properties.

## Remarks

An equivalent characterisation uses a one-point extension property.
Let $A \subseteq \mathbb{U}$ be finite, and let $A \cup \set{x}$ carry a metric extending that of $A$.
Then there is a point $u \in \mathbb{U}$ with $d(u, a) = d(x, a)$ for all $a \in A$.
A complete separable metric space with this property is isometric to $\mathbb{U}$.

The space $C[0, 1]$ with the sup norm is also universal for separable metric spaces, but it is not ultrahomogeneous.

## Explanations

### L1 human

The Urysohn space is one separable metric space that contains a perfect copy of every other one.
It is also as symmetric as possible: any finite configuration of points can be moved onto any other congruent configuration by a symmetry of the whole space.
It is built step by step, adding at each stage every point that a finite metric extension asks for.