Urysohn Sphere

The Urysohn sphere is the unique complete metric space, up to isometry, that is separable, has all distances at most 1, contains an isometric copy of every separable metric space with distances at most 1, and has the following extension property. Every isometry between two finite subsets, with their induced metrics, extends to an isometry of the entire space onto itself (Sabok 2016).

The notation denotes the bounded version of the universal Urysohn metric space, rather than an ordinary Euclidean sphere. Its extension property has a useful equivalent description in terms of its metric . If is a finite set and satisfies

for every , then some satisfies for every . These are exactly the triangle inequalities needed to adjoin one more point, allowing to coincide with an existing point when a prescribed distance is 0.