Urysohn universal space

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.