Urysohn universal space

A metric space (U,d) is called Urysohn universal[1] if it is separable and complete and has the following property:

given any finite metric space X, any point x in X, and any isometric embedding f : X\{x} → U, there exists an isometric embedding F : X → U that extends f, i.e. such that F(y) = f(y) for all y in X\{x}.

If U is Urysohn universal and X is any separable metric space, then there exists an isometric embedding f:X → U. (Other spaces share this property: for instance, the space l^∞ of all bounded real sequences with the supremum norm admits isometric embeddings of all separable metric spaces ("Fréchet embedding"), as does the space C[0,1] of all continuous functions [0,1]→R, again with the supremum norm, a result due to Stefan Banach.)

Furthermore, every isometry between finite subsets of U extends to an isometry of U onto itself. This kind of "homogeneity" actually characterizes Urysohn universal spaces: A separable complete metric space that contains an isometric image of every separable metric space is Urysohn universal if and only if it is homogeneous in this sense.

Urysohn proved that a Urysohn universal space exists, and that any two Urysohn universal spaces are isometric. This can be seen as follows. Take ${\displaystyle (X,d),(X',d')}$, two Urysohn universal spaces. These are separable, so fix in the respective spaces countable dense subsets ${\displaystyle (x_{n})_{n},(x'_{n})_{n}}$. These must be properly infinite, so by a back-and-forth argument, one can step-wise construct partial isometries ${\displaystyle \phi _{n}:X\to X'}$ whose domain (resp. range) contains ${\displaystyle \{x_{k}:k<n\}}$ (resp. ${\displaystyle \{x'_{k}:k<n\}}$). The union of these maps defines a partial isometry ${\displaystyle \phi :X\to X'}$ whose domain resp. range are dense in the respective spaces. And such maps extend (uniquely) to isometries, since a Urysohn universal space is required to be complete.