The Urysohn universal space is a certain metric space that contains all separable metric spaces in a particularly nice manner.
This mathematics concept is due to Pavel Urysohn.
A metric space (U,d) is called Urysohn universal[1] if it is separable and complete and has the following property: 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.)
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.
, two Urysohn universal spaces.
These are separable, so fix in the respective spaces countable dense subsets
These must be properly infinite, so by a back-and-forth argument, one can step-wise construct partial isometries
The union of these maps defines a partial isometry
range are dense in the respective spaces.
And such maps extend (uniquely) to isometries, since a Urysohn universal space is required to be complete.