Urysohn universal space

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, who presented an explicit construction. Another construction has been subsequently developed by Felix Hausdorff and a more general notion was discussed by Miroslav Katětov.

Definition

A metric space is called Urysohn universal if it is separable and complete and has the following property:

Properties

If U is Urysohn universal and X is any separable metric space, then there exists an isometric embedding f:''X → U''.
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.

Existence and uniqueness

Urysohn proved that a Urysohn universal space exists, and that any two Urysohn universal spaces are isometric. This can be seen as follows. Take, 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 whose domain contains . The union of these maps defines a partial isometry whose domain resp. range are dense in the respective spaces. And such maps extend to isometries, since a Urysohn universal space is required to be complete.