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