Facts about the Urysohn Space – Some useful, some cool

(This is almost verbatim the talk I gave recently (Feb 23, 2012) at the Toronto Student Set Theory and Topology Seminar. I will be giving this talk again on April 5, 2012)

I have been working on a problem involving the Urysohn space recently, and I figured that I should fill people in with the basic facts and techniques involved in this space. I will give some useful facts, a key technique and 3 cool facts. First, the definition!

Definition: A metric space U has the Urysohn property if

  • U is complete and separable
  • U contains every separable metric space as an isometric copy.
  • U is ultrahomogeneous in the sense that if A,B are finite, isometric subspaces of U then there is an automorphism of U that takes A to B .

You might already know a space that satisfies the first two properties – The Hilbert cube [0,1]^\omega or C[0,1] the continuous functions from [0,1] to [0,1] . However, these spaces are not ultrahomogeneous. Should a Urysohn space even exist? It does, but the construction isn’t particularly illuminating so I will skip it.

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