(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 has the Urysohn property if
- is complete and separable
- contains every separable metric space as an isometric copy.
- is ultrahomogeneous in the sense that if are finite, isometric subspaces of then there is an automorphism of that takes to .
You might already know a space that satisfies the first two properties – The Hilbert cube or the continuous functions from to . 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.