(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.
Continue reading Facts about the Urysohn Space – Some useful, some cool