Boise Set Theory Seminar, Boise, February 2013

Abstract: Is there a single separable metric space which contains all the others? This question was answered in the 1920’s by Banach and Mazur, who showed that $C[0,1]$ is such a space. But around the same time Urysohn gave another example (now called Urysohn space $U$) which additionally exhibits strong symmetry properties. Recently Urysohn’s construction has found numerous generalizations and applications. I’ll give a (modern) presentation of the construction, and briefly mention a couple of these recent results.