Boise Set Theory Seminar
Thursday, March 21 from 1:30 to 2:30pm
Title: Of the Urysohn Space $U$, Space-Filling Curves, and Isometric Embeddings of $U$ in $C[0,1]$
Speaker: M. Randall Holmes
Dr Coskey recently discussed the universal separable metric space of Urysohn. If you missed his talk, do not fear to come to mine: the essential properties of this space $U$ that are needed for my development will be reviewed in my talk. It is also known that $C[0,1]$, the space of continuous functions on the unit interval with the sup metric, is a universal separable metric space. I will prove this result in my talk: the proof uses space-filling curves. $U$ and $C[0,1]$ are both universal separable metric spaces up to isometry. Of course, since both are universal, $U$ can be embedded isometrically into $C[0,1]$. Sierpinski asked about this (to all appearances rather casually): of course $U$ can be embedded into $C[0,1]$ using his general method involving space-filling curves, but can it be embedded in some other way? I answered this question, and the answer is No. Embeddings of $U$ in $C[0,1]$ always involve space-filling curves in a certain sense. Further, the structure of an isometric copy of $U$ in $C[0,1]$ containing the zero function is exactly determined: if one of the points of $U$ is designated as 0, the norm of any linear combination of points of $U$ when isometrically embedded into $C[0,1]$ (and so into any Banach space) is as large as the distances among the points (and the point designated as 0) permits, and so uniquely determined. Any metric copy of U in a Banach space containing 0 has a uniquely determined Banach space as its linear closure! This is not true for familiar spaces!