Topology lecture this Friday

Nipissing University’s department of Computer Science and Mathematics welcomes Dr. Jan van Mill, professor of Topology at the University of Amsterdam, to campus for a special lecture on Friday, October 13, at 1p.m. in room A129. The lecture is titled Homogeneity in Erdos type spaces.

Here’s an abstract:

For subsets $X$ of the real line we investigate homogeneity properties of the Erd\H{o}s type space $E(X) = \{p\in \ell^2 : (\forall\, n)(p_n\in X)\}$. This space has much in common with the countable infinite product of copies of $X$, which is homogeneous by the result of Lawrence. Continuous families of coordinate permutations form an important ingredient in his proof. The Erd\H{o}s space $E(X)$ is 1-dimensional in many cases which is an obstacle in homogeneity issues. On the other hand, $E(X)$ is invariant under coordinate permutations of $\ell^2$ which suggests to investigate whether the Lawrence ideas are applicable. Our main result is the construction of a subset $X$ of the real line such that every homeomorphism $f$ of $E(X)$ is norm-preserving. That is, $\| f(p) \| = \| p\|$ for every $p\in E(X)$. Hence $E(X)$ need not be homogeneous. Since for every $\varepsilon > 0$, the sphere $\{p\in E(X) : \| p \| = \varepsilon\}$ is zero-dimensional, a natural question is whether spheres are always homogeneous in Erd\H{o}s type spaces. We prove that they are not.
This is joint work with K. P. Hart.