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Topology lecture this Friday

Dr. Jan van Mill

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.

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