Description
A MAGYAR TUDOMÁNYOS AKADÉMIA MATEMATIKAI TUDOMÁNYOK OSZTÁLYA
tisztelettel meghívja Önt Jan van Mill -az MTA tiszteleti tagja- Erdős spaces címmel tartott előadására.
Abstract:
Erdős space $E$ is the `rational' Hilbert space, that is the set of vectors in $\ell^2$ which coordinates are all rational. Here $\ell^2$ is the familiar Hilbert space of all sequences $x=(x_1,x_2,x_3,\dots)\in\mathbb R^\infty$ such that $\sum_{i=1}^\infty x_i^2 < \infty$. Erdős space was introduced by Hurewicz who asked to compute its topological dimension. This problem was solved by Erdős in 1940; he showed that $E$ is 1-dimensional by establishing that every nonempty clopen subspace is unbounded. This result, in combination with the obvious fact that $E$ is homeomorphic to $E\times E$, lends the space its importance in topological dimension theory. Complete Erdős space is the `irrational' Hilbert space, that is the set of vectors in $\ell^2$ which coordinates are all irrational. This space from 1940 surfaced later in topological dynamics as the `endpoint set' of several interesting objects. The author in collaboration with Dijkstra obtained several increasingly powerful topological characterizations of the Erdős spaces. As an application it follows that if $M$ is an at least 2-dimensional manifold (with or without boundary) and $D$ is a countable dense subset of $M$ then the group of homeomorphisms of $M$ that fix $D$ is homeomorphic to $E$. Homeomorphism groups are given the compact-open topology. The Erdős spaces started their careers as curious examples in topological dimension theory. It turned out however that they are fundamental objects that surface in many places.