Erdős spaces - Inaugural lecture by Honorary Member Jan van Mill

Erdős space E is the “rational” Hilbert space, that is, the set of vectors in l2 whose coordinates are all rational. Here l2 is the familiar Hilbert space of all sequences x = (x1, x2, x3,…) in R∞ which are square summable. Erdős space was introduced by Hurewicz, who was asked to compute its topological dimension. This problem was solved by Erdős in 1940; he showed that E is one-dimensional by establishing that every non-empty clopen subspace is unbounded. This result, combined with the obvious fact that E is homeomorphic to E x E, gives the space its importance in topological dimension theory. Complete Erdős space is the “irrational” Hilbert space, that is, the set of vectors in l2 whose coordinates are all irrational.

Jan van Mill
(For more pictures click on the photo)Photo: mta.hu / Tamás Szigeti

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 characterisations of Erdős spaces. As an application, it follows that if M is a, at a minimum, two-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 set-wise is homeomorphic to E. Homeomorphism groups are given the compact-open topology. Erdős spaces started out as curious examples in topological dimension theory. However, they turned out to be fundamental objects that surface in many places.