From Countable Ordinals to Infinite Hamming Cubes

Abstract: Countable Stone spaces—compact, Hausdorff, totally disconnected spaces familiar from Stone duality— are completely classified up to homeomorphism by countable ordinals. In this talk, we ask what the groups of homeomorphisms of these spaces look like from very far away.

The answer turns out to be surprisingly combinatorial. For most successor ordinals, their large-scale geometry is described by an infinite Hamming cube: vertices are binary sequences with finite support, and moving one step means changing a single coordinate. We'll explore how this Hamming cube arises naturally by looking at clopen partitions of a countable Stone space, and how the Cantor–Bendixson derivative lets us move between different ordinal ranks.

This is joint work with George Domat and Robert Alonzo Lyman.

general topologylogic

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic