Filters, topologies, the Rado graph and the Urysohn space

TL;DR

This paper explores the automorphism groups of the Rado graph and homeomorphism groups of the Urysohn space, discussing their connections to topologies and filters on countable sets.

Contribution

It provides new insights into the structure of these groups and their relation to topological and filter-based concepts on countable sets.

Findings

Analysis of automorphism groups of the Rado graph

Connections between homeomorphism groups of the Urysohn space and filters

Further theoretical insights into topologies on countable sets

Abstract

My work with Anatoly Vershik concerned automorphism groups of the Rado graph and homeomorphism groups of the Urysohn space. This paper contains some further thoughts on these issues, together with connections to topologies and filters on countable sets.