Topological infinite gammoids, and a new Menger-type theorem for infinite graphs

Abstract

Answering a question of Diestel, we develop a topological notion of gammoids in infinite graphs which, unlike traditional infinite gammoids, always define a matroid. As our main tool, we prove for any infinite graph $G$ with vertex sets $A$ and $B$ that if every finite subset of $A$ is linked to $B$ by disjoint paths, then the whole of $A$ can be linked to the closure of $B$ by disjoint paths or rays in a natural topology on $G$ and its ends. This latter theorem re-proves and strengthens the infinite Menger theorem of Aharoni and Berger for `well-separated' sets $A$ and $B$. It also implies the topological Menger theorem of Diestel for locally finite graphs.