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

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.