Topological infinite gammoids, and a new Menger-type theorem for infinite graphs (Q1671662)

Summary: 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-subsets \(A\) and \(B\), 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 implies the topological Menger theorem of \textit{R. Diestel} [J. Comb. Theory, Ser. B 87, No. 1, 145--161 (2003; Zbl 1021.05063)] for locally finite graphs. It also implies a special case of the infinite Menger theorem of \textit{R. Aharoni} and \textit{E. Berger} [Invent. Math. 176, No. 1, 1--62 (2009; Zbl 1216.05092)].