The Erdős–Menger conjecture for source/sink sets with disjoint closures

Erdős conjectured that, given an infinite graph G and vertex sets A , B ⊆ V ( G ) , there exist a set P of disjoint A–B paths in G and an A–B separator X ‘on’ P , in the sense that X consists of a choice of one vertex from each path in P . We prove the conjecture for vertex sets A and B that have disjoint closures in the usual topology on graphs with ends. The result can be extended by allowing A, B and X to contain ends as well as vertices.