End spaces and tree-decompositions

We present a systematic investigation into how tree-decompositions of finite adhesion capture topological properties of the space formed by a graph together with its ends. As main results, we characterise when the ends of a graph can be distinguished, and characterise which subsets of ends can be displayed by a tree-decomposition of finite adhesion. In particular, we show that a subset Ψ of the ends of a graph G can be displayed by a tree-decomposition of finite adhesion if and only if Ψ is G_δ (a countable intersection of open sets) in |G|, the topological space formed by a graph together with its ends. Since the undominated ends of a graph are easily seen to be G_δ, this provides a structural explanation for Carmesin's result that the set of undominated ends can always be displayed.