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 $\Psi$ of the ends of a graph $G$ can be displayed by a tree-decomposition of finite adhesion if and only if $\Psi$ is $G_\delta$ (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 $G_\delta$, this provides a structural explanation for Carmesin's result that the set of undominated ends can always be displayed.