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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 treedecomposition 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.

A subgraph‐universal graph/a topological minor‐universal graph in a class of graphs is a graph in which contains every graph in as a subgraph/topological minor. We prove that the class of all countable planar graphs does not contain a topological minor‐universal graph. This answers a question of Diestel and Kühn and strengthens a result of Pach stating that there is no subgraph‐universal graph in . Furthermore, we characterise for which subdivided stars there is a topological minor‐universal graph in the class of all countable ‐free graphs.