We prove that the edge-end space of an infinite graph is metrizable if and only if it is first-countable. This strengthens a recent result by Aurichi, Magalhaes Jr. \ and Real (2024). Our central graph-theoretic tool is the use of tree-cut decompositions, introduced by Wollan (2015) as a variation of tree decompositions that is based on edge cuts instead of vertex separations. In particular, we give a new, elementary proof for Kurkofka's result (2022) that every infinite graph has a tree-cut decomposition of finite adhesion into its ω-edge blocks. Along the way, we also give a new, short proof for a classic result by Halin (1984) on K₊, ⏙-subdivisions in k-connected graphs, making this paper self-contained.
Max Pitz (Tue,) studied this question.