We show that every connected graph has a spanning tree that displays all its topological ends. This proves a 1964 conjecture of Halin in corrected form, and settles a problem of Diestel from 1992. arXiv:1409.6640v5 [math.CO] 6 Feb 2018 Observation 2.5. For proper separations (A, B) and (C, D) the following are equivalent.