The classification of compact 1-manifolds
A Lean 4 (mathlib) proof of the classification of 1-manifolds, specifically that every compact, connected, Hausdorff topological space charted on the half-line [0,∞) is homeomorphic to the circle S¹ or to the closed unit interval [0,1]. The proof follows David Gale's outline: shrink to a finite atlas of interval charts, then induct on the atlas size, merging two overlapping charts at each step; a connected overlap glues two charts into one (two boundary charts glue onto [0, 1]), while a disconnected overlap has exactly two components and yields the circle.
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