nLab Brown-Stallings lemma

Statement

Let $X$ be a differentiable manifold (paracompact). If every compact subspace of $X$ is contained in an open subset diffeomorphic to Euclidean space (with its standard smooth structure), then all of $X$ is diffeomorphic to Euclidean space: $X \simeq \mathbb{R}^n$.

See for instance Milnor 64, lemma 3 (p. 168)

Related entries

• Reeb sphere theorem

References

• John Milnor, Differenial topology, chapter 6 in T. L. Saaty (ed.) Lectures On Modern Mathematic II 1964 (web, pdf)