Brown-Stallings lemma

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)

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

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