nLab
Brown-Stallings lemma

Contents

Statement

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

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

References

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

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