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I'm very stuck on the following exercise in the book "A Comprehensive Introduction to Differential Geometry V.1" by Michael Spivak: Let $M^m$ be a smooth connected non-compact manifold. Show that $M$ is the union of a sequence of open sets $U_n$ with the following properties:

  1. $U_n \cap U_{n+1}$ is non-empty for all $n$
  2. For every compact set $C \subset M$ there is $N$ such that $U_n \cap C$ is empty for all $n>N$
  3. $U_n$ is diffeomorphic to $\mathbb{R}^m$ for all $n$.

Any help would be greatly appreciated. Thank you.

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    @user89499: Right, so that starting point is useless. Your observation means that all but one of the ends of the manifold must be covered by non-compact $U_n$... still seems feasible but I have no idea how to attack it.2013-08-20

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Proof idea: First choose an increasing sequence of connected compact subsets ${C_n}$ so that $\cup C_n = M.$ Then choose a cover $U=${$U_\alpha$} of M by open sets diffeomorphic to $\mathbf{R}^m$ which has no finite subcover. First find a finite cover for $C_1$ by sets in U to start your sequence of sets. Then find finite covers of $C_2 \setminus C_1, C_3\setminus C_2,$ etc to continue your sequence.

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    Ah actually this still doesn't help. Even in that case the sets can become disconnected.2012-11-20