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Problem is this: suppose a manifold $$M=\bigcup_{n\in\mathbb{N}} U_n,$$ where each $U_n$ is diffeomorphic to Euclidean space, and $U_n$ is contained in $U_{n+1}$. Then please show that $M$ is diffeomorphic to Euclidean space.

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I suppose you meant to say that each $U_i$ is an open subset of $M$? –  Hurkyl Nov 24 '12 at 17:50
    
Where is this question from? –  levap Nov 24 '12 at 19:45
    
@Hurkyl, aren't they automatically open for being diffeomorphic to R^n? –  lee Nov 25 '12 at 5:38
    
@levap GTM33, page 21. –  lee Nov 25 '12 at 5:39
    
You can check out this article. It shows it under weaker hypothesis that doesn't involve smoothness, but it also seems that the original article which Hirsch refers to is also about topological spaces, with no mention of a smooth structure. It is interesting whether the smoothness assumption can possibly simply things further. –  levap Nov 25 '12 at 6:08
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