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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 $M$?2012-11-24
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    Where is this question from?2012-11-24
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    @Hurkyl, aren't they automatically open for being diffeomorphic to R^n?2012-11-25
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    @levap GTM33, page 21.2012-11-25
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    You can check out [this](http://arxiv.org/pdf/math/0404372.pdf) 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.2012-11-25
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    @lee: Mainly, I wanted confirmation that the choice of $U_n$'s is constrained by the hypothesis that $\bigcup U_n$ is actually a manifold. When I first read your question, my first thought was to set $U_n = \mathbb{R}^n$, and the union wouldn't have been a Euclidean space at all!2012-11-25

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