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As mentioned in the title, it's well know that boundary of the boundary of a manifold is empty. That is, if $M$ is the boundary of a manifold $N$, i.e. $M=\partial N$, then $M$ is a manifold without boundary, i.e. $\partial M=\varnothing$. For example, the sphere $S^n$ has no boundary because $S^n=\partial B^{n+1}$ where $B^{n+1}$ is the closed unit ball in $\mathbb{R}^{n+1}$. What I would like to ask is that: is there an easy proof or a short proof for this statement?

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    What definition of "boundary of a manifold" are you using?2012-01-02
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    Are there several definitions of "boundary of a manifold"? I didn't know that. But the one I am using is this: http://en.wikipedia.org/wiki/Manifold#Manifold_with_boundary2012-01-02
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    You should probably get a better reference... that blurb in Wikipedia is not exactly a piece of great exposition!2012-01-02
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    Of course I know that. However, since you have asked me "What definition of "boundary of a manifold" are you using?", I just quoted the definition which is available from wiki.2012-01-02

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