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?
Boundary of the boundary of a manifold is empty
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manifolds
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3Of 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
1 Answers
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Let us define a (topological) $n$-manifold with boundary to be a (Hausdorff, second-countable) topological space $M$ locally homeomorphic to the closed half space $H$ in $\mathbb R^n$, and the boundary $\partial M$ of $M$ to be the subset of $M$ of points which do not have a neighborhood homeomorphic to an open set in $\mathbb R^n$.
Then:
show that the claim that $\partial\partial M=\emptyset$ follows from the observation that $\partial\partial H=\emptyset$;
show that $\partial\partial H=\emptyset$.
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1@Mariano: Can you give a hint as to how to proceed further? – 2015-10-24