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We want to show that the boundary $\partial M$ of an $n-$manifold M is a closed subset of the manifold. We show that its complement $M\setminus\partial M$ is open in $M$. Indeed, each point $x \in M\setminus\partial M$ has an open neighborhood $V_x\subseteq M$ homeomorphic to $\mathbb R^n$. It remains only to show that $V_x$ lies entirely in $M\setminus \partial M$ which means that $V_x\cap \partial M=\emptyset$. Thank you for your help!

EDIT: In a manifold with boundary each point has an open neighborhood that is homeomorphic to $\mathbb R^n$ or to $\mathbb R^n_+=\{(x_1,\cdots,x_n)\in\mathbb R^n\;|\; x_n\ge 0\}$, the points who have open neighborhoods homeomorphic to $\mathbb R^n_+$ form the boundary of the manifold.

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  • $\begingroup$ How do you find your $V_x$? $\endgroup$ – user99914 Nov 25 '17 at 6:51
  • $\begingroup$ It is by definition of a point $\endgroup$ – palio Nov 25 '17 at 7:53
  • $\begingroup$ Can you also include the definition of "manifold with boundary" in your question? $\endgroup$ – user99914 Nov 25 '17 at 7:55
  • $\begingroup$ I included a definition, thank you! $\endgroup$ – palio Nov 25 '17 at 8:04
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    $\begingroup$ Your definition is not completely correct. The boundary point are those points lying in $x_n = 0$. (there are points in $\mathbb R^n_+$ which are not boundary points). $\endgroup$ – user99914 Nov 25 '17 at 8:07

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