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Let $S$ be a subset of $R^n$. Show that $S$ is closed if and only if the boundary of $S$ belongs to $S$.

I was thinking about using definitions of closed points and boundary points to prove this, but failed. Can anyone show me how to do this?

How can I prove that? Thanks!

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    What are your definitions of *boundary point* and *closed set*?2012-10-28

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Here I'm asumming $\partial E = \{ x : \text{ every open ball around $x$ contains points of $E$ and $E^c$}\}$

Suppose $\partial E \subseteq E$. Then let $x\in E^c$, then since $\partial E\subset E$ we must have some open ball which contains only points of $E^c$ around $x$, so $E^c$ is open, and hence $E$ is closed.

Now suppose that $E$ is closed. Then $E^c$ is open, so for every $x\in E^c$ we have an open ball around $x$ which is contained completely in $E^c$. This means that $E^c \cap \partial E = \emptyset$, and hence $\partial E \subset E$.

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    Oh sorry just looking you have a typo I think check it!! It is an ok proof. Yet you might want to state why the intersect of E(c) is and the boundary of E is empty. Then state why that fact dictates or shows that delta E is a subset of E. I have proven something similar and the last questions I gave are the answers I am pondering. I believe it is more appropriate to state that delta E(c) intersect delta E = the empty set, since all sets have some kind of defined boundary.2016-04-16