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I'm trying to show that a point is in the boundary of $E$ if and only if it belongs to the closure of both $E$ and its complement.

Let $\overline{E}$ denote the closure of $E$ and $E^\circ$ be the interior of $E$. Then the boundary is defined to be $\overline{E} \setminus E^\circ$.

So what I need to show is $\overline{E} \setminus E^\circ = \overline{E} \cap \overline{(S \setminus E)}$.

After looking at this link: A point is in the boundary of $E$ if and only if it belongs to the closure of both $E$ and its complement. I am confused about how the OP concluded that he/she needed to show $S \setminus E^\circ = \overline{(S \setminus E)}$.

Could someone explain why?

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Student
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1 Answers1

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$\newcommand{\cl}{\operatorname{cl}}\newcommand{\int}{\operatorname{int}}$Suppose that you can show that $S\setminus\int E=\cl(S\setminus E)$. Then

$$(\cl E)\setminus\int E=\big((\cl E)\cap S\big)\setminus\int E=(\cl E)\cap(S\setminus\int E)=(\cl E)\cap\cl(S\setminus E)\;,$$

which is exactly what you want.

To see where the idea comes from, note that both $(\cl E)\setminus\int E$ and $(\cl E)\cap\cl(S\setminus E)$ are subsets of $\cl E$. If $A$ is any set disjoint from $\cl E$, therefore,

$$(\cl E)\setminus\int E=(\cl E)\cap\cl(S\setminus E)\quad\text{iff}\quad A\cup\Big((\cl E)\setminus\int E\Big)=A\cup\Big((\cl E)\cap\cl(S\setminus E)\Big)\;,$$

and a clever choice of $A$ might give us sets that are easier to think about than $(\cl E)\setminus\int E$ and $(\cl E)\cap\cl(S\setminus E)$. In particular, if we let $A=S\setminus\cl E$, we can compare the set,

$$(S\setminus\cl E)\cup\Big((\cl E)\setminus\int E\Big)\;,$$

which is simply all of $S$ except $\int E$, or $S\setminus\int E$, with the set

$$(S\setminus\cl E)\cup\Big((\cl E)\cap\cl(S\setminus E)\Big)\;,$$

which is simply $\cl(S\setminus E)$, and try to show that these are equal.

Brian M. Scott
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  • Thank you very much! I follow the first 3 lines. It will take me sometime to digest the rest. – Student May 25 '13 at 02:37
  • @Student: You’re very welcome; feel free to leave a comment if you get stuck on any of it. – Brian M. Scott May 25 '13 at 02:38
  • Dear @BrianM.Scott I'm reading the following proof here on MathStack Exchange, could you give some help to understand it? I think that we need to use this question, but I don't know how to do it. – Math Mar 29 '23 at 11:13
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    @Math23: I see that you accepted the answer by Oliver Diaz; did that answer your questions satisfactorily? – Brian M. Scott Mar 29 '23 at 20:49
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    @BrianM.Scott Yes thanks! You are an expert in topology, I was not able to use the results so I asked for your help. – Math Mar 29 '23 at 21:49