The boundary of a domain without a closed subset Code Answer

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Let $D$ be a bounded domain in $mathbb{R}^{N}$ ($Ngeq2$) and $E$ a closed subset of $D$ with empty interior. Show that the boundary of $Dsetminus E$ is the union of $E$ and the boundary of $D$:
$$partial(Dsetminus E)=partial Dcup E.$$

Answer

Let $A:=Dsetminus E$. Then
begin{align*}
partial A&=overline{A}setminus A\
&=overline{D}setminus (Dsetminus E)\
&=overline{D}cap D^{complement}cup (overline{D}cap E)\
&=partial Dcup S,
end{align*}
where the bar over the set indicates its closure and the top “C” is for the complement.

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