Let $X$ be a topological space, $A\subset X$ and $B=Fr(A)$.
I have two questions.
How do you prove that if $A$ is open or closed, then $B^\circ=\emptyset$?
Why can this not be proven for a set that is both open and closed?
Prove that the interior of the boundary of an open or closed set is empty.
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general-topology
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0Just to be clear, what is your definition of boundary? – 2017-01-09
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0@Exodd. $ Fr(A)=\overline{A}\cap \overline{(A^c)} $ – 2017-01-09
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0By your definition, $Fr(X) = Fr(\emptyset) = \emptyset$, so even if $A$ is both open and closed, the interior of $B$ is empty.. – 2017-01-09
1 Answers
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$$(Fr(A))^\circ=(\overline{A}\cap \overline{A^c})^\circ=(\overline{A})^\circ\cap (\overline{A^c})^\circ=A^\circ\cap (A^c)^\circ=A^\circ\cap (\overline{A})^c=\emptyset$$
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1It’s not necessarily true that $\operatorname{int}\operatorname{cl}A=\operatorname{int}A$: consider the set $A=(0,1)\cup(1,2)$ in $\Bbb R$. – 2017-01-09