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Suppose $X$ is a metric space and $S$ and $A$ are subsets of $X$.

If $S \subset A \subset Cl(S)$ , then $Cl(A) = Cl(S)$.

Also if $Int(S) \subset A \subset S$, then $Int(A) =Int(S)$.

What if, $Int(S) \subset A \subset Cl(S)$ , then would $Cl(A) = Cl(S)$ and $Int(A) =Int(S)$ ?

johny
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1 Answers1

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What if $X=\Bbb R$, $S=(0,1)\cup\{2\}$, and $A=(0,1)$? Or $S=(0,1)\cup(1,2)$ and $A=(0,2)$?

If $S$ is nice enough so that $\operatorname{cl}\operatorname{int}S=\operatorname{cl}S$, you can get one of the equalities that you want, and if $\operatorname{int}\operatorname{cl}S=\operatorname{int}S$, you can get the other; see if you can see which is which and then prove the results.

Brian M. Scott
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