Idempotence of the interior of the closure I'm reading the Complex Analysis text by Ahlfors. I'm stuck on exercise 5 on chapter 3: Prove that $\overline{\overline{\overline{{\overline{X}}^c}^c}^c}^c=\overline{{\overline{X}}^c}^c$. I've manged to rephrase the question as $ Int(Cl(Int(Cl(X))))=Int(Cl(X))$. I've proven one inclusion like so: $Int(Cl(X)) \subseteq Cl(Int(Cl(X)))$, taking interior of both sides gives $LHS \supseteq RHS$. How can I prove the other inclusion? Is there a way of proving both at once? Thank you! P.S. in the book, topological spaces haven't been defined yet. So it might be true for metric spaces only.

Provided that you know the following:

For any set $A$, $Int(A)\subset A \subset Cl(A)$ and for any sets $A \subset B$, $Int(A) \subset Int(B)$ and $Cl(A) \subset Cl(B)$

As you said, $Int(Cl(A))\subset Cl(Int(Cl(A)))$ and taking interiors gives $Int(Int(Cl(A))) \subset Int(Cl(Int(Cl(A))))$ so $Int(Cl(A)) \subset Int(Cl(Int(Cl(A))))$.

For the other direction, note that $Int(Cl(A)) \subset Cl(A)$ which gives $Cl(Int(Cl(A))) \subset Cl(Cl(A))=Cl(A)$ and this $Int(Cl(Int(Cl(A)))) \subset Int(Cl(A))$.

This is in fact true for any topological space, as is the related fact: $Cl(Int(Cl(Int(A)))) = Cl(Int(A))$. These two statements tell us that for a given set $A$ we cant obtain a maximum of $7$ sets (including $A$) by taking interiors and closures. A good exercise is to find some $A \subset \mathbb{R}$ where all of the $7$ possible sets are distinct!