Prove that $Fr(Fr(Fr(A))) = Fr(Fr(A))$

Question

I want to learn topology and I found this excercise in my book: Prove that $Fr(Fr(Fr(A))) = Fr(Fr(A))$ (in any topological space). By definition $Fr(Fr(Fr(A))) = Cl(Fr(Fr(A))) \setminus Int(Fr(Fr(A)))$. What is more $Cl(Fr(Fr(A)) = Fr(Fr(A))$. So I need to prove that $Int(Fr(Fr(A))) = \emptyset $, but here I'm stuck. Please help me? :)

Answer 1

The boundary of any closed set is nowhere dense. $\operatorname{Fr}(A)$ is a closed set. So that interior is empty.