Prove that $IntCl(A)$ is regular open

Question

A set $A$ is said to be regular open if $IntCl(A)=A$

I am trying to prove that $IntCl(A)$ is regular open for any $A\subseteq X$

One of the inclusions is direct,

$Int(Cl(A))\subseteq Int(Cl(Int(Cl(A))))$

I have problems proving the inverse inclusion.

Answer 1

As Daniel Schepler marked, $Int(Cl(A))\subseteq Cl(A)$. This is due to the fact that $Int(A)\subseteq A\subseteq Cl(A)$. So, if we substitute $Cl(A)$ by $A$, $$Int(Cl(A))\subseteq Cl(A)\subseteq Cl(Cl(A))=Cl(A)$$ As a consequence,$$Int(Cl(Int(Cl(A))))\subseteq Int(Cl(Cl(A)))=Int(Cl(A)),$$ as you wanted to prove.

Answer 2

$$Int(Cl(A))=Int(Int(Cl(A)))\subseteq Int(Cl(Int(Cl(A))))\subseteq Int(Cl(Cl(A)))=Int(Cl(A))$$