For any topological space, we have

$$ Int(Cl(A\cap B))=Int(Cl(A))\cap Int(Cl(B)), $$

where $A,B$ are open sets. As a corollary, for any $A,B$, we have

$$ Int(Cl(Int(A\cap B)))=Int(Cl(Int(A)))\cap Int(Cl(Int(B))). $$

A related inclusion diagram, where $\to$ means $\subseteq$.