Duality Between Interior and Closure Under Complementation

Let $(X,\mathcal{T})$ be a topological space, and let $A\subseteq X$. Write $X\setminus S$ for the complement relative to $X$ of a subset $S\subseteq X$, and let $\operatorname{int}_X$ and $\operatorname{cl}_X$ denote the interior and the closure in $X$.

Then the following hold.

1. $X\setminus\operatorname{int}_X(A)=\operatorname{cl}_X(X\setminus A)$.

2. $X\setminus\operatorname{cl}_X(A)=\operatorname{int}_X(X\setminus A)$.