The boundary of the set is the boundary of its complement

up:: Boundary (Topology)

Let $S\subseteq X$ be a set in a Topological Space $(X,\tau )$. Denote the Closure by $(\cdot {)}^{-}$ and the Interior as $(\cdot {)}^{\circ }$.

Then its Boundary is

$$\partial S=S^{-}\setminus S^{\circ }=S^{-}\cap {S^{\circ }}^C$$

where ${\cdot }^C$ is the set’s Complement.

Using that The closure is the complement of the interior of the complement and flipping the intersection, we have

$$\partial S={S^{\circ }}^C\cap {{S^C}^{\circ }}^C={S^{\circ }}^C\setminus {(S^C)}^{\circ }$$

Using that The interior is the complement of the closure of the complement yields

$$\partial S={S^{\circ }}^C\cap {{S^C}^{\circ }}^C={{{S^C}^{-}}^C}^C\setminus {(S^C)}^{\circ }={(S^C)}^{-}\setminus (S^C{)}^{\circ }=\partial (S^C)$$

Thus, $\partial S=\partial (S^C)$.