Complement of Interior equals Closure of Complement

Theorem
Let $T$ be a topological space.

Let $H \subseteq T$.

Let $H^-$ denote the closure of $H$ and $H^\circ$ denote the interior of $H$.

Let $\complement \left({H}\right)$ be the complement of $H$ in $T$:


 * $\complement \left({H}\right) = T \backslash H$

Then:
 * $\complement \left({ H^\circ}\right) = \left({\complement \left({H}\right)}\right)^-$

and similarly:


 * $\left({\complement \left({ H}\right)}\right)^\circ = \complement \left({H^-}\right)$

These can alternatively be written:


 * $T \setminus H^\circ = \left({T \setminus H}\right)^-$


 * $\left({T \setminus H}\right)^\circ = T \setminus H^-$

which is arguably easier to follow.