Boundary is Intersection of Closure with Closure of Complement

Theorem
Let $T$ be a topological space.

Let $X \subseteq T$.

Let $\partial X$ be the boundary of $X$.

Let $\overline X = T \setminus X$ be the complement of $X$ in $T$.

Let $X^-$ be the closure of $X$.

Then:
 * $\partial X = X^- \cap \left({\overline X}\right)^-$