Boundary of Boundary is not necessarily Equal to Boundary

Theorem
Let $T$ be a topological space.

Let $H \subseteq T$.

Let $\partial H$ denote the boundary of $H$.

While it is true that:
 * $\map \partial {\partial H} \subseteq \partial H$

it is not necessarily the case that:
 * $\map \partial {\partial H} = \partial H$

Proof
From Boundary of Boundary is Contained in Boundary, we have that:


 * $\map \partial {\partial H} \subseteq \partial H$

It remains to be proved that the equality does not always hold.

Proof by Counterexample:

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.

Let $H \subseteq S$ such that $H \ne \O$ and $H \ne S$.

From Boundary of Subset of Indiscrete Space:


 * $\partial H = S$

From Boundary of Boundary of Subset of Indiscrete Space:


 * $\map \partial {\partial H} = \O$

The result follows.