Boundary of Boundary of Subset of Indiscrete Space

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

Let $H \subseteq S$.

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

where $\partial H$ denotes the boundary of $H$.

Proof
From Boundary of Subset of Indiscrete Space, either $\partial H = S$ or $\partial H = \O$, depending on whether $H = \O$ or $H = S$ or not.

From Open and Closed Sets in Topological Space, $\O$ and $S$ are both closed and open in $T$.

So from Set Clopen iff Boundary is Empty:
 * $\map \partial {\partial H} = \O$