Boundary of Subset of Discrete Space is Null

Theorem
Let $S$ be a set.

Let $\vartheta$ be the discrete topology on $S$.

Let $A \subseteq S$.

Then:
 * $\partial A = \varnothing$

where:
 * $\partial A$ is the boundary of $A$.

Proof
Let $T = \left({S, \vartheta}\right)$ be the discrete space on $S$.

Then by definition $\vartheta = \mathcal P \left({S}\right)$, that is, is the power set of $S$.

Let $A \subseteq S$.

Then from All Sets in Discrete Topology are Clopen it follows that $A$ is both open and closed in $T$.

The result follows from Set Clopen iff Boundary is Empty.