Boundary of Subset of Discrete Space is Null

Theorem
Let $T = \left({S, \tau}\right)$ be a discrete topological space.

Let $A \subseteq S$.

Then:
 * $\partial A = \varnothing$

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

Proof
Let $A \subseteq S$.

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

The result follows from Set Clopen iff Boundary is Empty.