Boundary of Empty Set is Empty
Let $T$ be a topological space.
- $\partial_T \O = \O$
where $\partial_T \O$ denotes the boundary in topology $T$ of $\O$.
- $\partial_T \O = \O^- \cap \relcomp T \O^-$
where $\O^-$ denotes the closure of $\O$.
- $\O^- = \O$
Thus the result follows by Intersection with Empty Set.
The result follows from Set is Clopen iff Boundary is Empty.