Boundary of Empty Set is Empty/Proof 1

Proof
By Boundary is Intersection of Closure with Closure of Complement:
 * $\partial_T \O = \O^- \cap \relcomp T \O^-$

where $\O^-$ denotes the closure of $\O$.

By Closure of Empty Set is Empty Set:
 * $\O^- = \O$

Thus the result follows by Intersection with Empty Set.