Boundary of Empty Set is Empty
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Theorem
Let $T$ be a topological space.
Then:
- $\partial_T \O = \O$
where $\partial_T \O$ denotes the boundary in topology $T$ of $\O$.
Proof 1
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.
$\blacksquare$
Proof 2
From Open and Closed Sets in Topological Space, $\O$ is clopen in $T$.
The result follows from Set is Clopen iff Boundary is Empty.
$\blacksquare$