Boundary of Empty Set is Empty

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Let $T$ be a topological space.


$\partial_T \O = \O$

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.


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.