Boundary of Empty Set is Empty/Proof 1

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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

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.

$\blacksquare$


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