# 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.

$\blacksquare$

## Sources

- Mizar article TOPGEN_1:4