Set is Clopen iff Boundary is Empty

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Theorem

Let $T$ be a topological space, and let $H \subseteq T$.


Then $H$ is both closed and open in $T$ if and only if:

$\partial H = \varnothing$

where $\partial H$ is the boundary of $H$.


Proof

From Set is Open iff Disjoint from Boundary we have that:

$H$ is open in $T$ if and only if $\partial H \cap H = \varnothing$


From Set is Closed iff it Contains its Boundary we have that:

$H$ is closed in $T$ if and only if $\partial H \subseteq H$


From the above it follows that $\partial H = \varnothing$ if and only if $H$ is closed and $H$ is open.

$\blacksquare$


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