Topological Closure is Closed

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Theorem

Let $T$ be a topological space.

Let $H \subseteq T$.


Then the closure $\cl H$ of $H$ is closed in $T$.


Proof

From Closure of Topological Closure equals Closure:

$\cl {\cl H} = \cl H$

From Set is Closed iff Equals Topological Closure, it follows that $\cl H$ is closed.

$\blacksquare$


Also see


Sources