Set is Closed iff Equals Topological Closure/Proof 2

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Let $T$ be a topological space.

Let $H \subseteq T$.

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

$H = \map \cl H$


Let $H^\complement$ denote the relative complement of $H$ in $T$.

By definition, we have that $H$ is closed in $T$ if and only if $H^\complement$ is open in $T$.

By Set is Open iff Neighborhood of all its Points, this is equivalent to:

$\forall x \in H^\complement: \exists U \in \tau: x \in U \subseteq H^\complement$

By Empty Intersection iff Subset of Complement, we have that:

$U \subseteq H^\complement \iff U \cap H = \O$

By Condition for Point being in Closure, it follows that $H^{\complement}$ is open in $T$ if and only if:

$\forall x \in H^\complement: x \notin \map \cl H$

By the Rule of Transposition, this is equivalent to $\map \cl H \subseteq H$.

From Set is Subset of its Topological Closure, we have that $H \subseteq \map \cl H$.

By definition of set equality, this is equivalent to:

$H = \map \cl H$