Topological Closure is Closure Operator

Theorem

The topological closure operator is a closure operator.

Proof

Extensive

Follows from Set is Subset of its Topological Closure.

Increasing

Follows immediately from Topological Closure of Subset is Subset of Topological Closure.

Idempotent

Follows immediately from Closure of Topological Closure equals Closure.

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$\blacksquare$