##  [Closure Operator](/index.php/closure-operator) 

  ##  [Closure Operator](https://puremath.quantumdictionary.io/closure-operator-0) 

  

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Definition

A closure operator on a set X is a map cl: P(X) → P(X) that assigns to each subset A its closure cl(A) and satisfies three axioms: extensivity (A ⊆ cl(A)), monotonicity (A ⊆ B ⇒ cl(A) ⊆ cl(B)), and idempotence (cl(cl(A)) = cl(A)). In topology cl(A) is the smallest closed superset of A.