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Closure operator

In mathematics, a closure operator on a setS is a functioncl:P(S)→P(S){\displaystyle \operatorname {cl} :{\mathcal {P}}(S) ightarrow {\mathcal {P}}(S)} from the power set of S to itself that satisfies the following conditions for all sets X,Y⊆S{\displaystyle X,Y\subseteq S}Closure operators are determined by their closed sets, i.e., by the sets of the form cl(X), since the closure cl(X) of a set X is the smallest closed set containing X. Such families of "closed sets" are sometimes called closure systems or "Moore families".

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