Closure operator

In mathematics, a closure operator on a set S is a function ${\displaystyle \operatorname {cl} :{\mathcal {P}}(S)\rightarrow {\mathcal {P}}(S)}$ from the power set of S to itself that satisfies the following conditions for all sets ${\displaystyle X,Y\subseteq S}$

${\displaystyle X\subseteq \operatorname {cl} (X)}$	(cl is extensive),
${\displaystyle X\subseteq Y\Rightarrow \operatorname {cl} (X)\subseteq \operatorname {cl} (Y)}$	(cl is increasing),
${\displaystyle \operatorname {cl} (\operatorname {cl} (X))=\operatorname {cl} (X)}$	(cl is idempotent).

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".^[1] A set together with a closure operator on it is sometimes called a closure space. Closure operators are also called "hull operators", which prevents confusion with the "closure operators" studied in topology.