Closure operator

In mathematics a closure operator is a unary operator or function on subsets of a given set which maps a subset to a containing subset with a particular property.

A closure operator on a set X is a function C on the power set of X, $C:{\mathcal {P}}X\rightarrow {\mathcal {P}}X$, satisfying:

$A\subseteq B\Rightarrow CA\subseteq CB;\,$

$A\subseteq CA;\,$

$CCA=CA.\,$

A topological closure operator satisfies the further property

$C(A\cup B)=CA\cup CB.\,$