Closure operator

In mathematics, a closure operator on a set S is a function cl : P ( S ) → P ( S ) {displaystyle operatorname {cl} :{mathcal {P}}(S) ightarrow {mathcal {P}}(S)} from the power set of S to itself which satisfies the following conditions for all sets X , Y ⊆ S {displaystyle X,Ysubseteq S} In mathematics, a closure operator on a set S is a function cl : P ( S ) → P ( S ) {displaystyle operatorname {cl} :{mathcal {P}}(S) ightarrow {mathcal {P}}(S)} from the power set of S to itself which satisfies the following conditions for all sets X , Y ⊆ S {displaystyle X,Ysubseteq 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 'Moore families', in honor of E. H. Moore who studied closure operators in his 1910 Introduction to a form of general analysis, whereas the concept of the closure of a subset originated in the work of Frigyes Riesz in connection with topological spaces. Closure operators are also called 'hull operators', which prevents confusion with the 'closure operators' studied in topology. A set together with a closure operator on it is sometimes called a closure space.