In mathematics, a closure operator on a Set (mathematics)|set S is a Function (mathematics)|function \operatorname{cl}: \mathcal{P}(S)\rightarrow \mathcal{P}(S) from the power set of S to itself which satisfies the following conditions for all sets 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 Moore families, in honor of E. H. Moore who studied closure operators in 1911. Closure operators are also called hull operators, which prevents confusion with the closure operators studied in point-set topology|topology. A set together with a closure operator on it is sometimes called a closure system.
Closure operators have many applications:In topology, the closure operators are Kuratowski closure axioms|topological closure operators, which must satisfy
: \operatorname{cl}(X_1 \cup\dots\cup X_n) = \operatorname{cl}(X_1)\cup\dots\cup \operatorname{cl}(X_n)
for all n\in\N (Note that for n=0 this gives \operatorname{cl}(\varnothing)=\varnothing).In algebra and logic, many closure operators are finitary closure operators, i.e. they satisfy
: \operatorname{cl}(X) = \bigcup\left\{\operatorname{cl}(Y) : Y\subseteq X \text{ and } Y \text{ finite} \right\}.
In universal logic, closure operators are also known as consequence operators.In the theory of partially ordered sets, which are important in theoretical computer science, closure operators have an alternative definition.
In mathematics, a closure operator on a Set (mathematics)|set S is a Function (mathematics)|function \operatorname{cl}: \mathcal{P}(S)\rightarrow \mathcal{P}(S) from the power set of S to itself which satisfies the following conditions for all sets 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 Moore families, in honor of E. H. Moore who studied closure operators in 1911. Closure operators are also called hull operators, which prevents confusion with the closure operators studied in point-set topology|topology. A set together with a closure operator on it is sometimes called a closure system.
Closure operators have many applications:In topology, the closure operators are Kuratowski closure axioms|topological closure operators, which must satisfy
: \operatorname{cl}(X_1 \cup\dots\cup X_n) =...