# Closure Operator

In mathematics, a closure operator on a set S is a function from the power set of S to itself which satisfies the following conditions for all sets

 (cl is extensive) (cl is increasing) (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 "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 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 topological closure operators, which must satisfy

for all (Note that for this gives ).

In algebra and logic, many closure operators are finitary closure operators, i.e. they satisfy

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.

### Other articles related to "closure operator, closure, operator":

Closure Operator - History
... The concept of a closure is due to E ... form of general analysis, whereas that of a closure subset originated in the work of Frigyes Riesz in connection with topological spaces ...
Closure (mathematics) - Closure Operator
... Given an operation on a set X, one can define the closure C(S) of a subset S of X to be the smallest subset closed under that operation that contains ... For example, the closure of a subset of a group is the subgroup generated by that set ... The closure of sets with respect to some operation defines a closure operator on the subsets of X ...
Consequence Operator - Closure Operators On Partially Ordered Sets
... a partial order P to itself is called a closure operator if it satisfies the following axioms for all elements x, y in P ... k ≤ idP is called a kernel operator, interior operator, or dual closure ... B, then the self-map on the powerset of B given by μA(X) = A ∪ X is a closure operator, whereas λA(X) = A ∩ X is a kernel operator ...
Antimatroid - Convex Geometries
... See also Convex set, Convex geometry, and Closure operator If F is the set system defining an antimatroid, with U equal to the union of the sets in F, then ... To be a closure operator, τ should have the following properties τ(∅) = ∅ the closure of the empty set is empty ... The family of closed sets resulting from a closure operation of this type is necessarily closed under intersections ...
Consequence Operator - Closed Sets
... The closed sets with respect to a closure operator on S form a subset C of the power set P(S) ... X of S the smallest set Y ∈ C such that X ⊆ Y is a closure operator ... A closure operator on a set is topological if and only if the set of closed sets is closed under finite unions, i.e ...