# 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 *F* on the power set of *X*, , satisfying:

A *topological closure operator* satisfies the further property

A *closed* set for *F* is one of the sets in the image of *F*

## Closure system

A **closure system** is the set of closed sets of a closure operator. A closure system is defined as a family of subsets of a set *X* which contains *X* and is closed under taking arbitrary intersections:

The closure operator *F* may be recovered from the closure system as

## Examples

In many algebraic structures the set of substructures forms a closure system. The corresponding closure operator is often written and termed the substructure "generated" or "spanned" by *A*. Notable examples include

- Subsemigroups of a semigroup
*S*. The semigroup generated by a subset*A*may also be obtained as the set of all finite products of one or more elements of*A*. - Subgroups of a group. The subgroup generated by a subset
*A*may also be obtained as the set of all finite products of zero or more elements of*A*or their inverses. - Normal subgroups of a group. The normal subgroup generated by a subset
*A*may also be obtained as the subgroup generated by the elements of*A*together with all their conjugates. - Submodules of a module (algebra) or subspaces of a vector space. The submodule generated by a subset
*A*may also be obtained as the set of all finite linear combinations of elements of*A*.

The principal example of a topological closure system is the family of closed sets in a topological space. The corresponding closure operator is denoted . It may also be obtained as the union of the set *A* with its limit points.