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
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
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.