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, $F:{\mathcal {P}}X\rightarrow {\mathcal {P}}X$, satisfying:

$A\subseteq B\Rightarrow FA\subseteq FB;\,$

$A\subseteq FA;\,$

$FFA=FA.\,$

A topological closure operator satisfies the further property

$F(A\cup B)=FA\cup FB.\,$

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 ${\mathcal {C}}$ of subsets of a set X which contains X and is closed under taking arbitrary intersections:

${\mathcal {S}}\subseteq {\mathcal {C}}\Rightarrow \cap {\mathcal {S}}\in {\mathcal {C}}.\,$

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

$FA=\bigcap _{A\subseteq C\in {\mathcal {C}}}C.\,$

Examples

In many algebraic structures the set of substructures forms a closure system. The corresponding closure operator is often written $\langle A\rangle$ 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 ${\overline {A}}$. It may also be obtained as the union of the set A with its limit points.