Closure operator

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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, ${\displaystyle F:{𝒫}X\to {𝒫}X}$, satisfying:

${\displaystyle A\subseteq B\Rightarrow FA\subseteq FB;}$

${\displaystyle A\subseteq FA;}$

${\displaystyle FFA=FA.}$

A topological closure operator satisfies the further property

${\displaystyle F(A\cup B)=FA\cup FB.}$

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

Closure system[edit]

A closure system is the set of closed sets of a closure operator. A closure system is defined as a family ${\displaystyle {𝒞}}$ of subsets of a set X which contains X and is closed under taking arbitrary intersections:

${\displaystyle {𝒮}\subseteq {𝒞}\Rightarrow \cap {𝒮}\in {𝒞}.}$

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

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

Examples[edit]

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