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, $F : \mathcal{P}X \rarr \mathcal{P}X$ , 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 stuctures 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.