Closure

In topology, the closure of a set <math>A</math> is the intersection of all closed sets containing <math>A</math>. Equivalently, the closure of <math>A</math> is the union of <math>A</math> and all limit points of <math>A</math>

A closure operator is an abstract (category theory) form of the topological notion of closure which can be applied to any set <math>A</math>. It is a function <math>cl</math> from <math>A</math> to the power set of <math>A</math> satisfying the following conditions:

• <math>A\in cl(A)</math> (augmentation)
• If <math>B\subset A</math> then <math>cl(B)\subset cl(A)</math> (monotonicity)
• <math>cl(A)\in cl(A)</math> (idempotence)

In the category of topological spaces, this operator is isomorphic to the standard topological one.