# Closure (topology)

From Citizendium

In mathematics, the **closure** of a subset *A* of a topological space *X* is the set union of *A* and *all* its limit points in *X*. It is usually denoted by . Other equivalent definitions of the closure of A are as the smallest closed set in *X* containing *A*, or the intersection of all closed sets in *X* containing *A*.

## Properties

- A set is contained in its closure, .
- The closure of a closed set
*F*is just*F*itself, . - Closure is idempotent: .
- Closure distributes over finite union: .
- The complement of the closure of a set in
*X*is the interior of the complement of that set; the complement of the interior of a set in*X*is the closure of the complement of that set.