# Interior (topology)

From Citizendium

In mathematics, the **interior** of a subset *A* of a topological space *X* is the union of all open sets in *X* that are subsets of *A*. It is usually denoted by . It may equivalently be defined as the set of all points in *A* for which *A* is a neighbourhood.

## Properties

- A set contains its interior, .
- The interior of a open set
*G*is just*G*itself, . - Interior is idempotent: .
- Interior distributes over finite intersection: .
- 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.