Interior (topology)

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 $$A^{\circ}$$. 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, $$A^{\circ} \subseteq A$$.
 * The interior of a open set G is just G itself, $$G = G^{\circ}$$.
 * Interior is idempotent: $$A^ = A^{\circ}$$.
 * Interior distributes over finite intersection: $$(A \cap B)^{\circ} = A^{\circ} \cap B^{\circ}$$.
 * 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.