Interior (topology)

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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 ${\displaystyle A^{\circ }}$. It may equivalently be defined as the set of all points in A for which A is a neighbourhood.

Properties[edit]

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

${\displaystyle (X-A)^{\circ }=X-{\overline {A}};~~{\overline {X-A}}=X-A^{\circ }.}$