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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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: $A^{{\circ }{\circ }}=A^{\circ }$.
• 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.

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