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 $A^{\circ}$ . 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.