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 . 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.