Interior

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Definition

Suppose X is a topological space and A is a subset of X. The interior of A in X is defined in the following equivalent ways:

  1. It is the unique largest open subset of X that is contained in A.
  2. It is the union of all open subsets of X contained in A.
  3. It is the set of all points a \in A for which there exists an open subset U \ni a that is completely contained in A.
  4. It is the set-theoretic complement in X of the closure of the complement of A in X.

Related notions

  • Open subset: A subset is open iff it equals its own interior.
  • Regular open subset: A subset is regular open iff it equals the interior of its closure.