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∘. 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, A∘⊆A.
* The interior of a open set G is just G itself, G=G∘.
* Interior is idempotent: A∘∘=A∘.
* Interior distributes over finite intersection: (A∩B)∘=A∘∩B∘.
* 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)∘=X−A‾;X−A‾=X−A∘.