Interior (topology)

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This editable Main Article is under development and subject to a disclaimer.

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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 ∘ {\displaystyle A^{\circ }} . 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 {\displaystyle A^{\circ }\subseteq A} .
* The interior of a open set G is just G itself,  G = G ∘ {\displaystyle G=G^{\circ }} .
* Interior is idempotent:  A ∘ ∘ = A ∘ {\displaystyle A^{{\circ }{\circ }}=A^{\circ }} .
* Interior distributes over finite intersection:  ( A ∩ B ) ∘ = A ∘ ∩ B ∘ {\displaystyle (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 ) ∘ = X − A ¯ ; X − A ¯ = X − A ∘ . {\displaystyle (X-A)^{\circ }=X-{\overline {A}};~~{\overline {X-A}}=X-A^{\circ }.}