Closure (topology)

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

[edit intro]

In mathematics, the closure of a subset A of a topological space X is the set union of A and all its limit points in X. It is usually denoted by  A ¯ {\displaystyle {\overline {A}}} . Other equivalent definitions of the closure of A are as the smallest closed set in X containing A, or the intersection of all closed sets in X containing A.

## Properties[edit]

* A set is contained in its closure,  A ⊆ A ¯ {\displaystyle A\subseteq {\overline {A}}} .
* The closure of a closed set F is just F itself,  F = F ¯ {\displaystyle F={\overline {F}}} .
* Closure is idempotent:  A ¯ ¯ = A ¯ {\displaystyle {\overline {\overline {A}}}={\overline {A}}} .
* Closure distributes over finite union:  A ∪ B ¯ = A ¯ ∪ B ¯ {\displaystyle {\overline {A\cup B}}={\overline {A}}\cup {\overline {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 ∘ . {\displaystyle (X-A)^{\circ }=X-{\overline {A}};~~{\overline {X-A}}=X-A^{\circ }.}