Closure (topology)

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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‾. 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‾.
* The closure of a closed set F is just F itself, F=F‾.
* Closure is idempotent: A‾‾=A‾.
* Closure distributes over finite union: 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∘.