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 $\overline{A}$ . 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

A set is contained in its closure, $A \subseteq \overline{A}$ .
The closure of a closed set F is just F itself, $F = \overline{F}$ .
Closure is idempotent: $\overline{\overline A} = \overline A$ .
Closure distributes over finite union: $\overline{A \cup B} = \overline A \cup \overline B$ .
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.