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 ${\displaystyle \bar{A}}$. 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, ${\displaystyle A\subseteq \bar{A}}$.
• The closure of a closed set F is just F itself, ${\displaystyle F=\bar{F}}$.
• Closure is idempotent: ${\displaystyle \overline{\stackrel{‾}{A}}=\bar{A}}$.
• Closure distributes over finite union: ${\displaystyle \overline{A\cup B}=\bar{A}\cup \bar{B}}$.
• 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.

${\displaystyle (X-A{)}^{\circ }=X-\bar{A};\overline{X-A}=X-A^{\circ }.}$