Closure (topology)

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.

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