Closure (topology)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Error in widget AddThis: unable to write file /var/www/html/citiz/public_html/wiki/extensions/Widgets/compiled_templates/wrt63858b11bec615_79524966

Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
Citable Version  [?]

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

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 ${\displaystyle {\overline {A}}}$. 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, ${\displaystyle A\subseteq {\overline {A}}}$.
• The closure of a closed set F is just F itself, ${\displaystyle F={\overline {F}}}$.
• Closure is idempotent: ${\displaystyle {\overline {\overline {A}}}={\overline {A}}}$.
• Closure distributes over finite union: ${\displaystyle {\overline {A\cup B}}={\overline {A}}\cup {\overline {B}}}$.
• 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.
${\displaystyle (X-A)^{\circ }=X-{\overline {A}};~~{\overline {X-A}}=X-A^{\circ }.}$