Closure (topology)

From Citizendium, the Citizens' Compendium
Jump to: navigation, search
This article is a stub and thus not approved.
Main Article
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
This editable Main Article is under development and not meant to be cited; by editing it you can help to improve it towards a future approved, citable version. These unapproved articles are 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 . 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.


  • A set is contained in its closure, .
  • The closure of a closed set F is just F itself, .
  • Closure is idempotent: .
  • Closure distributes over finite union: .
  • 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.