Closed set

From Citizendium
Jump to: navigation, search
This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

In mathematics, a set , where is some topological space, is said to be closed if , the complement of in , is an open set. The empty set and the set X itself are always closed sets. The finite union and arbitrary intersection of closed sets are again closed.

Examples

  1. Let X be the open interval (0, 1) with the usual topology induced by the Euclidean distance. Open sets are then of the form
    where and is an arbitrary index set (if then the open interval (a, b) is defined to be the empty set). The definition now implies that closed sets are of the form
    .
  2. As a more interesting example, consider the function space (with a < b). This space consists of all real-valued continuous functions on the closed interval [a, b] and is endowed with the topology induced by the norm
    In this topology, the sets
    and
    are open sets while the sets
    and
    are closed (the sets and are the closure of the sets and respectively).