Difference between revisions of "Closed set"

Revision as of 21:59, 28 February 2010

In topology, a closed set is one which contains all of its limit points. Equivalently, a set in some topological space (including, for example, any metric space) is closed if and only its complement is an open set, or alternatively if its closure is equal to itself.

One common example of a closed set is a closed interval $[a, b] = \{x \mid a \leq x \leq b\}$ of the real numbers.

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-In topology, a '''closed set''' is the [[complement]] of an [[open set]].
+In topology, a '''closed set''' is one which contains all of its [[limit point]]s.  Equivalently, a [[set]] in some [[topological space]] (including, for example, any [[metric space]]) is closed if and only its [[complement]] is an [[open set]], or alternatively if its [[closure]] is equal to itself.
-Equivalently, a set is closed if it contains all of its [[limit point]]s, or if its [[closure]] is equal to itself.
+One common example of a closed set is a closed interval <math>[a, b] = \{x \mid a \leq x \leq b\}</math> of the [[real number]]s.
-Under the [[standard topology]] of the real line, a closed set is the union of a number of disjoint closed intervals and closed rays.
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 [[Category:Topology]]