Difference between revisions of "Closed set"
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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. | 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. | ||
| − | 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. | + | 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. However, closed subsets of <math>\mathbb{R}</math> can take a variety of more complicated forms. For example, the set <math>\left\{\frac{1}{n} \mid n \in \mathbb{Z}_{> 0} \right\} \cup \{0\}</math> is closed, as is the [[Cantor set]]. |
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[[Category:Topology]] | [[Category:Topology]] | ||
Revision as of 22:01, 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 ![$[a, b] = \{x \mid a \leq x \leq b\}$](http://latex.artofproblemsolving.com/1/3/9/13960dbd81a2bcac0fa24c031283321ec49b7365.png)
of the real numbers. However, closed subsets of 
can take a variety of more complicated forms. For example, the set 
is closed, as is the Cantor set.
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