Difference between revisions of "Closed set"
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| − | In topology, a '''closed set''' is | + | In [[topology]], a '''closed set''' is a set 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 if]] its [[complement]] is an [[open set]], or alternatively if its [[closure]] is equal to itself. |
| − | One common example of a closed set is | + | In any topological space, the empty set and the entire space are both closed sets. The intersection of any collection of closed sets is a closed set. Also, the union of any two closed sets (or any finite number of closed sets) is a closed set. Note, however, that the union of an arbitrary collection of closed sets does not have to be closed. For example, every one-point set is closed in <math>\mathbb{R}</math> (or in any <math>T_1</math> space), but clearly the union of any arbitrary number of one-point sets does not need to be closed. |
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| + | One common example of a closed set is the 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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| + | The pre-image of a closed set under a [[continuous]] function is also closed. | ||
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[[Category:Topology]] | [[Category:Topology]] | ||
Latest revision as of 18:33, 1 March 2010
In topology, a closed set is a set 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 if its complement is an open set, or alternatively if its closure is equal to itself.
In any topological space, the empty set and the entire space are both closed sets. The intersection of any collection of closed sets is a closed set. Also, the union of any two closed sets (or any finite number of closed sets) is a closed set. Note, however, that the union of an arbitrary collection of closed sets does not have to be closed. For example, every one-point set is closed in 
(or in any 
space), but clearly the union of any arbitrary number of one-point sets does not need to be closed.
One common example of a closed set is the 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.
The pre-image of a closed set under a continuous function is also closed.
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