Difference between revisions of "Compact set"
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| − | + | '''Compactness''' is a [[topology | topological]] property that appears in a wide variety of contexts. In particular, it is a "tameness property" that tells you that the objects you are dealing with are in some sense well-behaved. | |
==Definition== | ==Definition== | ||
| − | Let <math>X</math> be a [[ | + | Let <math>X</math> be a [[topological space]] and let <math>S\subset X</math>. |
| − | + | A [[set]] of [[open set]]s <math>G_{\alpha}\subset X</math> is said to be an ''open [[cover]]'' of <math>S</math> if <math>S\subset\bigcup_{\alpha}G_{\alpha}</math>. | |
| − | + | The set <math>S</math> is said to be '''compact''' if and only if for every open cover <math>\{G_{\alpha}\}</math> of <math>S</math>, there exists a [[finite]] set <math>\{\alpha_1,\alpha_2,\ldots,\alpha_n\}</math> such that <math>\{G_{\alpha_k}\}_{k=1}^{n}</math> is also an open cover of <math>S</math>. This is often expressed in the sentence, "A set is compact if and only if every open cover admits a finite subcover." | |
| − | |||
| − | The set <math>S</math> is said to be ''' | ||
[[Category:Topology]] | [[Category:Topology]] | ||
{{stub}} | {{stub}} | ||
Latest revision as of 17:15, 26 February 2008
Compactness is a topological property that appears in a wide variety of contexts. In particular, it is a "tameness property" that tells you that the objects you are dealing with are in some sense well-behaved.
Definition
Let 
be a topological space and let 
.
A set of open sets 
is said to be an open cover of 
if 
.
The set is said to be compact if and only if for every open cover 
of , there exists a finite set 
such that 
is also an open cover of . This is often expressed in the sentence, "A set is compact if and only if every open cover admits a finite subcover."
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