Difference between revisions of "Compact set"

(New page: The notion of '''Compact sets''' is very important in the field of topology ==Definition== Let <math>X</math> be a metric space Let <math>S\subset X</math> A set of open set...)
 
 
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The notion of '''Compact sets''' is very important in the field of [[topology]]
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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 [[metric space]]
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Let <math>X</math> be a [[topological space]] and let <math>S\subset X</math>.
  
Let <math>S\subset X</math>
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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>.
  
A set of [[open set]]s <math>G_{\alpha}\subset X</math> is said to be an '''open cover''' of <math>S</math> iff <math>S\subset\displaystyle\cup_{\alpha}G_{\alpha}</math>
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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 '''Compact''' if and only if for every <math>\{G_{\alpha}\}</math> that is an open cover 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>
 
  
 
[[Category:Topology]]
 
[[Category:Topology]]
  
 
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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 $X$ be a topological space and let $S\subset X$.

A set of open sets $G_{\alpha}\subset X$ is said to be an open cover of $S$ if $S\subset\bigcup_{\alpha}G_{\alpha}$.

The set $S$ is said to be compact if and only if for every open cover $\{G_{\alpha}\}$ of $S$, there exists a finite set $\{\alpha_1,\alpha_2,\ldots,\alpha_n\}$ such that $\{G_{\alpha_k}\}_{k=1}^{n}$ is also an open cover of $S$. 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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