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...)
 
m (Definition)
Line 6: Line 6:
 
Let <math>S\subset X</math>
 
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> iff <math>S\subset\displaystyle\cup_{\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\bigcup_{\alpha}G_{\alpha}</math>
  
 
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>
 
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>

Revision as of 05:10, 23 February 2008

The notion of Compact sets is very important in the field of topology

Definition

Let $X$ be a metric space

Let $S\subset X$

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

The set $S$ is said to be Compact if and only if for every $\{G_{\alpha}\}$ that is an open cover 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 article is a stub. Help us out by expanding it.