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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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\ | + | 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 06:10, 23 February 2008
The notion of Compact sets is very important in the field of topology
Definition
Let 
be a metric space
Let 
A set of open sets 
is said to be an open cover of 
iff 
The set is said to be Compact if and only if for every 
that is an open cover of , there exists a finite set 
such that 
is also an open cover of
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