Difference between revisions of "Cover"

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A [[set]] <math>\{S_i \mid i \in I\}</math> of sets is said to '''cover''' another set <math>S</math> if <math>S \subset \bigcup_{i \in I} S_i</math>.  
 
A [[set]] <math>\{S_i \mid i \in I\}</math> of sets is said to '''cover''' another set <math>S</math> if <math>S \subset \bigcup_{i \in I} S_i</math>.  
  
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The notion of covering is extremely broad, and mathematicians are often interested in covers where particular restrictions are placed on the <math>S_i</math>.  For example, if we have only [[finite]]ly many of the <math>S_i</math> (the index set <math>I</math> is finite), we have a ''finite cover.''  If <math>I</math> is [[countable]], we have a ''countable cover.''  In [[topology]], one may be interested in the case that the <math>S_i</math> are [[open]] sets, in which case we have an ''open cover.''
  
 
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Latest revision as of 17:12, 26 February 2008

A set $\{S_i \mid i \in I\}$ of sets is said to cover another set $S$ if $S \subset \bigcup_{i \in I} S_i$.

The notion of covering is extremely broad, and mathematicians are often interested in covers where particular restrictions are placed on the $S_i$. For example, if we have only finitely many of the $S_i$ (the index set $I$ is finite), we have a finite cover. If $I$ is countable, we have a countable cover. In topology, one may be interested in the case that the $S_i$ are open sets, in which case we have an open cover.

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