A bit more topology ... for now

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(Difference between revisions)
(→General Definition of Compactness)
(→General Definition of Compactness)
 
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   A set <m>A</m> is compact <m>\\leftright</m> every open "covering" of  
   A set <m>A</m> is compact <m>\\leftright</m> every open "covering" of  
   <m>A</m> contains a finite "sub-covering". i.e <br> if <m> A \\subset bigcup{i \\in I}{}{U_i} </m>, with all sets <m> U_i</m> open sets, then there exists<br> a finite set of indices  
   <m>A</m> contains a finite "sub-covering". i.e <br> if <m> A \\subset bigcup{i \\in I}{}{U_i} </m>, with all sets <m> U_i</m> open sets, then there exists<br> a finite set of indices  
-
<m> J = \\( i_1,...,i_m \\) \\subset I</m> such that <m> A \\subset U_{i_1} bigcup{}{} ...  bigcup{}{} U_{i_m} </m>
+
<m> J = \\delim{lbrace} i_1,...,i_m \\delim{rbrace} \\subset I</m> such that <m> A \\subset U_{i_1} bigcup{}{} ...  bigcup{}{} U_{i_m} </m>

Current revision as of 18:57, 11 February 2006

Contents

[edit] Compact Sets

[edit] Speaking Loosely

  • A compact set in <m>\\bbR</m> is any finite union of disjoint closed intervals
  • <m> A \\subset \\bbR^n</m> is compact <m>\\leftright~A</m> is closed and bounded.

[edit] Speaking Tightly :)

[edit] General Definition of Compactness

  A set <m>A</m> is compact <m>\\leftright</m> every open "covering" of 
  <m>A</m> contains a finite "sub-covering". i.e 
 if <m> A \\subset bigcup{i \\in I}{}{U_i} </m>, with all sets <m> U_i</m> open sets, then there exists
 a finite set of indices

<m> J = \\delim{lbrace} i_1,...,i_m \\delim{rbrace} \\subset I</m> such that <m> A \\subset U_{i_1} bigcup{}{} ... bigcup{}{} U_{i_m} </m>

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