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Revision as of 15:12, 10 February 2006

Contents 1 Compact Sets 1.1 Speaking Loosely 1.2 Speaking Tightly :) 1.2.1 General Definition of Compactness

Compact Sets

Speaking Loosely

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

Speaking Tightly :)

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 
 <m> A \\subset bigcup{i \\in I}{}{U_i} </m>
  if <m>A</m> is not closed.