From Vectorcalcumb

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 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.

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 
 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 = \\( i_1,...,i_m \\) \\subset I</m> such that <m> A \\subset U_{i_1} bigcup{}{} ... bigcup{}{} U_{i_m} </m>