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Product of compact sets compact in box topology?

  1. Jul 20, 2011 #1
    So Tychonoff theorem states products of compact sets are compact in the product topology.

    is this true for the box topology? counterexample?
     
  2. jcsd
  3. Jul 21, 2011 #2

    micromass

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    A counterexample is [itex]\prod_{n\in \mathbb{N}}{[0,1]}[/itex]. Can you show why?
     
  4. Jul 21, 2011 #3
    if S_n is the set with empty sets in each index except n where for index n you have [0,1], then {S_n} is an open cover with no finite subcover...i think
     
  5. Jul 21, 2011 #4

    micromass

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    Such a sets will always be empty. Try to consider a cover by all sets of the form

    [tex]\prod_{n\in \mathbb{N}}{A_i}[/tex]

    Where Ai=[0,0.6[ or Ai=]0.5,1]
     
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