Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
    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
    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]
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook