Product of compact sets compact in box topology?

  • #1

Main Question or Discussion Point

So Tychonoff theorem states products of compact sets are compact in the product topology.

is this true for the box topology? counterexample?
 

Answers and Replies

  • #2
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A counterexample is [itex]\prod_{n\in \mathbb{N}}{[0,1]}[/itex]. Can you show why?
 
  • #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
 
  • #4
22,097
3,282
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
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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