# Product of compact sets compact in box topology?

by spicychicken
Tags: compact, product, sets, topology
 P: 3 So Tychonoff theorem states products of compact sets are compact in the product topology. is this true for the box topology? counterexample?
 Mentor P: 18,330 A counterexample is $\prod_{n\in \mathbb{N}}{[0,1]}$. Can you show why?
 P: 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
Mentor
P: 18,330
Product of compact sets compact in box topology?

 Quote by spicychicken 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

$$\prod_{n\in \mathbb{N}}{A_i}$$

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

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