product of compact sets compact in box topology?

spicychicken

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

is this true for the box topology? counterexample?

micromass

A counterexample is [itex]\prod_{n\in \mathbb{N}}{[0,1]}[/itex]. Can you show why?

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

micromass

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 A_i=[0,0.6[ or A_i=]0.5,1]