# Product of compact sets compact in box topology?

1. Jul 20, 2011

### spicychicken

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

is this true for the box topology? counterexample?

2. Jul 21, 2011

### micromass

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

3. Jul 21, 2011

### 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

4. Jul 21, 2011

### micromass

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]