For a second-countable space $X$, we have $${\rm Bor}(X\times X) = {\rm Bor}\, X \otimes {\rm Bor}\, X,$$ that is the Borel $\sigma$-algebra of the product is the product $\sigma$-algebra. Some [silly counterexamples to this statement can be produced for uncountable discrete spaces][1], *etc*. However, I was wondering what happens in this particular example of 

$X = \beta N$, 

the Stone-Cech compactification of the integers? Does this formula still hold?


  [1]: https://mathoverflow.net/questions/39882/product-of-borel-sigma-algebras