  M. Talagrand showed that, for the Cech-Stone compactification \beta\omega\ of
the space of natural numbers, the norm and the weak topology generate different
Borel structures in the Banach space C(\beta\omega). We prove that the Borel
structures in C(\beta\omega) generated by the weak and the pointwise topology
are also different.
  We also show that in C(\omega*), where \omega*=\beta\omega - \omega, there is
no countable family of pointwise Borel sets separating functions from
C(\omega*).
