On Borel structures in the Banach space C(βω)C(βω)

M. Talagrand showed that, for the Čech–Stone compactification βω of the space of natural numbers ω  , the norm and the weak topology generate different Borel structures in the Banach space C(βω)C(βω). We prove that the Borel structures in C(βω)C(βω) generated by the weak and the pointwise topology are also different. We also show that in C(ω⁎)C(ω⁎), where ω⁎=βω∖ωω⁎=βω∖ω, there is no countable family of pointwise Borel sets separating functions from C(ω⁎)C(ω⁎).