On Borel structures in the Banach space C(\beta\omega)

TL;DR

This paper investigates the differences between Borel structures generated by various topologies in the Banach space C(eta ext{ω}) and shows that certain topologies produce distinct Borel structures, with implications for function separation.

Contribution

It proves that the Borel structures generated by the weak and pointwise topologies in C(eta ext{ω}) are different, extending Talagrand's results and analyzing the structure in C( ext{ω}*).

Findings

Borel structures from weak and pointwise topologies differ in C(eta ext{ω}).

No countable family of pointwise Borel sets can separate functions in C( ext{ω}*).

The results extend understanding of topological and measure-theoretic properties in Banach spaces.

Abstract

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*).