A note on weak-star and norm Borel sets in the dual of the space of continuous functions.

Let Bo(T,τ) be the Borel σ-algebra generated by the topology τ on T. In this paper we show that if K is a Hausdorff compact space, then every subset of K is a Borel set if and only if Bo(C*(K), w* = Bo(C*(K),|⋅|), where w* denotes the weak-star topology and |⋅| is the dual norm with respect to the sup-norm on the space of real-valued continuous functions C(K). Furthermore, we study the topological properties of the Hausdorff compact spaces K such that every subset is a Borel set. In particular, we show that if the axiom of choice holds true, then K is scattered. [ABSTRACT FROM AUTHOR]