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

Let Bo(T, tau) be the Borel sigma-algebra generated by the topology tau 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 ifBo(C*(K),w*) = Bo(C*(K), parallel to.parallel to),where w* denotes the weak-star topology and parallel to.parallel to 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.