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

Abstract

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.