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

Abstract

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