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

Abstract:

Let $\operatorname {Bo}(T,\tau )$ 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 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 $C(K)$. 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.