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

Abstract

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 if, $$Bo(C^*(K),w^*)=Bo(C^*(K),\|\cdot\|);$$ where $w^*$ denotes the weak-star topology and $\|\cdot\|$ 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.