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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-startopology and $\|\cdot\|$ is the dual norm with respect to the sup-norm on thespace of real-valued continuous functions $C(K)$. Furthermore we study thetopological properties of the Hausdorff compact spaces $K$ such that everysubset is a Borel set. In particular we show that, if the axiom of choice holdstrue, then $K$ is scattered.

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