  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.
