  We prove that if $K$ and $L$ are compact spaces and $C(K)$ and $C(L)$ are
isomorphic as Banach spaces then $K$ has a $\pi$-base consisting of open sets
$U$ such that $\bar{U}$ is a continuous image of some compact subspace of $L$.
This gives some information on isomorphic classes of the spaces of the form
$C([0,1]^\kappa)$ and $C(K)$ where $K$ is Corson compact.
