  We construct a ZFC example of a nonmetrizable compact space $K$ such that
every totally disconnected closed subspace $L\subseteq K$ is metrizable. In
fact, the construction can be arranged so that every nonmetrizable compact
subspace may be of fixed big dimension. Then we focus on the problem if a
nonmetrizable compact space $K$ must have a closed subspace with a
nonmetrizable totally disconnected continuous image. This question has several
links with the the structure of the Banach space $C(K)$, for example, by
Holszty\'nski's theorem, if $K$ is a counterexample, then $C(K)$ contains no
isometric copy of a nonseparable Banach space $C(L)$ for $L$ totally
disconnected. We show that in the literature there are diverse consistent
counterexamples, most eliminated by Martin's axiom and the negation of the
continuum hypothesis, but some consistent with it. We analyze the above problem
for a particular class of spaces. OCA+MA however, implies the nonexistence of
any counterexample in this class but the existence of some other absolute
example remains open.
