  We investigate computable metrizability of Polish spaces up to homeomorphism.
In this paper we focus on Stone spaces. We use Stone duality to construct the
first known example of a computable topological Polish space not homeomorphic
to any computably metrized space. In fact, in our proof we construct a
right-c.e. metrized Stone space which is not homeomorphic to any computably
metrized space. Then we introduce a new notion of effective categoricity for
effectively compact spaces and prove that effectively categorical Stone spaces
are exactly the duals of computably categorical Boolean algebras. Finally, we
prove that, for a Stone space $X$, the Banach space $C(X;\mathbb{R})$ has a
computable presentation if, and only if, $X$ is homeomorphic to a computably
metrized space. This gives an unexpected positive partial answer to a question
recently posed by McNicholl.
