  We begin to study classical dimension theory from the computable analysis
(TTE) point of view. For computable metric spaces, several effectivisations of
zero-dimensionality are shown to be equivalent. The part of this
characterisation that concerns covering dimension extends to higher dimensions
and to closed shrinkings of finite open covers. To deal with zero-dimensional
subspaces uniformly, four operations (relative to the space and a class of
subspaces) are defined; these correspond to definitions of inductive and
covering dimensions and a countable basis condition. Finally, an effective
retract characterisation of zero-dimensionality is proven under an effective
compactness condition. In one direction this uses a version of the construction
of bilocated sets.
