  We investigate conditions under which a co-computably enumerable closed set
in a computable metric space is computable and prove that in each locally
computable computable metric space each co-computably enumerable compact
manifold with computable boundary is computable. In fact, we examine the notion
of a semi-computable compact set and we prove a more general result: in any
computable metric space each semi-computable compact manifold with computable
boundary is computable. In particular, each semi-computable compact
(boundaryless) manifold is computable.
