Compact manifolds with computable boundaries

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 semicomputable compact manifold with computable boundary is computable. In particular, each semi-computable compact (boundaryless) manifold is computable.