Semicomputable manifolds in computable topological spaces

We study computable topological spaces and semicomputable and computable sets in these spaces. In particular, we investigate conditions under which semicomputable sets are computable in ambient spaces which are second countable Hausdorff. We prove that a semicomputable compact manifold M is computable if its boundary ∂M is computable. We also show how this result combined with a certain construction which compactifies a semicomputable set leads to the conclusion that some noncompact semicomputable manifolds in computable metric spaces are computable.