Computable neighbourhoods of points in semicomputable manifolds

We examine conditions under which a semicomputable set in a computable metric space contains computable points. We prove that computable points in a semicomputable set S are dense if S is a manifold (possibly with boundary) or S has the topological type of a polyhedron. Moreover, we find conditions under which a point in some set has a computable compact neighbourhood in that set. In particular, we show that a point x in a semicomputable set has a computable compact neighbourhood if x has a neighbourhood homeomorphic to Euclidean space.