Isn't this applicable to all topological n-manifolds including differentiable n-manifolds?

If that is the case,

By the definition of a topological manifold, every point in n-manifold M has a neighborhood which is homeomorphic to Euclidean space . That implies that there is an embedding of when k = n, where M is a topological n-manifold. The remaining steps are established by composing two embeddings of g and h if we define g and h such as

, , where .