Let be a non-empty differentiable manifold of dimension . Prove there exists an embedding for all .
For the sphere it's clear that the stereographic projection is an embedding, but I can't prove the general case. Any ideas would be apreciated.
Let be a non-empty differentiable manifold of dimension . Prove there exists an embedding for all .
For the sphere it's clear that the stereographic projection is an embedding, but I can't prove the general case. Any ideas would be apreciated.
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 .