Final answer:
To answer the question, one must demonstrate that around any point p where an immersion ϕ is injective, there exists a local neighborhood V in which ϕ restricts to an embedding, utilizing the conceptual framework of the Inverse Function Theorem to prove the existence of this neighborhood V.
Step-by-step explanation:
The question pertains to showing the existence of a neighborhood V within a manifold M at a point p such that the restriction of an immersion ϕ: M → N to that neighborhood is an embedding. An immersion is a differentiable map between differentiable manifolds that is injective at the level of tangent spaces. We need to prove that, locally around p, this immersion behaves like an injective map overstating that ϕ is not only injective on the tangent spaces but also injective within a neighborhood of p.
To show this, we rely on the fact that an immersion is locally an embedding around any point where it is injective. We can use the Inverse Function Theorem which states that if ϕ is an immersion at point p, then there is a neighborhood V of p in which ϕ is a diffeomorphism onto its image; hence, it is an embedding on V. Therefore, there exists such a neighborhood where the mapping is both open and injective; this neighborhood is what we need to demonstrate the claim.