# multivariable calculus – Corollary of The Immersion Theorem

Here’s what I’m trying to prove;

Let $$D subseteq mathbb{R}^d$$ be a non-empty open set. Let $$d < n$$ and $$k in mathbb{N}_{infty}$$. Let $$phi: D to mathbb{R}^n$$ be a $$C^k$$ embedding. Then, $$phi(D)$$ is a $$C^k$$ submanifold in $$mathbb{R}^n$$ of dimension $$d$$.

The corollary is from Duistermaat and Kolk’s Multidimensional Analysis I so I am using definitions and so on from that book.

Attempted Proof:

Observe that $$phi: D to phi(D)$$ is a homeomorphism and that $$Dphi(y)$$ is injective for every $$y in D$$. Let $$x in phi(D)$$. Then, there exists a unique $$y in D$$ such that $$phi(y) = x$$. By the Immersion Theorem, there exists an open set around $$y$$, call it $$U_y$$, such that $$phi(U_y)$$ is a $$C^k$$ submanifold in $$mathbb{R}^n$$ of dimension $$d$$.

Since $$x in phi(U_y)$$, there exists an open set $$V_x$$ around $$x$$ such that $$phi(U_y) cap V_x$$ is the graph of some $$C^k$$ function $$f: mathbb{R}^d supsetto mathbb{R}^{n-d}$$ defined on an open set. Since $$phi$$ is a homeomorphism, $$phi(U_y)$$ is open and so, $$phi(U_y) cap V_x$$ is open around $$x$$. Then, observe that $$phi(D) cap (phi(U_y) cap V_x) = phi(U_y) cap V_x$$ is equal to the graph of some $$C^k$$ function (the same one as above). This proves that $$phi(D)$$ is a $$C^k$$ submanifold in $$mathbb{R}^n$$ of dimension $$d$$ at $$x$$. Since $$x$$ was arbitrary, we have our result. $$Box$$

Does the argument work? If it doesn’t, then why? How can I fix it?