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?