Let $M$ be a $k$-dimensional topological submanifold of $\mathbb{R}^n$. Does there exist a homeomorphism $F$ from $\mathbb{R}^n$ to $\mathbb{R}^n$ such that $F(M)\subset \mathbb{R}^k$?
I know that for every $p\in M$, there exists an open sets $U_p\subset\mathbb{R}^n$, $V_p\subset\mathbb{R}^k$, and $W_p\subset\mathbb{R}^{n-k}$ and a homeomorphism $\phi_p:U_p\to V_p\times W_p\subset\mathbb{R}^n$ such that $\phi_p(U_p\cap M)=V_p\times\{0\}\subset\mathbb{R}^k$.
Using the fact $M=\bigcup\limits_{p\in M}U_p$, I am hoping that there is a way to combine the $\phi_p$'s into a single function $F$ which, like the $\phi_p$'s, would be a homeomorphism from $\mathbb{R}^n$ to $\mathbb{R}^n$ such that $F(M)\subset \mathbb{R}^k$.