Setup/Notation: Let $n,m\in \mathbb{N}$ and let $C(\mathbb{R}^n,\mathbb{R}^m)$ be the space of continuous functions from $\mathbb{R}^n$ to $\mathbb{R}^m$ equipped with the compact-open topology. Let $\mathcal{I}(\mathbb{R}^n,\mathbb{R}^m)$ be the subset of $C(\mathbb{R}^n,\mathbb{R}^m)$ consisting of injective functions.
Observations - Effect of Dimension:
- If $n\leq m$: The subset $\mathcal{I}(\mathbb{R}^n,\mathbb{R}^m)$ need not be closed. To see this note that the family $$ f_{n}(x):=\frac1{n}\cdot x, $$ converges to $0$ (in the compact-open topology).
- If $n>m$: Then Brouwer's Invariance Theorem implies that $\mathcal{I}(\mathbb{R}^n,\mathbb{R}^m)=\emptyset$. In particular, $\overline{\mathcal{I}(\mathbb{R}^n,\mathbb{R}^m)}=\emptyset\neq C(\mathbb{R}^n,\mathbb{R}^m)$.
Question: Is there a critical $m^{\star}$ (depending on $n$) such that if: $$ \begin{cases} \overline{\mathcal{I}(\mathbb{R}^n,\mathbb{R}^m)} = C(\mathbb{R}^n,\mathbb{R}^m) &: m\geq m^{\star}\\ \overline{\mathcal{I}(\mathbb{R}^n,\mathbb{R}^m)} \neq C(\mathbb{R}^n,\mathbb{R}^m) & : m<m^{\star} \end{cases}? $$ If so, more precisely, what is $m^{\star}$ and how does it depend on $n$?
Partial Thoughts/Observations'' - (Edit): If we embed $C(\mathbb{R}^n,\mathbb{R}^m)$ into $C(\mathbb{R}^n,\mathbb{R}^{m+n})$ via $f\mapsto \tilde{f}_{\infty}(x):=[x\mapsto (f(x),0)]$ then, for any $f\in C(\mathbb{R}^n,\mathbb{R}^{m})$ we can define the maps: $$ \tilde{f}_{n}(x):= (f(x),\frac1{n}\cdot x). $$ Moreover, for each $n\in \mathbb{N}$, the map $f\mapsto \tilde{f}_n$ is an embedding. So, then in this way... an affirmative pseudo-answer to the question below is ``kind of'' since: $\lim\limits_{n\to \infty}\,\tilde{f}_n =\tilde{f}_{\infty}$ and each $\tilde{f}_n \in \mathcal{I}(\mathbb{R}^n,\mathbb{R}^{m+n})$ and is the image of some $C(\mathbb{R}^n,\mathbb{R}^m)$.
