**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](https://en.wikipedia.org/wiki/Invariance_of_domain) 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 $C(\mathbb{R}^n,\mathbb{R}^m)$.



The closure of the set of injective continuous functions
Kuratowski proved in [Sur les théorèmes du „plongement" dans la théorie de la dimension. *Fundamenta Mathematicae* 28.1 (1937): 336-342](http://matwbn.icm.edu.pl/ksiazki/fm/fm28/fm28137.pdf) that the set of embeddings of an at most $n$-dimensional separable metrizable space into $\mathbb{R}^{2n+1}$ contains a dense $G\_\delta$-set.
Hence $m^\*\le 2m+1$.


