Another question from Exercise 6d in section 50 in Munkres' textbook in Topology. I have a question regarding exercise 6d in section 50 from Munkres' Topology textbook: Exercise 6c in section 50 Munkres' Topology textbook.
Show that if $N=2m+1$, then $U_\epsilon(C)$ is dense in $\mathcal{C}(X,\mathbb{R}^N)$.
I am given the following hint:
Given $f\in \mathcal{C}(X,\mathbb{R}^N)$ and $\delta,\epsilon >0$ choose $g:C\to \mathbb{R}^N$ so that: $d(f(x),g(x))<\delta$ for $x\in C$, and $\Delta(g)<\epsilon$. Extend $f-g$ to $h: X \to [-\delta,\delta]^N$ using the Tietze theorem.
Where does he use the fact that $N=2m+1$?, we have: $f|_C = g+ h|_C$, so $\Delta(f|_C) = \Delta(g+h|_C)< \epsilon + (2\delta)^N$
We need to show that $f\in U_\epsilon(C)$ or that it's a limit point of $U_\epsilon(C)$, how exactly I don't see it.
Thanks in advance.
 A: So the hint you gave gives the full proof modulo a proof that such a function $g$ exists. This is because if we let $G=f-h$, then $G|_C=g$, and since $\Delta(g)<\epsilon$ we have that $G\in U_{\epsilon}(C)$. Moreover, because of how we constructed $h$, we have that $\|G-f\|=\|h\|<\delta$. Thus we have shown that we can construct an element of $U_{\epsilon}(C)$ arbitrarily close to any given $f\in \mathcal{C}(X,\mathbb{R}^N)$, making it a dense subset.
Proving the existence of such a $g$ is rather difficult, but we are given the tools to do it in the proof of the imbedding theorem given by Munkres on pp 312 (the exercise is only merely meant to be a generalization of this). Here Munkres shows that $U_{\epsilon}(X)$ is dense in $\mathcal{C}(X,\mathbb{R}^N)$ when $X$ is a compact metrizable space with topological dimension at most $m$. In exercise 6 we are dealing with a locally compact, second-countable Hausdorff space $X$ such that every compact subspace of $X$ has topological dimension at most $m$. Since $C$ is a compact Hausdorff subspace of $X$ we know that $C$ is regular, and by the other assumptions on $X$, $C$ is also second-countable with topological dimension at most $m$. By Urysohn's metrization theorem, $C$ is metrizable, so by Munkres proof we have that $U_{\epsilon}(C)$ is dense in $\mathcal{C}(C,\mathbb{R}^N)$. Since $f\in\mathcal{C}(X,\mathbb{R}^N)$, we have that $f|_C\in\mathcal{C}(C,\mathbb{R}^N)$, and using this density fact we can invoke such a $g:C\to\mathbb{R}^N$ so that $g\in U_{\epsilon}(C)$, i.e. $\Delta(g)<\epsilon$, and $\|f|_C-g\|<\delta$, i.e. $d(f(x),g(x))<\delta$ for all $x\in C$. With this $g$ in hand we can now do as the hint says, and complete the proof.
Note: For completeness, I could include Munkres' proof that $U_{\epsilon}(X)$ is dense in $\mathcal{C}(X,\mathbb{R}^N)$ when $X$ is a compact metrizable space with topological dimension at most $m$, which is where the fact that $N=2m+1$ comes up. However, the book is online here with the proof on page 312.
