I have been trying to work through John Lee's "Introduction to Topological Manifolds" and things were going alright until I reached Theorem 4.88 in the book, where a part of the proof of has had me stumped for several days now.
The statement of the theorem is the following:
Theorem 4.88 (Zero Sets of Continuous Functions). Suppose M is a topological manifold, and B $\subseteq$ M is any closed subset. Then there exists a continuous function f : M $\rightarrow$ $[0,\infty)$ whose zero set is exactly B.
Lee's proof begins with the following:
$\>\>\>\>\>$First, consider the special case in which M $={\Bbb{R}^n}$ and B $\subseteq{\Bbb{R}^n}$ is a closed subset. It is straightforward to check that $$u(x)=\inf\{|x-y|:y\in{B}\}$$ does the trick. (This function u is called the distance to B.)
$\>\>\>\>\>$Now, let M be an arbitrary n-manifold and let B be a closed subset of M. Let $\mathcal{U}=(U_\alpha)_{\alpha\in{A}}$ be a cover of M by open subsets homeomorphic to ${\Bbb{R}^n}$, and let $(\psi_\alpha)_{\alpha\in{A}}$ be a subordinate partition of unity. For each $\alpha$, the construction in the preceding paragraph yields a continuous function $u_\alpha:U_\alpha\rightarrow[0,\infty)$ such that $u_{\alpha}^{-1}(0)=B\cap{U_\alpha}$.
The part of the proof that I seem to be unable to motivate for myself is the sentence I bolded at the end. The continuation of the proof afterwards is relatively straight-forward and so I will omit it.
The nature of the functions $u_\alpha$ is left unspecified with the only lead present being that a construction similar to the one we employed in the special case works. That seems to point to the utilization of some sort of function similar to the $u(x)$ one that worked for the special case. The problem I seem to encounter here is that although $\Bbb{R}^n$ is taken with its natural Euclidean norm and metric, we have never specified such a metric for an arbitrary manifold. Moreover, the fact that n-manifolds are metrizable was mentioned earlier in the text, however, it was left unproven, along with a statement that " since we have no need for this, we do not pursue it any further." This has only led me to believe that assuming a metric on M is not an intended part of Lee's proof.
Without a metric on M however, defining the functions $u_\alpha$ in a similar way to u seems to become impossible, at least to me. The conclusion I've come to is that the construction of the aforementioned functions must somehow employ the fact that we have chosen each $U_\alpha$ to be homeomorphic to $\Bbb{R}^n$ and make use of the natural metric there. I have been trying to find a way to use those facts to construct the needed functions, however, the problem I keep encountering is that B and $U_\alpha$ may well be disjoint, in which case I seem to have no way of getting information about the values that $u_\alpha$ takes on $U_\alpha$.
Any help is deeply appreciated.
When two topological spaces are homeomorphic, we can use that homeomorphism to transport continuous functions on one space to continuous functions on the other space with similar topological relations by using composition with the homeomorphism. The fact that one space has extra structure (like a metric) that makes it easier to do something there doesn't mean anything like this structure must be used in the other space too. Just simply compose the homeomorphism between the spaces (in the appropriate direction) with a nice real-valued continuous function on one space to get a nice real-valued continuous function on the other space.