The existence of a smooth functions taking values $0$ and $1$ on two given closed sets

Question

In theorem 5.1 on page 39 Boothby's book(An introduction to Differentiable Manifolds By William M. Boothby), he prove that:

Let $F\subset \mathbb R^n$ be a closed set and $K\subset \mathbb R^n$ compact, $F\cap K=\varnothing$. Then there is a $C^{\infty}$ function $\sigma(x)$ whose domain is all of $\mathbb R^n$ and whose range of values is the closed interval $[0,1]$ such that $\sigma(x)\equiv1$ on $K$ and $\sigma(x)\equiv0$ on $F$.

In exercise 8 on page 40 he ask that:

In theorem 5.1, assume only that $K$ is closed. Does the theorem still hold?

I guess that the answer is yes. How can I prove this exercise (chapter II, section 5 exercise 8) throughout an elementary way without using of partition of unity?

Of course, he has expressed only the theorem 5.1 without saying about partition of unity. Is there any elementary way for answering to the exercise?

Answer 1

Do you know the following fact?

If $C \subset \mathbb R^n$ is closed there is a $C^\infty$ function $\phi_C$ with the property that $\phi_C \ge 0$ and $\phi_C(x) = 0$ if and only if $x \in C$?

You can define $$f(x) = \frac{\phi_K(x) - \phi_F(x)}{\phi_K(x) + \phi_F(x)}.$$ This function is $C^\infty$, equals $1$ on $K$, $-1$ on $F$, and is otherwise strictly in between $-1$ and $1$.

Now use $\sigma = \dfrac 12 (f+1)$.