Tietze–Urysohn's lemma in $\mathbb{R}^n$

Question

Let $F_1$ and $F_0$ be closed subsets in $\mathbb{R}^n$, $F_0\cap F_1=\varnothing$. How to build a $C^{\infty}$- function $f:\mathbb{R}^n\to \mathbb{R}$, such that $f|_{F_1}=1$, $f|_{F_0}=0$ and $0<f(x)<1$ for $x\notin F_1\cup F_2$?

The only useful fact I know is Tietze–Urysohn's lemma, which states that only
continuous function exists.
Could you give me any hints?

Answer 1

Tietze extension theorem and Urysohn lemma(the version on normal spaces at least) does not say anything about smoothness. But there is some sort of generalization via differential geometry, the result is known as The smooth Urysohn lemma. Details of the proof and how to construct the function can be found on the book Functions of Several Real Variables by Martin A. Moskowitz page 281.