Smooth function, which separates between a closed and a open set.

Question

Let $M$ be a smooth manifold, $O\subseteq M$ an open subset and $B\subseteq M$ a closed subset, such that $closure(O)\subseteq interior(B)$

I think there must exist a smooth function $f\colon M\rightarrow \mathbb{R}$ such that $0\le f\le 1$, $f|_{M-B}\equiv 0$ and $f|_O\equiv 1$, but was not able to find a reference in the general case. (There are many references if $B$ is compact.)

Can someone please point me in the right direction?

Answer 1

One reference was supplied by John in a comment, with a correction by Jack Lee:

Proposition 2.26 in Introduction to Smooth Manifolds by John M. Lee's, first edition. (Also, Proposition 2.25 of the second edition).

Note that you could just as well consider the disjoint closed sets $\overline{O}$ and $\overline{M\setminus B}$, since the level sets of a continuous function are closed. That is, you are asking for a smooth function separating two disjoint closed sets. The search for "smooth Urysohn lemma" and "disjoint closed sets" brings up a few additional results, such as Lemma 1.3.2 in Manifold Theory by Peter Petersen.