Every closed subset of a manifold is the zero set of a smooth nonnegative function

If $M\setminus A=\mathrm{\varnothing }$, take $g=0$. This includes the empty manifold. Otherwise $M$ is nonempty; all subsequent constructions are for this case.

Apply [L1] to the cover of $M$ by its smooth chart domains. Enumerate the resulting relatively compact balls $B_i$, repeating one if there are only finitely many, and choose chart witnesses $(U_i,{\phi }_i)$ with $K_i:=\overline{B_i}\subseteq U_i$. Countable Choice supplies these witnesses. Each $K_i$ is compact, and their interiors cover $M$ because they contain the $B_i$. Coordinate derivatives on $K_i$ therefore make sense in the containing chart, rather than on a ball whose boundary might leave that chart.

Apply [L1] and [L2] on the open manifold $M\setminus A$ to obtain an at-most-countable family $W_k⋐V_k$ whose $W_k$ cover $M\setminus A$, with compact $\overline{V_k}$ there. Compact subsets of this open manifold remain compact in $M$ by continuity of the inclusion, and they are closed in the Hausdorff manifold $M$; hence $\overline{W_k}$ is compact in $M$ and lies in $V_k$. By [L3] and [A1], choose global smooth $b_k:M\to [0,1]$ equal to one near $\overline{W_k}$ and supported in $V_k$. Enumerate them by positive integers, padding by zero functions in the finite case. Each $b_k$ vanishes on an open neighbourhood of every point of $A$, and some $b_k$ equals one at each point of $M\setminus A$. Local finiteness on $M\setminus A$ is not used to assert smoothness across $A$.

For each $k\ge 1$ set $C_k:=\max (\{0\}\cup \{{\sup }_{x\in {\phi }_i(K_i)}\mid D_{\sigma }(b_k\circ {\phi }_i^{-1})(x)\mid :1\le i\le k,\mathrm{\mid }\sigma \mathrm{\mid }\le k\}),c_k:=\frac{2^{-k}}{1+C_k}>0.$ There are finitely many derivative words of length at most $k$ in each fixed finite-dimensional chart, so [L6] makes $C_k$ finite. In dimension zero the empty word is the only word. Put $S_N:={\sum }_{k=1}^N{c}_k{b}_k$. These are smooth nonnegative functions, and the coefficients require no additional family choice.

Fix $i$ and a derivative word $\sigma$. For every $k\ge \max \{i,\mathrm{\mid }\sigma \mathrm{\mid },1\}$, ${\sup }_{{\phi }_i(K_i)}\mid D_{\sigma }(c_k{b}_k\circ {\phi }_i^{-1})\mid \le c_k{C}_k\le 2^{-k}\mathrm{.}$ The finitely many earlier terms have finite bounds by [L6]. Thus [L4] gives uniform convergence of every derivative series on ${\phi }_i(K_i)$ to a continuous function $H_{i,\sigma }$. For the empty word the limit defines the same pointwise function $g:={\sum }_{k\ge 1}{c}_k{b}_k$ in every chart. Every point belongs to some $K_i$, so this series is everywhere finite and well defined.

On the open set ${\phi }_i(\mathrm{int}{K}_i)$, we prove that each $H_{i,\sigma }$ has coordinate derivative ${\mathrm{\partial }}_j{H}_{i,\sigma }=H_{i,\sigma j}$, where $\sigma j$ denotes appending the differentiation in coordinate $j$. Fix a point there and a small nondegenerate closed segment in direction $j$ through it, wholly in that open set. The restrictions of $D_{\sigma }(S_N\circ {\phi }_i^{-1})$ to this segment converge at every point by step 4.1, and their derivatives in its varying coordinate converge uniformly to the restriction of $H_{i,\sigma j}$. Applying [L5] identifies the derivative of their already known limit with $H_{i,\sigma j}$. The point and coordinate were arbitrary. Starting with $H_{i,\mathrm{\varnothing }}=g\circ {\phi }_i^{-1}$ and repeating this argument for every finite word proves all its iterated partial derivatives exist and are the continuous limits of step 4.1. Therefore $g$ is smooth on $\mathrm{int}{K}_i$ by [L6]. In dimension zero smoothness is automatic in the singleton charts. Since these interiors cover $M$, $g$ is smooth on all of $M$, including every point of $A$.

All summands are nonnegative. At a point of $A$ every $b_k$ is zero, so $g=0$; at a point of $M\setminus A$, some $b_k=1$ by step 2.2 and its coefficient $c_k$ is positive, so $g\ge c_k>0$. Hence $g^{-1}(0)=A$. In fact every coordinate derivative of $g$ at $A$ is zero: each individual bump vanishes near that point, so every derivative term is zero there, and step 5.1 identifies the derivative of the sum with their limit.

The nonempty-complement case is proved by steps 5.1 and 6.1; the empty-complement case was step 1.1. Thus every closed subset is the zero set of a smooth nonnegative function under exactly the stated ${\mathrm{A}\mathrm{C}}_{\omega }$.