The Stone-Weierstrass theorem has an analog for the algebras of smooth functions, called  

> **Naсhbin's theorem**: An involutive subalgebra $A$ in the algebra ${\mathcal C}^\infty(M)$ of smooth functions on a smooth manifold $M$ is dense in ${\mathcal C}^\infty(M)$ if and only if $A$ separates the points and the tangent vectors of $M$.

See details in: "L.Nachbin. Sur les algèbres denses de fonctions diffèrentiables sur une variètè, C.R. Acad. Sci. Paris 228 (1949) 1549-1551", or in  [J.G.Llavona's monograph][3], or [here][4]. 


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**Remark.** The easiest way to describe the topology on ${\mathcal C}^\infty(M)$ is, I think, the following. 

1. For each function $f\in {\mathcal C}^\infty(M)$ let us define its *support* as the set of points in $M$ where $f$ has non-zero [germs][1]:
$$
\text{supp}f=\{x\in M:\ f\not\equiv 0\ (\mod x)\}
$$

2. Let us define *differential operators* (see e.g. [S.Helgason's book][2]) on $M$ as linear mappings $D:{\mathcal C}^\infty(M)\to {\mathcal C}^\infty(M)$ which do not extend the support of functions:
$$
\text{supp}Df\subseteq \text{supp}f,\quad f\in{\mathcal C}^\infty(M).
$$

3. Then we say that a sequence of functions $f_n$ converges to a function $f$ in ${\mathcal C}^\infty(M)$
$$
f_n\overset{{\mathcal C}^\infty(M)}{\underset{n\to\infty}{\longrightarrow}}f
$$
if and only if for each differential operator $D:{\mathcal C}^\infty(M)\to {\mathcal C}^\infty(M)$ the sequence of functions $Df_n$ converges to $Df$ in the space ${\mathcal C}(M)$ of continuous functions with the usual *topology of uniform convergence on compact sets* in $M$:
$$
Df_n\overset{{\mathcal C}(M)}{\underset{n\to\infty}{\longrightarrow}}Df
$$

Of course, this is equivalent to the convergence in ${\mathcal C}^\infty(U)$  for each smooth local chart $\varphi:U\to V$, $U\subseteq\mathbb{R}^m$, $V\subseteq M$.

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This is strange, I can't find an analog for the algebras of holomorphic functions (on complex manifolds). Did anybody think about this?

> **Question**: let $A$ be a subalgebra in the algebra ${\mathcal O}(M)$ of holomorphic functions on a complex manifold $M$ (as a first approximation, we can think that $M$ is just an open subset in ${\mathbb C}^n$). Which conditions should $A$ satisfy for being dense in ${\mathcal O}(M)$? 


  [1]: https://en.wikipedia.org/wiki/Germ_(mathematics)
  [2]: http://www.amazon.com/Geometric-Invariant-Differential-Operators-Spherical/dp/0821826735
  [3]: https://books.google.ru/books/about/Approximation_of_Continuously_Differenti.html?id=C7auJPRK5nAC&redir_esc=y
  [4]: http://arxiv.org/abs/1303.2424