Stone-Weierstrass theorem. Let $X$ be a compact Hausdorff space, let $\mathscr{A}$ be the algebra of real valued continuous functions on $X$ and let the topology in $\mathscr{A}$ of uniform convergence. Then a necessary and sufficient condition that a subalgebra $\mathscr{F}$ of $\mathscr{A}$ be dense in $\mathscr{A}$ is that
(1) for each $x$ in $X$ there is an $f$ in $\mathscr{F}$ with $f(x)\neq 0$,
(2) for $x\neq y$ in $X$ there is an $f$ in $\mathscr{F}$ with $f(x)\neq f(y)$,
Nachbin's theorem gives an analog for Stone–Weierstrass theorem for algebras of real valued $C^r$ functions on a $C^r$ manifold.
Nachbin's theorem. Let $M$ be a manifold of class $C^r$, let $\mathscr{A}$ be the algebra of real valued functions of class $C^r$ on $M$ and let the topology in $\mathscr{A}$ be that of uniform convergence on compact sets up to the derivatives of order $r$. Then a necessary and sufficient condition that a subalgebra $\mathscr{F}$ of $\mathscr{A}$ be dense in $\mathscr{A}$ is that
(1) for each $x$ in $M$ there is an $f$ in $\mathscr{F}$ with $f(x)\neq 0$,
(2) for $x\neq y$ in M there is an $f$ in $\mathscr{F}$ with $f(x)\neq f(y)$,
(3) for $x$ in $M$ and a tangent vector $v$ in $T_x(M)$ there is an $f$ in $\mathscr{F}$ such that $Df(x)v\neq 0$.
The only references (with the respective proof) that I found for Nachbin theorem were the two below:
Llavona, José G. Approximation of continuously differentiable functions. North-Holland Mathematics Studies, 130. Notas de Matemática [Mathematical Notes], 112. North-Holland Publishing Co., Amsterdam, 1986. xiv+241 pp. ISBN: 0-444-70128-1
L.Nachbin. Sur les algèbres denses de fonctions diffèrentiables sur une variètè, C.R. Acad. Sci. Paris 228 (1949) 1549-1551
I would like more accessible references to quote them in a future article.
By "accessible" I mean articles that can be downloaded by online databases or books published in the last two decades which are possible to buy online.
Thank you in advance.
They digitized most of Comptes Rendus, here is Nachbin's article: https://gallica.bnf.fr/ark:/12148/bpt6k31801/f1549.item