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I finally gave this statement its own entry, in order to be able to conveniently point to it:
embedding of smooth manifolds into formal duals of R-algebras
Such a statement is true for manifolds with finitely many or countably many connected components.
As Theo Johnson-Freyd once pointed out to me, and later expanded in this answer on MO: http://mathoverflow.net/a/91445 it is false for arbitrary paracompact Hausdorff manifolds, in particular, given two uncountable (discrete) sets S and T one can find a morphism of real algebras C^∞(T)→C^∞(S) that is not induced by a function S→T.
However, the construction is a very subtle set-theoretical argument that uses measurable cardinals.
Thanks. I have made the standard regularity assumptions explicit in the entry now and added pointer to this MO discussioon.
Ah, I was just wondering what sort of things break for uncountable disjoint unions of second countable manifolds.
EDIT: I was thinking continuum-many summands, which is still better behaved than for general uncountable coproducts.
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