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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 10th 2015

    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

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 10th 2015

    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: 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.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 10th 2015

    Thanks. I have made the standard regularity assumptions explicit in the entry now and added pointer to this MO discussioon.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 11th 2015
    • (edited Mar 11th 2015)

    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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