  We show that for any smooth Hausdorff manifolds M and N, which are not
necessarily second countable, paracompact or connected, any isomorphism from
the algebra of smooth (real or complex) functions on N to the algebra of smooth
functions on M is given by composition with a unique diffeomorphism from M to
N. An analogous result holds true for isomorphisms of algebras of smooth
functions with compact support.
