The Polish Lie ring of vector fields on a smooth manifold is algebraically determined

Let L be a Polish (i.e., complete separable metrizable) Lie ring. L is said to be algebraically determined if, whenever R is a Polish Lie ring and φ:R→L is an algebraic isomorphism of Lie rings, then φ is a topological isomorphism. The purpose of this paper is to prove that the Lie ring of vector fields on a smooth manifold is an algebraically determined Polish Lie ring. A new fact about the ring of real numbers plays a crucial role in the proof of the general theorem. An application of the main theorem will be described to prove that certain algebraic objects are complete invariants for classifying smooth manifolds up to diffeomorphism.