Nash Functions - Nash Manifolds

Along with Nash functions one defines Nash manifolds, which are semialgebraic analytic submanifolds of some Rn. A Nash mapping between Nash manifolds is then an analytic mapping with semialgebraic graph. Nash functions and manifolds are named after John Forbes Nash, Jr., who proved (1952) that any compact smooth manifold admits a Nash manifold structure, i.e., is diffeomorphic to some Nash manifold. More generally, a smooth manifold admits a Nash manifold structure if and only if it is diffeomorphic to the interior of some compact smooth manifold possibly with boundary. Nash's result was later (1973) completed by Alberto Tognoli who proved that any compact smooth manifold is diffeomorphic to some affine real algebraic manifold; actually, any Nash manifold is Nash diffeomorphic to an affine real algebraic manifold. These results exemplify the fact that the Nash category is somewhat intermediate between the smooth and the algebraic categories.