On embedding of subcartesian differential space and application

Consider a locally compact, second countable and connected subcartesian differential space with finite structural dimension. We prove that it admits embedding into a Euclidean space. The Whitney embedding theorem for smooth manifolds can be treated as a corollary of embedding for subcartesian differential space. As applications of our embedding theorem, we show that both smooth generalized distributions and smooth generalized subbundles of vector bundles on subcartesian spaces are globally finitely generated. We show that every algebra isomorphism between the associative algebras of all smooth functions on two subcartesian differential spaces is the pullback by a smooth diffeomorphism between these two spaces.