TITLE: Differentiable manifolds $\mathscr C^k$ vs. $\mathscr C^\infty$
QUESTION [5 upvotes]: I noticed that there exists a (in some sense) better definition of the tangent space via the dual of a certain quotient algebra which is easier to work with in some cases. This however only works for smooth manifolds. My question is, is there a substantial loss of generality if I assume that a manifold is smooth rather than $\mathscr C^k$? What do I really lose by making such an assumtion? I noticed that most books simply work with smooth manifolds right from the start, barely mentioning the (not as nice) $\mathscr C^k$ manifolds.
Thanks

REPLY [3 votes]: As others have already pointed out, there is no substantial loss. Any $C^r$-manifold admits a unique $C^s$-structure (up to $C^s$-diffeomorphism) for $r \leq s \leq \infty$. This result and its proof can for instance be found in these lecture notes: http://www.math.ist.utl.pt/~martinez/files/difftopDMT.pdf.