Given a manifold $M$, one can define the $C^{\infty}$ topology on $C^{\infty}(M)$ by the seminorms associated to compact sets $K$, $P_{K}^{s}(f) = \sup_{x \in K} \max_{|\alpha| \leq s} |D^{\alpha}f|$. However, this definition of seminorms is going to be dependent on the choice of the coordinates. Presumably, the resulting topology will not be dependent on the choice of the coordinates, but I wonder how to prove it.
The topology is induced by the metric $d(x,y) = \sum_{n(K,s)} 2^{-n(K,s)} \frac{p_{K}^{s}(x,y)}{1 + p_{K}^{s}(x,y)}$. Different seminorms are equivalent because they differ by a smooth transition function on a compact set, but this metric is some sort of sum of all the seminorms. How to control the equivalence of all the seminorms uniformly?