A friend told me that the $\mathbf{C^1}$-topology on the set $C^\infty(M,N)$ of smooth functions between two smooth manifolds $M$ and $N$ can be defined as the coarsest topology making the map $$ C^\infty(M,N) \to C(TM,TN), \qquad f \mapsto df $$ a homeomorphism onto its image, where $C(TM,TN)$ is the space of continuous functions between the tangent bundles $TM$ and $TN$ equipped with the compact-open topology.

Is this definition correct, i.e. equivalent to the usual one by jet spaces? Wikipedia and the other references I found have very long definitions for the C1-topology.