I am studying transversality. There is a result that I would intuitively say is true (for example by locally identifying the manifold with its tangent space), but I can't get around the Differential Geometry formalism and write a formal proof.
The setting is the following:
Let $f:M\to N$, $g:V\to N$ be smooth maps between smooth manifolds, $\dim(M) + \dim(V) = \dim(N)$. Assume $f\pitchfork g$. How can I prove (or disprove) that $Im(f) \cap Im(g) \subseteq N$ is a discrete set?
I think I could endow my manifold with an arbitrary Riemannian metric and use the exponential map to locally identify tangent space and manifold to make my intuition precise, but I would like a cleaner proof.
Thank you