Transverse a single point

Question

I got very confused with understanding this theorem. So $\{y\}$ is a point, how could it be transversed by $f$?

Proof: Given any $y \in Y.$ alter $f$ homotopically to make it transversal to $\{y\}$.

Thank you for your help~~

Answer 1

The tangent space to a point is trivial, so to say that $f$ is transverse to $\{y\}$ is just to say that $y$ is a regular value of $f$, i.e. that $f'(x):T_xX\to T_yY$ is surjective for all $x\in f^{-1}(y)$.