I am currently sitting and staring at a proof of the following theorem.
Theorem. Let $X$ be a manifold with boundary and $f_0:X \to Y$ a smooth map to a manifold $Y$. Let $g:Z \to Y$ be a closed smooth map from the manifold $Z$, and suppose $\partial f_0$ is transverse to $g$. Then there exists a smooth map $f_1 : X \to Y$ transverse to $g$ such that $f_1$ is homotopic to $f_0$ and $\partial f_1 = \partial f_0$.
I am confused about the closed condition. Why is it necessary?
Some context: The proof starts by claiming that, because $g$ is closed, there is an open neighborhood $U$ of $\partial X$ on which $f$ is transverse to $g$. I do not feel that this step is as clear as the author thinks it is.