I am interested in the following claim, which is a bit counter-intuitive to me. I am wondering whether it is correct.
Claim. Let $Y$ be an open (path-)connected subset of $\mathbb{R}^d$, and let $\{Z_i\}_{i\in I}$ be a countable family of closed submanifolds of $Y$. Then, for any path $\gamma:[0,1]\to Y$ there exists a smooth path $\gamma':[0,1]\to Y$ such that (1) $\gamma$ and $\gamma'$ are path-homotopic and (2) $\gamma'$ is transversal to $Z_i$ for every $i\in I$.
For the case of a single submanifold $Z_i$, the question has an answer here: Tranversal paths between two points
There, a user refers to the following result in Guillemin and Pollack's Differential Topology (p.73), which immediately implies the claim for a singleton $Z_i$.
Corollary. If, for $f:X\to Y$, the boundary map $\partial f:\partial X\to Y$ is transversal to $Z$, then there exists a map $g:X\to Y$ homotopic to $f$ such that $\partial g=\partial f$ and $g \pitchfork Z$.
My (very rough) understanding is as follows: The corollary follows from the Transversality Homotopy Extension Theorem (p.72), which is proven from the Transversality Theorem. The constructed homotopy in the Extension Theorem is independent of the particular submanifold $Z$, thus all proofs should still go through with a countable family because a countable intersection of full measure subsets (of the unit ball '$S$') has full measure.
Is this reasoning correct?
