In Guillemin-Pollack's book Differential Topology, the Transversality theorem states that
The transversility Theorem. Suppose that $F:X \times S \to Y$ is a smooth map of manifolds, where only $X$ has boundary, and let $Z$ be any boundaryless submanifold of Y. If both $F$ and $\partial F$ are transversal to $Z$, then for almost every $s\in S$ , both $f_s$ and $\partial f_s$ are transversal to $Z$
(Note that the notation $\partial f$ is just the restriction to the boundary $\partial X$)
However, I am wondering what if $X$ also has empty boundary? and how to define $\partial F$ in this case.
For example, I want to use transversility theorem to show the following problem:
Problem. Let X and Y be submanifolds of $\mathbb R^N$. Show that for almost every $a \in \mathbb R^N$, the translate $X+a$ intersects $Y$ transversally.
My sketch of proof: construct $F:X \times \mathbb R^N \to \mathbb R^N$ by $(x,a) \to x+a$. Now let $Z=Y$ in this case and "apply” the above theorem. However, I am not sure whether I could apply the theorem since here $X$ could be boundaryless.