# A question on Parametric Transversality Theorem

While reading the book Introduction to Smooth Manifolds by John M. Lee I get confused by the statement of the Parametric Transversality Theorem (see last paragraph). Let $F:\mathbb R\times \mathbb R\to \mathbb R^2$ be defined by $F(x,s) = (0,s)$. Then $F$ is smooth, which I believe is transverse to the submanifold $X =\mathbb R\times \{0\}$. However, for the smooth family of maps $\{F_s\}$ defined by $F_s(x) = F(x,s)$, there is no $s\in \mathbb R$ such that $F_s$ is transverse to the embedded submanifold $X$. This contradicts the Parametric Transversality Theorem. So there must be something that I missed. Any help is appreciated.

I give below a relevant definition and a theorem in the book to standardize the discussion.

$F$ transverse to $S$. If $F:N\to M$ is a smooth map and $S\subseteq M$ is an embedded submanifold, we say $F$ is transverse to $S$ if for every $x\in F^{-1}(S)$, the spaces $T_{F(x)}S$ and $dF_x(T_xN)$ together span $T_{F(x)}M$.

Parametric Transversality Theorem. Suppose $N$ and $M$ are smooth manifolds, $X\subseteq M$ is an embedded submanifold, and $\{F_s:s\in S\}$ is a smooth family of maps from $N$ to $M$. If the map $F:N\times S\to M$ is transverse to $X$, then for almost every $s\in S$, the map $F_s:N\to M$ is transverse to $X$.

Au contraire. It's transverse (vacuously) for all $s\ne 0$.
I realize this question is quite old but I've been asking essentially the same question since I ran into transversality in the course of some research I'm doing. It's nice to know I'm not alone. I would just add the following: First, as stated in Ted Shifrin's answer, $$F_s$$ is (trivially) transverse to $$X$$ for all $$s \neq 0$$. This is because $$F_s^{-1}(X) = \emptyset$$ and this satisfies the definition of transverse as given in Lee. My assumption is that then in Theorem 6.30(a) we are to assume the condition "if $$F^{-1}(S)$$ is nonempty". Second, for $$s = 0$$, $$F^{-1}_s(X) \neq \emptyset$$ and as a result $$F_s$$ is not transverse to $$X$$, but that is OK since this is the only $$s$$ for which this happens and $$\{0\}$$ has measure zero.
Finally, I think it's interesting that, if $$\dim N < \dim M - \dim X$$ then there must be a set $$G \subset S$$ with $$S \setminus G$$ of measure zero such that $$F_s^{-1} (X) = \emptyset$$ for every $$s \in G$$.