The question is let $f: X \rightarrow \mathbb{R}^2$, show that for almost every $c \in \mathbb{R}$, we have that $f^{-1}(\{c\}\times\mathbb{R})$ is a smooth submanifold of $X$.
I want to apply Sard's theorem. But the precise statement I plan to use is:
For any smooth map $f$ of a manifold $X$ with boundary into a boundaryless manifold $Y$, almost every point of $Y$ is regular value of both $f: X \rightarrow Y$ and $\partial f: \partial X \rightarrow Y$.
To my understanding, $\mathbb{R}^2$ is boundaryless under this situation according to Shaun Ault's respons at Is $\mathbb{R}^2$ boundaryless?
But the bigger problem is, I don't know if $X$ is with boundary. I hope the condition that $X$ is with boundary is to apply the later part of the theorem, $\partial f: \partial X \rightarrow Y$. So when we only need almost every point of $Y$ is regular value of $f: X \rightarrow Y$, we don't need to know if $X$ is with boundary.
This sounds make sense, but I really don't want to assume anything when I am starting to learn differential topology. So could anyone point me some directions?
Thank you very much