This problem comes from Hirsch's differential topology in page 67.
"Generically" a $C^1$ immersion $S^1\to \mathbb{R}^2$ has only a finite number of crossing points.
Then I want to ask if in general a $C^1$ immersion $I\to M$, where $I$ is an interval in $\mathbb{R^1}$ and $M$ is a $C^1$ manifold, has finite self-intersections.
Intuitively, since the curve is $C^1$, it can not change drastically in a small enough neighborhood, then we can perturbate it a little near the limit points of self-intersections to obtain a immersed curve with finite self-intersections. I don't know how to prove it rigorously, even for the case $I\to \mathbb{R}^2$. Since this problem occurs in chapter 3, the proof may need transversality theorems.