Number of points in the intersection of two differential curves

Question

If $c_1, c_2$ are two (differents) piecewise differential simple closed curves in $\mathbb R^2$ then the number of points in the intersection of these curves is necessarily a finite set?

Thanks you all for any suggestion.

Answer 1

Without any further restriction, $c_1$ could be equal to $c_2$, and then the statement would be false. Even if the equality is not the case, analogous situations where the curves are equal for a range of the parameter $t$, though not equal over the whole domain, would be counterexamples. Edit for explicit example: Take for instance the unitary circumference centered at the origin as $c_1$, and then the following construction for $c_2$: for $y>0$, it's just the upper half of the circumference; for $y \leq 0$, it is just part of a square of side 2. Then each curve is piecewise differentiable, closed, and their intersection is the whole upper circle (not a finite set), though they are different for $y<0$.