Suppose you have points $A$ and $B$ in Euclidean plane. We can define the notion of a path joining $A$ and $B$ putting these, say, as continuous functions $\gamma\colon[0,1]\to\mathbb{R}^{2}$ with $\gamma(0)=A$ and $\gamma(1)=B$.
Now we consider pairs of such paths joining $A$ and $B$. There are some pairs which intersect in an infinite number of points - e.g., the pair consisting of any two equal paths - and some which intersect only a finite number of times. I guess this is easy to picture.
Now, the question is:
Can we find two different paths such that they intersect an infinite number of times?
This somewhat puzzled my head because on the one hand, the 'boundness' of the path and the 'curvature' involved in picking at least one non-straight path makes me think that this had to be finite, but the somewhat unrelated picture of a sequence bounded between $0$ and $1$ converging to $1$ makes me have hope that it is indeed possible to construct such curves.
So is this possible? I can't really write formally something which would justify this.
Does this kind of problem ring a bell for some area of mathematics?
Thanks
You can start with your two paths being the same, and then perturb just the first half of one of them. This way, you get two different paths that coincide on the interval $[\frac{1}{2},1]$.
Edit: And if you are looking for two curves with infinitely many intersection points, but not as many as in the previous paragraph, you can take the two following curves:$$\gamma_1(t)=(t,0),\quad\gamma_2(t)=\left(t,t\cos\left(\frac{\pi}{2t}\right)\right).$$