Is a path that has shape "$P$", a Jordan's curve?
If no, state why?
Anyway, what i'm trying to say is a path, that has no intersection (i'm not sure), but its initial and end point didn't meet at a point.
That path can be described as a letter "$P$".
Consider the question: is "P" homeomorphic to $S^1$? Try to consider the connected components you get by removing a point from the two spaces. Alternatively, if you don't want to use topology, consider possible parametrizations of "P", let's say a continuous map from $[0,1]$ to $\mathbb R^2$ such that $f(0)=f(1)$ and injective in $[0,1)$: in order to be injective you need f(1) to be the point of "auto-intersection" of $P$. Can you now conclude?