Is a P-shaped path a Jordan's Curve?

Question

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 "$P$".

Lorenzo Cecchi · Answer 1 · 2019-10-28T10:48:42.473

0

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?

edited Oct 28 '19 at 10:48

answered Oct 28 '19 at 10:34

Lorenzo Cecchi

442

I've not learned topology anyway, Is that mean, Letter P is homeomorphic with Letter S? So it's not a Jordan's Curve then? – user516076 Oct 28 '19 at 10:41
$S^1$ is a circle. Or in other words a letter $O$ – Vasily Mitch Oct 28 '19 at 10:44
Ok, last question. So it's a Jordan's Curve right? – user516076 Oct 28 '19 at 11:22
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If you agree with the definition I give in my answer, P is not a Jordan curve. – Lorenzo Cecchi Oct 28 '19 at 11:25

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1 Answers1