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For sure every curve that is homeomorphic to a circle is a simple closed curve, but is every simple closed curve homeomorphic to a circle?

Is there a proof for that, or is there some topological invariant that is not not shared with simple closed curve and a curve that is homeomorphic to a circle?

"Simple closed curve" = "non-self-intersecting continuous closed curve in plane."

Hulkster
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    What, precisely, is a continuous curve? Seems to me that your answer to this question should go most of the way to answering your main question. – Lubin Feb 04 '15 at 01:07
  • I don't know and I don't care; it's enough for me if we can do all the various calculations with arbitrary rational precision $\epsilon >0$. – Hulkster Feb 04 '15 at 02:26
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    Well, you should care. As @lhf says in his response, a continuous curve is precisely the image of an injective continuous map from the circle to the plane. It all drops out from that. – Lubin Feb 04 '15 at 04:00
  • Well, we can stretch and bend the continuum as much as we wish, and also remember for curve $\gamma[0,1] \rightarrow \mathbb{R}^2$, $\gamma[0]=\gamma[1]$, that $\gamma(x)=\gamma(y) \iff x=y$. "What is the continuum?" is a fundamental question and I don't have the answer - I take the pragmatic approach. – Hulkster Feb 04 '15 at 04:45
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    Imagine person A asks a question: For sure every prime number is a primordial number but is every primordial number a prime? Then you ask them in the comments: what do you actually mean by a primordial number? and they reply: I don't know and I don't care, it's enough for me that we know how to add and multiply them. Would you be able to help that person? My point is, nobody here will ask you for the precise definition just to split hairs. They are trying to find out what you're actually asking and it's impossible if you refuse to define the terms you're using to pose the question. – Adayah Jul 28 '18 at 18:02

2 Answers2

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"non-self-intersecting continuous closed curve in plane" is the same as "image of a continuous injective function from the circle to the plane".

Since the circle is compact, such a function has a continuous inverse and so is a homeomorphism between the circle and the curve.

lhf
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A non-self-intersecting continuous closed curve in the plane should be equivalent to a circle at least topologically however still bounded to the plane there are curves that are not circles which may not be smoothly transformed into a circle furthermore a circle is not equivalent to itself turned inside out if you distinguish between the inside and the outside. Whether or not two curves bounded to an euclidean plane are the same can be determined by counting the total number of signed revolutions about the curve circle=1 inside out circle=-1 for more information see https://www.youtube.com/watch?v=R_w4HYXuo9M

user421813
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