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So I understand what the cantor set is geometrically when referencing a line, but I am currently working on a problem that deals with the Cantor set on a circle, and for the life of me I can't visualize or conceptualize what this means.

The way my book define the cantor set was as follows: $$\bigcap_{n=0}^\infty C^n$$ Where: $$C^0=[0,1], C^1=[0,\frac{1}{3}]\cap[\frac{2}{3},1],....$$ And that makes sense to me. But what I don't get is this idea of the cantor set on a circle. So I guess I'm basically asking what each iteration of the cantor set on a circle looks like, and what it means to for the cantor set to be on something like a circle.

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Due to lack of context, I can ony guess, but I suppose that the author of that text has in mind something like $\varphi(C)$, where $\varphi\colon[0,1]\longrightarrow S^1$ is the map defined by $\varphi(x)=\bigl(\cos(2\pi x),\sin(2\pi x)\bigr)$. Although $\varphi$ is not a homeomorphism, it turns out that $\varphi(C)$ and $C$ are homeomorphic.

Under this approach, that set will be again an intersection of a decreasing sequence of sets, the first three of which can be seen in the picture below.

enter image description here

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  • $\begingroup$ The statement of the full problem is :Draw a Cantor set C on the circle and consider the set A of all chords between points of C, and prove A is compact. There is a similar question here, but I my issue is more with actually visualizing what this set is. $\endgroup$ Oct 17 '20 at 8:53
  • $\begingroup$ Why “similar”? It's the exact same question. $\endgroup$ Oct 17 '20 at 9:06
  • $\begingroup$ You are right it is. However, that question doesn't really help with visualizing the cantor set on the circle or even explain what that means. That's what I'm having trouble with. $\endgroup$ Oct 17 '20 at 9:09
  • $\begingroup$ @Christopher Quinn La Fond Jr.: visualizing the cantor set on the circle --- Visualize the points of the usual Cantor set in the unit interval $[0,1]$ colored yellow, and the complementary open intervals colored black. Now visualize picking up the unit interval and bending it into a circle so that the points $0$ and $1$ are merged into one point. The part that's colored yellow on the circle will be a Cantor set on the circle. $\endgroup$ Oct 17 '20 at 9:27
  • $\begingroup$ I have added a picture to my answer. $\endgroup$ Oct 17 '20 at 9:51

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