nowhere dense set on circle

Let $$X$$ be a nowhere dense set of circle $$S^1$$. Here $$S^1$$ is equipped with the standard topology and measure.

Q Can we say that $$X$$ is a finite point set? Is there a counterexample?

• map the cantor set on the unit interval to the disc? – Sean Nemetz Sep 18 at 4:51
• Have you thought of "inserting" $[0,1]$ into $S^1$? Take a closed arc of $S^1$ which is not the whole of $S^1$, and see how it is homeomorphic to $[0,1]$. Consider the image of some nowhere dense set in $[0,1]$, on $S^1$. Do you think it is nowhere dense? Do you have an example of such a nowhere dense set? – астон вілла олоф мэллбэрг Sep 18 at 4:55

Well, you can say it, but it would no be true. Just consider the set$$\left\{\left(\cos\left(\frac1n\right),\sin\left(\frac1n\right)\right)\,\middle|\,n\in\mathbb N\right\}\subset S^1.$$