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Is it possible to extract a neighborhood around any point on a simple closed curve such that the boundary of this neighborhood intersects the curve at only two points?

For a simple closed curve $\gamma$, for any $z_0$ on $\Gamma$, there exists a neighborhood $U$ containing $z_0$ such that $\partial U$ has only two intersection points with $\Gamma$.

In the context of complex analysis, this assumption is often referred to as a 'free boundary arc'.

Considering the complex plane only.

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    $\begingroup$ Are you familiar with the Schoenflies theorem? $\endgroup$
    – Moishe Kohan
    Oct 21 at 7:34
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    $\begingroup$ Are $\gamma$ and $\Gamma$ the same thing? $\endgroup$
    – Greg Friedman
    Oct 21 at 8:38

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The answer is positive. For any Jordan curve, there is a homeomorphism of the Riemann sphere, which sends this curve to a circle. Your statement immediately follows.

One proof of this statement involves the Riemann mapping theorem and the boundary correspondence theorem of Caratheodory. But there are also purely topological proofs, of course.

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