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Jordan curves and arcs and related material such as the Schoenflies theorem in dimension two.

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A Jordan arc in the unit disk

It's certainly the case that $\mathbb{R}^2\setminus J$ is path connected. So any two points in $D\setminus J$ are joined by a path in $\mathbb{R}^2$ missing $J$. If this path isn't in $D$ it hits the …
Robin Chapman's user avatar