Let $\gamma$ be a closed path in $\mathbb{R}^2$ ($\gamma : [0,1] \to \mathbb{R}^2$ is continuous and satisfys $\gamma(0)=\gamma(1)$). Are the following statements true?
- $\gamma$ divides the plane into (at most countably infinite, since every component would contain a rational point) connected components $Q_1, Q_2, Q_3,…$:
$$\mathbb{R}^2 \setminus \gamma = \bigcup_{i=1}^{\infty} Q_i$$
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There is exactly one unbounded component (which we call) $Q_1$. $Q_i$ is bounded for every $i\geq 2$.
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The boundary of the union of all bounded components is equal to $\gamma$:
$$\partial \bigcup_{i=2}^{\infty} Q_i = \gamma$$
If 3 doesn't hold, do we at least know that the boundary is a subset of $\gamma$?
Essentially I am asking if a generalization of the Jordan curve theorem for not necessarily simply connected paths is true.
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