Jordan Curve Theorem says that any plane continuum homeomorphic to $\mathbb{S}^1$ separates the plane into exactly two components.


"Let $\alpha$ and $\beta$ be two homeomorphic plane continua. Is is true that $\alpha$ separates the plane into exactly two components if and only if $\beta$ separates the plane into two components?"

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    $\begingroup$ It is indeed true that the number of connected components of the complement of a compact subsets in $\mathbb{R}^n$ is a topologic invariant. It is a consequence of the Alexander duality. $\endgroup$ Nov 30, 2014 at 11:54
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    $\begingroup$ Complementing Pietro Majer's comment: we require a statement of Alexander duality expressing the homology of the complement of a subspace of Euclidean space as the Alexander–Spanier cohomology (or the Čech cohomology) of the subspace. See sections 2 and 9 of chapter 6 of Spanier's "Algebraic topology", specifically 6.2.16, 6.2.17, 6.9.9, 6.9.10. Alternatively, the topological invariance is also a consequence of the Klee trick and the Künneth theorem. The Klee trick is stated in theorem 3.2 of Steve Ferry's geometric topology notes (math.uchicago.edu/~shmuel/tom-readings/Ferrynotes.pdf). $\endgroup$ Nov 30, 2014 at 17:26


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