# Generalization of Jordan Curve Theorem

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

Now

"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?"

• 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. – Pietro Majer Nov 30 '14 at 11:54
• 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). – Ricardo Andrade Nov 30 '14 at 17:26