2
$\begingroup$

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?"

$\endgroup$
  • 3
    $\begingroup$ 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. $\endgroup$ – Pietro Majer Nov 30 '14 at 11:54
  • 1
    $\begingroup$ 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). $\endgroup$ – Ricardo Andrade Nov 30 '14 at 17:26

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.