[ColinWright]

There is an obvious enhancement of the Jordan Curve Theorem, which is to say that the interior is effectively a disk, and the exterior is effectively a plane with a hole. There is then an obvious 3D version, with a (topological) sphere separating space into a topological ball[0] and a topological R^3 with a hole. This enhanced version, although natural and obvious, is false.

I use this as an example of why "obvious" things aren't always true, need proving, and understanding their proofs can lead to useful insights.

[0] Edit: corrected "sphere" to "ball"

[gohrt]

(I think) this is a case were lay intuition assumes objects are piece-wise differentiable, but the formal math definitions don't.

That is, most people think of physical objects, where there is a minimum "planck length", and no fractal-ish structures with infinitely fine-scale structure.

It is still abstractly interesting that even 2-D fractals satisfy the Jordan Curve therorem, whie 3-D faractals do not satisfy the 3-D version of the conjecture.

[tehwalrus]

Ooh, OK, you've tickled my interest. Can you provide an explanation (or a link to something similar) with the disproof of the 3D version?

[ColinWright]

In three dimensions there is an embedding of S^2 (which is the surface of a sphere[0]) into R^3 such that the exterior is not homeomorphic to R^3 with B^3 missing.

http://en.wikipedia.org/wiki/Alexander_horned_sphere (search for "jordan")

In essence, the "outside" gets very "tangled" and can't be smoothly converted into a "proper" exterior.

[0] S^2 = { (x,y,z) : , x, y, z, in R with x^2+y^2+z^2 = 1 }

[tehwalrus]

Hmm, so because part of the surface is a fractal, it can't be simply connected?

I'd be interested to learn whether a linear version (straight pipes instead of curved/broken torus ones) also has the same topological properties. It is clearly fractal, and clearly also homotopy identical to S^2, but obviously the two ends are no longer interlocking, and I wonder if this is as crucial to the result as both 'ends' being fractal clearly is.

In any case, cool! thanks :)

[ColinWright]

The surface is simply connected, it's the outside that ends up not being simply connected. This works equally well with piece-wise linear embeddings. And the current version of the Alexander Horned Sphere is not actually interlocking - the embedding is contractable back to S^2.

It takes a while to get your head around what this really is.

[tehwalrus]

OK, "proximate" rather than interlocking. I did understand the geometry of it, I just chose the wrong word. Thanks for your explanations! :)