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Warning: My level of understanding of topology is very low. Small words would be appreciated. :)

Browsing Wikipedia, I came to crumpled cube, defined as "a 2-sphere together with its interior". Intuitively, I would think that "the interior of a 2-sphere" is exactly synonymous to "an (open) 3-ball", and "together with" means "union", and therefore the whole definition is synonymous with "a closed 3-ball". But if that were true, then we would have no need for the term crumpled cube.

What's the difference between "3-ball" and "crumpled cube"?

Quuxplusone
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Odd as it may sound, not every subset of $\Bbb{R}^3$ that is homeomorphic to a 2-sphere is the boundary of something homeomorphic to a 3-ball. One famous example of this is the Alexander horned sphere. In the terms of that article, the solid Alexander horned sphere (the exterior of the Alexander horned sphere, together with the point at infinity) is a particular crumpled cube that is not a 3-ball.

Micah
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  • I'm willing to believe that, but I could use more details, like, is it true that every(thing homeomorphic to a) 2-sphere is the boundary of something? is this the same thing as having an "interior"? and how do we show that a particular 2-sphere's interior isn't (homeomorphic to) a 3-ball?

    Also, that Wikipedia article currently claims "The horned sphere, together with its inside, is a topological 3-ball…", which seems like the opposite of what you just said!

     – Quuxplusone Jan 15 '14 at 09:26
  • Every topological 2-sphere in $\Bbb{R}^3$ is the boundary of something. This is the 3-dimensional Jordan-Brouwer separation theorem. But we don't get very much information about what it's the boundary of (unlike in 2 dimensions, where the Schoenflies theorem tells us that nothing weird can happen). The proofs of all of these theorems are absurdly hard. – Micah Jan 15 '14 at 09:40
  • The solid Alexander horned sphere is actually the exterior of the horned sphere, as usually presented. (As contrasted with the Alexander horned ball, which is the interior, and which is a ball.) – Micah Jan 15 '14 at 09:43
  • aha, I see. So the Wikipedia article was correct, but it would also be correct to add that "The horned sphere, together with its outside, is not* a topological 3-ball…"*. Both "sides" are crumpled cubes, but only one of them is a 3-ball. Right? – Quuxplusone Jan 15 '14 at 19:34
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    Yes, that's exactly right (at least, as long as you're thinking about the horned sphere as a subset of $S^3$, so the exterior is actually compact). – Micah Jan 15 '14 at 23:26