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

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.