Can Unbounded 3D Space Truly Exist?

[touqra]

Can there be any unbounded 3 dimensional space? For example, for a 2-dimensional space, we have an unbounded surface that resides on a sphere.
How about three-dimensional space?

 

[Hurkyl]

Sure. For example, the surface of a 4-dimensional sphere!


(PS: I'm not sure if "unbounded" is the word you're looking for... though it might be)

 

[robphy]

R³ is unbounded.
Maybe he is looking for "finite but unbounded".
Another example is the analog of a torus (The Asteroids topology ).

 

[jcsd]

Now this is what has confused me, surely an n-sphere is bounded as a metric space, I think the correct matehamtical term is 'boundaryless' i.e. a manifold without boundaries.

 

[HallsofIvy]

The example he gives (surface of a sphere) is what I would call (perhaps "paradoxically) "bounded but having no boundary".

That is, the set of all possible distances between points has an upper bound but there is no boundary: points such that every neighborhood contains some points in the set and some points not in the set.

Of course, there exist 3 dimensional bounded sets that have no boundary- but you have to imagine them embedded in 4 dimensional space. The surface of a 4-sphere is an example.

 

[jcsd]

Actually now I think a little more, boundaryless and compact is probably what the OP was looking for.

 

[mathwonk]

it always helps to define your terms. i.e. does "bounded" mean not very big, or having an edge?