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Unbounded space

  1. Feb 24, 2005 #1
    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?
     
  2. jcsd
  3. Feb 24, 2005 #2

    Hurkyl

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    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)
     
  4. Feb 25, 2005 #3

    robphy

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    R3 is unbounded.
    Maybe he is looking for "finite but unbounded".
    Another example is the analog of a torus (The Asteroids topology :tongue2: ).
     
  5. Feb 25, 2005 #4

    jcsd

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    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.
     
  6. Feb 26, 2005 #5

    HallsofIvy

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    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.
     
  7. Feb 26, 2005 #6

    jcsd

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    Actually now I think a little more, boundaryless and compact is probably what the OP was looking for.
     
  8. Feb 26, 2005 #7

    mathwonk

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