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Are open sets in R^n always homeomorphic to R^n?

  1. Jan 19, 2013 #1
    I know that open intervals in R are homeomorphic to R. But does this extend to any dimension of Euclidean space? (Like an open 4-ball is it homeomorphic to R^4?)

    My book doesn't talk about anything general like that and only gives examples from R^2.
     
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  3. Jan 19, 2013 #2

    Bacle2

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    No for general open sets; look at, e.g., an open annulus, or any disconnected open set. But an open n-disk D:={x in R^n : ||x||<1 } (or any translation of it) is homeomorphic to R^n.
     
  4. Jan 22, 2013 #3

    disregardthat

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    Any connected open set in R^n is homeomorphic to R^n, for any n. An open set in R^n is homeomorphic to the disjoint union of equally many R^n's as connected parts of your open set.
     
  5. Jan 22, 2013 #4

    Bacle2

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    Actually, an open annulus is open and connected, but not homeomorphic to R^n, since it is not simply-connected.
     
  6. Jan 22, 2013 #5

    disregardthat

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    thanks for the correction, i meant simply connected :)
     
  7. Jan 22, 2013 #6

    micromass

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    Also not true, then open set [itex]B(0,1)\setminus\{0\}[/itex] of [itex]\mathbb{R}^3[/itex] is simply connected but not homeomorphic to [itex]\mathbb{R}^3[/itex].
     
  8. Jan 23, 2013 #7

    mathwonk

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