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Isomorphic between plane and line

  1. Aug 21, 2009 #1
    Hi,

    I understand that the open interval (0,1) is isomorphic to the real line [tex]\mathbb{R}^1[/tex]

    May i know whether there is also isomorphism from [tex]\mathbb{R}^2[/tex] to [tex]\mathbb{R}^1[/tex]

    thanks a lot!!!!
     
  2. jcsd
  3. Aug 22, 2009 #2
    What do you mean by an isomorphism in this case?
     
  4. Aug 22, 2009 #3
    hmm.... the usual sense of isomorphism.... where there is a structure preserving bijective map from R^2 to R^1. sorry is my question still vague?
     
  5. Aug 22, 2009 #4

    Hurkyl

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    The question is which structure are you interested in! It's structure as a set? As an additive group? As a topological space?

    I imagine it's the topology you care about. In that case, the answer is "no". The common proof (AFAIK) is to consider the image of a subset of R2 in the shape of the letter Y -- if such an isomorphism existed, it must be injective on the Y. Can you derive a contradiction from that?
     
  6. Aug 22, 2009 #5
    There is a Borel isomorphism between [itex]\mathbb{R}^2[/itex] and [itex]\mathbb{R}[/itex].

    There is a group somorphism between [itex]\mathbb{R}^2[/itex] and [itex]\mathbb{R}[/itex] (both with addition).

    BUT: There is no map between [itex]\mathbb{R}^2[/itex] and [itex]\mathbb{R}[/itex] that is both a Borel isomorphism and a group isomorphism.
     
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