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Homework Help: Cantor's (0,1]~[0,1]

  1. Jun 18, 2012 #1
    1. Can anybody elxplain to me (or point me to a URL of an explanation) how Cantor proved the existence of a bijection (0,1]~[0,1]?



    2. It's not for homework. I have to understand it generally for a paper I am writing.



    3. I think it has someting to do with transfinites but I can't get it.
     
    Last edited: Jun 18, 2012
  2. jcsd
  3. Jun 18, 2012 #2

    LCKurtz

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    I don't know if it is how Cantor did it but it is easy enough. To map [0,1] to (0,1] try this:
    ##0\rightarrow \frac 1 2##
    ##\frac 1 2 \rightarrow \frac 1 3##
    ##\frac 1 3 \rightarrow \frac 1 4##
    ##\frac 1 4 \rightarrow \frac 1 5##
    ...
    ##\frac 1 n \rightarrow \frac 1 {n+1}##
    Map all other points into themselves. That gives a 1-1 correspondence between the two intervals.
     
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