Cantor's (0,1]~[0,1]

  • #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.
 
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  • #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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