Re: bijection of R: R <--> Rx.....xR


"Timothy Golden http://www.BandTechnology.com"; <tttpppggg@xxxxxxxxx> wrote
in message news:1126096850.166195.157500@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> Dirk Once wrote:
>>there is also a bijection
>> R <--> RxRxR
>>and in fact between R and the product set of any countable
>>number of R's:
>> R <--> Rx.....xR
>
> Could someone please expound on this?
> How does R map to RxR?
>
> -Tim
>

Let a1, a2, a3, b1 ,b2, b3 etc be digits "0" .. "9"

(0.a1a2a3a4... , 0.b1b2b3b4...) <--> 0.a1b1a2b2a3b3....

Note that we "disallow" representation like 0.4999... for 0.5, so that every
real has a unique representation.

This sort of thing will work for any finite R <--> Rx...xR. I can't quite
see how to extend it to a countably infinite number of R terms though.



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