Re: Peano's space-filling curve

From: G. A. Edgar (edgar_at_math.ohio-state.edu.invalid)
Date: 06/29/04 

Date: Tue, 29 Jun 2004 14:35:24 -0400

>
> Here is one that is not too complicated -- and it leaves explicit
> decimal representations out of the picture:
>
> A bijection between the unit interval and the unit square.
> ----------------------------------------------------------
>
> Define, for x non-negative rational:
> g(0) = 0
> g(x+1) = (g(x)+3)/4
> g(1/x) = 1 - g(x)
> Extend to irrationals by limits.
>
> Now define f(x,y) of two real variables, based on g as defined above:
>
> f(x,y) = (4g(x)+8g(y))/9
>
> This function appears to provide a bijection of the closed unit square
> to the closed unit interval. It is monotonic in each coordinate (but
> not continuous).
>
> Michel.

This claim seems quite unlikely to me. If g is discontinuous, then
"extend to irrationals by limits" needs explanation, too.
Are you sure you don't have the same problem as the decimal
construction of countably many exceptions? 

-- 
G. A. Edgar                               http://www.math.ohio-state.edu/~edgar/

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