Re: Peano's space-filling curve

From: Dave Seaman (dseaman_at_no.such.host)
Date: 06/07/04 

Date: Mon, 7 Jun 2004 14:53:11 +0000 (UTC)

On Mon, 7 Jun 2004 14:42:44 +0200, John Morgan wrote:

> David C. Ullrich <ullrich@math.okstate.edu> wrote in message
> news:l354c0974tb47j6gtbn56lk09gmdopqfgk@4ax.com...

>> >This must have an inverse f^-1:, a bijection, and I
>> >wonder why this inverse won't map [0,1] to [1,0]^2.
>>
>> That map does have an inverse, and the inverse does
>> map [0,1] onto [0,1]^2.

> So if there is a bijection from[0,1] to [0,1]^2 why did
> nobody say so at the beginning of all of this. I was told
> there was a surjection, but no mention of a bijection.

As far back as May 19, I wrote the following:

Dave Seaman <dseaman@no.such.host> wrote in message
news:c8flao$5d6$1@mozo.cc.purdue.edu...
>
> If you are talking about a simple bijection, it's easy. Cantor showed
> that R^n has the same cardinality for every n > 0. The complex plane is
> really just R^2 with a certain algebraic structure.

If you check Google groups, you will surely find other similar statements
quite early in the thread. Everyone has been telling you all along that
bijections exist, but not every injection is a bijection (the Peano curve
being an example). 

-- 
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>

Relevant Pages

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