Re: Peano's space-filling curve

From: John Morgan (john.morgan_at_REMOVECAPSataraxie.fr)
Date: 06/07/04 

Date: Mon, 7 Jun 2004 14:17:58 +0200

David C. Ullrich <ullrich@math.okstate.edu> wrote in message
news:fa16c0dm9nh04mpb4ra1c89q00did2mrd1@4ax.com...
> On Sat, 5 Jun 2004 19:40:36 +0000 (UTC), Dave Seaman
> <dseaman@no.such.host> wrote:
>
> >On Sat, 05 Jun 2004 13:53:18 -0500, David C Ullrich
wrote:
> >> On Sat, 5 Jun 2004 13:50:34 +0200, "John Morgan"
> >><john.morgan@REMOVECAPSataraxie.fr> wrote:
> >
> >>> I have a problem applying "onto" for the Peano
> >>> construct. If the line passes arbitrarily close to
> >>> a point, so that all neighbourhoods of that point
> >>> contain point(s) through which the line passes,
> >>> does that make it "onto".
> >
> >> No. To be onto it has to actually pass through
> >> every point. Which it does.
> >
> >Actually, since f is continuous, f([0,1]) is compact and
> > therefore closed. Hence, the fact that f([0,1]) is
dense
> > in [0,1]^2 is a sufficient condition for f to be onto.
>
> We all understand that. Well, most of us do - in a context
> where one of us is having so much trouble.
> understanding simple definitions

If you mean me , why don't you write "John Morgan". After
all, I think I have explained sufficiently often for you to
know it by now, that I am not one of you.

Here I read that f is continuous and onto. Elsewhere a
poster tells me that f cannot be all three of '1 to
1',continuous and onto. So f is not '1 to 1'? Re-arrange the
following words to make a well-known phrase, or saying.
"together., Get, act, your,"

> (as evidenced by _repeated_ confusion over how I and I^2
> can have the same cardinality even though space-filling
> curves are not bijections) it seems like pointing this out
> is going to cause more confusion than claification.

After reading the post, the response, and the response to
the response, I think I can see more clearly than ever why I
am experiencing confusion. If the numerous clever people on
this group had discussed the problem of explaining Peano to
a non-mathy before setting out to do so, they might have
saved all of us a lot of time and headaches. I still only
have part of the picture, a full three and a half weeks and
a certain amount of rancour later

Cheers,

John

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    (sci.math)