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Re: Perfectly Uncountable
Posted:
Jun 29, 2012 6:06 AM
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On Jun 29, 3:55 am, William Elliot <ma...@panix.com> wrote:
> On Thu, 28 Jun 2012, David C. Ullrich wrote:
> > >On Tue, 26 Jun 2012, David C. Ullrich wrote:
> > >> On Tue, William Elliot <ma...@panix.com> wrote:
> > >> >Finally, does C continuously map onto [0,1]?
>
> > >> Come on now. Read the wikipedia article on the Cantor
> > >> set. I haven't looked, but this must be in there, it's
> > >> a basic property of the Cantor set. (If it's not there
> > >> let us know and it will be there soon.)
>
> > >I don't recall seeing much that's that useful for this problem
> > >or similar problems beyond the base Cantor set stuff.
>
> > Well look again - the answer to the question you asked is
> > right there in black and white. (I looked the other day.)
>
> Still didn't find it. Nothing about scattered
Why would you expect to find something about "scattered" in an article
on the Cantor set? The Cantor set is not scattered. The Cantor set is
famous for being a nowhere dense perfect set, and "perfect" set is
more or less the opposite of "scattered".
> and I couldn't find C maps onto [0,1].
Sheesh.
[BEGIN QUOTE]
The Cantor set is sometimes regarded as "universal" in the category of
compact metric spaces, since any compact metric space is a continuous
image of the Cantor set; however this construction is not unique and
so the Cantor set is not universal in the precise categorical sense.
The "universal" property has important applications in functional
analysis, where it is sometimes known as the representation theorem
for compact metric spaces.[8]
[END QUOTE]
Notice where it says "any compact metric space is a continuous image
of the Cantor set"? Reference [8], by the way, is Willard's "General
Topology".
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