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Topic: Perfectly Uncountable
Replies: 27   Last Post: Jul 2, 2012 4:17 AM

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 David C. Ullrich Posts: 21,553 Registered: 12/6/04
Re: Perfectly Uncountable
Posted: Jun 26, 2012 11:10 AM

On Tue, 26 Jun 2012 01:52:08 -0700, William Elliot <marsh@panix.com>
wrote:

>On Mon, 25 Jun 2012, David C. Ullrich wrote:
>> On Mon, 25 Jun 2012 09:28:24 -0500, David C. Ullrich
>> <ullrich@math.okstate.edu> wrote:

>
>> >Clarify something first: Do you have some reason to think that
>> >this is actually true?

>
>> Forget I asked that. This notion of "scattered" is new to me.
>> If S is compact Hausdorff and not scattered then S has a
>> perfect subset. QED, by the result above and Tietze.
>>

>To prove that a not scattered compact Haudorff space S
>can be continuously mapped to [0,1].
>
>S has a not empty perfect kernel K.
>Case K has a multi-point component.
> There's a Urysohn function f that maps K onto [0,l].
> f can be extended to all of S by Tietze's extension theorem.
>
>Case K has no multi-point component.
> K is totally disconnected. K is uncountable.
> K can be continuously mapped onto the Cantor set C.
>As easy as you claim that to be, I don't see it.

Write K = K_0 union K_1, where each K_j is nonempty, closed,
and K_0 and K_1 are disjoint.

Since K_j is open and K has no isolated points, K_j must be
infinite.

Since K_j is infinite, we can repeat the argument. Write
K_0 = K_00 union K_01 and K_1 = K_10 union K_11,
such that [etc.].

And now repeat again, splitting each K_jk into two pieces.

Now, for each x in K there is an infinite string b_0 b_1 ...
of 0's and 1's such that x lies in K_[b_0 b_1 ... b_n] for
every n. Define f(x) to be the element of the Cantor
set corresponding to the string b_0 b_1 ... .

The fact that each K_alpha is open for every
finite sequence alpha of 0's and 1's shows that f is continuous.

>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.)

Date Subject Author
6/24/12 William Elliot
6/24/12 David C. Ullrich
6/24/12 William Elliot
6/25/12 Butch Malahide
6/25/12 William Elliot
6/25/12 Butch Malahide
6/26/12 William Elliot
6/25/12 David C. Ullrich
6/25/12 David C. Ullrich
6/26/12 William Elliot
6/26/12 Richard Tobin
6/26/12 David C. Ullrich
6/28/12 William Elliot
6/28/12 David C. Ullrich
6/29/12 William Elliot
6/29/12 quasi
6/30/12 William Elliot
6/29/12 Butch Malahide
6/29/12 David C. Ullrich
6/30/12 William Elliot
6/30/12 David C. Ullrich
7/2/12 William Elliot
6/26/12 Frederick Williams
6/26/12 quasi
6/26/12 Frederick Williams
6/26/12 quasi
6/26/12 quasi
6/26/12 Frederick Williams