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Topic: Onto [0,1]
Replies: 40   Last Post: Apr 29, 2013 10:16 PM

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David C. Ullrich

Posts: 3,024
Registered: 12/13/04
Re: Onto [0,1]
Posted: Apr 22, 2013 5:17 PM
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On Mon, 22 Apr 2013 13:16:57 -0700 (PDT), Butch Malahide
<fred.galvin@gmail.com> wrote:

>On Apr 22, 9:56 am, dullr...@sprynet.com wrote:
>> On Sun, 21 Apr 2013 18:55:29 -0700, William Elliot <ma...@panix.com>
>> wrote:

>> >Can a perfect compact Hausdorff space be continuously mapped onto [0,1]?
>> [. . .]
>> The answer to the question you meant to ask is also yes, it seems
>> to me. Say K is a perfect compact Hausdorff space. Either
>> K has a connected subset containing more than one point or not.
>> If C is a subset of K containing p and q, p <> q, then it follows
>> from Tietze that there is a continuous f : K -> [0,1] with
>> f(p) = 0 and f(q) = 1; now f(C) must be connected, qed.
>> On the other hand, if K has no connected subsets with
>> more than one point: It's easy to construct a continuous
>> map from K onto the Cantor set, qed. (Say K is the disjoint
>> union of the closed set A_0 and A_1. Map A_0 to the
>> left half of the Cantor set and A_1 to the right half.
>> Now A_0 is the disjoint union of the closed sets
>> A_00 and A_0,1... repeat countably many times
>> and you've defined a continuous map onto the Cantor set.)

>Nice. I think you can combine the two cases into one.
>Let A_0 and A_1 be disjoint perfect subsets of K.

[smacks forehead] Duh, the only reason I had the
first case was because I wanted A_0 and A_1 to have
union K. But there's no need for that - one could
just do what I did, decreasing the union a bit at
each stage, constructing a continuous function
from a Cantor-ish subset of K onto the Cantor
set, then invoke Tietze to map K onto the Cantor

>Construct a
>continuous f_1 : K -> [0,1] with f_1(A_0) = 0 and f_1(A_1) = 1.
>Let A_00 and A01 be disjoint perfect subsets of A_0, and let A_10 and
>A_11 be disjoint perfect subsets of A_1. Construct a continuous f_2 :
>K -> [0,1] with f_2(A_00 \/ A_10) = 0 and f_2(A_01 \/ A_11) = 1.
>Continuing in this way, define a continuous function f_n : K -> [0,1]
>for each natural n. Finally, the function
>f(x) = f_1(x)/2 + f_2(x)/4 + f_3(x)/8 + . . . + f_n(x)/2^n + . . . .
>maps K continuously onto [0,1]. Right?

Not that I looked at the details, but the idea is certainly
right. That bit with the sum at the end is exactly
equivalent to my "map this to the left half of
the Cantor set, etc", followed by the standard map
of the Cantor set onto [0,1].

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