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

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Butch Malahide

Posts: 894
Registered: 6/29/05
Re: Onto [0,1]
Posted: Apr 22, 2013 4:16 PM
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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. 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?



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