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

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Bacle H

Posts: 283
Registered: 4/8/12
Re: Onto [0,1]
Posted: Apr 21, 2013 11:03 PM
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On Sunday, April 21, 2013 12:40:44 PM UTC-7, Butch Malahide wrote:
> On Apr 21, 1:42 pm, baclesb...@gmail.com wrote:
>

> > On Sunday, April 21, 2013 12:56:53 AM UTC-7, William Elliot wrote:
>
> > > Can an uncountable compact Hausdorff be continuously mapped onto [0,1]?
>
> >
>
> > More specifically, use the representation of x in C Cantor set in base 3
>
> >
>
> > with only 0's and 2's in the expansion of 3, and map
>
> >
>
> > f: x=0.a1a2.....   ---> 0.b1b2.......
>
> >
>
> > Wheref(bi)= 0 , if ai=0 , f(bi)=1 , if ai=2 .
>
>
>
> Mapping the Cantor set continuously onto [0,1] is easy, as you showed.
>
> Mapping the Cantor set continuously onto a general compact metric
>
> space (as stated in your previous message) is somewhat harder. But I'm
>
> not sure what this has to do with the original poster's question,
>
> which I interpreted as: Can EVERY uncountable compact Hausdorff space
>
> be continuously mapped onto [0,1]? (The answer, of course, is
>
> negatory.) The alternative reading, "Can SOME uncountable compact
>
> Hausdorff space be continuously mapped onto [0,1]?", would be silly,
>
> seeing as [0,1] is a compact Hausdorff space and is continuously
>
> mapped onto itself by the identity map.


You're right; I was trying to exclude trivial cases like maps from

closed intervals to [0,1]. I thought Elliot meant to ask if these types

of non-trivial examples existed.



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