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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 21, 2013 3:40 PM
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On Apr 21, 1:42 pm, 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.

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