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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 2:42 PM
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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 .

This f is continuous, but not absolutely continuous ( which

preserves sets of measure zero. )

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