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Bacle H
Posts:
283
Registered:
4/8/12


Re: Onto [0,1]
Posted:
Apr 21, 2013 11:03 PM


On Sunday, April 21, 2013 12:40:44 PM UTC7, Butch Malahide wrote: > On Apr 21, 1:42 pm, baclesb...@gmail.com wrote: > > > On Sunday, April 21, 2013 12:56:53 AM UTC7, 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 nontrivial examples existed.



