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.