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quasi
Posts:
9,917
Registered:
7/15/05


Re: Onto [0,1]
Posted:
Apr 25, 2013 11:09 PM


Butch Malahide wrote: >Butch Malahide wrote: >>quasi wrote: >> > >> > Prove or disprove: >> > >> > If X is a topological space and f: X > [0,1] is a >> > continuous surjection, then X has a subspace homeomorphic >> > to the Cantor set. >> >> No, X could have the discrete topology. > >Here's a less trivial (because it's nonconstructive) >counterexample. Let C be the Cantor set. >Let c = C = 2^{aleph_0}. The product space CxC contains just >c subsets homeomorphic to C; of course, each of those subsets >has cardinality c. By transfinite induction, we can construct >a subset X of CxC which meets each vertical line {x}xC (x in >C) while containing no homeomorph of C. Of course there is a >continuous surjection from X to C.
How do you ensure that X contains no homeomorph of C?
quasi



