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quasi
Posts:
10,741
Registered:
7/15/05


Re: Onto [0,1]
Posted:
Apr 25, 2013 7:19 PM


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 C homeomorphic >> to the Cantor set and such that f(C) = [0,1]. > >If X = [0,1] and f(x) = x, then X has a subspace C homeomorphic >to the Cantor set, but f(C) is not equal to [0,1].
Yes and f(C) is not equal to [0,1] for any such subspace C homeomorphic to the Cantor set.
That pretty much kills the idea.
Thanks.
quasi



