Date: Apr 25, 2013 7:19 PM
Subject: Re: Onto [0,1]
Butch Malahide 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.