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Topic: Onto [0,1]
Replies: 40   Last Post: Apr 29, 2013 10:16 PM

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Butch Malahide

Posts: 894
Registered: 6/29/05
Re: Onto [0,1]
Posted: Apr 25, 2013 6:39 PM
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On Apr 25, 5:19 pm, quasi <qu...@null.set> 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].



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