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

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quasi

Posts: 10,208
Registered: 7/15/05
Re: Onto [0,1]
Posted: Apr 25, 2013 7:19 PM
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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



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