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

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Posts: 12,067
Registered: 7/15/05
Re: Onto [0,1]
Posted: Apr 25, 2013 6:33 PM
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quasi wrote:
>quasi wrote:
>>William Elliot wrote:
>>>Can an uncountable compact Hausdorff be continuously mapped
>>>onto [0,1]?

>>This has already been answered by David Ullrich (with
>>improvements Butch Malahide)

>Sorry -- I hit post button too early.
>Continuing ...
>A followup question.
>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.

Here's a stronger version ...

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].


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