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

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William Elliot

Posts: 2,637
Registered: 1/8/12
Re: Onto [0,1]
Posted: Apr 21, 2013 9:55 PM
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On Sun, 21 Apr 2013, Butch Malahide wrote:

> On Apr 21, 2:56 am, William Elliot <> wrote:
> > Can an uncountable compact Hausdorff be continuously mapped onto [0,1]?
> Let X be the ordinal omega_1 + 1 with its order topology. X is an
> uncountable compact Hausdorff space. X can not be continuously mapped
> onto [0,1]. (Hint: X is scattered.) Whether X can be discontinuously
> mapped onto [0,1] is independent of ZFC.

Whoops, wrong question.
Can a perfect compact Hausdorff space be continuously mapped onto [0,1]?

BTW, countable, (locally) compact Hausdorff spaces are imperfect.

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