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Math Forum » Discussions » sci.math.* » sci.math

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 21, 2013 4:40 AM
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On Apr 21, 2:56 am, William Elliot <ma...@panix.com> 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.

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