Date: Apr 21, 2013 9:55 PM
Author: William Elliot
Subject: Re: Onto [0,1]

On Sun, 21 Apr 2013, Butch Malahide wrote:

> 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.


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

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