> 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.