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].