Date: Apr 25, 2013 11:09 PM
Author: quasi
Subject: Re: Onto [0,1]
Butch Malahide wrote:

>Butch Malahide wrote:

>>quasi wrote:

>> >

>> > 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.

>>

>> No, X could have the discrete topology.

>

>Here's a less trivial (because it's nonconstructive)

>counterexample. Let C be the Cantor set.

>Let c = |C| = 2^{aleph_0}. The product space CxC contains just

>c subsets homeomorphic to C; of course, each of those subsets

>has cardinality c. By transfinite induction, we can construct

>a subset X of CxC which meets each vertical line {x}xC (x in

>C) while containing no homeomorph of C. Of course there is a

>continuous surjection from X to C.

How do you ensure that X contains no homeomorph of C?

quasi