Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Onto [0,1]
Replies: 40   Last Post: Apr 29, 2013 10:16 PM

 Messages: [ Previous | Next ]
 Butch Malahide Posts: 894 Registered: 6/29/05
Re: Onto [0,1]
Posted: Apr 25, 2013 9:21 PM

On Apr 25, 5:46 pm, Butch Malahide <fred.gal...@gmail.com> wrote:
> On Apr 25, 5:12 pm, quasi <qu...@null.set> 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.

Date Subject Author
4/21/13 William Elliot
4/21/13 Butch Malahide
4/21/13 William Elliot
4/22/13 David C. Ullrich
4/22/13 Butch Malahide
4/22/13 David C. Ullrich
4/21/13 Bacle H
4/21/13 William Elliot
4/21/13 Bacle H
4/21/13 Bacle H
4/21/13 Butch Malahide
4/21/13 Bacle H
4/25/13 William Elliot
4/25/13 Butch Malahide
4/27/13 William Elliot
4/27/13 Butch Malahide
4/27/13 Butch Malahide
4/29/13 Butch Malahide
4/29/13 William Elliot
4/25/13 William Elliot
4/25/13 quasi
4/25/13 Butch Malahide
4/25/13 quasi
4/25/13 quasi
4/25/13 quasi
4/25/13 quasi
4/25/13 David C. Ullrich
4/21/13
4/25/13 quasi
4/25/13 quasi
4/25/13 quasi
4/25/13 Butch Malahide
4/25/13 quasi
4/25/13 Butch Malahide
4/25/13 Tanu R.
4/25/13 quasi
4/25/13 Butch Malahide
4/25/13 quasi
4/26/13 Butch Malahide
4/26/13 quasi