The Math Forum

Search All of the Math Forum:

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

Math Forum » Discussions » sci.math.* » sci.math

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]

Posts: 12,067
Registered: 7/15/05
Re: Onto [0,1]
Posted: Apr 25, 2013 11:09 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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?


Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.