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

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quasi

Posts: 10,450
Registered: 7/15/05
Re: Onto [0,1]
Posted: Apr 25, 2013 11:09 PM
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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



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