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

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Butch Malahide

Posts: 894
Registered: 6/29/05
Re: Onto [0,1]
Posted: Apr 25, 2013 9:21 PM
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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.

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