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




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.



