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Re: Onto [0,1]
Posted:
Apr 25, 2013 3:10 PM


On Apr 25, 2:22 am, William Elliot <ma...@panix.com> wrote: > On Sun, 21 Apr 2013, Butch Malahide wrote: > > Mapping the Cantor set continuously onto a general compact metric > > space (as stated in your previous message) is somewhat harder.
Correctly: every *nonempty* compact metric space is a continuous image of the Cantor set. (Likewise, every nonempty separable complete metric space is a continuous image of the space of irrational numbers.)
> How would you do that or equivalently, continuously > map {0,1}^N onto a given compact metric space?
Let Y be a nonempty compact metric space. Then, for some natural n_1, Y is the union of n_1 nonempty closed sets of diameter < 1. Next, for some natural n_2, each of those n_1 sets is the union of n_2 (not necessarily distinct) nonempty closed sets of diameter < 1/2. Next, for some natural n_3, each of the previously chosen n_1*n_2 sets is the union of n_3 nonempty closed sets of diameter < 1/n. And so on. Use these coverings in the obvious way to define a continuous surjection from the infinite product space X = D(n_1)xD(n_2)x. . . to Y, where D(n) is a discrete space of cardinality n. Finally, observe that X is a continuous image (in fact a homeomorph but we don't need that) of the Cantor set.



