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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 3:10 PM
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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.

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