On Sep 8, 2015, at 9:27 AM, James Propp <jamespropp@gmail.com> wrote:
Good point, Dan!
For all but countably many points (namely the dyadic rationals), we use the binary representation of the original point in [0,1].
For the dyadic rationals, we need to know which of the two "clones" we're using.
E.g., when we split [0,1] into two pieces and give each piece an endpoint (creating a new point out of thin air), we give the right endpoint of the left piece the label .0111... and the left endpoint of the right piece the label .1000...
In this way, we get a labelling of the points of my set using infinite strings of bits, where each string corresponds to a unique point in the set, and vice versa.
Hopefully that's clear.
Not to me it isn't. I don't even know how you are defining your set, since you never stated that. (Yes, you stated what the stages are, but not how the final set is defined in terms of the stages. And you don't even seem to be defining your bijection above in terms of the final set, but only in terms of the stages. —Dan
Jim
On Tue, Sep 8, 2015 at 12:22 PM, Dan Asimov <asimov@msri.org <mailto:asimov@msri.org>> wrote:
To evaluate whether the bijection is a homeomorphism, it would be immensely helpful if you defined it.