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. Jim On Tue, Sep 8, 2015 at 12:22 PM, Dan Asimov <asimov@msri.org> wrote:
To evaluate whether the bijection is a homeomorphism, it would be immensely helpful if you defined it.
—Dan
On Sep 8, 2015, at 9:20 AM, James Propp <jamespropp@gmail.com> wrote:
I still don't see why the bijection between my original set and the product set {0,1} x {0,1} x ... fails to be a homeomorphism. Can someone explain that to me?
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