Possibly dumb question: Do you know that the various intervals created (or at least defined) never overlap? 'Cause if they did, then some incarnations of a point x could at a later stage be asked to slide by one amount, and other incarnations by a different amount (or direction). —Dan
On Sep 8, 2015, at 12:32 PM, James Propp <jamespropp@gmail.com> wrote:
Part of what I had in mind is that points in the final set arise from points in the original interval [0,1] through a process of repeated sliding (and, in the case of dyadic rationals, "cloning"). If we look at the sequence of slide operation (Left or Right) that a point undergoes, e.g., Left, Right, Right, Left, Right, Right, ... and replace each Left by a 0 and each Right by a 1, we get an infinite string of bits; this is the intended bijection between my set of points and the product set {0,1} x {0,1} x {0,1} x ...
Clear now? (Apologies in advance if still murky.)
Jim
On Tue, Sep 8, 2015 at 12:35 PM, Dan Asimov <dasimov@earthlink.net> wrote:
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.
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