Here's a definition of Jim's set. For simplicity, I'll define a slightly different set, that when it splits an interval in half, just moves the right half a bit to the right, rather than both doing this and moving the left half to the left. For any subset S of Z+ (think of S as "the set of places that are 1 in the binary representation of the number before we start to move it"), sum_{s \in S} (1/2^k + 1/3^k) To get Jim's original set, add sum{s not in S} -1/3^k to this. You can get the K'th "stage" of Jim's construction by adding 1/2^k + 1/3^k for k<=K, and 1/2^k for k > K. I'm pretty sure the set is a Cantor set, but I'm still confused as to its measure. But having a precise definition of the set, without defining it as a limit, should make it easier to figure this out. Andy 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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