Timeline for For which classes of metric spaces can we prove that quasi-isometry is an equivalence relation in ZF?
Current License: CC BY-SA 4.0
11 events
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| Oct 10, 2020 at 18:42 | comment | added | Erik Walsberg | In general CC is needed to develop analysis. The result on Baire category is in core.ac.uk/download/pdf/82313129.pdf | |
| Oct 10, 2020 at 18:42 | comment | added | Erik Walsberg | I have no idea how much you can get in ZF, but you should be able to go quite far with weak forms of choice. I would guess that symmetry of $\mathcal{R}$ over separable metric spaces follows from a weaker form of choice, possibly countable choice (CC). CC doesn't imply the pathologies of full choice, in particular it is consistent with the axiom of determinacy (AD rules out all choice pathologies). And ZF + CC is equivalent to ZF + "countable products of compact pseudometric spaces are Baire", so it looks like CC is necessary to develop metric geometry over separable spaces. | |
| Apr 30, 2020 at 13:25 | comment | added | Carl-Fredrik Nyberg Brodda | @YCor Thank you, you are right of course. I have edited the question to reflect this. | |
| Apr 30, 2020 at 13:24 | history | edited | Carl-Fredrik Nyberg Brodda | CC BY-SA 4.0 |
Edited to accommodate YCor's comments.
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| Apr 30, 2020 at 13:22 | history | edited | Asaf Karagila♦ | CC BY-SA 4.0 |
Titles be titling...
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| Apr 30, 2020 at 13:22 | comment | added | Asaf Karagila♦ | Well, interesting is subjective. So I'd daresay "none of them"? :P | |
| Apr 30, 2020 at 13:11 | history | edited | YCor |
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| Apr 30, 2020 at 13:07 | comment | added | YCor | Also your definition (even in ZFC) is not the usual one: the identity map $(\mathbf{R},d)\to(\mathbf{R},\sqrt{d})$ is coarsely Lipschitz (even large-scale Lipschitz), coarsely surjective (= essentially surjective), but not a quasi-isometry. Either you want to stick to coarse equivalence, or restrict to spaces for which these notions are equivalent (that is to say, large-scale geodesic spaces). I note that the definition used by Corson in the paper you link is the usual one. | |
| Apr 30, 2020 at 13:05 | comment | added | YCor | Clearly, "be quasi-isometric" is not an appropriate term if it fails to be an equivalence relation. You should better refer to it as the relation $X\,\mathcal{R}\,Y$ "there exists a large-scale Lipschitz essentially surjective map $X\to Y$". Corson says "the QI-relation", which is OK. | |
| Apr 30, 2020 at 13:03 | history | edited | YCor |
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| Apr 30, 2020 at 13:02 | history | asked | Carl-Fredrik Nyberg Brodda | CC BY-SA 4.0 |