Timeline for Making compact subsets "parallel"
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Oct 29, 2017 at 20:30 | comment | added | Taras Banakh | @user116515 In my next post I wrote a correct (and not very difficult) proof of the general case. | |
| Oct 29, 2017 at 20:29 | history | edited | Taras Banakh | CC BY-SA 3.0 |
I deleted the proof of the general case as it contained a gap.
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| Oct 29, 2017 at 17:23 | comment | added | Taras Banakh | @user116515 I started to write down the proof of general case and found a gap (namely in condition (3) of Claim). So I deleted the general case from my answer and leaved only zero-dimensional case, which is correct. | |
| Oct 28, 2017 at 19:56 | comment | added | user116515 | I admit it will take me some time to digest your sketch, especially since it uses terminology with which I am not familiar. But I'm very impressed if you produced some new mathematics from my somewhat idle question. Very nice! | |
| Oct 28, 2017 at 18:41 | comment | added | Taras Banakh | @user116515 I added to my answer a sketch of the construction of a parallel metric in the general case (of lower and upper semicontinuous partitions). I plan to write a text with a complete proof and post it to Arxiv. | |
| Oct 28, 2017 at 18:37 | history | edited | Taras Banakh | CC BY-SA 3.0 |
Added a solution of the general problem
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| Oct 28, 2017 at 0:15 | history | edited | Taras Banakh | CC BY-SA 3.0 |
Added a reduction to the original problem to the case of open perfect maps.
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| Oct 27, 2017 at 23:52 | comment | added | Taras Banakh | @user116515 A partition is upper (resp. lower) semicontinuous if for any closed (resp. open) set its saturation is closed (resp. open). A saturation of a subset $A$ by a partition is the union of all cells of the partition that intersect $A$. But maybe it is more convenient to think of partitions as fibers of the corresponding qutient maps. | |
| Oct 27, 2017 at 23:48 | comment | added | Taras Banakh | @user116515 On the other hand, if a partition of a space into compact sets is both lower and upper semicontinuous, the the quotient map induced by this partition is perfect and open. So, we fall into the framework of my answer. So, now we should try to exclude the zero-dimensionality, which was essentially used for the construction of the parallel ultrametric. | |
| Oct 27, 2017 at 23:48 | comment | added | user116515 | Yes, MTyson's example indicated to me that (*) needed to be modified to be more symmetric -- though I did not know the terms upper/lower semicontinuity of a partition. Condition (*) has been changed, I think, to rule out examples like MTyson's, though perhaps there are other counterexamples. | |
| Oct 27, 2017 at 23:45 | comment | added | Taras Banakh | @user116515 It seems that the second statement in Will Brian's example should be adjusted: one can take a sequence of pairwise disjoint clopen sets of diameter $>1$ in a suitable Cantor set and take other sets be singletons. Then the obtained partition of the Cantor set satisfies $(*)$ but does not admit a "parallel" metric. It seems that in case of a partition one should assume that the partition is both lower semicotinuous (which is the condition (*)) and upper semicontinuous. The absence of the upper semicontinuity yields counterexamples like that of MTyson. | |
| Oct 27, 2017 at 23:31 | comment | added | user116515 | Thanks. This is a nice partial answer, which seems to be closely related to Will Brian's example on the Cantor set. | |
| Oct 27, 2017 at 20:56 | history | edited | Taras Banakh | CC BY-SA 3.0 |
added 12 characters in body
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| Oct 27, 2017 at 20:40 | history | answered | Taras Banakh | CC BY-SA 3.0 |