Timeline for Retracting to a bigger compact
Current License: CC BY-SA 4.0
26 events
| when toggle format | what | by | license | comment | |
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| May 5, 2020 at 2:04 | comment | added | erz | I've made a question that specifically asks whether ANR implies CBR mathoverflow.net/questions/359384/… | |
| May 4, 2020 at 19:33 | comment | added | erz | I agree that on principle the definition of ANR is even more complicated than RBC. However, there is so much information about them available, and in particular a lot of "nice" spaces belong to that class. | |
| May 4, 2020 at 16:26 | comment | added | Iosif Pinelis | @erz : Thank you for your continued interest. ANR certainly seems a more studied property than RBC, so that even a partial reduction of RBC to ANR may be helpful. However, ANR seems to be of the same logical structure as RBC: "$\exists$ a neighborhood ... such that $\exists$ a retraction ..." for ANR versus "$\exists$ a bigger compact than $K$ such that $\exists$ a retraction ..." for RBC. The two remaining $\exists$ quantifiers in the definition of ANR don't by themselves look entirely satisfying, as this leaves the job of finding the existing objects to the user. | |
| May 4, 2020 at 7:25 | comment | added | erz | The proof that I thought i had that ANR's have CBR property turned out to be incorrect (i don't know if the claim is correct). Perhaps that could be a natural thing to ask? This class contains manifolds and is stable with respect to taking open subsets. Note that CBR is not stable with respect to taking neither open nor closed subsets. | |
| May 4, 2020 at 3:10 | comment | added | Iosif Pinelis | @erz : Sorry, I missed the convexity condition there. This is now corrected. | |
| May 4, 2020 at 3:09 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| May 4, 2020 at 1:23 | comment | added | erz | could you please elaborate on "Using certain known results, it is not hard to see that every closed topological subspace of a locally convex metrizable topological vector space has the RBC property."? | |
| May 4, 2020 at 0:41 | comment | added | Iosif Pinelis | @Anonymous : Thank you very much for your latest comment as well. I have updated the post accordingly. | |
| May 4, 2020 at 0:40 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| May 3, 2020 at 17:24 | comment | added | Anonymous | I don't have a reference handy, but I think that Mazurkiewicz gave an example of a non-trivial connected, completely metrizable subset of the plane such that no compact subset with more than one point is connected. | |
| May 3, 2020 at 14:31 | comment | added | Iosif Pinelis | Thank you erz, Anonymous, and Taras Banakh for your very helpful comments. I have updated the post accordingly. Also, now I have the specific question: Is it true that all Polish spaces have the RBC property? Plus, I have now provided the motivation. | |
| May 3, 2020 at 14:29 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| May 2, 2020 at 11:57 | comment | added | Anonymous | Yes, that is one of many known examples. | |
| May 2, 2020 at 7:30 | comment | added | Taras Banakh | @Anonymous The Bernstein subset of the plane is connected but contains no infinite compact subsets, so fails to have RBC. | |
| May 1, 2020 at 17:01 | comment | added | Anonymous | Yes, I should have mentioned that I was considering infinite spaces. I think that spaces with the property that no compact subset with more than one point is connected are called punctiform spaces. | |
| May 1, 2020 at 15:48 | comment | added | erz | Knaster-Kuratowski fan is a connected subset of the plane such that removing a single point makes it totally disconnected. If it has a non-singleton connected compact subset, that subset would be compact, connected and such that removal of a single point makes it totally disconnected. That is impossible, and so, as @Anonymous suggests, the Knaster-Kuratowski fan is not RBC | |
| May 1, 2020 at 13:08 | comment | added | Iosif Pinelis | @Anonymous : Thank you for your comment. A trivial remark, though: If e.g. $X$ has $\le1$ element, then it obviously has your property: "$X$ is a connected space such that no compact subset having at least two elements is connected". However, then $X$, being compact, also obviously has the RBC property. So, you need your $X$ to be of cardinality $\ge2$. Then, to me, the question becomes this: Is there an example of a space of cardinality $\ge2$ having your property? (I apologize in advance if this question has an obvious answer.) | |
| May 1, 2020 at 12:57 | comment | added | Iosif Pinelis | @erz : Thank you for your comments, especially the first one, suggesting (to me) that non-RBC spaces are rare (if exist at all). | |
| May 1, 2020 at 11:49 | comment | added | Anonymous | For the second question, let X be a connected space such that no compact subset having at least two elements is connected. Then X obviously fails to have the RBC property. | |
| May 1, 2020 at 5:31 | comment | added | erz | As for your first question, ANR's seem to have RBC (not sure if that is valuable) | |
| May 1, 2020 at 3:30 | comment | added | erz | Concerning your second question: take a strongly rigid space $X$ (i.e. such that if $f:X\to X$ is continuous, then either $f$ is a constant map, or the identity. If $X$ is non-compact, and $f:X\to X$ has a relatively compact image, it must be a constant map. Hence, if you take $K$ to be a two-point set, there is no retraction on a bigger compact set containing $K$. The only remaining part is to find a non-compact strongly rigid space. So far everywhere I looked, the spaces are either compact, or it is not specified (and it is not easy to see) whether they are compact or not. | |
| Apr 30, 2020 at 21:14 | comment | added | Denis Nardin | @IosifPinelis Apologies I've misread the question. | |
| Apr 30, 2020 at 19:52 | comment | added | Wojowu | My comments were saying that the infinite discrete space and the long line do not satisfy your stated property for deformation retracts, though they have pretty obvious retracts onto large compact sets. I'm afraid I don't really see a way to adapt these to the question at hand. | |
| Apr 30, 2020 at 19:49 | comment | added | Iosif Pinelis | @Wojowu : I did not see your previous comments, but I think for me, with my total inexperience in this field, almost any comments could be useful. | |
| Apr 30, 2020 at 19:44 | comment | added | Wojowu | Please ignore my previous comments, I was thinking of deformation retracts. | |
| Apr 30, 2020 at 19:01 | history | asked | Iosif Pinelis | CC BY-SA 4.0 |