Timeline for diameter as a Morse function
Current License: CC BY-SA 3.0
18 events
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| Feb 8, 2016 at 14:55 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| May 15, 2014 at 21:01 | comment | added | Neil Strickland | @katz: it would be better if you edited your question to incorporate these clarifications, rather than leaving them as comments. | |
| May 15, 2014 at 18:39 | review | Close votes | |||
| May 15, 2014 at 21:50 | |||||
| May 15, 2014 at 18:14 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| May 15, 2014 at 17:41 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| May 15, 2014 at 17:30 | history | edited | Mikhail Katz |
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| May 15, 2014 at 17:27 | comment | added | Mikhail Katz | As far as your question "I don't see how you get different 'connected components'", think of the vertices of an equilateral triangle. This set cannot be deformed to a single point without passing through a set containing a pair of antipodal points. Therefore they lie in different components of $X_1$. | |
| May 15, 2014 at 17:25 | comment | added | Mikhail Katz | Any two-point set (not the antipodal pair) can be naturally deformed to a single point, so it lies in the component of a single point (the "regular 1-gon"). | |
| May 15, 2014 at 17:24 | comment | added | Mikhail Katz | As far as finite sets are concerned, it turns out that the minima are necessarily finite, so it does not really matter whether we consider closed sets or finite sets. For the purposes of the formal properties of the space it may be a bit more convenient to consider all closed sets (with the appropriate diameter restriction, of course). | |
| May 15, 2014 at 17:22 | comment | added | Mikhail Katz | @Willie, thanks, that was a typo and should have been $X_1$ in the first paragraph. These spaces have a natural metric defined by the Hausdorff distance between subsets, so the underlying topology is automatic. | |
| May 15, 2014 at 17:21 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| May 15, 2014 at 10:23 | comment | added | Willie Wong | Lastly, what do you mean by "1-1 correspondence"? What about the subset of 2 points in $X_1$? On that component there is no extrema of the Morse function, because you exclude existence of antipodal points. Similar for all even numbers. | |
| May 15, 2014 at 10:16 | comment | added | Willie Wong | Also, the paper you linked to specified finite subsets, not arbitrary closed subsets as in your question statement, can you please clarify? (I ask because, relative to the Hausdorff metric if you allow arbitrary closed subsets I don't see how you get different "connected components"). | |
| May 15, 2014 at 10:09 | comment | added | Willie Wong | This may be a bit of a dumb question: what is the topology you are putting on $X_*$? And should the first appearance of the symbol $X_2$ be in fact $X_1$? | |
| May 15, 2014 at 5:13 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| May 14, 2014 at 13:42 | history | edited | Mikhail Katz |
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| May 14, 2014 at 11:32 | history | edited | Mikhail Katz |
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| May 14, 2014 at 8:31 | history | asked | Mikhail Katz | CC BY-SA 3.0 |