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May
15
revised diameter as a Morse function
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May
15
revised diameter as a Morse function
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May
15
revised diameter as a Morse function
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May
15
comment diameter as a Morse function
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
comment diameter as a Morse function
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
comment diameter as a Morse function
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
comment diameter as a Morse function
@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
revised diameter as a Morse function
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May
15
revised diameter as a Morse function
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May
14
revised diameter as a Morse function
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May
14
revised diameter as a Morse function
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May
14
asked diameter as a Morse function
May
14
revised Levi's book on Leibnizian calculus
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May
14
revised Levi's book on Leibnizian calculus
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May
14
comment Levi's book on Leibnizian calculus
@quid, thanks very much for looking this up. In case anybody has access to the library, I provide the relevant link to the 1776 entry: opac.tib.uni-hannover.de/DB=1/XMLPRS=N/PPN?PPN=031333311
May
13
revised Levi's book on Leibnizian calculus
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May
13
comment Levi's book on Leibnizian calculus
@quid, Thanks for pointing out my error.
May
13
comment Levi's book on Leibnizian calculus
Thanks. There seems to be nothing but tables there. Does this include both books? I was expecting to find a calculus book but did not find any mention of "Calculus differentialis" either in the title or the contents.
May
13
comment Levi's book on Leibnizian calculus
Do you have access to the pdf?