Why "separable"?
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From: Robert Israel on 16 Mar 2010 22:28 > The definition I get from my book for "separable" is clear enough: > > "A space X is *separable* if there exists a countable dense subset > of X." > > > But, like with many definitions in mathematics, there's "cognitive > dissonance" between the word being defined and the definition. In > this case, there is no obvious reason to choose the word "separable" > to refer to the property given in the definition. As far as I can > tell, "viscous", "brackish", and "non-refundable" are equally > suitable. Why one earth "separable"? I don't know, but I suspect the term "separable" originated with Frechet in his doctoral thesis (see <http://www-history.mcs.st-andrews.ac.uk/Obits2/Frechet_RSE_Obituary.html>). -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
From: William Elliot on 17 Mar 2010 02:34 On Tue, 16 Mar 2010, kj wrote: > > The definition I get from my book for "separable" is clear enough: > > "A space X is *separable* if there exists a countable dense subset > of X." > > But, like with many definitions in mathematics, there's "cognitive > dissonance" between the word being defined and the definition. 1st and 2nd countable are simple names. Completely separable for 2nd countable defies semantics, and conflicts with separable, for completely separable would, by common mathematical usage, be a stronger form of separable and not a distinctly different concept. Since 2nd countable is a stronger form of 1st countable, why isn't 1st countable accordingly called separable? > In this case, there is no obvious reason to choose the word > "separable" to refer to the property given in the definition.
From: David Bernier on 17 Mar 2010 06:41 Robert Israel wrote: >> The definition I get from my book for "separable" is clear enough: >> >> "A space X is *separable* if there exists a countable dense subset >> of X." >> >> >> But, like with many definitions in mathematics, there's "cognitive >> dissonance" between the word being defined and the definition. In >> this case, there is no obvious reason to choose the word "separable" >> to refer to the property given in the definition. As far as I can >> tell, "viscous", "brackish", and "non-refundable" are equally >> suitable. Why one earth "separable"? > > I don't know, but I suspect the term "separable" originated with Frechet > in his doctoral thesis (see > <http://www-history.mcs.st-andrews.ac.uk/Obits2/Frechet_RSE_Obituary.html>). I don't know the details, but I've found accounts from around 1927. From what appears below, it seems that Tychonoff and Vedenissof used "separable" for "has a countable dense set" in a 1926 article in Bulletin des Sciences Math�matiques. E. W. Chittenden wrote in a 1927 survey article that: << Tychonoff and Vedenissof have called them separable spaces.[Dagger mark] In a letter to me Fr�chet calls attention to the fact that the word separable is already in use in a more general sense and suggests the term perfectly separable. >> [Quote:] 5. Perfectly Separable Spaces. Among the spaces which were considered by Hausdorff are those whose points of accumulation are definable in terms of an enumerable family of neighborhoods.* Such spaces are said to satisfy the second axiom of enumerability. Tychonoff and Vedenissof have called them separable spaces.[Dagger mark] In a letter to me Fr�chet calls attention to the fact that the word separable is already in use in a more general sense and suggests the term perfectly separable. [Unquote] [Dagger mark]: Bulletin des Sciences Math�matiques, vol. 50 (1926), p. 17. E. W. CHITTENDEN, "THE METRIZATION PROBLEM", http://www.ams.org/bull/1927-33-01/S0002-9904-1927-04295-1/S0002-9904-1927-04295-1.pdf (Bulletin of the AMS, publication year 1927 around Jan., Feb.). David Bernier
From: David Bernier on 23 Mar 2010 01:42
David Bernier wrote: > Robert Israel wrote: >>> The definition I get from my book for "separable" is clear enough: >>> >>> "A space X is *separable* if there exists a countable dense subset >>> of X." >>> >>> >>> But, like with many definitions in mathematics, there's "cognitive >>> dissonance" between the word being defined and the definition. In >>> this case, there is no obvious reason to choose the word "separable" >>> to refer to the property given in the definition. As far as I can >>> tell, "viscous", "brackish", and "non-refundable" are equally >>> suitable. Why one earth "separable"? >> >> I don't know, but I suspect the term "separable" originated with Frechet >> in his doctoral thesis (see >> <http://www-history.mcs.st-andrews.ac.uk/Obits2/Frechet_RSE_Obituary.html>). >> [...] > From what appears below, it seems that > Tychonoff and Vedenissof used "separable" > for "has a countable dense set" in a 1926 article > in Bulletin des Sciences Math�matiques. > > E. W. Chittenden wrote in a 1927 survey article that: > > << Tychonoff and Vedenissof have called > them separable spaces.[Dagger mark] In a letter to me > Fr�chet calls attention to the fact that the word > separable is already in use in a more general sense > and suggests the term perfectly separable. >> [...] > [Dagger mark]: Bulletin des Sciences Math�matiques, vol. 50 (1926), p. 17. Conclusion: Tychonoff and Vedenisoff attribute to Frechet the naming "separable space", or "separable abstract space". But I'm confused when Chittenden writes: "Fr�chet calls attention to the fact that the word separable is already in use in a more general sense and suggests the term perfectly separable." What might that "more general sense" be? On the first page of their 1926 article Melanges, Tychonoff and Vedenisoff refer the reader to: M. Frechet, "Esquisse d'une theorie des ensembles abstraits", (Sir Asustosk Mookerjee Commemoration volumes, vol. 2; the Baptist Mission Press, Calcutta, 1922). I'm guessing the "Sir" above is "Ashutosh Mukherjee": < http://en.wikipedia.org/wiki/Ashutosh_Mukherjee > Snippet view of vol. 1 (1921) of the Silver Jubilee volumes at Google Books: < http://books.google.com/books?id=-GNFAAAAYAAJ > . It would be sensible for volume 2 to have appeared in 1922. Reference for the "Melanges" article by Tychonoff and Vedenisoff: < http://gallica.bnf.fr/ark:/12148/bpt6k4862772.image.f16.langEN > and following pages stored in Gallica. Tychonoff and Vedenisoff attribute to Frechet the notion of a set M with a system of neighborhoods. For each x in M, there is a set V(x) such that A is an element of V(x) if and only if A is a neighborhood of x. There seems to be one axiom for system of neighborhoods: if A is an element of V(x), then x is an element of A. They also define the notion of equivalent systems of neighborhoods: if V and W are two systems of neighborhoods for a set/space M, and E is a subset of M, V and W are equivalent if and only if the derived set of E in V is equal to the derived set of E in W (for any subset E of M). Tychonoff and Vedenisoff also attribute to Frechet the naming "separable space", or "separable abstract space". David Bernier > E. W. CHITTENDEN, "THE METRIZATION PROBLEM", > http://www.ams.org/bull/1927-33-01/S0002-9904-1927-04295-1/S0002-9904-1927-04295-1.pdf |