Re: Scattered sets are G-delta

On May 27, 2:39 pm, "Dave L. Renfro" <renfr...@xxxxxxxxx> wrote:
Dave L. Renfro wrote (in part):

Last night [...] I found some additional references,
all later than Hobson's and Young's 1902 or 1903 to
1907 references I've already cited. [...] I'll post
these later today, when I get a chance, probably
within 6 hours from now.

Below are some references for results involving
scattered sets being G_delta. The URLs are
where you can find bibliographic details about
the items I cite. In a few cases, the URL will
take you to a digital file for the item.

In Kuratwoski's "Une méthode d'élimination des
nombres transfinis des raisonnements mathématiques",
Fundamenta Mathematica 3 (1922), 76-108, the last
line on p. 96 before the footnotes is: "Tout
ensemble clairseme est un G_delta." [All scattered
sets are G_delta.]http://matwbn.icm.edu.pl/ksiazki/fm/fm3/fm3114.pdf

Both "scattered implies G_delta" and "countable
and G_delta implies scattered" are explicitly
stated (in different terms) and proved on
pp. 628-629 of Arthur Harold Blue, "On the
structure of sets of points of classes one,
two, and three", Mathematische Annalen 102
(1929), 624-632 <http://tinyurl.com/6dxfmx>.
Both results can also be found in Blue's
July 1928 Ph.D. Dissertation under Edwin W.
Chittenden (Section 10, pp. 19-21).

Both "scattered implies G_delta" and "countable
and G_delta implies scattered" are explicitly
stated and proved in Lusin's 1930 book, pp. 106-107:
<http://tinyurl.com/3u4hgz>.

Both "scattered implies G_delta" and "countable
and G_delta implies scattered" are essentially
in Volume I of Kuratowski's 1966 topology book,
<http://tinyurl.com/4svccy>. See: Section 12.II,
Remark, p. 96; Section 12.VI, Theorem 4, p. 101;
Section 34.V - 34.VI, pp. 417-419. I suspect both
parts are at least implicit in the 1933 first edition
also. For example, see Section 19.III, p. 112 of
<http://matwbn.icm.edu.pl/kstresc.php?tom=3&wyd=10>.

In the 3'rd edition (1935) of Hausdorff's book,
<http://tinyurl.com/4uwgoe>, Section 30.4
(first half, pp. 194-195, especially Theorem V)
proves that scattered sets (called "reducible
sets" in this reference) in a separable metric
space are G_delta. The result is almost certainly
in the 1'st edition (1914), <http://tinyurl.com/3tzck6>,
but I couldn't find it after about 20 minutes
of searching. [The fact that I can't read a bit
of German means it still might be there.]

Both "scattered implies G_delta" and "countable
and G_delta implies scattered" are explicitly
stated and proved in Thomson's 1994 book on
symmetric properties of sets and functions,
Appendix A.1, Theorem A.3, pp. 404-405:
<http://books.google.com/books?id=BMWk0X8rl_YC>

In Bruckner/Bruckner/Thomson's 1996 text "Real Analysis",
Exercise 1:1.27 (p. 7) is: "Show that every scattered
set is of type G_delta." <http://tinyurl.com/5qsov6>

A useful survey for scattered sets, especially for
literature references, is Z. Semadeni's 1971 book
"Banach Spaces of Continuous Functions" (Section 8.5,
pp. 147-151; especially Section 8.5.11, pp. 150-151).
<http://tinyurl.com/49yszs>

Finally, for the record, here are some of the places
I looked where one might think either of the results
could be found, but I wasn't able to find them. In two
cases (i.e. Hahn's books) it's probably there, but my
inability to read German prevented me from finding it.

1912 -- Pierpont, Volume 2 <http://tinyurl.com/4svccy>

1921 -- Hahn <http://tinyurl.com/4svccy>

1932 -- Hahn <http://tinyurl.com/4svccy>

1952/2000 - Sierpinski, "General Topology"http://books.google.com/books?id=rlDCcGC6Xz0C&pg=PR4

1964/1998 -- Vaidyanathaswamy, "Set Topology"http://books.google.com/books?id=yDMipybQ64kC

1980 -- Moschovakis, "Descriptive Set Theory"http://tinyurl.com/52omgj

1995 -- Kechris, "Classical Descriptive Set Theory"http://tinyurl.com/4svccy

1998 -- Srivastava, "A course on Borel Sets"http://tinyurl.com/49yszs

Dave L. Renfro

Wow. Fantastic. Thanks again.
.

Relevant Pages

  • Re: Genius of Charles Darwin
    ... propaganda than science. ... which contained references to "creator"? ... "Darwin himself, as a young biologist aboard H.M.S. Beagle, was ... did not appear in the first edition of Origin of Species. ...
    (talk.origins)
  • Re: The Dual space of l_infty
    ... any more references? ... space of continuous functions Con some compact space X and then ... are those of the Statistics Department or of Purdue University. ... Herman Rubin, Department of Statistics, Purdue University ...
    (sci.math)
  • Re: The Dual space of l_infty
    ... See e.g. J. Diestel, "Sequences and Series ... any more references? ... Any idea of the dual space of c (the space of all sequences that have ... space of continuous functions Con some compact space X and then ...
    (sci.math)
  • Re: Scattered sets are G-delta
    ... Dave L. Renfro wrote: ... Below are some references for results involving ... "Banach Spaces of Continuous Functions" (Section 8.5, ...
    (sci.math)
  • Measurable fileds of Hilbert spaces
    ... Is there a generalization of measurable fields of Hilbert ... spaces to, let's say, measurable fields of Banach spaces? ... References? ... (if online so much the better). ...
    (sci.math.research)