Chain of strictly decreasing closed sets is countable
igo...@gmail.com
Baire:
Let U_1 contains U_2 contains ... contains U_alpha be a strictly
decreasing chain of open subsets of the reals, alpha is an ordinal.
Then alpha is at most countable.
Let F_1 contains F_2 contains ... contains F_alpha be a strictly
decreasing chain of closed subsets of the reals, alpha is an ordinal.
Then alpha is at most countable.
The second of these results is used in proving the Cantor-Bendixson
theorem. I'm just wondering if there is a commonly accepted short name
for
these results.
Thanks in advance.
Igor
Dave L. Renfro
If you're only interested in a mathematical reply,
skip past the following rant.
***********************************************************
I posted a detailed reply to Igor on February 14, the same
day Igor posted his question, but I notice that my reply
isn't in the google archive. In fact, I see that several
(but not all) posts I've made during the past few months,
some of which were very lengthy and detailed replies
to people, are not in the google archive. They *are*
in the Math Forum archive, however, which I'd expect
since I made my posts through the Math Forum.
I've been posting through the Math Forum because for
over a year my posts haven't been getting through
when I post with google. I wrote google about my
problem (this was before the new beta-groups change
over, by the way, so that change had nothing to do
with my posts not getting through), but because the
reply I got about a week later -- not the immediate
automated reply, but the follow-up human reply -- was
so inappropriate that I just gave up on google and
decided to take my chances with Math Forum.
Basically, the person at google suggested
a couple of things which, if they had read
my e-mail, they'd already know that I'd done
what they suggested. Then they said that in
any event, it seems that my posts were now
getting through since some recent posts of
mine were in the archive. Again, if they had
read my letter, they would have seen that the
very same posts they were pointing to were
the posts I mentioned in my letter as being
made through the Math Forum, not google.
(My e-mail dealt with two issues, one which
I expected they could do something about,
and one (the issue of Math Forum posts sometimes
getting archived by google and sometimes not)
which I didn't expect they could do anything
about, but which thought I'd bring up anyway
just in case they could.)
Then comes the beta-groups change, which made things
worse because google's already too cluttered web pages
got even more cluttered. Not to mention that now their
line lengths are something like 50 characters, so
the posts are all broken up, variable font size is
used which is stupid for a group like this, etc.
So I've been sticking to Math Forum.
But now the Math Forum has, for some reason, changed
their site so that posts there are also virtually
unreadable. Reading posts there is now like trying
to read a book where 40% of the left margin of
_every_single_page_ is reserved for the table of
contents, publishing information, and other books
by the same publisher. Plus, their URL's are now
ridiculously long (like google's, although in each
case they can be shortened, but the real issue is
if this is supposed to be an improvement, why are
we now having to do this), and the over 100 URL's
I have for posts of mine that are especially detailed
in providing references or information about something
no longer work, not to mention the many useful
cross-references I've made in my posts using URL's
to other posts.
This is the second time I've seen a lot of my work
go down the tubes, the first time being when Math Forum
went from Swarthmore to Drexel. For example, just two
weeks ago I gave my calculus students the first URL below
http://mathforum.org/discuss/sci.math/m/403695/403696
http://groups.google.co.uk/groups?th=e88d9cd407adfd13
which was supposed to take them to a large list of
links I put together a while back on useful calculus
links and, a few days after I gave it to them, it no
longer works. The second URL above will take you to
the same list, but who knows how long that's going
to last. This post also contains a link to another
post of mine, a huge list of worked out max/min
problems, but this link also doesn't work, and worse
yet, it happens to be a post that for some reason
never got archived at google. If anyone's interested,
here's the latest URL for it
WEB PAGES FOR CALCULUS MAX/MIN PROBLEMS
http://mathforum.org/kb/thread.jspa?threadID=75952
I say "latest", because there's no telling how long
this URL will work. Someone coming across the present
post in the archives two or three years from now will
probably have to do a phrase search at Math Forum to
find it.
Also, there are several very lengthy posts that
I've been working on over the past year that update
and greatly extend some of my most detailed posts
(e.g. my essays on the continuity of derivatives
and on normal numbers), and many of these include
URL's to every reasonably significant UseNet post
on the particular topic that I've been able to find
through numerous searches. But, of course, now every
one of those URL's that goes to a Math Forum post
no longer works.
This problem extends well past me, even for my
own posts, because there are many people who
have links to some of my sci.math posts on their
web pages and now these links no longer work.
To bring this rant to a close, I've pretty much
lost all desire to continue providing lengthy
lists of references and statements of results
for the purpose of having it availible in the
sci.math archives. I know that the vast majority
of posts here are not of this type, but it seems
that both google and Math Forum have arranged
their archives so that these kinds of posts
have much less value than they used to have.
As for this post, maybe with google's beta-groups
change my post will get through (because my old
account at google probably no longer exists), but
if not, I guess I'll post it through the Math Forum
and take my chances on whether anyone who doesn't
read sci.math there will see it.
***********************************************************
Dave L. Renfro wrote (on Feb. 14):
> For the record, the results are in Section 24.II (pp. 257-258)
> of the 1966 Academic Press edition. Also, as you probably
> know but this is for those who might be interested, the
> Cantor-Bendixson theorem can be proved less constructively
> using Lindelof's 1903 notion of a condensation point.
> Moreover, there is at least one other proof due to Hausdorff
> (first appeared in his 1914 topology book, I think) in which
> you consider what's left after removing the union of all
> dense-in-themselves subsets of the given set.
>
> Regarding the results you stated, Baire might be responsible
> for the open set version, but I thought the closed set version
> was proved by Cantor and Bendixson around 1884 (see pp. 90-92
> of Michael Hallett's book and pp. 111-115 of Joseph Dauben's
> book, pp. 203-208 of José Ferreirós' book). However, I notice
> that Kamke (p. 132 of the English version published by Dover
> in 1950) also credits the closed set version to Baire. Maybe
> what Baire did was prove the result for an arbitrary decreasing
> ordinal-sequence of closed sets, while Cantor and Bendixson
> didn't generalize out of the realm of ordinal-sequences obtained
> by applying the derived set operation (for non-limit ordinals)
> and intersections (for limit ordinals).
>
> I'm pretty sure I've heard the closed set version referred
> to as "Cantor's Intersection Theorem" or "Cantor's Stationary
> Theorem" before. Unfortunately, the term "Cantor's Intersection
> Theorem" almost always refers to the result that a decreasing
> w-sequence of nonempty compact sets has a nonempty intersection.
>
> The result doesn't appear to be named in Kechris' "Classical
> Descriptive Set Theory" (both open and closed set version
> are proved on p. 33), Sierpinski's "General Topology" (the
> 1956 2'nd English edition, pp. 66-67), Harbacek/Jech's
> "Introduction to Set Theory" (3'rd edition, pp. 192-193),
> Jech's "Set Theory" (1978 edition, p. 34).
>
> Well, I thought I'd be able to answer your question easily
> since this is something I've looked at quite a bit, as can
> easily be guessed from these posts:
>
> http://mathforum.org/discuss/sci.math/m/270118/270130
> http://mathforum.org/discuss/sci.math/m/384280/384286
> http://mathforum.org/discuss/sci.math/m/475274/475791
>
> However, I'm not sure what an appropriate name for the
> result is. I don't think you should use "Cantor Intersection
> Theorem", and I can't find anything searching for "Cantor*
> Stationary Theorem" (or even "Cantor* stationary") in Math.
> Reviews, the Euler database, JSTOR, google, or google-groups.
I came across the following footnote on p. 146 of John C.
Morgan's book "Point Set Theory" (Marcel Dekker, 1990):
"Although commonly called the Cantor Intersection Theorem,
Cantor established this result only for a sequence of
successive derived sets of a bounded set. The theorem
for an arbitrary sequence of compact sets was first
established by Baire. [snip some citations] "
This isn't exactly the same thing the original poster
asked about, but it adds support to my guess as to why
Kamke credits Baire.
Dave L. Renfro
Dave Rusin
Dave L. Renfro <renf...@cmich.edu> wrote:
>If you're only interested in a mathematical reply,
>skip past the following rant.
I would like to chime in here because the rant is very important to me too,
and I think it affects anyone else who, like me, sees value in the
permanent accessibility of (a small portion of) old sci.math posts.
To recap a few of Dave's points, things are bad because of three things
to have hit USENET recently:
1. Google Groups has a "New and Improved" interface which suffers
several new deficiencies.
2. Google is starting to lose articles from its archives.
3. Math Forum has changed its interface too and old URLs no longer work.
The upshot is that in a very short time period we have gone from having
two sources of archival information to having none in some cases.
Here is my own example. I needed to find a 1999 article of mine, with
Message-ID: <7cofdk$ib0$1...@gannett.math.niu.edu>
I was unable to find it at google by typing some keywords into its
the search engine. I did succeed at Math Forum, finding it at
http://www.mathforum.org/epigone/sci.stat.math/phimplilchol
so I'm not hallucinating when I claim it really existed.
Seeing Dave Renfro's comments, I went to find the article in the
"New" math forum but was unable to do so from the old URL.
In fact, although I had saved all the data you see here, I hadn't
made a note of what the article was about, so I couldn't today do
a keyword search either!
In fact, what I did today was to dig up a May 2002 file I had made
when I was bored one day: I saved the results of a search on my name
at google. THAT file I could grep for the message ID and so I had
the few words that show up in a google search. Those I searched in
Math Forum, so I now know the file still exists there, but under
a completely different URL:
http://mathforum.org/kb/message.jspa?messageID=1525844
So this is what we have now:
Google, which _used to have_ a copy of the article, on public display,
but which now has no trace of the article in its groups archives
(though a search of the _web_ through Google still finds the article
at Math Forum -- pointing to the old URL, unfortunately, but thanks to
the Google Cache I was able to read it)
Math Forum, which has the article and a searchable index, but which
offers no way to find the article given only its old URL.
Both of the archives now serve up old articles in the proportional font
that makes a mess of our beloved ASCII graphics. Yuk.
On top of all this, although Math Forum has thoroughly revamped their
interface, they didn't make the one change I have been asking them to
make for some time: they still don't turn off the "reply-to" feature
for threads that have been dormant for a decade or more!
Is there a point to my rant? I'm not sure. If you want "action items",
try these:
1. If you have old posts that you value, find them and make copies.
Recordable CDs are cheap. If you've got your own web site, you might
want to put things there; if not, save the CDs until you do have your
own site.
2. Keep an eye on these archives to stay abreast of changes, in case
you are in the habit of using them or encouraging others to do so.
3. I believe it to be true that the keepers of these archives read
their email and take requests and complaints to heart. If you notice
that something of yours has gone missing, or if you want to make
a suggestion, do so. One letter might get ignored, but then again,
maybe not. Multiple letters would be harder to ignore.
dave
James Dolan
dave rusin <ru...@vesuvius.math.niu.edu> wrote:
|Both of the archives now serve up old articles in the proportional
|font that makes a mess of our beloved ASCII graphics. Yuk.
google groups does at least still have a "show original" option using
a fixed font. my experience is that the "show original" option
currently works rather well (in contrast to some times in the past
when what it showed was often something quite different from the
original).
(it would be nice though if users could select "show original" as
their permanent default option in the "preferences" section of their
user profile. in general, google has a number of nice options which
could easily compensate for many of the defects in google, if only
users were allowed to select those options as their permanent default
preferences.)
(it would also be nice if user permanent default preferences really
were permanent rather than subject to being reset to the always
maximally evil standard settings on random frequent occasions. this
complaint is directed against pretty much every system in the world
rather than just against google.)
i've heard rumors that google isn't especially eager to continue
supporting the "show original" option; perhaps it would be a good idea
for people who offer their opinions to google to let them know that
they do at least appreciate the support of the "show original" option,
and would be disappointed if it were watered down or removed (and very
pleased if it were improved by being made selectable as a permanent
user preference).
--
[e-mail address jdo...@math.ucr.edu]
Jesse F. Hughes
> in article <d0ab1l$sch$1...@news.math.niu.edu>,
> dave rusin <ru...@vesuvius.math.niu.edu> wrote:
>
> |Both of the archives now serve up old articles in the proportional
> |font that makes a mess of our beloved ASCII graphics. Yuk.
>
> google groups does at least still have a "show original" option using
> a fixed font. my experience is that the "show original" option
> currently works rather well (in contrast to some times in the past
> when what it showed was often something quite different from the
> original).
"Show original" still shows something different from the original.
They mangle the headers and body of the posts to remove any email
addresses, so that if one wants to respond via email, one must have a
Google account. I find this unacceptable.
Worse, this mangling destroys message id references in the body of the
post. Other mangling seems to be currently repeated in the "Show
original" option, as seen here:
http://groups-beta.google.com/group/google-labs-groups2/msg/c25ab19f32f771b7?dmode=source
It seems to me that if Google wants to make available a public archive
of Usenet posts, they ought to do so without editing the contents of
the posts more than necessary. If I post using my email address, then
by gum, they ought to present it that way in the archive.
--
"I'm the guy. I have always been the guy. Your post will sit here for
a while, soon be ignored, except for people coming to read my reply,
and your satisfaction will fade as you move on, and I'll still be the
guy." -- James S. Harris will *always* be the guy. Duh.
James Dolan
jesse f. hughes <je...@phiwumbda.org> wrote:
|jdo...@math-cl-n03.math.ucr.edu (James Dolan) writes:
|
|> in article <d0ab1l$sch$1...@news.math.niu.edu>,
|> dave rusin <ru...@vesuvius.math.niu.edu> wrote:
|>
|> |Both of the archives now serve up old articles in the proportional
|> |font that makes a mess of our beloved ASCII graphics. Yuk.
|>
|> google groups does at least still have a "show original" option
|> using a fixed font. my experience is that the "show original"
|> option currently works rather well (in contrast to some times in
|> the past when what it showed was often something quite different
|> from the original).
|
|"Show original" still shows something different from the original.
|They mangle the headers and body of the posts to remove any email
|addresses, so that if one wants to respond via email, one must have a
|Google account. I find this unacceptable.
|
|Worse, this mangling destroys message id references in the body of
|the post. Other mangling seems to be currently repeated in the "Show
|original" option, as seen here:
|
|http://groups-beta.google.com/group/google-labs-groups2/msg/c25ab19f32f771b7?dmode=source
you're right; i'd forgotten about a few of the ways in which google's
current "show original" fails to show the actual original. it would
be much better if "show original" showed the true original.
--
[e-mail address jdo...@math.ucr.edu]
Dave Rusin
> 2. Google is starting to lose articles from its archives.
> Here is my own example. I needed to find a 1999 article of mine, with
> Message-ID: <7cofdk$ib0$1...@gannett.math.niu.edu>
> I was unable to find it at google by typing some keywords into its
> the search engine.
> Google, which _used to have_ a copy of the article, on public display,
> but which now has no trace of the article in its groups archives
I think I need to retract these remarks. David Cantrell pointed out to
me that the missing article is in fact easily found at Google.
It seems I had the wrong Message ID. I cannot explain how that
happened; that string seems to be nowhere in my files.
My apologies to the people at Google.
I guess maybe I got too exercised by Dave Renfro's rant; sorry.
Still, I think this point is probably good advice:
> 1. If you have old posts that you value, find them and make copies.
> Recordable CDs are cheap. If you've got your own web site, you might
> want to put things there; if not, save the CDs until you do have your
> own site.
dave
PS -- Someone once ended a sci.math article by remarking how many
Dave's there were in this newsgroup, but I can't find that article either.
Ullrich? Seaman? Anybody got a link?
mensa...@aol.com
James Dolan wrote:
> in article <d0ab1l$sch$1...@news.math.niu.edu>,
> dave rusin <ru...@vesuvius.math.niu.edu> wrote:
>
> |Both of the archives now serve up old articles in the proportional
> |font that makes a mess of our beloved ASCII graphics. Yuk.
>
> google groups does at least still have a "show original" option using
> a fixed font. my experience is that the "show original" option
> currently works rather well (in contrast to some times in the past
> when what it showed was often something quite different from the
> original).
>
> (it would be nice though if users could select "show original" as
> their permanent default option in the "preferences" section of their
> user profile.
You can also click on the [Fixed Font] link found at the
upper right corner above the adverts. This did not exist
in the early versions, so they do appear to be listening
to criticism. As far as I can tell, once clicked it stays
that way.
Robert Israel
Dave Rusin <ru...@vesuvius.math.niu.edu> wrote:
>PS -- Someone once ended a sci.math article by remarking how many
>Dave's there were in this newsgroup, but I can't find that article either.
>Ullrich? Seaman? Anybody got a link?
<http://groups-beta.google.com/group/sci.math/msg/70e0d6fef1a47216>
Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Dave Rusin
Robert Israel <isr...@math.ubc.ca> wrote:
>In article <d0amj7$no7$1...@news.math.niu.edu>,
>Dave Rusin <ru...@vesuvius.math.niu.edu> wrote:
>
>>PS -- Someone once ended a sci.math article by remarking how many
>>Dave's there were in this newsgroup, but I can't find that article either.
>>Ullrich? Seaman? Anybody got a link?
>
><http://groups-beta.google.com/group/sci.math/msg/70e0d6fef1a47216>
That's it! (Well, that and its follow-ups.) Then there was this one too:
<http://groups-beta.google.com/group/sci.math/msg/172a2a9a1f2498bc>
Jesse F. Hughes
It would be better still if they just showed the bloody original to
begin with, minus some headers.
I don't like that they edit my post when they present it. That seems
a fairly clear violation of my copyright (but one that is fairly
trivial, too, since my posts aren't big money-makers for me).
--
"There are people [...] who think it's socially acceptable to level
accusations of mental illness in insulting exchanges to make
points[...] [They] are rather sick [them]selves, and in reality, are
sociopathic." --- James Harris, evidently a self-described sociopath
igo...@gmail.com
> Igor wrote:
>
> >> In his book Topology, Kuratowski attributes the following
> >> theorems to Baire:
> >>
> >> Let U_1 contains U_2 contains ... contains U_alpha be
> >> a strictly decreasing chain of open subsets of the reals,
> >> alpha is an ordinal. Then alpha is at most countable.
> >>
> >> Let F_1 contains F_2 contains ... contains F_alpha be
> >> a strictly decreasing chain of closed subsets of the
> >> reals, alpha is an ordinal. Then alpha is at most countable.
> >>
> >> The second of these results is used in proving the
> >> Cantor-Bendixson theorem. I'm just wondering if there
> >> is a commonly accepted short name for these results.
> Dave L. Renfro wrote (on Feb. 14):
>
> > For the record, the results are in Section 24.II (pp. 257-258)
> > of the 1966 Academic Press edition.
It is strange... I either missed it, or this post did not appear on my
local ISP's news server either.
> > Well, I thought I'd be able to answer your question easily
> > since this is something I've looked at quite a bit, as can
> > easily be guessed from these posts:
> >
> > http://mathforum.org/discuss/sci.math/m/270118/270130
> > http://mathforum.org/discuss/sci.math/m/384280/384286
> > http://mathforum.org/discuss/sci.math/m/475274/475791
Hmm... Unfortunately, as you've already pointed out these links are
broken due to the changes in the Math Forum archive.
> > However, I'm not sure what an appropriate name for the
> > result is. I don't think you should use "Cantor Intersection
> > Theorem", and I can't find anything searching for "Cantor*
> > Stationary Theorem" (or even "Cantor* stationary") in Math.
> > Reviews, the Euler database, JSTOR, google, or google-groups.
In any case. Thank you very much for taking the time to dig up all this
useful, if perhaps not completely relevant, information.
Igor
David Bernier
[...]
> PS -- Someone once ended a sci.math article by remarking how many
> Dave's there were in this newsgroup, but I can't find that article either.
> Ullrich? Seaman? Anybody got a link?
http://groups.google.ca/groups?selm=9k9dcr$gm8$1...@tornews.torolab.ibm.com
I can't quote a message that hasn't yet reached my ISP's newsserver...
Apologies to whoever found that article.
http://groups.google.ca/ uses the old system by default.
One thing about the new system that I really don't like
is that some links to archived messages at google.com
appear inside posts at http://groups.google.com/ as:
http://groups.google.com...something/
and clicking on the link doesn't work.
David Bernier
David Bernier
Google has web sites ending in .ca, .uk, etc.
For the .ca google, the default for usenet messages is old style;
see for example:
http://groups.google.ca/groups?selm=d0ab1l$sch$1...@news.math.niu.edu
David Bernier