Scattered sets are G-delta
vinay....@yahoo.com
A subset S of real numbers is called scattered if every nonempty
subset A of S has an isolated point. Show that every scattered set is
a G-delta set.
Any help will be appreciated. Thanks in advance.
Tomáš Procházka
1) your scattered set S should be countable (because every compact interval <-K;K> of R can contain only finite number of isolated points); you can intersect S with <-K;K> and take isolated points until there are some
2) now when S is countable collection of isolated points x_n, for every n there is eps_n such that (x_n-eps_n;x_n+eps_n) intersects S only at point x_n; for every natural m define G_m as union over n of open sets (x_n-eps_n/m,x_n+eps_n/m). For every m you get open set and intersection of all of G_m gives S.
I don't know if it is correct, but I hope it can help you somehow.
Tomas
tango
A subset S of R is scattered iff there is some map f:R -> R such that
S = {x| lim as y->x f(y) is + infinity}.
tango
This is okay.
> because every compact interval <-K;K> of R can contain only finite number of isolated points
No. Consider {1/n| n = 1,2,3..} as a subset of [0, 1].
> now when S is countable collection of isolated points x_n,
Take a break.
Tomáš Procházka
But this shouldn't change the argument. In every <-K;K> subset there is only countable number of isolated points and by diagonal argument there is only countably points in whole S. :-)
Tomas
mamapapa...@gmail.com
> In every <-K;K> subset there is only countable number of
> isolated points and by diagonal argument there is only countably points in whole S.
How do you get from "in every <-K;K> subset there is only countable
number of isolated points" to "there is only countably points in whole
S"?
Dave L. Renfro
It takes a bit more work than this to prove a scattered set
is countable. For example, if you take any countable ordinal,
then any subset of the reals whose order type (when the points
in the subset are ordered as they're ordered on the real line)
is scattered, and this doesn't exhaust all scattered sets
(because, among other things, you can insert in various places
"reverse well ordered sets of reals").
If no one has said much by tomorrow, I'll post an outline of
at least one way you can show scattered sets are countable.
For now, however, the following posts might be of use:
http://groups.google.com/group/sci.math/msg/d2c88b689bd79a93
http://groups.google.com/group/sci.math/msg/1b7439a55bf25eb9
http://groups.google.com/group/sci.math/msg/743f6ff391a9b843
http://groups.google.com/group/sci.math/msg/ccb2413d81b46b08
http://groups.google.com/group/sci.math/msg/0fb6110e86c6e017
Dave L. Renfro
Tomáš Procházka
I thought (A1=S intersect<-1;1>, A2=S intersec<-2;2>\<-1;1>...):
take 1 number from A1
take 1 number from A1 and one number from A2
etc.
Isn't this the 'diagonal' argument
123...
456...
789...
..
..
and going like 124753...?
Tomas
vinay....@yahoo.com
Is closure of a scattered set countable?
William Elliot
>
> If no one has said much by tomorrow, I'll post an outline of
> at least one way you can show scattered sets are countable.
Isn't the problem to show a scattered subset of R is G_delta?
To show that they're countable just shows that they're F_sigma.
If S scattered subset R, then int S = nulset.
The converse is false.
I've not been able to show S is nowhere dense,
. . int cl S = nulset.
I suppect nowhere dense and scattered are equivalent within R.
Dave L. Renfro
> If no one has said much by tomorrow, I'll post
> an outline of at least one way you can show
> scattered sets are countable.
Ooops, I meant G_delta. However, one of the
key issues involves showing they're countable,
at least in the proofs I'm aware of.
There are several mathematically equivalent
definitions of "scattered set". The version
used by the original poster is: "A subset S of
real numbers is called scattered if every nonempty
subset A of S has an isolated point." This is
ambiguous, since we're not told whether the
notion "isolated point" is with respect to the
original set or with respect to the set A.
It should be with respect to A. Otherwise,
we're looking at a strictly stronger notion
of smallness, a set of points each of which
is isolated from all the other points in the
set. (To see this, choose A = S in the "incorrect"
interpretation of the original posters's definition.)
This definition of scattered is equivalent to
Hausdorff's 1914 version of the Cantor-Bendixson
theorem, one of the three ways in which the
Cantor-Bendixson theorem can be approached
(Cantor's iterating the derived set function
in 1883, Lindelof's method using condensation
points in 1905, and Hausdorff's 1914 method),
which I outlined in the following post:
http://groups.google.com/group/sci.math/msg/1b7439a55bf25eb9
I just realized that I brought the wrong notes
with me, so I'll have to look at home for my
other notes and outline how to show G_delta
tomorrow. I did think about it some just now,
but nothing immediately evident comes to mind
and I'd rather not waste time on a problem whose
solution I know exactly where it is, especially since
it's not an area I'm currently working on. [I will
mention, however, that a certain poster who goes
by Butch Malahide has published a proof that a set
is scattered if and only if it's countable and
G_delta. Maybe he'll jump in now that I've mentioned
his name. The result itself is due, I believe, to
William H. Young and can be found on pp. 65 & 298
of his 1906/1972 book. Also, E. Hobson published an
article about this in 1904.]
Given some other comments in this thread, I thought
I'd outline some subclasses of the countable sets
of the real line that are useful in describing notions
of smallness (the following are all hereditary) when
the set is countable:
A. Sets with countable closure
B. Isolated sets
C. Scattered sets
D. Sets that are countable
We have:
A proper subset of B proper subset of C proper subset D
SOME COUNTEREXAMPLES
B, but not A: Take any Cantor set (perfect nowhere dense
set of the reals) and consider the collection of midpoints
of the bounded complementary intervals. The closure of
this set will be the union of these midpoints along with
the Cantor set that was used. By choosing a Cantor set
with positive measure, we can even get an isolated set
whose closure has positive measure.
C, but not B: {0, 1, 1/2, 1/3, 1/4, 1/5, ...}
D, but not C: the set of rational numbers
William Elliot, in this thread, suggested that maybe
"scattered" is equivalent to "nowhere dense" for the
real line. This is not true, since every scattered
set is countable and there exist uncountable nowhere
dense sets (e.g. a Cantor set). Indeed, as some of
the posts I cited earlier in this thread say, scattered
sets are in a certain sense "maximally nowhere dense":
A scattered set is nowhere dense relative to every
nonempty perfect subset of R [*]. This means, for
instance, that not only the set itself but also the
closure of the set fails to be dense in any portion of
any perfect set, regardless of how small/thin the perfect
set is. I believe this way of looking at scattered sets
is due to A. Denjoy (around 1914-1917, I think).
[*] That is, if S is scattered and P is a perfect set,
then 'S intersect P' is a nowhere dense subset of
P (where P is given the subspace topology it inherits
from the reals).
Dave L. Renfro
Dave L. Renfro
> Given some other comments in this thread, I thought
> I'd outline some subclasses of the countable sets
> of the real line that are useful in describing notions
> of smallness (the following are all hereditary) when
> the set is countable:
>
> A. Sets with countable closure
>
> B. Isolated sets
>
> C. Scattered sets
>
> D. Sets that are countable
>
> We have:
>
> A proper subset of B proper subset of C proper subset D
Darn! This isn't completely correct. I only have a few
moments (I'm in a university library to pick some volumes
up and they close in a few minutes), so I'll say more
tomorrow, but here's the correct relationship:
.
(use fixed font style and, if reading at Math Forum, choose
"Plain Text" option in upper left of page)
.
countable nowhere dense
\ /
\ /
\ /
\ /
\ /
\ /
\ /
scattered
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
countable closure isolated
.
Strictly stronger than 'isolated and has countable closure'
is what one might call "uniformly isolated", where there is
a positive lower bound on the distances between distinct
points (equivalent, in R^n at least, to being locally finite,
or to being locally singleton). Strictly stronger than being
uniformly isolated is being a finite set.
.
Dave L. Renfro
Tim Little
> I suppect nowhere dense and scattered are equivalent within R.
The Cantor set is nowhere dense, but has no isolated points
and so certainly is not scattered.
- Tim
William Elliot
The Cantor set is closed, compact Hausdorff, zero-dimensional, without
isolated points, uncountable and nowhere dense. If nonnul open U subset
Cantor set C, then some r,s with nonnul (r,s) subset U subset C. Some x
in (r,s) with trinary expression requiring a 1. x not in C,
contradiction.
Butch Malahide
> [. . .]
> [I will
> mention, however, that a certain poster who goes
> by Butch Malahide has published a proof that a set
> is scattered if and only if it's countable and
> G_delta. Maybe he'll jump in now that I've mentioned
> his name. The result itself is due, I believe, to
> William H. Young and can be found on pp. 65 & 298
> of his 1906/1972 book. Also, E. Hobson published an
> article about this in 1904.]
Just recently I was trying to convince somebody not to cite Davies et
al., Real Anal. Exchange 1976 as the source for that characterization,
insisting that it was several decades older than that, but I was sadly
unable to find the real source. I'm grateful for the references you
just provided, cryptic as they are, and will be even more grateful for
any further information you might be able to provide. Do you by any
chance know the titles of the two works you mentioned?
Dave L. Renfro
> Just recently I was trying to convince somebody not to
> cite Davies et al., Real Anal. Exchange 1976 as the
> source for that characterization, insisting that it
> was several decades older than that, but I was sadly
> unable to find the real source. I'm grateful for the
> references you just provided, cryptic as they are,
> and will be even more grateful for any further
> information you might be able to provide. Do you by
> any chance know the titles of the two works you mentioned?
I looked over some references I have and it appears
Hobson is responsible for showing scattered sets are
G_delta. Section 3 (pp. 319-320) of Hobson [1904]
shows that any scattered set is G_delta and, on
p. 323, there is a statement (proved prior to this)
of the characterization that, for countable sets,
the property of being scattered is equivalent to
being G_delta. Without looking very carefully into the
issue (but this is something I will be doing in the
next year or two, for reasons unrelated to the present
sci.math discussion), it appears that "countable and
G_delta implies scattered" is due to William H. Young.
However, as is often the case with Young's work, it
might be difficult to pin down a definitive first
proof of the result. In any event, the idea of G_delta
sets (called "inner limiting sets" by Young) first
appeared in Young's works, beginning around 1902,
so 1902 is a lower bound on a date for the result
(that countable and G_delta implies scattered).
I believe Young/Young [1906/1972] (pp. 64-65) gives
a proof of "countable and G_delta implies scattered".
It is also stated there that scattered sets are G_delta
(in the "necessary and sufficient" result stated as
Corollary 2 on p. 65), but on p. 298 of the Notes at
the end of the book (the additional notes included in
the 1972 Chelsea edition), an excerpt of a letter to
Young by L.E.J. Brouwer (dated April 4, 1913) appears
to indicate that the proof of "scattered sets are
G_delta" is not given.
The first edition of Hobson's book on real functions,
Hobson [1907] (full text on the internet), appears to
reproduce the results from Hobson [1904] that I stated
above (see Section 98, pp. 131-133 of Hobson [1907]).
I haven't looked at Hobson's 2'nd edition, but the
same results can be found in the 3'rd edition,
Hobson [1927/1957], but I forgot to write down
the specific Section and page references before
I left home this morning.
Ernest W. Hobson, "On inner limiting sets of points in a
linear interval", Proceedings of the London Mathematical
Society (2) 2 (1904), 316-326.
Grace C. Young and William H. Young, "The Theory of Sets
of Points", Chelsea, 1906/1972.
http://tinyurl.com/4wls5u
http://tinyurl.com/4cuquz
Ernest W. Hobson, "The Theory of Functions of a Real
Variable and the Theory of Fourier's Series", Cambridge
University Press, 1907.
http://books.google.com/books?id=LLJdAECDOwQC
Ernest W. Hobson, "The Theory of Functions of a Real
Variable and the Theory of Fourier's Series", Volume I,
3'rd edition, Dover Publications, 1927/1957.
One or both results have likely been published by
other people who wrote on various aspects of the
Cantor-Bendixson theorem, such as: Maurice Frechet
(his 1906 Dissertation, a long paper in Volume 10
of Fund. Math., and others), Denjoy, Brouwer (in the
1910s, maybe early 1920s also, he wrote some papers
on certain "constructive proofs" of results involving
perfect sets and the CB theorem), Sierpinski (in the
late 1910s and early 1920s he wrote some papers about
"effective" proofs of the CB theorem and of other results
about scattered sets), T. Viola (an Italian mathematician
who wrote some papers in the early 1930s in which scattered
sets, and even the idea of unilaterally scattered sets,
were applied to unilateral derivative and Dini derivate
results). However, at this time, I haven't looked into
these, but it's something I intend to do so at some
point.
For anyone who's interested, here are a couple of
interesting results I came across that I don't believe
I've previously mentioned in my posts about scattered
sets or the Cantor-Bendixson theorem.
1. Let X be a complete separable metric space and
E be a subset of X. Then E is scattered if and
only if every function f: X --> reals is the
pointwise limit of a sequence {f_n} of functions
f_n: X --> reals such that each f_n is continuous
at each point of E. [Theorem 2, p. 181 of R. Daniel
Mauldin, "Some examples of sigma-ideals and related
Baire systems", Fundamenta Mathematica 71 (1971),
179-184.]
2. There exists a countable set in the real line that
is not scattered, but every homeomorphic image of
the set is nowhere dense. [Theorem 3, p. 353 of
John C. Morgan, "On sets every homeomorphic image
of which has the Baire property", Proceedings of
the American Mathematical Society 75 (1979), 351-354.
Finally, another poster mentioned "A subset S of R is
scattered iff there is some map f:R -> R such that
S = {x| lim as y->x f(y) is + infinity}.", a result
I've previously posted about. In case someone wants
references for this (and it's likely the result
appears in papers by one or more of the authors I
mentioned above where Frechet is named) --->
Burnett Meyer, "On restricted functions", American
Mathematical Monthly 62 #1 (January 1955), 29-30.
Robert Judson Bumcrot and Mark Sheingorn, "Variations
on continuity: Sets of infinite limit", Mathematics
Magazine 47 #1 (Jan./Feb. 1974), 41-43.
Janos T. Toth and Laszlo Zsilinszky, "On the class
of functions having infinite limit on a given set",
Colloquium Mathematicum 67 (1994), 177-180.
Tomasz Natkaniec, "On sets determined by limits
of a real function", Scientific Bulletin of Lodz
Technical University [= Zeszyty Naukowe Politechniki
Lodzkiej], Matematyka 27 #719 (1995), 95-104.
Dave L. Renfro
Dave L. Renfro
> I haven't looked at Hobson's 2'nd edition, but
> the same results can be found in the 3'rd edition,
> Hobson [1927/1957], but I forgot to write down
> the specific Section and page references before
> I left home this morning.
>
> Ernest W. Hobson, "The Theory of Functions of a Real
> Variable and the Theory of Fourier's Series", Volume I,
> 3'rd edition, Dover Publications, 1927/1957.
Actually, I can give the Section and page numbers.
It's written on a yellow post-it note I have on
one of the papers I brought with me from home today,
although I didn't realize this until now.
It's Section 100 (pp. 143-144) of Hobson [1927/1957].
Dave L. Renfro
Dave L. Renfro
> I just realized that I brought the wrong notes
> with me, so I'll have to look at home for my
> other notes and outline how to show G_delta
> tomorrow.
I thought I'd find a proof that didn't explicitly
involve the transfinite sequence of Cantor-Bendixson
derivatives of a set, but the proof I was thinking
of uses this. The specific reference is:
Fred Galvin, "Characterize the countable G_delta
subsets of R" [solution to reader query; also
solved by Roy O. Davies], Real Analysis Exchange
2 (1976-77), 74-75.
I may write up an edited version of the proof,
but right now I'm post'ed out.
Dave L. Renfro
Butch Malahide
Many thanks for all the info!
Butch Malahide
part):
> I thought I'd find a proof that didn't explicitly
> involve the transfinite sequence of Cantor-Bendixson
> derivatives of a set,
Let me try to give a proof like that. Here goes nothing . . .
We're talking about subsets of a separable metric space, e.g. the real
line. To save typing, I'll write A+B for the union and AB for the
intersection of sets A and B.
LEMMA. For any set X, if U is the union of all open sets V such that
XV is a G-delta, then XU is a G-delta.
PROOF. {V: V is open & XV is G-delta} is an open cover of the
separable metric space U. By some well-known theorem from general
topology, this cover has an open point-finite refinement, which
moreover is countable; call it {B_1, B_2, ...}. For each n, the set
D_n = XB_1 + XB_2 + ... + XB_n + (B_{n+1} + B_{n+2} + ...)
is a G-delta, since XB_1, ..., XB_n are G-deltas and (B_{n+1} + ...)
is open. Using the fact that the collection of B_n's is point-finite,
it's easy to see that the intersection of the D_n's is equal to XU,
which is therefore a G-delta.
THEOREM. If X is a scattered set, then X is a G-delta.
PROOF. With U as above, if X = XU we're done. Otherwise, X-U is a
nonempty subset of X, and has an isolated point y. Choose an open set
V with V(X-U) = {y}. Then XV = XUV + {y} is the union of two G-deltas,
so XV is a G-delta, so V is a subset of U, so y is in U, a
contradiction.
Is this OK?
Dave L. Renfro
> I looked over some references I have and it appears
> Hobson is responsible for showing scattered sets are
> G_delta. Section 3 (pp. 319-320) of Hobson [1904]
> shows that any scattered set is G_delta [...]
Last night I did a bit of digging into some books and
(copies of) papers I have for more early appearances
of these two results (scattered implies G_delta;
countable and G_delta implies scattered) and I found
some additional references, all later than Hobson's
and Young's 1902 or 1903 to 1907 references I've
already cited. [Actually, I didn't cite anything
by Young before his 1906 book, but I think all the
relevant papers are cited in Hobson's 1904 paper.]
However, as they're from the 1920s and 1930s, they
may also be of historical interest. I'll post these
later today, when I get a chance, probably within
6 hours from now.
Dave L. Renfro
Dave L. Renfro
> 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
Butch Malahide
Wow. Fantastic. Thanks again.
Dave L. Renfro
at the Math Forum's sci.math web pages. However, my post
hasn't shown up at the google sci.math archive yet, so
in the event that my post didn't get sent through usenet
channels so that those who don't read sci.math at Math
Forum would see it, I'm posting it again, via google-groups.
Dave L. Renfro wrote (in part):
>> I thought I'd find a proof that didn't explicitly
>> involve the transfinite sequence of Cantor-Bendixson
>> derivatives of a set, [...]
Butch Malahide wrote:
I printed out a copy of this yesterday and looked it
over this morning. It seems to me that this is fine.
Below is a version that I wrote (on paper) this
morning while trying to follow it. Most of the
notation I use is similar to your notation, but
there are some notational differences.
The idea behind the lemma (but not the rationale for
its use in this situation) is that, while countable
unions of G_delta sets do not have to be G_delta,
arbitrary unions of G_delta sets will be G_delta
if each of the G_delta sets is a relatively open
subset of a fixed set.
The following three facts are used:
1. A finite union of G_delta sets is a G_delta set.
2. A countable intersection of G_delta sets is
a G_delta set.
3. If X is a subset of R (the reals) and C* is a
collection of open-in-X sets, then there exists
a countable subcollection {C_1, C_2, ...} of C*
that has the same union as C*. This follows from
the separability of R (more specifically, the fact
that R is a hereditarily Lindelof space). Moreover,
there exists a subcollection {D_1, D_2, ...}
of {C_1, C_2, ...} with the same union as C* and
which has the property that no point of the union
belongs to infinitely many of the D_n's. This
follows from the fact that R is a paracompact space.
LEMMA: Let X be a subset of R and let U* be the collection
of all subsets of X that are both (G_delta)-in-R
and open-in-X. Then the union of the collection U*
is (G_delta)-in-R.
PROOF: By Fact 3, there exists a subcollection
{U_1, U_2, ...} of U* with the same union and
such that no point of the union belongs to
infinitely many of the U_n's. For each positive
integer n, choose an open-in-R set V_n such that
U_n = X intersect V_n (definition of open-in-X).
Let W_n = U_1 + U_2 + ... + U_n + [V_(n+1) + V_(n+2) + ...],
where '+' stands for 'union'. By Fact 1,
U_1 + U_2 + ... + U_n is (G_delta)-in-R. Also,
V_(n+1) + V_(n+2) + ..., being a union of open-in-R
sets, is open-in-R. Therefore, each W_n is a
union of two (G_delta)-in-R sets, and hence
is (G_delta)-in-R. We now show that
W_1 intersect W_2 intersect W_3 intersect ...,
an intersection of countably many (G_delta)-in-R
sets and hence (G_delta)-in-R, is equal to
the union of the collection U*, which will
complete the proof. Clearly, the union of U*
is a subset of this intersection, since each
of the sets U_k belongs to every one of the
sets W_n. For the opposite inclusion, use
the fact that each point in the union of U*
belongs to at most finitely many of the U_n's:
Show that if a point belongs to one of the
sets V_m and not to the union of U*, then that
point cannot belong to each of the sets W_n.
THEOREM: Each scattered set in the real line is G_delta.
PROOF: Let X be a scattered subset of R, let U* be the
collection of sets associated with X in the
Lemma, and let U be the union of U*. I claim that
X = U, and thus from the Lemma we have that X
is G_delta. For a later contradiction, assume
U is a proper subset of X. Since X-U is a nonempty
subset of X, there exists a point y in X-U
such that y is isolated-in-(X-U). In particular,
{y} is open-in-(X-U), and thus there exist an
open-in-R set V such that {y} = V intersect X-U.
Then XV = X(VU) + X(V-U) [V consists of two parts:
the part of V that belongs to U and the part of V
that doesn't belong to U] and X(V-U) = V(X-U)
[look at a general Venn diagram for the sets
U, V, and X], so XV = XUV + V(X-U) = XUV + {y}.
Since this is a union of two G_delta sets,
it follows that XV is G_delta. That is, XV is
(G_delta)-in-R and open-in-X. Hence, XV is a
subset of U, which implies that y belongs to U.
This contradicts the fact that y belongs to X-U.
REMARK: The same proof goes through for separable
metric spaces.
Dave L. Renfro