From: dlrenfro@gateway.net (Dave L. Renfro)
Subject: Re: Question from Rudin
Date: 15 May 2000 14:04:16 -0400
Newsgroups: sci.math
Summary: Cantor-Bendixson rank of a closed set

ullrich <ullrich@math.okstate.edu>
[sci.math Sun, 14 May 2000 18:30:17 GMT]
<http://forum.swarthmore.edu/epigone/sci.math/flarzhozal>

wrote (in part)

> It's a more interesting question than I realized at first.
> Since you've appeared I don't have to worry about trying to
> figure it out<g>: I imagine it's easy to construct examples
> of compact subsets of the line such that the K_alpha become
> constant at precisely any given alpha_0?
>     Assuming yes, that's sort of interesting - makes the
> countable ordinals a little more "real"...

What you're getting at is something called the Cantor-Bendixson
rank of a closed set. Define the derived set of a set A to be

  A' = {x in R: x is a limit point of A}  [R = the reals].

Now define by transfinite induction the b'th derived set of A,
denoted here by A^b, for any ordinal b as follows.

Let A^0 = A. If b is a successor ordinal, let A^b = [A^(b-1)]'.
If b is a nonzero limit ordinal, let A^b be the intersection of
all the sets A^c such that c < b.

Note that if A is a closed set, then A^b is a closed set for 
each ordinal b. In this case, we get a non-increasing (with respect
to set inclusion) transfinite sequence of closed sets:

A^0 -> A^1 -> A^2 -> ... -> A^w -> A^(w+1) -> ... -> A^(w^w) -> ...

It is easy to see that any non-increasing transfinite sequence of
closed sets, whether of the above form or not, is eventually
stationary. [[ It suffices to prove that if {F_c : c < b} is a
strictly decreasing b-sequence of closed sets, then b < w_1. To
show this, for each c < b choose x_c in F_c and not F_{c+1}.
Since the complement of F_{c+1} is open, there exists an open
interval (r_c, s_c) with rational endpoints containing x_c that
belongs to the complement of F_{c+1}. Because distinct values
of c give rise to distinct rational intervals, the map from
the ordinal b = {c : c < b} to the collection of rational open
intervals in R defined by c |--> (r_c, s_c) is injective. Hence,
b must be countable since there are only countably many rational
open intervals in R. ]] This continues to hold in any second
countable topological space.

Using this fact it is not difficult to prove the Cantor-Bendixson
theorem: Given any closed set F in the reals, F is the union of
a perfect set and a countable set. [This is how Cantor and 
Bendixson (independently) originally proved the C-B theorem
in 1883.]

I'm not sure what Bendixson was motivated by, although I do know
that he was closely following Cantor's work and that they exchanged
a number of letters during this time, but Cantor was motivated
by questions involving trigonometric sets of uniqueness. In 1870
Cantor proved that if two convergent trigonometric series are
equal, then their coefficients must be term-wise equal. Two years
later Cantor managed to improve this result by showing that
"equality everywhere" of the series expansions can be weakened
to "equality except for a closed countable set having finite
Cantor-Bendixson rank". Cantor's further researches into the
nature of closed sets (e.g. the C-B theorem) led him into those
areas for which he is better known for (cardinal and ordinal
numbers) and, as a result of the successes he had there, he
never returned to questions involving trigonometric sets of
uniqueness. [At least, if Cantor did return to these questions,
there is no record of any further results by him.]

[[ Call a set of real numbers a U-set if the agreement of any
   two convergent trigonometric series on the complement of 
   this set implies that their coefficients are term-wise equal.
   (i.e. An "interpolating" set for trigonometric series, to
   use the terminology employed by complex analysts.) W. H. Young
   improved Cantor's result in 1909 by showing that any countable
   set is a U-set. It was known that any measurable U-set has
   measure zero. (There exist U-sets with positive outer measure:
   Take any set of positive outer measure not containing a perfect
   subset. Then a short argument--due to W. H. Young--on page 344
   of Zygmund's treatise TRIGONOMETRIC SERIES shows this set is
   a U-set.) It was thought the converse was true, that any set
   of measure zero is a U-set. However, the Russian D. E. Men'shov
   proved in 1916 that the symmetric Cantor set whose n'th stage
   dissection ratio is 1/(n+1), which is easily shown to have
   measure zero, is NOT a U-set. At this point it still wasn't
   known if a Borel U-set could be uncountable. This was settled
   by Nina Bary (Russia, 1922) and A. Rajchman (Poland, 1923),
   who showed that uncountable closed U-sets exist. Bary also
   improved Young's result by showing that any countable union
   of closed U-sets is a U-set. (As far as I know, it is still
   an unsolved problem as to whether the union of two Borel, or
   even just two G_delta, U-sets is a U-set.) Recently (1987),
   G. Debs and J. Saint-Reymond proved that any U-set having the
   Baire property is a first category set in the reals. Their
   proof was rather involved and used descriptive set-theoretic
   notions. A more elegant proof using functional analysis ideas
   was given by A. S. Kechris and A. Louveau (see their monograph,
   or see Colloq. Math. 59 (1990), pp. 63-79). Very recently,
   Miroslav Zeleny ["Sets of extended uniqueness and
   sigma-porosity", Comment. Math. Univ. Carolinae 38 (1997),
   337-341] has proved there exists a closed U-set that isn't
   sigma-porous (a slightly stronger property than being
   simultaneously measure zero and first category). ]]

Note that the Cantor-Bendixson theorem implies any closed set
of real numbers is either countable or has cardinality of the
continuum. Thus, the continuum hypothesis is true for closed sets.
This is as far as Cantor got in settling CH. (Cantor had proved
that any nonempty perfect set has cardinality of the continuum.)
Of course, once you know this for closed sets, you automatically
have it for F_sigma sets. However, I don't think these notions
(F_sigma and/or G_delta sets) were employed by Cantor. W. H. Young
extended CH to G_delta sets in 1903, and then Hausdorff proved
that the continuum hypothesis holds for any Borel set in 1916.
This was further extended to any continuous image of a Borel set
(the analytic sets) by Suslin in 1917 (announcement only; I don't
believe a proof was published until Lusin's 1930 monograph on
analytic sets). Further up the projective set hierarchy the
problem becomes very difficult, since it is independent of the
ZFC axioms, and so the results vary depending on which large
cardinal hypotheses (and/or other non-ZFC hypotheses) are assumed.

Back to the Cantor-Bendixson (C-B) rank of a closed set . . .

It is easy to show that for any ordinal b < w_1, there exists a
closed subset of the rational numbers having C-B rank equal to b.

THEOREM: Let (A, <) be an unbounded and dense linearly ordered
         set and let (B, <') be a countable linearly ordered set.
         Then (B, <') is order-isomorphic to a subset of (A, <).

PROOF: Re-order B as b_1, b_2, ... . We will define an order
       isomorphism f from (B, <') to (A, <) by (ordinary)
       induction. Let f(b_1) be any element of A, say a_1. Now
       consider the <' relation of b_2 to b_1. If b_2 >' b_1,
       choose a_2 in A such that a_2 > a_1. If b_2 <' b_1, choose
       a_2 in A such that a_2 < a_1. Such an a_2 can be found in
       either case because (A, <) is unbounded. Now assume that
       a_1, a_2, ..., a_n have been chosen in A having the same
       <-order relationship as b_1, b_2, ..., b_n have in (B, <').
       If b_{n+1} <' b_n or b_{n+1} >' b_n, choose a_{n+1} < a_n
       or a_{n+1} > a_n, respectively, using the fact that (A, <)
       is unbounded. If b_{n+1} lies between (in the <' ordering)
       two elements of {b_1, b_2, ..., b_n}, choose a_{n+1} to be
       an element of A lying between (in the < ordering) the two
       corresponding elements of {a_1, a_2, ..., a_n}. This is
       possible because (A, <) is dense. Now define f: B --> A
       by f(b_k) = a_k. It is straightforward to check that f
       is an injective order preserving map from (B, <') onto
       a subset of (A, <).

Since the rational numbers are unbounded and dense in their usual
ordering, any countable linear order type can be represented
by some subset of the rationals. In particular, any countable
WELL ordering (i.e. a countable ordinal) can be represented by
a subset of the rationals. To be explicit, if you take a
subset of the rationals having order type w^b + 1 (ordinal
exponentiation), then its b'th derived set is nonempty while its
(b+1)'st derived set is empty. {See page 197 of Hausdorff [1],
5.2(3) on page 78 of Rosenstein [2], or 8.6.6 on page 154 of
Semandeni [3] for this specific construction. Semandeni's book,
by the way, gives some interesting applications in functional
analysis of these ideas in his Chapter II.8 "Compact 0-dimensional
spaces".}

[1] Felix Hausdorff, SET THEORY, 3'rd edition, Chelsea, 1935/1978.
    [MR 19 (p. 111)]

[2] Joseph G. Rosenstein, LINEAR ORDERINGS, Pure and Applied
    Mathematics 98, Academic Press, 1982.
    [MR 84m:06001; Zbl 488.04002]

[3] Zbigniew Semadeni, BANACH SPACES OF CONTINUOUS FUNCTIONS,
    Monografie Matematyczne 55, PWN-Polish Scientific Publishers,
    1971. [MR 45 #5730; Zbl 225.46030]

The Cantor-Bendixson rank of a closed set is often used in
certain areas. [It plays a nontrivial role in one of my papers,
in fact.] Moreover, the ideas can be generalized to the setting
of a monotone operator defined on the power set of a given set.
This setting, where the monotone operator is transfinitely
iterated until a fixed point is obtained, allows one to
"constructively obtain" things such as: the linear span of
a set of vectors in a vector space, the closure of a set in a
topological space, the sigma-algebra generated by a collection
of subsets of a fixed set, the set of well-formed formulas in 
first order logic, etc. In fact, one of the common proofs of
the Schroder-Berstein theorem can even be put into this
context. {See the exercises on page 69 of Harbacek/Jech [4].}

[4] Karel Hrbacek and Thomas Jech, INTRODUCTION TO SET THEORY,
    3'rd edition, Pure and Applied Mathematics 220, Marcel
    Dekker, 1999.

There are two other methods of proving the C-B theorem that I
know of, and each of these methods has given rise to a number
of papers exploiting their generalizations. [It remains to be
seen, however, whether there is a proof of the C-B theorem using
Clifford algebras.] There is the condensation point proof due to
Lindelof (1903), which has been generalized to the topological
localization of ideals [See Kuratowski's TOPOLOGY treatise,
Henry Blumberg's papers, and the numerous papers by Troy R. Hamlett
and/or Dragan Jankovic (e.g. "New topologies from old via ideals",
Amer. Math. Monthly 97 (1990), 295-310).], and there is a
"maximal dense-in-itself set" proof due to Hausdorff, which
leads to the generalization to hereditarily Lindelof topological
spaces that one can find in some topology texts (e.g. Kelly's
text, page 57; Willard's text, page 114).

Dave L. Renfro