Discrete sets, free sequences and cardinal properties of topological spaces
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee. This
dissertation does not include proprietary or classi ed information.
Santi Spadaro
Certi cate of Approval:
Stewart Baldwin
Professor
Mathematics and Statistics
Gary Gruenhage, Chair
Professor
Mathematics and Statistics
Michel Smith
Professor
Mathematics and Statistics
George Flowers
Dean
Graduate School
Discrete sets, free sequences and cardinal properties of topological spaces
Santi Spadaro
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Ful llment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
August 10, 2009
Discrete sets, free sequences and cardinal properties of topological spaces
Santi Spadaro
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
Vita
Santi Spadaro, son of Agatino Natale Spadaro and Maria Luisa Ceccio, was born on
June 18, 1981, in Messina, Italy, on the northeastern tip of Sicily, which according to
somebody is the actual birthplace of The Bard. After two years as a Physics major in
Messina he moved to the University of Catania, where he graduated in Mathematics with
highest honors in April of 2005. He entered the Graduate Program at Auburn University
in August of 2006. Besides Mathematics, he enjoys Music, Literature and Cold Weather.
iv
Dissertation Abstract
Discrete sets, free sequences and cardinal properties of topological spaces
Santi Spadaro
Doctor of Philosophy, August 10, 2009
(Laurea, University of Catania, Catania, Italy, 2005)
55 Typed Pages
Directed by Gary Gruenhage
We study the in uence of discrete sets and free sequences on cardinal properties of
topological spaces. We focus mainly on the minimum number of discrete sets needed to
cover a space X (denoted by dis(X)) and on re ection of cardinality by discrete sets, free
sequences and their closures. In particular, we o er several classes of spaces such that the
minimum number of discrete sets required to cover them is always bounded below by the
dispersion character (i.e., minimum cardinality of a non-empty open set). Two of them are
Baire generalized metric spaces, and the rest are classes of compacta. These latter classes
o er several partial positive answers to a question of Juh asz and Szentmikl ossy. In some
cases we can weaken compactness to the Baire property plus some other good property.
However, we construct a Baire hereditarily paracompact linearly ordered topological space
such that the gap between dis(X) and the dispersion character can be made arbitrarily
big. We show that our results about generalized metric spaces are sharp by constructing
examples of good Baire generalized metric spaces whose dispersion character exceeds the
minimum number of discrete sets required to cover them. With regard to discrete re ection
of cardinality we o er a series of improvements to results of Alan Dow and Ofelia Alas.
We introduce a rather weak cardinal function, the breadth, de ned as the supremum of
cardinalities of closures of free sequences in a space, and prove some instances where it
v
manages to re ect cardinality. We  nish with a common generalization of Arhangel?skii
Theorem and De Groot?s inequality and its increasing chain version.
vi
Acknowledgments
First of all I?d like to thank my advisor, Dr. Gary Gruenhage, for his great patience and
his constant encouragement and support. Thanks are also due to the committee members:
Dr. Stewart Baldwin, for teaching two interesting courses in Set Theory and providing
helpful feedback on preliminary versions of the results from this dissertation and Dr. Michel
Smith, for his useful comments at the seminar, his wise advice regarding teaching matters,
his patience and availability. I would also like to thank Dr. Georg Hetzer, whom I had the
pleasure of having both as a teacher and former committee member, but was eventually
replaced due to schedule con icts. Last but not least, I would like to thank Carolyn, Gwen,
Lori and Maritha (and Debby): the friendliest and most e cient secretaries I have ever
met.
vii
Style manual or journal used Journal of Approximation Theory (together with the style
known as \auphd"). Bibliograpy follows van Leunen?s A Handbook for Scholars.
Computer software used The document preparation package TEX (speci cally LATEX)
together with the departmental style- le auphd.sty.
viii
Table of Contents
1 Introduction 1
2 Notation and background 4
3 Covering Baire generalized metric and linearly ordered spaces by dis-
crete sets 8
3.1 Generalized metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Good spaces with bad covers . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 Linearly ordered spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Covering compact spaces by discrete sets 22
4.1 Hereditary separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 The shadow of a metric space . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.3 Homogeneity and beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5 Closures of discrete sets, closures of free sequences and cardinality 31
5.1 A crash course on elementary submodels . . . . . . . . . . . . . . . . . . . . 31
5.2 Depth, spread, free sequences and cardinality . . . . . . . . . . . . . . . . . 33
6 Arhangel?skii, De Groot, free sequences and increasing chains 40
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.2 A common generalization of Arhangel?skii?s Theorem and De Groot?s inequality 41
6.3 The increasing strenghtening . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Bibliography 45
ix
Chapter 1
Introduction
The world is discrete. It is made up of quanta, quarks, atoms, elements, separate
entities. It should come as no surprise then, that discrete sets play an important role even
in an eminently continuous area of mathematics like Topology. In Topology a set is called
discrete if each of its points can be separated from the others by an open set. This de nition
matches with the meaning of the word discrete in any other discipline. The spread of a
space is the supremum of the cardinalities of its discrete sets. So, if a space has countable
spread, each of its discrete sets is at most countable. A classical result of De Groot says
that the cardinality of a space is bounded above by the power of the power of its spread.
So a space of countable spread has cardinality at most 22!. This is only one of many results
showing the great in uence that discrete sets have on cardinal properties of topological
spaces.
This dissertation deals with two very natural problems involving discrete sets. How
many discrete sets are needed to cover a good space? When do closures of discrete sets re ect
the cardinality of a space? The latter question goes back to an old paper of Arhangel?skii,
where the author asks: is it true that in every compact space there is a discrete set whose
closure has the cardinality of the whole space? Although the answer is known to be no, at
least consistently, re ection properties of discrete sets have became an active area of research
in Topology, as shown by the papers [1], [2], [6] and [21]. The depth of X (indicated with
g(X)) is the supremum of cardinalities of closures of discrete sets in X. This is usually
bigger than the spread, and closer to the cardinality: for example, the unit interval has
countable spread but depth continuum. Alan Dow [5] proved that if X is a compact space
of countable tightness where g(X)  c then jXj c. Ofelia Alas [1] proved that if X is a
compact space where discrete sets have all size less than continuum and g(X)  c, then,
1
under Martin?s Axiom, jXj c. These two results are partial answers to a special case of
Arhangel?skii?s Problem that was studied by Alas, Tkachuk and Wilson in [2]. In the  nal
chapter of this dissertation we prove a common generalization of Alas and Dow?s theorems
that removes compactness from their assumption. Compactness was essential in both Alas
and Dow?s result. This suggests that the role of compactness in discrete re ection might be
less crucial than previously thought. Moreover, we give a series of other improvements to
their results, and from there ask a couple of natural questions. It looks like the construction
of counterexamples to these questions would require completely new methods than those
used for Arhangel?skii?s original problem.
The cardinal function dis(X) was introduced by Juh asz and Van Mill in the paper
[19], as the minimum number of discrete sets required to cover a space X. The authors
were especially interested in its behavior on compact spaces. In particular, they asked: is
it true that dis(X)  c for every compact space X without isolated points. This is true
for the unit interval, since it has countable spread and size continuum, so their question
appears like a very natural and fundamental one. Juh asz and Van Mill proved it to be true
for compact hereditarily normal spaces and had some other partial answers that showed
a counterexample to their question must have been a very weird compact space. In fact,
the answer to their question was positive, as proved by Gruenhage in [14]. By exploiting a
Lemma of Gruenhage but using a completely di erent approach, Juh asz and Szentmikl ossy
[20] proved that in every compact space X where every point has character at least  ,
dis(X)  2 . This generalizes both Gruenhage?s result and the classical  Cech-Pospi sil
Theorem.
One of the main questions in [20] is the following. When dealing with a space X, call a
cardinal small if it is less than the cardinality of every non-empty open set in X. Is it true
that no compact space can be covered by a small number of discrete sets? A positive answer
would generalize their theorem, since in a compact space where every point has character
at least  , every open set has cardinality at least 2 . However, a solution to their question
would seem to require completely di erent methods than those used to study dis(X) so
2
far, since there is no direct reference to character, so one cannot lean on  Cech-Pospi sil-like
tecniques. Here we provide several partial positive answers to their question, that suggest a
possible counterexample would be a rather pathological compact space. Moreover we obtain
some results outside of the compact realm: for example we determine the least number of
discrete sets required to cover a  -product, or in some cases we can replace compactness
with a much weaker property, like the Baire property. Finally we give a systematic study of
the Juh asz and Szentmikl ossy?s problem on two classes of Baire generalized metric spaces,
inspired by our new result that no Baire metric space can be covered by a small number
of discrete sets. Our study leads to two examples of very good Baire spaces that are very
close to metric and yet can be covered by a small number of discrete sets. Also, we show
a family of nice looking Baire linearly ordered topological spaces that can be covered by a
really small number of discrete sets (see chapter 3 for a precise de nition of really small).
This shows that compactness cannot be relaxed to the Baire property in our result about
compact LOTS.
In the  nal chapter we prove a common generalization of two basic theorems in the
theory of cardinal functions -Arhangel?skii Theorem and De Groot?s inequality- that involves
the size of free sequences. The theorem has been proved by Juh asz independently in 2003,
but this is the  rst time it appears in print. Moreover, using and streamlining some ideas
of Juh asz from [18], we prove the increasing version of our theorem.
3
Chapter 2
Notation and background
The cardinality of a countable set is indicated with ! or @0. The Greek letter !
also stands for the set of all non-negative integers. The symbol @1 stands for the  rst
uncountable cardinal, and c stands for the cardinality of the continuum. For a cardinal  ,
the symbol  + indicates the least cardinal bigger than  . If S is a set then P(S) stands
for the power set of S. If S is a set and  is a cardinal we set [S] = fA S : jAj =  g
and [S]  = fA S : jAj  g. The continuum hypothesis, or CH, is the statement that
c =@1. The generalized continuum hypothesis, or GCH, is the statement that 2 =  + for
every cardinal  .
A space is called crowded if it has no isolated points. The letter I will denote the closed
unit interval. A G -set in a space X is an intersection of  many open sets. G!-sets are
more commonly known as G -sets. We will need several classical cardinal functions, whose
de nitions are recalled below.
De nition 2.1. The character of the point x in X ( (x;X)) is the least cardinality of a
local base at x. The character of the space X is de ned as  (X) = supf (x;X) : x2Xg.
A space of countable character is also called  rst-countable.
De nition 2.2. The spread of X is de ned as s(X) = supfjDj: D X and D is discrete
g. We also de ne a related cardinal function as ^s(X) = minf : if A X and jAj=  then
A is not discrete g.
De nition 2.3. The tightness of X (t(X)) is de ned as the least cardinal number  such
that for every A X and x2AnA there is B A such that jBj  and x2B.
A setfx :  < gis called a free sequence iffx :  < g\fx :    g=;for every
 < . Every free sequence is a discrete set.
4
De nition 2.4. We set F(X) = supfjFj : F  X and F is a free sequence g. Also
^F(X) = minf : if A X and jAj=  then A is not a free sequence g.
The above cardinal function allows an elegant characterization of the tightness of a
compact space.
Theorem 2.5. (Arhangel?skii) Let X be a compact Hausdor space. Then t(X) = F(X).
Proof. See [18], 3.12.
A cellular family is a family of pairwise disjoint open sets.
De nition 2.6. The cellularity of X is de ned as c(X) = supfjCj : C is a cellular family
of open subsets of Xg.
De nition 2.7. The weight of X (w(X)) is the least cardinality of a base for X.
Since in a discrete set any point can be separated from the others by a basis element,
it is clear that s(X) w(X).
De nition 2.8. The dispersion character of X is de ned as the least cardinality of a non-
empty open set in X.
De nition 2.9. A family of open sets U is said to be a local  -base at the point x2X if
for every open set V  X such that x2V there is a set U 2U such that U  V. The
 -character of the point x in X (  (x;X)) is the least cardinality of a local  -base at x.
The  -character of the space X is de ned as   (X) = supf  (x;X) : x2Xg.
The  -character plays a fundamental role in Set-theoretic Topology, since it character-
izes the compact spaces that can be mapped onto Tychono cubes.
Theorem 2.10. (Shapirovskii?s Theorem on maps onto Tychono Cubes) Let X be a com-
pact space. Then X can be mapped onto I if and only if there is a closed set F  X such
that   (p;F)  for every p2F.
5
De nition 2.11. A set G X is called regular open if Int(G) = G. The number of regular
open sets in X is indicated with  (X).
Regular open sets generate the topology of a regular space. A simple, yet very e ective
lower bound on the number of regular open sets is due to F. Burton Jones.
Lemma 2.12. (Jones? Lemma) If X is a hereditarily normal space and D X is a discrete
set then  (X) 2jDj.
Proof. See [18], 3.1 b)
One of the nicest consequences of Shapirovskii?s Theorem on maps onto Tychono 
cubes is the following upper bound on the number of regular open sets.
Theorem 2.13. (Shapirovskii) Let X be a compact hereditarily normal space. Then  (X) 
2c(X).
Theorem 2.14. (  Cech-Pospi sil) Let X be a compact space such that  (x;X)  for every
x2X then jXj 2 .
Proof. see [18], 3.16.
De nition 2.15. A map between topological spaces is called perfect if it is closed and has
compact point inverses.
De nition 2.16. A space is called Lindel of if every open cover has a countable subcover.
Recall that a re nement V for a cover U of a space X is a family of subsets of X such
that for every U 2U there is V 2V such that U  V and U still covers X. A family of
subsets of a space X is called point-countable if every point of X is in at most countably
many members of that family.
De nition 2.17. A space X is called meta-Lindel of if every open cover for X has a point-
countable open re nement.
6
De nition 2.18. A space is called collectionwise Hausdor if every closed discrete set
expands to a disjoint family of open sets.
De nition 2.19. A space is called Baire if every intersection of countably many dense open
sets is dense.
Some of the most fruitful strengthenings of Baire involve topological games.
De nition 2.20. Let X be a non-empty topological space. The strong Choquet game is
de ned as follows. Player I chooses an open set U0 and a point x02U0. Player II chooses
an open set V0  U0 such that x0 2V0. Then player one chooses an open set U1  V0 and
a point x12U1. Player II proceeds as before. Player II wins the game if Tn2!Vn6=;.
De nition 2.21. A space X is called strong Choquet if player II has a winning strategy in
the strong Choquet game for X.
We think that the meaning of winning strategy is rather intuitive. See [23], page 43,
for a more precise de nition.
Theorem 2.22. Every strong Choquet space is Baire.
Proof. See for example [23], Theorem 8.11.
De nition 2.23. Let (X; ) be a space. Then the dispersion character of X is de ned as
 (X) = minfjUj: U2 nf;gg.
Other more specialized notions and results will be recalled as the need arises.
7
Chapter 3
Covering Baire generalized metric and linearly ordered spaces by discrete
sets
Our interest in the cardinal functiondis(X) was sparked by the discovery thatdis(X) 
 (X) was true for Baire metric spaces. This suggested it might be interesting to look at
Juh asz and Szentmikl ossy?s question in the class of generalized metric spaces. Generalized
metric spaces can be described as spaces that resemble metric spaces in some sense, yet
can deviate from them a lot. For example,  -spaces are a popular generalized metric class
inspired by the Bing metrization theorem, that even contains spaces that fail to be  rst-
countable. One such space is one of the main counterexamples in this dissertation, being a
Baire  -space for which dis(X) <  (X). The question of whether a  rst-countable example
having all those features exists remains open in ZFC, while we do have a consistent  rst
countable  -space for which dis(X) <  (X). Such space is even a normal Moore space,
and normal Moore spaces are known to be metric in some models of set theory. So it
can be described as our strongest example, even if it relies on additional axioms. We also
prove that dis(X)  (X) is true for two Baire generalized metric classes satisfying a mild
covering-type property.
The  rst class of compact spaces for which we found Juh asz and Szentmikl ossy?s con-
jecture to be true is that of compact linearly ordered spaces. This made us wonder whether
compact could be replaced by Baire. We were able to construct a family of hereditarily
paracompact Baire linearly ordered spaces for which dis(X) <  (X) and the gap between
dis(X) and  (X) can be made arbitrarily large.
8
3.1 Generalized metric spaces
Given a collectionG of subsets of X, set st(x;G) = SfG2G : x2Ggand ord(x;G) =
jfG2G : x2Ggj. Recall that a sequence fGn : n2!g of open covers of X is said to be a
development if fst(x;Gn) : n2!g is a local base at x for every x2X. A space is called
developable if it admits a development. A regular developable space is called a Moore space.
We say that a set A X expands to a collection C P(X) if for every x2A there is
C2C such that x2C.
De nition 3.1. Let  be a cardinal. We call a space  -expandable if every closed discrete
set expands to a collection of open sets G such that ord(x;G)  for every x2X.
The following theorem is new even for all complete metric spaces.
Theorem 3.2. Let X be a Baire !1-expandable developable space. Then dis(X)  (X).
Proof. Fix a development fGn : n 2 !g for X and suppose by contradiction that  =
dis(X) <  (X). Since the inequality dis(X)  !1 is true for every crowded Baire space
X we can assume that   !1. Set X = S < D , where each D is discrete. De ne
D ;n =fx2D : st(x;Gn)\D =fxgg and set Xn = S 2 D ;n.
Claim: For every x2Xk there is a neighbourhood G of x such that jG\Xkj  .
Proof of Claim. Let G2Gk be such that x2G. Then G hits each D ;k in at most one
point: indeed, if y;z 2G\D ;k with y 6= z, we?d have both st(y;Gk)\D ;k = fyg and
z2st(y;Gk)\D ;k, which is a contradiction. 4
Now X = Sn2!Xn, so, by the Baire property of X, there is k2! such that U  Xk
for some non-empty open set U. By the claim we can assume that jU\Xkj  . So
jU\(XknXk)\D ;jj> for some  < and j2!.
Notice that the set D ;j is actually closed discrete: indeed suppose y =2D ;j were some
limit point. Let V 2Gj be a neighbourhood of y and pick two points z;w2V \D ;j. By
9
de nition of D ;j we have st(z;Gj)\D ;j =fzg. But w2V  st(z;Gj), which leads to a
contradiction.
Observe now that also S := U\(XknXk)\D ;j is closed discrete and hence we can
expand it to a collection U =fUx : x2Sg of open sets such that ord(y;U) !1 for every
y2X. Set Vx = Ux\st(x;Gj)\U and observe that Vx6= Vy whenever x6= y and if we put
V =fVx : x2Sg then we also have that ord(y;V) !1 for every y2X. For every x2S
pick f(x) 2Vx\Xk: the mapping f has domain of cardinality >  , range of cardinality
  and  bers of cardinality  !1, which is a contradiction.
Corollary 3.3. dis(X)  (X), for every Baire collectionwise Hausdor (or meta-Lindel of)
developable space X.
Corollary 3.4. dis(X)  (X), for every Baire metric space X.
Recall that a network is a collection N of subsets of a topological space such that for
every open set U  X and every x2U there is N 2N with x2N  U. A  -space is a
space having a  -discrete network.
Our next aim is proving that dis(X)   (X) for every regular Baire !1-expandable
 -space. We could give a more direct proof, but we feel that the real explanation for that
is the following probably folklore fact, a proof of which can be found in [4].
Lemma 3.5. Every regular Baire  -space has a dense metrizable G -subspace.
Call dis (X) the least number of closed discrete sets required to cover X. Clearly
dis(X)  dis (X). In a  -space, one can use a  -discrete network to split every discrete
set into a countable union of closed discrete sets. So the following lemma is clear.
Lemma 3.6. If X is a crowded  -space then dis(X) = dis (X).
The next lemma and its proof are essentially due to the anonymous referee of [28].
Lemma 3.7. Let X be an !1-expandable crowded Baire space such that dis (X)  , and
A X with jAj  . Then jAj  .
10
Proof. Since X is Baire crowded we can assume that   !1. Let X = S < D , where
each D is closed discrete. Let B = A\D . Then B is closed discrete, so we may expand
it to a family of open setsU such that ord(x;U ) !1 for every x2X. ThenjU j=jB j
and for all U2U , U\A6=;. Fix some well-ordering of A and de ne a function f :U !A
by:
f(U) = minfa2A : a2Ug:
We have that jf 1(a)j @1 for every a2A, and therefore jB j=jU j jAj @1  .
Since A = S 2 B it follows that jAj  .
The statement of the next theorem is due to the anonymous referee, and improves our
original theorem where X was assumed to be paracompact.
Theorem 3.8. Let X be a regular !1-expandable Baire  -space. Then dis(X)  (X).
Proof. Fix some dense metrizable G -subspace M X and suppose by contradiction that
dis (X) = dis(X) <  (X). Then Lemma 3:7 implies that  (M)   (X) and, since M
is Baire metric, by Corollary 3:4 we have dis(X) dis(M)  (M). So dis(X)  (X),
and we are done.
Corollary 3.9. For every paracompact Baire  -space X (in particular, for every strati able
Baire space), we have dis(X)  (X).
Notice that in the proofs of Theorems 3:2 and 3:8 all one needs is that X be dis(X)-
expandable.
Also, while we didn?t use any separation other than Hausdor in Theorem 3:2, regu-
larity seems to be essential in Theorem 3:8, since one needs a  -discrete network consisting
of closed sets to prove Lemma 3:5. This suggests the following question.
Question 3.10. Is there a collectionwise Hausdor or meta-Lindel of (non regular) Baire
 -space X such that dis(X) <  (X)?
11
3.2 Good spaces with bad covers
We now o er two examples to show that !1-expandability is essential in Theorem 3:8.
The  rst one is a modi cation of an example of Bailey and Gruenhage [3]. We will need
the following combinatorial fact which slightly generalizes Lemma 9.23 of [17]. It must be
well-known, but we include a proof anyway since we couldn?t  nd a reference to it.
Lemma 3.11. Let  be any in nite cardinal. There is a family A [ ]cf( ) of cardinality
 + such that jA\Bj<cf( ) for every A;B2A.
Proof. We begin by showing that there is a family F of functions from cf( ) to  such
that jFj =  + and jf 2 cf( ) : f( ) = g( )gj < cf( ), for any f;g 2 F. Indeed,
suppose we have constructedff :  < gwith the stated property. Let  = sup <cf( )  .
De ne f : cf( ) ! in such a way that f( ) 6= f ( ), for every  <   and  2cf( ).
Fix  2  : if  < cf( ) is such that f( ) = f ( ) we must have     <  . Hence
jf 2cf( ) : f( ) = f ( )gj<cf( ).
Now for A we can take (on cf( )  ) the family of graphs of functions in F.
Example 3.12. (ZFC) A regular Baire  -space P for which dis(P) <  (P).
Proof. Fix an almost disjoint family A [c]cf(c) such that jAj = c+. For every partial
function  2 c<! such that dom( ) = k for some k 2! let L = ff ;A : A2Ag where
f ;A : cf(c) ! c<! is de ned as follows: dom(f ;A( )) = k + 1, f ;A( )  k =  for every
 2cf(c) and ff ;A( )(k) :  2cf(c)g is a faithful enumeration of A.
When f2L we will refer to  f =  as the root of f, and set kf = dom( ).
Let now L = S 2c<!L and B = c!. We are going to de ne a topology on P = B[L
that induces on B its natural topology. For every  2c<!, let [ ] =fg2B : g  g and
B( ) = [ ][ff2L :  f   g:
Let fAn : n2!g be a partition of c into sets of cardinality c.
For f2L,  2cf(c) and k2! let
12
B ;k(f) =ffg[
[
 > 
(
B(f( )) : f( )(kf)2
[
n>k
An
)
:
The set B =fB( );B ;k(f) :  2c<!; 2cf(c);k2!g is a base for a topology on P,
as items (2) and (3) in the following list of claims show.
1. For  1; 22c<!, B( 1)\B( 2) =; if and only if  1 and  2 are incompatible.
2. Suppose B( )\B ;k(f)6=;. Then    f or  f   . If    f then B( )\B ;k(f) =
B ;k(f). If  ) f, then the intersection is B( ).
3. If B ;j(f)\B 0;k(g)6=;and  g ( f then the intersection is either B ;j(f) or a set of
the form B( ), for some  2ff( );g( 0) :  > ; 0> 0g.
4. If B ;j(f)\B 0;k(g)6=;and  g =  f then the intersection is a union of less than cf(c)
sets of the form B( ) where  2ran(f)\ran(g).
Proof of items (1)-(4). Item (1) is easy. For item (2), observe that B ;k(f)  B( f), so
B( f)\B( ) 6= ; which implies that  f and  are compatible. If    f then for each
 > we have   f( ) and f2B( ), so B ;k(f) B( ).
If  ) f then let  > be the unique ordinal such that B( )\B(f( ))6=;. Since  
and f( ) are compatible we must have f( )  , from which B( ) B(f( )) follows, and
hence the claim.
To prove item (3) observe that ifB ;j(f)\B 0;k(g)6=;and g ( f theng =2B ;j(f) and,
as the range of f consists of pairwise incompatible elements we have that [g( )]\[ f]6=;
for at most one  2cf(c). Therefore, B ;j(f)\B 0;k(g) = B(g( ))\B ;j(f), and the rest
follows from item (2).
Item (4) follows from almost-disjointness of the ranges.
Claim 1: The base B consists of clopen sets.
Proof of Claim 1. To see that B ;j(f) is closed pick g2LnB ;j(f) and let  be large enough
so that f =2B ;j(g). Suppose that B ;j(f)\B ;j(g)6=;. Then there are  > and  > 
13
such that f( ) and g( ) are compatible. Now we must have  g =  f or otherwise we would
have either  f  g( ) and hence f2B ;j(g), or  g f( ), which would imply g2B ;j(f).
So, by item (4) we have B ;j(f)\B ;j(g) = S 2CB(g( )) where jCj<cf(c) and hence, if
we let  > sup(C), then B ;j(g)\B ;j(f) =;.
Now, let p 2 BnB ;j(f) and i = kf + 2. We claim that B(p  i)\B ;j(f) = ;.
Indeed, if that were not the case then f( ) and p  i would be compatible, for some  . So
f( ) p  i p, which implies p2B ;j(f), contradicting the choice of p.
To see that B( ) is clopen, observe that B is dense in P and the subspace base is clopen,
so we can restrict our attention to limit points of B( ) in L. Suppose that f2LnB( ) is
some limit point, then, for all  2cf(c) and all j2! we have B ;j(f)\B( ) 6= ;. So  f
and  are compatible; moreover  f ( or otherwise f2B( ). Now there is at most one  0
such that f( 0) and  are compatible, whence the absurd statement B 0+1;0(f)\B( ) =;.
4
Claim 2: P is a  -space.
Proof of Claim 2. For each  2c<! let h( )2!<! be de ned by  (i)2Aj i h( )(i) = j.
For every s2!<! put Bs = fB( ) : h( ) = sg. We claim that Bs is a discrete collection
of open sets. Notice that the elements of Bs are all disjoint. Now if x 2 BnSBs, let
j = dom(s); then either x  (j+1) extends (at most) one  such that h( ) = s or x  (j+1)
is incompatible with every such  . So B(x  (j + 1)) will hit at most one element of Bs.
If f 2L then let l = max(ran(s)): we claim that B0;l(f) hits at most one element of Bs.
Indeed, for  xed  such that f( )(kf)2Sn>lAn either f( ) is incompatible with every  
such that h( ) = s or there is exactly one such  which is compatible with f( ). In the
latter case we can?t have    f because f( )(kf) =2ran(s), hence we have    f, which
implies B0;l(f) B( ).
Now we claim that L is a  -closed discrete set. Indeed, for every s2!<!, set Ls =
ff 2L : h( f) = sg. If g2Ls then every fundamental neighbourhood of g hits Ls in the
single point g. If g =2Ls then either  g is incompatible with every  f such that f 2Ls,
14
in which case every fundamental neighbourhood of g misses Ls, or there is f 2 Ls such
that  g and  f are compatible. If  g (  f then let l = s(kg): we have B0;l(g)\Ls =;. If
 f   g, then the root of every function of L which is in a fundamental neighbourhood of
g has domain strictly larger than dom(s) and hence every fundamental neighbourhood of g
misses Ls. 4
Observe now that P is Baire, because B P is a dense Baire subset. Also, dis(P) =
c < c+ =  (P)
One of the properties of Bailey and Gruenhage?s example that was lost in the modi -
cation is  rst-countability. This suggests the following question.
Question 3.13. Is there in ZFC a  rst-countable regular  -space X for which dis(X) <
 (X)?
The reason why we insist on a ZFC example is that we already have a consistent
answer to the previous question. In fact, the space we are now going to exhibit is  rst-
countable, normal and shows that !1-expandability cannot be weakened to !2-expandability
in Theorem 3:2. Our original motivation for constructing this example was showing that
paracompactness could not be weakened to normality in Corollary 3:9.
Recall that a Q-set is an uncountable subset of a Polish space whose every subset is
a relative F , and a Luzin set is an uncountable subset of a Polish space P which meets
every  rst category set of P in a countable set. The existence of Q-sets and Luzin sets in
the reals is known to be independent of ZFC (see, for example, [25]). Fleissner and Miller
[8] constructed a model of ZFC where there are a Q-set of the reals of cardinality @2 and a
Luzin set of the reals of cardinality @1.
Lemma 3.14. Let C be some Polish space having a base B = fBn : n2!g such that Bn
is homeomorphic to C for every n2!. Given a Q-set of cardinality @2 in C, there is one
which is dense and has dispersion character @2. Given a Luzin set in C, there is one which
is locally uncountable and dense.
15
Proof. Let X be a Q-set in C. Let B0 = fB2B : jB\Xj<@2g. Then Y = XnSB0 is
a Q-set such that  (Y) =@2. Set n0 = 0 and let Z0 be a homeomorphic copy of Y inside
Bn0. Set Z = Z0 and let n1 be the least integer such that Bn1 \Z = ;: clearly n1 > n0.
Now let Z1  Bn1 be a homeomorphic copy of Y and set Z = Z0[Z1. Now suppose you
have constructed a Q-set Z such that Z\Bi6= 0 for every 1 i nk 1 and let nk be the
least integer such that Z\Bnk =;; let Zk Bnk be a homeomorphic copy of Y into Bnk.
At the end of the induction let Z = Sn2!Zn, then Z is a Q-set with the stated properties.
The second statement is proved in a similar way.
Example 3.15. A normal Baire Moore space X for which dis(X) <  (X).
Proof. Take a model of ZFC where there are a Luzin set L0 R and a Q-set Z R with the
properties stated in Lemma 3:14. Let f be any homeomorphism from the irrationals onto
their square. Then L = f(L0nQ) is a Luzin subset of (RnQ)2, and by Lemma 3:14 we can
assume that it is locally uncountable and dense. Let Q =fqn : n2!g be an enumeration
and set Zn = Z fqng. Set T = Sn2!Zn and de ne a topology on X = L[T as follows:
points of L have neighbourhoods just as in the Euclidean topology on the plane, while a
neighbourhood of a point of x2Zn is a disk tangent at x to Zn, and lying in the upper
half plane relative to that line. To see that X is Baire, observe that if X = Sn2!Nn, where
Nn is nowhere dense in X, then L = Sn2!L\Nn. From the fact that L is dense in X it
follows that L\Nn is nowhere dense in L. From the fact that L is dense in the plane it
follows that L\Nn is nowhere dense in the plane. Since L\Nn L we have that L\Nn is
countable. So the uncountable set L would be covered by countably many countable sets,
which is a contradiction.
Now observe that  (X) =@2 >@1 = dis(X).
To prove that X is normal let H and K be disjoint closed sets. It will be enough to
show that H has a countable open cover, such that the closure of every member of it misses
K (see Lemma 1.1.15 of [7]). Fix n2!. We have H\Zn = Sj2!Hj, where Hj is closed
in the Euclidean topology on Zn for every j2!. Fix j2!. For each x2Hj let D(x;rx)
16
be a disk tangent to Zn at x such that D(x;rx)\K = ; and rx = 1k for some k 2 !.
Let U = Sx2Hj D(x;rx). First of all, we claim that no point of K\Zn is in U: indeed if
x2K\Zn then let Ix be an interval containing x and missing Hj, then the closest that a
point of Hj can come to x is one of the endpoints of Ix so there is room enough to separate
x from U by a tangent disk.
Now U = Sn2!Un, where Un = SfD(x;rx) : rx = 1ng. Let Vn = SfD(x;rx2 ) : rx = 1ng.
We claim that Vn\KnZn = ;: indeed, if some point x2KnZn were limit for Vn then
we would have a sequence of disks of radius 12n clustering to it. But then x2Un, which
contradicts U\K =;.
To separate points of HnT from K just choose for each such point an open set whose
closure misses K and use second countabiliy of L. That shows how to de ne the required
countable open cover of H.
Finally, a development for X is provided by Gn =fD(x;n) : x2Xg where D(x;n) =
B(x; 1n)nSi<nZi if x2L, while if x =2L, D(x;n) is a tangent disk of radius less than 1n
which misses SfZi : i<n and x =2Zig.
The cardinal @2 can be replaced by any cardinal not greater than c, under proper
set theoretic assumptions (see [8]). So the previous example shows that the gap between
dis(X) and  (X) for normal Baire Moore spaces can be as big as the gap between the  rst
uncountable cardinal and the continuum.
Since normal Moore spaces are, consistently, metrizable, there is no chance of getting
in ZFC a space with all the properties of Example 3:15. Nevertheless, the following question
remains open.
Question 3.16. Is there in ZFC a normal Baire  -space X for which dis(X) <  (X)?
Using a Q-set on a tangent disk space to get normality is an old trick (see for example
[30]). Also, to get a regular Baire Moore space X for which dis(X) <  (X) it actually
su ces to assume the negation of CH along with the existence of a Luzin set.
17
A potential way of weakening the set theoretic assumption in Example 3:15 would be
to replace Luzin set with Baire subset of cardinality @1, but even such an object would be
inconsistent with MA+:CH, while the presence of CH would make the whole construction
worthless, so we have no clue even about the following.
Question 3.17. Is there, at least under MA+ : CH or under CH, a normal Baire  -space
X for which dis(X) <  (X)?
Also, notice that no regular Baire  -space X for which dis(X) <  (X) can be separable
under CH. That is because any regular separable space with points G has cardinality  c
( x any dense countable set D, then the map taking any regular open set to its intersection
with D is 1-to-1. So there are no more than c many regular open sets in the space, but
every point in a regular space with G -points is the intersection of countably many regular
open sets). Thus dis(X) =@1  (X) if CH holds.
3.3 Linearly ordered spaces
Recall that a space is called a GO space if it embeds in a LOTS. We denote by m(X)
the minimum number of metrizable spaces needed to cover X. The following result is due
to Ismail and Szymanski.
Lemma 3.18. [16] Let X be a locally compact Lindel of GO space. Then w(X) ! m(X)
Theorem 3.19. Let X be a locally compact Lindel of GO space. Then dis(X) =jXj.
Proof. Suppose by contradiction that there exists  <jXj such that X = SfD :  2 g
where each D is discrete. Then jD j w(X)  ! m(X)   , for every  2  . So
jXj supfjD j:  2 g    <jXj.
Corollary 3.20. Let X be a locally compact paracompact GO space. Then dis(X)  (X).
Proof. Every locally compact paracompact space contains a non-empty open set with the
Lindel of property (see [7], 5.1.27). Fix one such U  X. Then U is a locally compact
Lindel of GO space and hence dis(X) dis(U) jUj  (X).
18
In the previous corollary we cannot weaken locally compact paracompact to Baire
paracompact, as the following example shows. Recall that a space is called non-archimedean
if it has a base such that any two elements are either disjoint or one is contained in the
other. Every non-archimedean space has a base which is a tree under reverse inclusion (see
[26]), and from this it is easy to see that it is (hereditarily) paracompact.
Example 3.21. There is a Baire non-archimedean (and hence hereditarily paracompact)
LOTS X such that dis(X) <  (X).
Proof. Let  and  be in nite cardinals such that cf( )  but  < . Let W =f 1g[ .
De ne an order on W by declaring  1 to be less than every ordinal. Let X =ff2W + :
supp(f) < +g, where supp(f) = minf < + : f( ) = 0 for every    g. Now take the
topology induced on X by the lexicographic order.
Claim 1: X is a strong Choquet space (and hence Baire).
Proof of Claim 1. We are going to describe a winning strategy for player II in the strong
Choquet game. In his  rst move player I chooses any open set B1 and a point f1 2
B1. Player II then chooses points a1;b1 2 X such that f1 2 (a1;b1)  B1. Let now
 1 = maxfsupp(f1);supp(a1);supp(b1)g and f 1 = (f1( ) : 0   <  1). De ne f 1 =
f 1_( 1;0;:::0) and f+1 = f 1_(1;0;:::;0).
Clearly a1 <f 1 <f1 <f+1 <b1. Now in her  rst move player II chooses the open set
A1 = (f 1 ;f+1 ).
Player I responds by choosing any open set B2  A1 and a point f2 2B2. Player II
proceeds as before. Notice that fn+1 thus constructed agrees with fn up to  n and that the
point h =  Sf n _ (0;0;:::;0) is in Tn 1An. So II has a winning strategy. 4
Claim 2: X is the union of  + many discrete sets.
Proof of Claim 2. For every  2  +, let D = ff 2 X : supp(f) =  g. Then X =
S
 2 + D and each D is discrete. Indeed, let f2D and de ne:
19
f ( ) =
8>
>>>>
><
>>>>
>>:
f( ) If  < 
 1 If  =  
0 If  > 
(3.1)
Similarly de ne:
f+( ) =
8>
>>>>
><
>>>>
>>:
f( ) If  < 
1 If  =  
0 If  > 
(3.2)
Then (f ;f+)\D =ffg. 4
Claim 3: X is non-archimedean.
Proof of Claim 3. Let B =f[ ] :  2W for some  2 +g, where [ ] =ff2X :   fg.
Then B is a basis for our space. Every element of B is open: indeed, if f 2 [ ] then let
 = maxfdom( );supp(f)g and f+ and f be de ned as in the proof of Claim 2. Then
f2(f ;f+) [ ].
Now let c2(a;b). Then there are ordinals  and  such that a( ) <c( ), c( ) <b( ),
while a( ) = c( ) and c( ) = b( ) for every  < and every  < . Set  = maxf ; g+ 1.
We have that [c   ] (a;b).
Now given two elements ofB, either one is contained in the other, or they are disjoint.
Therefore X is non-archimedean. 4
To complete the proof observe that  (X)   > > + dis(X).
Since for  xed  there are arbitrarily big cardinals  having co nality  , the former
example shows that the gap between dis(X) and  (X) can be arbitrarily big for hereditarily
paracompact Baire LOTS.
Notice that the Lindel of number of the previous space is   , in particular X is never
Lindel of.
20
Question 3.22. Is dis(X)   (X) true for every (Lindel of, hereditarily paracompact)
 Cech complete LOTS X?
21
Chapter 4
Covering compact spaces by discrete sets
Besides inspiring our study of the inequality dis(X)   (X) for generalized metric
spaces, Corollary 3:4 allowed us to prove a lemma that was crucial to many of our partial
positive answers to Juh asz and Szentmikl ossy?s original question about compact spaces.
4.1 Hereditary separation
Testing a conjecture about compact spaces on compact hereditarily normal spaces is
quite a natural thing to try, and indeed, Juh asz and Van Mill already did that for the
inequality dis(X) c, before Gruenhage proved it to be true for every compact Hausdor 
space.
Theorem 4.1. ([14]) Let f : X!Y be a perfect map. Then dis(X) dis(Y).
Let  ! be the product of countably many copies of the discrete space  .
A cellular family is a family of pairwise disjoint open sets in X. The following lemma
is crucial to most of our results.
Lemma 4.2. Let X be a compact space whose every open set contains a cellular family of
cardinality  . Then dis(X)  !.
Proof. Use regularity of X to  nd a cellular family fU :  < g such that the closures of
its members are pairwise disjoint. Suppose you have constructed open setsfU :  2 <ng.
Then letfU _ :  2 gbe a cellular family inside U such that the closures of its members
are pairwise disjoint and contained in U .
For each f2 ! let Ff = Tn2!Uf n, which is a non-empty set because of compactness,
and set Z = Sf2 ! Ff.
22
We are now going to show a perfect map  from Z onto  !. Note that  ! is a complete
(and hence Baire) metric space and  ( !) =  !. So, by Theorem 4:1 and Corollary 3:4 we
will get that dis(X)  !.
De ne  simply as  (x) = f whenever x 2 Ff. It is easy to see that the Ffs are
pairwise disjoint, so  is well-de ned. Moreover,  is clearly continuous, onto and has
compact  bers.
The following characterization of closed maps is well-known (see [7], Theorem 1.4.13)
Fact 4.3. A mapping f : X!Y is closed if and only if for every point y2Y and every
open set U X which contains f 1(y), there exists in Y a neighbourhood V of the point y
such that f 1(V) U.
Let now f2 !, and U be an open set in Z such that   1(f) = Ff = Tn2!Uf n U.
By compactness, we can  nd an increasing sequence of integers fjk : 1 k ng such that
Uf jn = T1 k nUf jk  U.
So let B(f  jn) be the basic neighbourhood in  ! determined by f  jn. Then
  1(B(f  jn)) Uf jn  U, which proves  is closed.
Theorem 4.4. Let X be a hereditarily collectionwise Hausdor compact space. Then
dis(X)  (X).
Proof. Recall that cellularity and spread coincide for hereditarily collectionwise Hausdor 
spaces (see [18], 2.23 a)). So if c(G) <  (X), for some open set G  X we also have
s(G) <  (X)  (G). Hence dis(X)  (X).
Suppose now that c(G)   (X) for every open set G X. If  (X) is a successor
cardinal then every open set contains a cellular family of size  (X), and hence, in view of
Lemma 4:2 we have dis(X)  (X).
If  (X) is a limit cardinal then, again by Lemma 4:2, every open set contains a cellular
family of size  for every  <  (X). Hence dis(X)  for every  <  (X), which implies
dis(X)  (X) again.
23
The following corollary also follows from Theorem 3:19.
Corollary 4.5. For every compact LOTS X, dis(X)  (X).
Proof. Compact LOTS are monotonically normal, and monotone normality is hereditary
(see [12]).
From Theorem 4:4 it also follows that, under V=L, dis(X)  (X) for every compact
hereditarily normal space X. Indeed, Stephen Watson [33] proved that compact hereditarily
normal spaces are hereditarily collectionwise Hausdor in the constructible universe. We
can do better, and prove that dis(X)  (X) for X compact hereditarily normal under a
slight weakening of GCH.
Theorem 4.6. (for every cardinal  , 2 < 2 +) Let X be a compact T5 space. Then
dis(X)  (X).
Proof. Suppose  rst that c(G) <  (X) for some open set G. Since c(G) = c(G) and G is
compact T5 we can assume that X = G.
Let  = c(X). By Shapirovskii?s bound on the number of regular open sets (see [18],
3.21) we have  (X)  2 . Note that  +   (X). If dis(X) <  (X) then we would have
s(X)   (X) and hence we could  nd a discrete D X such that jDj  +. By Jones?
Lemma,  (X)  2 + > 2 , which contradicts our upper bound for the number of regular
open sets.
If c(G)   (X) for every open set G, then reasoning as in the last few lines of the
proof of Theorem 4:4 we can conclude that dis(X)  (X).
Question 4.7. Is it true in ZFC that dis(X)  (X) for every compact T5 space?
4.2 The shadow of a metric space
A trivial observation is that all compact metrizable spaces satisfy dis(X)  (X).
24
The two most popular generalizations of compact metrizable spaces are dyadic com-
pacta and Eberlein compacta. In fact, they are two somewhat opposite classes, as their
intersection is precisely the class of compact metrizable spaces (see Arhangelskii).
This made us wonder whether dis(X)  (X) was true for them. In fact, we are able to
prove that for the weaker classes of polyadic and Gul?ko compacta. To achieve that we  rst
need to prove that dis(X) is always bounded below by the tightness. Recall that a space is
called initially  -compact if every set of cardinality  has a complete accumulation point.
Lemma 4.8. ([11]) Let X be an initially  -compact space such that dis(X)  . Then X
is compact.
Lemma 4.9. If X is compact then dis(X) t(X).
Proof. Suppose by contradiction that  = dis(X) <t(X). Let A X be a non-closed set,
and [A] be its  -closure, that is, the union of the closures of its subsets of cardinality  . If
we could prove that this last set is closed then we would have t(X)  , which is what we
want.
If [A] is not closed then it cannot be initially -compact, or otherwise, sincedis([A] ) 
 , it would be compact by Lemma 4:8. So there is B [A] such that jBj  and B has
no point of complete accumulation in [A] ; then, by compactness, there is a point x =2[A] 
that is of complete accumulation for B. But this contradicts the well-known and easy to
prove fact that [[A] ] = [A] .
A compactum is called polyadic if it is the continuous image of some power of the
one-point compacti cation of some discrete set.
The following lemmas are due to Gerlits.
Lemma 4.10. [9] Let X be polyadic and A X. Then there is a polyadic P  X such
that A P and c(P) c(A).
Lemma 4.11. [10] If X is polyadic then w(X) = t(X) c(X).
25
Theorem 4.12. For a polyadic compactum X we have dis(X)  (X).
Proof. If c(U)  (X) for any open set U  X then we are done by Lemma 4:2. If there
exists some open U such that c(U) <  (X), then let P be a polyadic space such that
U  P and c(P)  c(U). Assume dis(P) <  (X). Then t(P) <  (X), which implies
s(P) w(P) <  (X), and we are done, since jPj  (X).
Recall that an Eberlein compactum is a compact space which embeds in Cp(Y) for some
compact Y. Equivalently, a space is an Eberlein compactum if and only if it is a weakly
compact subspace of a Banach space. A Gul?ko compactum is a compact space X such that
Cp(X) is a Lindel of  -space. A Corson compactum is a compact space with embeds in a
 -product of lines. The following chain of implications holds:
Eberlein)Gul?ko)Corson
Lemma 4.13. Let X be a hereditarily meta-Lindel of space such that dis(X)  . If A X
is such that jAj  then jAj  .
Proof. If  < ! then the statement is obviously true. Assume that  is in nite, and let
X = S < D , where each D is discrete. Let B = A\D . For every x2B , let Ux
be an open set such that Ux\B = fxg. Then Sx2B Ux is meta-Lindel of, and hence
fUx : x2B g has a point-countable open re nement V . Now for every x2B choose
Vx 2V such that x2Vx and let U = fVx : x2B g. Clearly jU j = jB j and for all
U2U , U\A6=;. Fix some well-ordering of A and de ne a function f :U !A by:
f(U) = minfa2A : a2Ug:
Point-countability of U implies that jf 1(a)j @0 for every a 2 A, and therefore
jB j=jU j jAj @0  .
Since A = S 2 B it follows that jAj  .
26
Theorem 4.14. Let X be a hereditarily meta-Lindel of space containing a dense Baire
metrizable subset. Then dis(X)  (X).
Proof. LetM X be a dense metrizable subset and suppose by contradiction thatdis(X) <
 (X). Then, by the previous lemma we have  (M) =  (X). So dis(X)  dis(M)  
 (M) =  (X), which is a contradiction.
Corollary 4.15. For every Gul?ko compactum X we have dis(X)  (X).
Proof. Yakovlev ([34]) proved that every Corson compactum is hereditarily meta-Lindel of
and Gruenhage ([13]) proved that every Gul?ko compactum contains a dense Baire metriz-
able subset.
We are sorry to admit that we haven?t been able to answer the following two questions.
Question 4.16. Is dis(X)  (X) for every Corson compact X?
Question 4.17. Is dis(X)  (X) for every compact space with a (Baire) dense metrizable
subset?
As an application of the results in this section we are now going to determine how
many discrete sets are needed to cover the  -product of a Cantor cube.
Theorem 4.18. dis( (2 )) =  !.
To prove that we will embed in  (2 ) an Eberlein compactum X for which  (X) =  !.
Recall that a family A of subsets of a set T is called adequate if:
1. For every A2A, P(A) A.
2. If [A]<! A then A2A.
It is easy to see thatAwith the topology inherited from the product space 2T is closed,
and hence compact. Such a space is called an adequate compactum. Adequate families are
one of the most useful tools for constructing Corson compacta: especially handy is the
27
adequate family of all chains of a partial order. If the partial order has no uncountable
chains, then the corresponding adequate compactum is Corson.
Leiderman and Sokolov characterized all adequate Eberlein compacta. For a point
x22T de ne the support of x as supp(x) =fa2T : x(a) = 1g.
Theorem 4.19. ([24]) Let X be an adequate compact embedded in 2T. Then X is an
Eberlein compact if and only if there is a partition T = Si2!Ti such thatjsupp(x)\Tij<@0
for each x2X and i2!.
The next example is a modi cation of an example due to Leiderman and Sokolov. Their
original space was a strong Eberlein compactum (a weakly compact subset of a Hilbert
space), and hence scattered. Our space is far from being scattered.
Example 4.20. Let  be any in nite cardinal. There is an Eberlein compactum, embedded
in 2 , such that  (X) =  !.
Proof. Let W0 = Lim( ) and let fx :  2 g be an increasing enumeration of W0. Let
Wi =fx +i :  2 g. Now let T = Si2!Wi (Wi[f ig). De ne an order on T as follows
: ( 1; 1) < ( 2; 2) if and only if  1 < 2 and  1 > 2. Then every chain in T is countable,
so the adequate compact X constructed from the adequate family consisting of all chains in
T is Corson. Moreover, the partition in the de nition of T, along with Theorem 4:19 shows
that X is Eberlein. It remains to check that  (X) =  !. To see that, let U be any basic
open set. Then U is the set of all chains containing some  xed  nite chainf( i; i) : i kg,
enumerated in increasing order, and missing a  xed  nite number of elementsf( j; j) : j 
rg. Let t be an integer such that f i : i kg[f j : j  rg Ss<tWs. Now, for every
chain of the form f s : s tg with  s 2Ws for every s t and  t >  k we have that
f( i; i) : i kg[f( s; s) : s tg2U. Now the set of all such chains has cardinality  !,
since there is a natural bijection between that set and the set of all countable increasing
sequences in  .
Every  -product of compact spaces is countably compact, which reminds us of the
following question.
28
Question 4.21. Is dis(X) c for X countably compact crowded?
4.3 Homogeneity and beyond
The starting point for our next pair of results is the following easy observation.
Theorem 4.22. Let X be a homogeneous compactum. Then dis(X)  (X).
Proof. Combining Arhangel?skii?s theorem with the Juh asz-Szentmikl ossy?s result cited in
the introduction we get dis(X) 2 (X)  (X).
A space is homogeneous with respect to character if  (x;X) =  (y;X) for any x;y2X.
A space X is power homogeneous if X is homogeneous for some  .
The following lemma is due to Juh asz and Van Mill.
Lemma 4.23. ([19]) Every in nite compactum contains a point x with  (x;X) <dis(X).
We are also going to need a couple of results from Guit Jan Ridderbos? PhD Thesis.
Lemma 4.24. ([27]) Let X be power homogeneous. If the set of all points of  -character
 is dense in X, then   (X)  .
Let    (X) = supf   (x;X) : x2Xg, where    (x;X) is the least cardinality of a
 -network at x consisting of G -sets.
Lemma 4.25. ([27]) Let X be a power-homogeneous space of pointwise countable type such
that    (X)  . Then either  (X)  or X is homogeneous with respect to character.
Theorem 4.26. (CH) Let X be a power-homogeneous compactum. Then the minimum
number of discrete sets required to cover X is at least minf (X);!3g.
Proof. Suppose that  ! does not embed in X, then X does not map onto I!1 (see the
proof of [18], 3.22) and hence, as a consequence of Shapirovskii?s Theorem on maps onto
Tychono Cubes, the set of all points of countable  -character is dense in X. Therefore,
by Lemma 4:24,   (X) !. If  (X) !, then jXj !1, by Arhangel?skii?s theorem, and
29
since dis(X) !1 holds for every compactum, we are done. Otherwise, X is homogeneous
with respect to character, and hencejXj 2 (X) dis(X), by Juh asz and Szentmikl ossy?s
result.
If  ! embeds in X then dis(X)  2!1. Suppose that dis(X) < !3, that is dis(X)  
!2. Then, by Lemma 4:23, X contains a dense set of G!1 points. If  (X)  !1, then
 (X) 2!1 and we are done. Otherwise, X is homogeneous with respect to character, and
dis(X)  (X) is true again.
Corollary 4.27. (CH) If X is a power-homogeneous compactum such that jXj !3 then
dis(X)  (X).
Question 4.28. Is dis(X)  (X) true for every power-homogeneous compactum?
The following proposition at least says that the gap between  (X) and dis(X) can?t
be too big for power-homogeneous compacta.
Proposition 4.29. Let X be a power homogeneous compactum. Then  (X) 2dis(X).
Proof. Suppose by way of contradiction thatdis(X)  butjUj> 2 for every openU X.
Then by Lemma 4:23 the set of all points of character less than  is dense in X, which implies
  (X)   . Thus, in particular,    (X)   . If  (X)   , then, by Arhangel?skii?s
Theorem, jXj 2 , which contradicts our initial assumption. Otherwise  (X)   + and
X is homogeneous with respect to character, which even implies dis(X)  2 +, again a
contradiction.
30
Chapter 5
Closures of discrete sets, closures of free sequences and cardinality
5.1 A crash course on elementary submodels
In this and the next chapter of our dissertation we will make use of a technique from
Model Theory, that is gradually becoming a standard tool in Set-theoretic Topology. Here
we provide some basics on elementary submodels and their applications to Topology, that
will make this chapter self-contained. None of the results cited in this section is our own,
we refer the reader to [5] for more information as well as the missing proofs.
Given a formula  (x1;x2;:::;xn) of Set Theory, having free variables fx1;x2;:::xng
and a set M we write Mj=  (x1;x2;:::xn) if the formula  (x1;x2;:::xn) is true when you
restrict all quanti ers to M. For example if  (X) = (9x)(x2X) then M j=  (X) if and
only if (9x2M)(x2X), that is, X\M is non-empty.
De nition 5.1. If fa1;a2;:::;ang M  N we say that the formula  (a1;a2;:::;an) is
absolute for M and N if Mj=  (a1;a2;:::;an) if and only if Nj=  (a1;a2;:::an).
De nition 5.2. We will say that M is an elementary submodel of N and write M N if
for all n<! and for all formulae  with at most n free variables and for allfa1;a2;:::;ang 
M we have that  is absolute for M and N.
In practice M can take the place of N as long as the formulae we are taking up have all
free variables in M, or, in other words, all the objects we are dealing with in our proof lie
in M. It would be nice if N could be taken to be the whole set-theoretic universe. This is
not feasible; however, before writing a proof, we already know the size of the largest object
we will be considering. Say this is  . Then we can take for N the set H( +), consisting of
all hereditary sets of size   , which is a portion of the set-theoretic universe that is known
31
to satisfy all set-theoretic axioms that we are going to need and contains all objects we are
going to take up.
Most proofs of cardinal inequalities by elementary submodels follow a common plan.
Suppose you want to prove the size of a good enough topological space is no more 2 .
1. Start with an elementary submodel of size 2 having as elements X, the topology
on X, all cardinals you will be dealing with in your proof and a few other things.
Theorem 5:3 below tells you there always exists such an elementary submodel.
2. Assume that there is a point p2XnM. Get a contradiction. Then X  M and
hence jXj jMj 2 .
3. Sometimes you are going to need your elementary submodel to be  -closed (that is,
each of its subsets of size  is an element of it). This is helpful, for example, if you
know that the size of a certain item in your space is at most  . So if you are trying
to get a contradiction by inductively constructing such an item of size  + inside M,
you know you can always continue because the inductive step is always an element of
M. For instance, if you know the spread of your space is at most  , you could try
and get a contradiction by constructing a discrete set of size  + inside M. Theorem
5:5 below says that you can always get a  -closed elementary submodel of size 2 .
We now list three theorems that are the backbones of the use of elementary submodels
in Topology.
Theorem 5.3. For any set H and A H there is an elementary submodel M  H such
that A M and jMj jAj !.
Theorem 5.4. If M  H( ), where  is a regular cardinal and  2M is a cardinal such
that   M then for all A2M with jAj  we have A M. In particular, each countable
element of M is a subset of M.
Theorem 5.5. For any regular   2 and for any A H( ) with jAj 2 there is an
M H( ) so that A M, jMj= 2 and M  M.
32
One of the reasons why elementary submodels are so useful in topology is that they
make ugly-looking arguments involving trans nite induction transparent. And their ability
to eat up a trans nite induction in a single bite is an outproduct of their nice behaviour
with respect to chains.
If  de nes a linear order on M then M is called an elementary chain.
Theorem 5.6. Let M be an elementary chain. Then M SM whenever M2M.
Corollary 5.7. A chain under inclusion of elementary submodels of H is an elementary
chain. Moreover its union is an elementary submodel of H.
Proof. Let M;N 2 M and suppose without loss that M  N  H. Fix n 2 !, let
 be a formula with at most n free variables and fa1;a2;:::;ang M. Suppose N j=
 (a1;a2;:::;an). ThenHj=  (a1;a2;:::;an). SinceM H we haveMj=  (a1;a2;:::;an).
So M N, so M is an elementary chain. Let now fa1;a2;:::;ang SM. Then there is
M 2M such that fa1;a2;:::;ang M  SM. Now M  H and Hj=  (a1;a2;:::;an)
imply that M j=  (a1;a2;:::;an). By M  SM we have SMj=  (a1;a2;:::;an). So
SM H.
5.2 Depth, spread, free sequences and cardinality
Alas, Tkachuk and Wilson [2] asked whether a compact space in which the closure of
every discrete set has size  c must have size  c.
In [1] Ofelia Alas proves the following theorem, by way of a partial positive answer.
Theorem 5.8. (MA) Let X be a Lindel of regular weakly discretely generated space such
that ^s(X) c and jDj c for every discrete D X. Then jXj c.
We are going to prove that regular, Lindel of and weakly discretely generated can all be
dropped from the above theorem. But,  rst of all let?s de ne two cardinal functions that
will be handy in our study of this and related problems. Recall that a set fx :  <  g
is called a free sequence if fx :  < g\fx :    g = ; for every  <  . Every free
sequence is a discrete set.
33
De nition 5.9. Set g(X) = supfjDj : D X is discrete g (the depth of X) and b(X) =
supfjFj: F  X is a free sequence g (the breadth of X).
The condition g(X)   appears to be a lot stronger than b(X)   . In fact, while
the former implies that jXj 2 (simply observe that the hereditarily Lindel of number is
discretely re exive [2] and use De Groot?s inequality jXj 2hL(X)), the latter alone does
not put any bound on the cardinality of X. For example, the one-point compacti cation of
a discrete set of arbitrary cardinality satis es b(X) = !.
Before proving our  rst theorem, we need an old lemma of Shapirovskii, and a lemma
about elementary submodels, which must be well-known, although we could not  nd a direct
reference to it.
Lemma 5.10. Suppose c is a regular cardinal. Let   (2<c)+ be a regular cardinal and
A H( ) be a set of size  2<c. Then there is an elementary submodel M  H( ) such
that A M, jMj= 2<c and M is  -closed for every  < c.
Proof. It follows from regularity of the cardinal c that (2<c)j j = 2<c for every  < c. Let
now M0  H( ) be such that A  M0 and jM0j 2<c. Suppose we have constructed
fM :  <  g such that for every  <  we have M  H( ), jM j 2<c. Then let
M  H( ) be such that M [[M ]j j  M for every  <  and jM j 2<c. Then
fM :  < cg is a chain under containment of elementary submodels of H( ) and hence
it is also an elementary chain, from which it follows that M = S <cM is an elementary
submodel of H( ).
To see that M is < c-closed let  < c and fx :  <  g M. Then, by regularity
of c there is  < c such that fx :  <  g M . We can certainly assume  >  . But
[M ]j j M +1 and therefore fx :  < g2M +1 M.
Lemma 5.11. (Shapirovskii, see [18], 2.13) LetU be an open cover for some space X. Then
there is a discrete D X and a subcover W U such that jWj=jDj and X = D[SW.
Theorem 5.12. (2<c = c) Let X be a space such that ^s(X) g(X) c. Then jXj c.
34
Proof. Let M be an elementary submodel of a large enough fraction of the universe such
that fX; g M, c[fcg M, jMj c and M is  -closed for every  < c.
We claim that X  M. Suppose not and  x p 2 XnM. We claim that for every
x 2 X\M we can choose an open U 2 M such that x 2 U and p =2 U. Indeed,  x
x2X\M and let V2M be the set of all open sets V  X such that x =2V. Then V
covers Xnfxg, so by Shapirovskii?s Lemma we can  nd a discrete D2M and a subfamily
W V such that W2M, jWj = jDj c and Xnfxg D[SW. Now W2M and
jWj c imply thatW M. Notice that, since D2M, also D2M which implies D M,
since jDj c. So p =2D and hence there is W 2W such that p2W. Let U = XnW.
Then U2M is a neighbourhood of x such that p =2U.
So for every x2X\M choose Ux2M such that p =2U. The family U = fUx : x2
X\Mgcovers X\M, so, by Shapirovskii?s Lemma there is a discrete set D X\M and
a set W U such that jWj=jDj< c with X\M  D[SW. Since M is < c-closed we
have that D2M andW2M, and hence Mj= X D[SW. Now p =2W for any W 2W
and p =2D, since D X\M, by the same reason as before. But that?s a contradiction.
Can we switch discrete sets with free sequences in the previous theorem? Clearly not,
and the one-point compacti cation of a discrete set is a counterexample. However there
are some cases where we can. Let?s start by proving a kind of free-sequence version of
Shapirovskii?s Lemma.
Lemma 5.13. Let X be a space such that the closure of every free sequence is Lindel of and
U be an open cover for X. Then there is a free sequence F  X and a subcollection V U
such that jVj=jFj and X = F[SV.
Proof. Suppose you have constructed, for some ordinal  , a free sequence fx :  <  g
and countable subcollections fU :  <  g such that fx :  < g S   SU for every
 < .
35
Let U be a countable subcollection of U covering the Lindel of subspace fx :  < g
and pick a point x 2XnS   SU . Let  be the least ordinal such that
fx :  < g[
[
 < 
[
U = X:
Then fx :  < g is a free sequence and for V = S < U we have jVj=  .
Theorem 5.14. (2<c = c) Let X be a Lindel of space such that  (X) c and ^F(X) b(X) 
c. Then jXj c.
Proof. Let M be a < c-closed elementary submodel such that c[fcg M andfX; g M.
Claim: The closure of every free sequence in X\M is Lindel of.
Proof of Claim. Let F  X\M be a free sequence in X\M well-ordered in type  (where
  c because jMj c). We claim that F is also a free sequence in X. Denote by F 
the initial segment of F determined by its  th element. Let  = supf <  : F is a free
sequence in X by the same well-ordering of Fg. Then F is a free sequence in X. If not,
there would be some  <  such that x 2 F \F nF and x =2 M. But F is a free
sequence in X and therefore jF j< c. Thus F 2M, and hence F 2M, which along with
jF j c implies that F  M. So x2M, which is a contradiction. But now F +1 is also
a free sequence in X, because you can?t spoil freeness by adding a single isolated point.
Therefore  =  , which proves that F is a free sequence in X. Proceeding as before we
get that F  X\M, which proves our claim, since closed subspaces of Lindel of spaces are
Lindel of. 4
We claim that X  M. Suppose not, and let p2XnM. For every x2X\M use
 (X)  c to pick a neighbourhood Ux 2M of x such that p =2Ux. Let U = fUx : x2
X\Mg. By Lemma 5:13, there are a free sequence F  X\M and a subcollectionV U
such thatjFj=jVj< c with X\M F[SV. NowjFj< c, so F 2M and hence F 2M,
which, along with jFj c implies that F  M. Also, V  M and jVj < c imply that
36
V2M. Therefore Mj= X F[SV and hence there is V 2V such that p2V, which is
a contradiction.
Pseudocharacter   is not discretely re exive, unless the space is compact (see [2]).
The following lemma shows that the pseudocharacter of a space never exceeds its depth.
Lemma 5.15. Let  be an in nite cardinal and X be a space where jDj  for every
discrete D X. Then  (X)  . If in addition X is regular then  (F;X)  , for every
closed F  X such that jFj  .
Proof. Let F  X be a  -sized closed set (or a point, if X is not regular). Now let
V = fV  X : V is open and V \F = ;g. Then V covers XnF and hence we can  nd a
discrete D XnF and a subcollection U V with jUj=jDj such that XnF  SU[D.
So (Tx2DnF Xnfxg)\(TU2UXnU) = F, which implies that  (F;X)  .
The following corollary is another improvement of Alas? Theorem.
Corollary 5.16. (2<c = c) Let X be a Lindel of space such that ^F(X) g(X)  c. Then
jXj c.
Proof. This follows from Lemma 5:15 and Theorem 5:14.
In the above corollary Lindel ofness can be removed, if one assumes the space to be
regular.
Theorem 5.17. (2<c = c) Let X be a regular space such that ^F(X)  c and jDj c for
every discrete D X. Then jXj c.
Proof. Let M be an elementary submodel as before. By Lemma 5:15 every c-sized closed
subset of X has pseudocharacter  c.
We claim that X M. Suppose not and  x p2XnM and suppose that for some  < c
we have constructed a free sequence fx :  < g M and open sets fU :  < g M.
We have p =2fx :  < g. Now use the claim to choose a sequence G2M of open sets
such that jGj c and fx :  < g= TG. We have G M, so we can choose an open set
37
U 2M with p =2U and fx :  < g U . Now use < c-closed and elementarity to pick
x 2 (XnS   U )\M. Thus fx :   cg is a c-sized free sequence in X, which is a
contradiction.
In Theorem 5:17 one can safely work in ZFC if free sequences are assumed to be
countable. So we have a common framework for Alas? Theorem and Dow?s result about
compact spaces of countable tightness mentioned in the introduction. We have only one
case left to exhaust all relationships between the four cardinal functions we have de ned
and cardinality.
Theorem 5.18. (2<c = c) Let X be a regular space such that ^s(X) b(X)  c. Then
jXj c.
Proof. Let F  X. We claim that  (F;X) c. Indeed, for every x =2F use regularity to
choose an open neighbourhood Vx of x such that Vx\F = ;. Then fVx : x =2Fg covers
XnF, so we can choose a discrete D XnF such that XnF  SfVx : x2Dg[D. Now
we claim that for every p2DnF we can choose an E D such that p2E and E\F =;.
Indeed, simply use regularity to  nd an open neighbourhood U of p such that U\F = ;
and set E = U\D. So F = TfXnE : E D and E\F = ;g\TfVx : x2Dg. This
implies that  (F;X) c sincejDj< c and hence 2jDj c, by the set-theoretic assumption.
Now, an argument similar to the proof of Theorem 5:17 will  nish the proof.
Regularity can be replaced by Lindel ofness. We leave the details to the reader.
Question 5.19. Is there in ZFC a Hausdor non-regular space such that free sequences
are countable (discrete sets are countable), jDj c for every discrete D X (for every free
sequence F  X) and yet jXj> c?
Question 5.20. Is there, in some model of set theory, some (compact) regular space X
such that every discrete set has size < c, the closure of every discrete set has size  c and
yet the space has size > c.
38
To  nd a Hausdor counterexample to the above question, take a model of !1 < c < 2!1
and let X = 2!1. Let  =fUnC : U is open in the usual topology on 2!1 and jCj !1g.
Then every discrete set in (X; ) is closed and has size !1 < c.
39
Chapter 6
Arhangel?skii, De Groot, free sequences and increasing chains
6.1 Introduction
In 1968 A.V. Arhangel?skii proved his famous theorem saying that the cardinality of
a compact  rst-countable Hausdor space does not exceed the continuum. This solved a
long-standing question of Alexandro and boosted an active line of research investigating
generalizations of it. Here are two highlights.
Theorem 6.1. (Arhangel?skii-Shapirovskii) Let X be Hausdor space. Then:
jXj 2t(X) L(X)  (X)
Theorem 6.2. (Bell-Ginsburgh-Woods) Let X be normal weakly Lindel of  rst-countable
space. Then jXj c.
Here a space is weakly Lindel of if every open cover has a countable subcollection
whose union is dense in the space. The question asking whether normal can be replaced
with regular in this last result is certainly one of the most interesting in this area. A good
survey of Arhangel?skii Theorem and its o springs is Hodel?s ([15]).
An important tool in Arhangel?skii?s proof of his theorem is the notion of a free se-
quence. We have already seen that t(X) = F(X) in compact T2 spaces. If X is Lindel of,
this is not true anymore. Indeed, assume CH and take a Luzin subspace of the real line
with the density topology. Then the tightness is uncountable, since every countable set is
closed discrete, but free sequences are countable because the space is hereditarily Lindel of.
However, we always have F(X) L(X) t(X) for every Hausdor space X. Here we prove
that if X is Hausdor then jXj 2 (X) F(X) L(X). This is a generalization of Theorem 6:1
40
in view of what we just said. Also, we prove the increasing strengthening of our theorem,
and the proof we give seems to be shorter and simpler than even the proof of the increasing
strengthening of Arhangel?skii?s theorem as given by Juh asz (see [18], 6.11), although it still
relies on some of his ideas.
6.2 A common generalization of Arhangel?skii?s Theorem and De Groot?s in-
equality
Istv an Juh asz has kindly informed us that he independently proved Theorem 6:4 and
presented it along with other results in a series of talks in Jerusalem in 2003, but never got
around to publish it.
De ne  (X) = supfL(Xnfxg) : x2Xg.
Lemma 6.3.  (X) = L(X)  (X).
Proof. Obviously L(X)   (X). Also, if L(Xnfxg)   then for every y6= x select Uy
such that x =2Uy. ThenU =fUy : y6= xgcovers Xnfxgand hence we can  nd a subcover
V having cardinality   . Then TfXnU : U2Ug=fxg, which proves that  (x;X)  .
So, taking sups we have that  (X)  (X), and hence  (X) L(X)  (X).
To prove the other direction suppose that L(X)  (X) =  and let U be an open
collection such that jUj  and TU = fxg. Then Xnfxg = SfXnU : U 2Ug and
L(XnU)  for every U2U. Thus L(Xnfxg)  .
The following generalizes both Theorem 6.1 and De Groot?s inequality saying that the
cardinality of every hereditarily Lindel of space does not exceed the continuum.
Theorem 6.4. If X is T2 then jXj 2 (X) L(X) F(X).
Proof. Let  =  (X) L(X) F(X),  be a large enough regular cardinal and M  H( )
be  -closed, jMj= 2 and 2 [fX; ;2 g M. We claim that X M. Suppose not and
choose p2XnM. Let x2X\M. Since  (x;X)  there is a familyU2M of open sets
41
such that TU =fxg and jUj  . Now every 2 -sized element of M is also a subset of M,
so U M and hence we can choose an open set U2M such that x2U and p =2U.
Let U be the set of all open U 2M such that p =2U. Then U covers X\M. Let U0
be any subcollection of U having cardinality   . Since p2XnSU0, by elementarity we
can choose x0 2X\MnSU0. Now suppose that for some  2 + we have constructed
a set fx :  <  g and subcollections fU :  <  g such that jU j  for every  <  
and fx :  < g SS   U and let U be a subcollection of U having cardinality   
such that fx :  < g SU and pick a point x 2X\MnS   U . If the induction
didn?t stop before reaching  + then fx :  < +g would be a free sequence of size  + in
X. So there is a subcollection V U such that jVj  and X\M  SV. Therefore
Mj= X SV and hence H( )j= X SV. So there is V 2V such that p2V, which is a
contradiction.
6.3 The increasing strenghtening
Suppose X = S < X where X  X whenever  < and we know that f(X )  
for every  <  for some cardinal function f. What can we conclude about f(X)? This
general question has been the object of systematic study by Juh asz, who dedicated the
whole chapter 6 of his book [18] to it, Juh asz and Szentmiklossy [22] and Tkachenko [31]
[32]. In particular, we talk of an increasing strengthening of a cardinal inequality when we
can extend a cardinal inequality from a single space to an increasing chain of spaces of any
length. Increasing strengthenings of cardinal inequalities often involve rather technical and
complicated arguments. This is the case with the increasing strengthening of Arhangel?skii?s
Theorem [18] and that of the Bell-Ginsburgh-Woods Theorem [22]. We are now going to
prove the increasing strengthening of Theorem 6:1.
Lemma 6.5. ([18], 6.11) If X is T2, Y is a subspace of X with L(Y)  and p2Y, then
for every open set U in X containing p there is a family R of regular closed neighbourhoods
42
of p in X such that jRj  and
U\Y  
\
R\Y
.
Theorem 6.6. Let X = S < X , where X  X whenever  <  and suppose that
F(X )  (X ) L(X )  for every  < . Then jXj 2 .
Proof. If   2 then we are done by Theorem 6:4, so we can assume that  = (2 )+. Call
a set A X bounded if jAj 2 . The following claim is contained in [18], 6.11 but we
include its proof for completeness.
Claim: If A2[X]  then A is bounded.
Proof of claim. LetA X be bounded. Since (A) 2jAj 2 , by [18], 2.6 d), it will su ce
to prove that if F is closed unbounded then  (F) > 2 . Let F = F\X . Then L(F )  
for every  <  . Fix x2F, then there is  0 <  such that x2F 0. We have x2F 
for every  2 n 0, so  (x;F )   and hence we can  nd families fU :  0 <  <  g
of open sets such that jU j  and TU \F = fxg for every  >  0. Fix now  >  0.
For every U 2U use the Lemma to select a family RU of regular closed sets such that
jRUj  and TRU\F  U\F . Let R = SfRU : U2U g. Then TR \F =fxg.
Suppose by contradiction that  (F)  2 , then, since  > 2 we can  nd a  -sized family
Rx consisting of regular closed sets and a set a2[ ] such that R =Rx for every  2a.
So TRx\X = fxg for co nally many  ?s, which can only be if TRx = fxg. Hence
we have found an injection from F into the family of all families of size   consisting of
regular closed sets, which implies jFj  (F)  2 . But that contradicts the fact that F
is unbounded. 4
Let  be a large enough regular cardinal and M  H( ) be  -closed, jMj = 2 and
2 [fX; ;2 g M. We claim that X M. Suppose not and choose p2XnM. We claim
that for every x2X\M we can choose a neighbourhood U 2M of x such that p =2U.
43
Indeed,  x x2X\M and let V be the set of all open sets V such that x =2V. Note that
V covers Xnfxg. Suppose we have constructed subcollections fV :  < g of V such that
jV j  for every  <  and a set fx :  <  g such that fx :  < g SS < V for
every  < , where the closure is meant in Xnfxg. By the Claim, the set fx :  < g is
bounded and hence there is  < such thatfx :  < g X  . HenceL(fx :  < g) 
 , so there is a subcollection V of V such that jV j  and fx :  < g SV . If the
induction didn?t stop before reaching  + then F =fx :  < +gwould be a free sequence
of length  + in Xnfxg. Now F cannot converge to x, because, since jFj   + and
L(Xnfxg)  , the set F has a complete accumulation point in Xnfxg. Therefore, there
is an open neighbourhood G of x which misses  + many points of F and FnG is a free
sequence in X of cardinality  +. Now FnG is bounded and hence there is  < such that
FnG X , but that contradicts F(X )  .
So there is a subcollection W  V such that jWj   and Xnfxg  SW. By
elementarity we can takeW2M and henceW M, sincejWj  . Let W 2W such that
p2W. Then the set U = XnW 2M is an open neighbourhood of x such that p =2U.
Now let U be the set of all open sets U2M such that p =2U. Then U covers X\M.
Let U0 be any subcollection of U having cardinality   . Since there is a point (namely p)
in XnSU0, by elementarity we can pick x02X\MnSU0. Suppose that for some  2 +
we have constructed a setfx :  < gand subcollections fU :  < gsuch thatjU j  
for every  <  and fx :  < g SS   U . Since fx :  < g is bounded we have
L(fx :  < g  and hence we can  nd a subcollection U of U having cardinality   
such that fx :  < g SU . If U is not a cover of X, as before, we can pick a point
x 2X\MnS   U . If we didn?t stop then fx :  < +g would be a free sequence of
size  + in X. But that can?t be sincefx :  < +gis bounded. So there is a subcollection
V U such that jVj  such that X\M  SV. Therefore M j= X  SV and hence
H( )j= X SV. Thus there is V 2V such that p2V, which is a contradiction.
44
Bibliography
[1] O. Alas, On closures of discrete subsets, Q and A in General Topology, Vol. 20 (2002),
85-89.
[2] O. Alas, V. Tkachuk, R. Wilson, Closures of discrete sets often re ect global properties,
Topology Proc. 25 Spring (2000).
[3] B. Bailey and G. Gruenhage, On a question concerning sharp bases, Topology Appl.
153 (2005), 90{96.
[4] E. van Douwen, An unbaireable strati able space, Proc. Amer. Math. Soc. 67 (1977),
324{326.
[5] A. Dow, An introduction to applications of elementary submodels to topology, Topology
Proc. (1988), 13(1), 17-72.
[6] A. Dow, Closures of discrete sets in compact spaces, Studia Math. Sci. Hung. 42 (2005),
227{234.
[7] R. Engelking, General Topology, second ed., Sigma Series in Pure Mathematics, no.6,
Heldermann Verlag, Berlin, 1989.
[8] W. Fleissner and A. Miller, On Q-sets, Proc. Amer. Math. Soc. 78 (1980), 280{284.
[9] J. Gerlits, On a problem of S. Mr owka, Period. Math. Hungar. 4 (1973), 71{80.
[10] J. Gerlits, On a generalization of dyadicity, Studia Sci. Math. Hungar. 13 (1978), 1{17.
[11] J. Gerlits, I. Juh asz and Z. Szentmikl ossy, Two improvements on Tkachenko?s addition
theorem, Comment. Math. Univ. Carolinae, 46, 4 (2005), 705-710.
[12] G. Gruenhage, Generalized metric spaces in Handbook of Set Theoretic Topology,
edited by K. Kunen and J.E. Vaughan, North-Holland, Amsterdam 1984.
[13] G. Gruenhage, A note on Gul?ko compact spaces, Proc. Amer. Math. Soc. 100 (1987),
371{376.
[14] G. Gruenhage, Covering compacta by discrete and other separated sets, preprint.
[15] R. Hodel, Arhangel skii?s Solution to Alexandro ?s Problem: A Survey, Topology Appl.
153, 2199{2217.
[16] M. Ismail and A. Szymanski, On the metrizability number and related invariants of
spaces. II, Topology Appl. 71 (1996), 179{191.
45
[17] T. Jech, Set Theory: The Third Millennium Edition, revised and expanded, Springer,
Berlin 2002.
[18] I. Juh asz, Cardinal Function in Topology - Ten Years Later, Math. Centre Tracts 123,
1980 Amsterdam.
[19] I. Juh asz, J. van Mill, Covering compacta by discrete subspaces, Topology Appl. 154
(2007), 283{286.
[20] I. Juh asz, Z. Szentmikl ossy, A strengthening of the  Cech-Pospi sil theorem, Topology
Appl. 155 (2008), 2102{2104 .
[21] I. Juh asz, Z. Szentmikl ossy, Discrete subspaces of countably tight compacta, Annals of
Pure and Applied Logic 140 (2006), 72{74.
[22] I. Juh asz, Z. Szentmikl ossy, Increasing strengthenings of cardinal function inequalities,
Fund. Math. 126 (1986), no. 3, 209{216.
[23] A. Kechris, Classical descriptive set theory, Springer, New York, 1994.
[24] A. Leiderman, G. Sokolov, Adequate families of sets and Corson compacts, Comment.
Math. Univ. Carolin. 25 (1984), no. 2, 233{246.
[25] A. Miller, Special subsets of the real line in Handbook of Set Theoretic Topology, edited
by K. Kunen and J.E. Vaughan, North-Holland, Amsterdam 1984.
[26] P.J. Nyikos, On some non-archimedean spaces of Alexandro and Urysohn, Topology
Appl. 91 (1999), 1{23.
[27] G.J. Ridderbos, Power homogeneity in Topology, Doctoral Thesis, Vrije Universiteit,
Amsterdam (2007).
[28] S. Spadaro, Covering by discrete and closed discrete sets, Topology Appl. 156 (2009),
721{727.
[29] S. Spadaro, A note on discrete sets, submitted.
[30] F.D. Tall, Normality versus Collectionwise Normality in Handbook of Set Theoretic
Topology, edited by K. Kunen and J.E. Vaughan, North-Holland, Amsterdam 1984.
[31] M.G. Tkachenko, Chains and cardinals, Soviet Math. Dokl. 19 (1978), no. 2, 382{385.
[32] M.G. Tkachenko, The behavior of cardinal invariants under the union operation for a
chain of spaces, Moscow Univ. Math. Bull. 33 (1978), no. 4, 39{46.
[33] S. Watson, Locally compact normal spaces in the constructible universe, Can. J. Math.,
vol. XXXIV, n.5, 1982, 1091{1096.
[34] N. Yakovlev, On bicompacta in  -products and related spaces., Comment. Math. Univ.
Carolin. 21 (1980), no. 2, 263{283.
46