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
(
B(f( )) : f( )(kf)2
[
n>k
An
)
:
The set B =fB( );B ;k(f) : 2c ; 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 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 2clAn 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! 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
>>>>
><
>>>>
>>:
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( ) > + 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 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) 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 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) 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 . 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?
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
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