Countably Compact, Countably Tight, Non-Compact Spaces
by
James Dabbs
A thesis submitted to the Graduate Faculty of
Auburn University
in partial ful llment of the
requirements for the Degree of
Master of Science
Auburn, Alabama
August 6, 2011
Keywords: Countable Compactness, Countable Tightness, Stone- Cech Compacti cation,
Monads
Copyright 2011 by James Dabbs
Approved by
Gary Gruenhage, Chair, Professor, Mathematics & Statistics
Michel Smith, Professor, Mathematics & Statistics
Stewart Baldwin, Professor, Mathematics & Statistics
Randall Holmes, Professor, Mathematics & Statistics
Abstract
In this work, we will study several examples of countably compact, countably tight, non-
compact spaces. After reviewing the important basic notions, we will examine a construction
of several such spaces  rst given by Manes in \Monads in Topology" and will then detail
how to construct such spaces using a more direct and explicit topological process. We will
then use this new framework to describe several new spaces and to prove several propositions
which are much more transparent from this new viewpoint.
ii
Acknowledgments
First I would like to thank my advisor, Dr. Gary Gruenhage, whose excellent classes
reinvigorated my interest in general topology and whose guidance has been invaluable in
preparing this work. I would also like to thank the rest of my committee members, Dr.
Michel Smith, Dr. Stewart Baldwin and Dr. Randall Holmes for their intellectual stimulation
and assistance both in and out of the classroom.
I must also o er my sincerest thanks to my family, especially my parents Ricky and Mary
Lou Dabbs. Their constant encouragements to focus on my studies and to be a lifelong learner
have become guiding principles in my personal development, and I am forever indebted to
the opportunities they have a orded me. I would also like to thank my grandparents for
their unfaltering encouragement.
Finally, I would like to thank my girlfriend McCall Langford and all my friends for their
support when I needed to focus on my work and their diversions when I needed not to.
iii
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Background De nitions and Theorems . . . . . . . . . . . . . . . . . . . . . . . 3
3 Monads of Ultra lters and Their Algebras . . . . . . . . . . . . . . . . . . . . . 13
3.1 Tp! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Topological Descriptions and Properties . . . . . . . . . . . . . . . . . . . . . . 17
4.1  pX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 UX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5 Applications of the Topological Viewpoint . . . . . . . . . . . . . . . . . . . . . 27
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
iv
Chapter 1
Introduction
A vast body of literature is devoted to the study of compact Hausdor spaces. The class
of compact Hausdor spaces is well behaved in many important ways, but there is at least
one sense in which compact Hausdor spaces may be quite poorly behaved: any metric space
is sequential, i.e. the topology is entirely determined by convergent sequences in the space,
but there are non-trivial compact Hausdor spaces with no non-trivial convergent sequences.
A natural question is: are there topological spaces satisfying a weaker form of compactness
but which are to some degree determined by sequences in the space? In this work, we will
single out countable compactness as the weaker form of compactness under consideration.
When considering countable compactness, there is a natural question: what can be said
about spaces which are countably compact but not compact? For instance, no such space can
be Lindel of. The speci c question \is every separable, countably compact, countably tight
space compact?" was posed as a \classic problem" by Nyikos in [2]. A series of examples
of countably compact, countably tight, non-compact spaces satisfying successively stronger
separation properties has been constructed [3] [4], but most such constructions require the
assumption of additional axioms beyond ZFC. The strongest such example currently known
to exist in ZFC was constructed by Manes in [1] using the category-theoretical concept of a
monad.
In this work, we will give a construction of the spaces  rst described by Manes using a
new and more explicit topological approach. In Chapter 2, we review the basic background
notions. In Chapter 3, we summarize Manes? construction using monads. In Chapter 4, we
will detail the new topological construction of Manes? spaces and prove, using our description,
that they give examples of countably compact, countably tight, non-compact spaces and
1
discuss their other properties. We will also use this topological framework to construct a new
larger class of spaces with many similar properties. Finally, in Chapter 5, we will discuss some
applications of this new framework by providing several propositions and constructions that
would be either much more opaque or outright impossible from the categorical perspective.
2
Chapter 2
Background De nitions and Theorems
The material in this section is standard and can be found in any introductory topology
book, but de nitions and proofs can be found in [5]. Basic set theory notions are assumed,
and whenever unspeci ed, notation matches that used in [5].
The following terms are well known and will be used without further comment: topol-
ogy, topological space, subspace topology,  ner topology, open set, neighborhood, closed set,
limit point, closure, sequence, continuous function, homeomorphism, compact, T1, Hausdor ,
regular, completely regular, connected, dense.
We will often consider two di erent topologies on the same space. For clarity, we make
the following de nition:
De nition 2.1. If  is a topology on a set X and A  X, cl (A) is the closure of A in
(X; ). More speci cally:
cl (A) =\fCjA C;XnC2 g
We may use cl(A) or simply A if  is clear from context.
Throughout this chapter, X will denote an arbitrary topological space. The following
propositions are well known:
Proposition 2.2. If C X is closed and X is compact, then C is compact.
Proposition 2.3. If C X is compact and X is Hausdor then C is closed in X.
Proposition 2.4. If C  X is compact and f : X !Y is continuous then f(C)  Y is
compact.
3
Proposition 2.5. If xn!x in X and f : X!Y continuous then f(xn)!f(x) in Y.
Proposition 2.6. If D  X is dense, Y is Hausdor , and f;g : X !Y are continuous
with fjD = gjD, then f = g.
The following de nitions are less widely known, and so they are included for complete-
ness:
De nition 2.7. X is Urysohn provided that for every pair of points x;y2X there are open
sets Ux;Uy X with x2Ux and y2Uy with Ux\Uy =;.
Note that any Urysohn space is clearly Hausdor .
De nition 2.8. X is extremally disconnected if for every open O X, O is open.
The next several results outline the basic construction of and results about the Stone-
 Cech compacti cation. Proofs are omitted, but may be found in [7].
De nition 2.9. A  lter F on a set S is a collection of nonempty subsets of S such that:
 If A;B2F then A\B2F.
 If A2F and C A then C2F.
De nition 2.10. An ultra lter u on a set S is maximal  lter on S.
Note that we must appeal to some form of the Axiom of Choice to show that ultra lters
exist. By using Zorn?s Lemma, any  lter can be extended to an ultra lter.
Proposition 2.11. Let F be a  lter on a set S. The following are equivalent:
1. F is an ultra lter.
2. For every A S, either A2F or SnA2F.
3. If A[B = S, then either A2F or B2F.
4
4. If A S and A\B6=; for every B2F, then A2F.
De nition 2.12. C  X is a zero-set in X if there is a continuous f : X ! R with
C = f 1(0).
De nition 2.13. A z- lter is a  lterF on X so that for every C2F, C is a zero-set in X.
De nition 2.14. A z-ultra lter is a  lter which is maximal among z- lters.
De nition 2.15. For x2X, prin(x) =fB XjB a zero-set, x2Bg.
A z-ultra lter of the form prin(x) is called a principal ultra lter.
De nition 2.16.  X is the set of all z-ultra lters on X. The standard topology on  X
is de ned as follows: if F  X is a zero-set, let F = fu 2 X jF 2 ug. A basis for the
standard topology consists of all sets of the form  XnF for F a zero-set in X.
We will primarily consider the case when X is discrete, in which case every subset is a
zero-set and  X can be described more simply as the set of all ultra lters on X with basic
open sets of the form O for any O X.
We will typically regard X as a subset of  X by identifying x2X with prin(x)2 X.
With this identi cation,
Proposition 2.17. X  X is dense.
Proposition 2.18.  X is a compact Hausdor space.
If X is a completely regular, Hausdor space, the inclusion map X ,! X is an embed-
ding and  X is often referred to as the Stone-  Cech compacti cation of X.
Proposition 2.19. If Y is any compact Hausdor space and f : X !Y is a continuous
function, there is a unique continuous function F :  X !Y so that FjX = f. Moreover,
this property uniquely characterizes  X.
5
Proposition 2.20. If C;D X are disjoint zero-sets, then cl X(C) and cl X(D) are dis-
joint.
De nition 2.21. ! is the  rst in nite ordinal.
Proposition 2.22. j !j= 22!.
Proposition 2.23. If X is discrete,  X contains no non-trivial convergent sequences. That
is, any sequence that converges in  X is eventually constant.
De nition 2.24. If f : X!Y and u2 X, then f(u) =fB Y jf 1(B)2ug=fC 
Y jC f(A) for some A2ug.
The notation fu is often used in place of f(u).
De nition 2.25. X =  XnX with the subspace topology inherited from  X.
Thus an element of X is a non-principal ultra lter. Such an ultra lter is often called
free.
Proposition 2.26. Let p2! . p contains no  nite subsets of !.
Proof. Suppose there is some  nite F 2p of smallest cardinality. If jFj> 1 and n2F,
then either fng2p or Fnfng2p by 2.11.2, contradicting the minimality of jFj. Thus
F =fngfor some n2! and so for every B !, with n2B, B2p since p is a  lter. Thus
p = prin(n) and so p62! .
De nition 2.27. A set S  X is a weak P-set in X if for every countable C  X with
C\S =;, C\S =;.
De nition 2.28. A point x2X is a weak P-point in X if fxg X is a weak P-set in X.
An important fact which we will use frequently is:
Proposition 2.29. There are weak P-points in ! .
6
In fact, ! contains a dense set of weak P-points [7], but we will not have need of this
stronger result.
A central notion in this work is that of a p-limit:
De nition 2.30. If p 2 ! and (xn) is a sequence in X, then a point y 2 X is the p-
limit of (xn), denoted y = p-limxn provided that for every open O  X containing y,
fn2!jxn2Og2p.
The de nition of a p-limit depends essentially on the topology of X. We may use p-
limXxn or p-lim xn to emphasize the space X or topology  with respect to which the limit
is taken.
Note that if xn !x in the usual sense then for any open O about x, fnjxn 2Og is
co nite in ! and so this set is in every p2! by 2.26. Thus the notion of a p-limit is a
weakening of the usual notion of a limit.
The importance of p-limits can be seen in the following proposition:
Proposition 2.31. Let (xn) be a sequence in X. y2fxng if and only if y = p-limxn for
some p2 !.
Proof. Suppose y = p-limxn. Then for any open O containing y,fnjxn2Og2p and since
every set in p is nonempty, O\fxng6=; and y2fxng.
Conversely, if y 2fxng, let F = ffn j xn 2 Ogj O a neighborhood of yg. For any
A;B2F, A\B2F and soF can be extended to an ultra lter p2 !. Then by de nition,
for any neighborhood O of y, fnjxn2Og2F p and so y = p-limxn.
Let us record a few basic facts about p-limits:
Proposition 2.32. Let (xn) be a sequence in X and p2! . If X is Hausdor , then p-limxn
is unique if it exists.
Proof. Suppose x and y are both p-limits of (xn). If X is Hausdor , there are open Ux;Uy
containing x and y respectively with Ux\Uy = ;. Since x is a p-limit of (xn), the set
7
Ax = fnjxn 2Uxg2p and similarly Ay = fnjxn 2Uyg2p. But Ax\Ay = ; since
Ux\Uy =;, which contradicts the fact that p is a  lter.
Thus we are justi ed in referring to \the" p-limit of a sequence.
Proposition 2.33. Let (xn) be a sequence in X, p 2 ! and x = p-limxn. For any
continuous f : X!Y, f(x) = p-limf(xn).
Proof. Let f : X !Y be continuous and O Y be an open neighborhood of f(x). Then
x2f 1(O), which is open since f is continuous. Since x = p-limxn, fnjxn2f 1(O)g =
fnjf(xn)2Og2p as required.
Proposition 2.34. Let (xn), (yn) be sequences in X and p2! . If fnjxn = yng2p then
p-limxn = p-limyn.
Proof. Suppose A = fnjxn = yng2p. For any open neighborhood O of p-limxn, B =
fnjxn2Og2p. Since p is a  lter, A\B2p and for every n2A\B, yn = xn2O. Thus
A\B fnjyn2Og2p since p is a  lter and so p-limxn = p-limyn.
De nition 2.35. For p2! , X is p-compact if for every sequence (xn) in X, p-limxn2X
(in particular, p-limxn exists).
p-compactness is a weakening of compactness:
Proposition 2.36. If X is compact, then X is p-compact for every p2! .
Proof. Let (xn) be a sequence in X. For B !, let xB =fxnjn2Bg. Then for any p2! ,
fxBjB2pgis a collection of closed subsets of X with the  nite intersection property. Since
X is compact, there is some x2\fxB jB2pg. Let O be an open neighborhood of x and
A = fnjxn 2Og. For every B2p, x2xB, so there is some n2B with xn 2O. Thus
A\B6=;for each B2p and since p is an ultra lter, A2p by 2.11.4. Thus x = p-limxn.
De nition 2.37. X is ultracompact if X is p-compact for every p2! .
8
p-compactness can be used to de ne an important partial order on  !:
De nition 2.38. For p;q2! , the Comfort preorder  C on  ! is de ned by p C q ()
every q-compact space is p-compact.
De nition 2.39. For p;q2! , the Rudin-Keisler preorder  RK on  ! is de ned by p RK
q () p = fq for some f : !!!.
De nition 2.40. X is countably compact if every countable open cover has a  nite subcover.
Proposition 2.41. If X is p-compact and T1, then X is countably compact.
Proof. If X is p-compact, then every in nite set has a limit point by 2.31. Since X is T1, it
is countably compact by [5].
Proposition 2.42. If X is countably compact and fCn jn2!g is a countable collection of
nonempty closed sets such that Cn+1 Cn, then \Cn6=;.
Proof. Let Un = XnCn. Each Un is open. If\Cn =;then[Un = Xn\Cn = X sofUngis a
cover. If X is countably compact, there is some N2! so that\n NCn = Xn[n NUn =;,
contrary to the assumption that \n NCn = CN 6=;.
De nition 2.43. X is sequential if for every non-closed A X, there is a sequence (xn) in
A with xn!x62A.
A sequential space is one in which the topology is entirely determined by convergent
sequences. Using p-limits we can weaken this de nition in the obvious way:
De nition 2.44. For p2! , a space X is p-sequential if for every non-closed A X, there
is a sequence (xn) in A with p-limxn62A.
We can weaken this even further to get the following de nition:
De nition 2.45. X is countably tight if for every set A X and x2A, there is some
countable C A with x2C.
9
That this is a weaker notion will be shown shortly, but  rst let us introduce some
notation to aid in the proof:
De nition 2.46. For Y  X and S ! , de ne AX (S;Y) for each   !1 by trans nite
induction as follows:
 AX0 (S;Y) = Y
 AX +1(S;Y) =fx2Xjx = p-limxn for some p2S and some sequence (xn) in AX (S;Y)g
for successor ordinals
 AX (S;Y) =[ < A (S;Y) for limit ordinals
We may write A (S;Y) if X is clear from context.
Proposition 2.47. Let p2! . If X is p-sequential and Y  X, then clX(Y) = AX!1(fpg;Y).
Proof. Fix Y  X. Suppose that AX!1(fpg;Y) is not closed. Then since X is p-sequential,
there is some sequence (xn) in AX!1(fpg;Y) with p-limxn 62AX!1(fpg;Y). But then by con-
struction, there is some  <!1 with fxng AX (fpg;Y) and so p-limxn2AX +1(fpg;Y) 
AX!1(fpg;Y), a contradiction. Thus AX!1(fpg;Y) is closed and so clX(Y) AX!1(fpg;Y).
Clearly A0(fpg;Y) clX(Y). Suppose A (fpg;Y) clX(Y) for all  < . If  is a limit
ordinal thenA (fpg;Y) clX(Y) trivially. If not, then =  +1 and for anya2A (fpg;Y),
a = p-limxn for some sequence (xn) in A (fpg;Y). Thus a2A (fpg;Y) clX(Y) by 2.31
and the induction hypothesis. Thus A!1(fpg;Y) clX(Y) by trans nite induction.
Proposition 2.48. If S  ! so that for any Y  X, clX(Y) = AX!1(S;Y), then X is
countably tight.
Proof. Suppose S  ! satis es the hypotheses. We claim that for any  < !1 and y 2
AX (S;Y) there is a countable B Y with y2B. If y2A1(S;Y) then this is clear from
2.31. Proceeding by trans nite induction, suppose y2AX (S;Y). If  =  + 1, then y = p-
limyn2fyng for some sequence (yn) in AX (S;Y) by 2.31. By induction, for each n there is
10
Bn Y with yn2Bn. Then y2[Bn. If  is a limit ordinal, then y2AX (S;Y) for some
 < already.
If Y  X and y 2 clX(Y) = AX!1(S;Y) then y 2 AX (S;Y) for some  < !1 and so
y2B for some countable B Y. Thus X is countably tight.
Corollary 2.49. Every p-sequential space is countably tight.
De nition 2.50. C(X) =ff : X!Rjf is continuousg.
De nition 2.51. C(X;Y) =ff : X!Y jf is continuousg.
De nition 2.52. A category C is a collection of objects ob(C), morphisms mor(C) and
a composition operator  so that for every collection of objects A;B;C and morphisms
f : A!B and g : B!C, g f : A!C with the following properties:
 If f;g;h are morphisms, (f g) h = f (g h), provided both expressions are de ned.
 For each object A there is a unique identity morphism idA : A!X such that for any
morphism f : A!B, f idA = f = idB f.
De nition 2.53. A (covariant) functor F is a function F : C!D between categories so
that for every A2ob(C), F(A)2ob(D), for every f : A!B in C, F(f) : F(A)!F(B) in
D and F(f g) = F(f) F(g) whenever the composition f g is de ned.
For convenience, F(A) and F(f) may be denoted FA and Ff, respectively.
De nition 2.54. The identity functor 1C : C!C is de ned by 1CA = A and 1Cf = f for
every object A and morphism f.
De nition 2.55. If F;G are two functors from C to D a natural transformation  is a
collection of morphisms  A - called the component functions of  - one for each A2ob(C) so
that for every f : A!B in C, the following diagram commutes:
11
FA Ff //
 A
  
FB
 B
  
GA Gf //GB
De nition 2.56. A semigroup operation on a set S is an associative binary operation,
usually denoted a b for a;b2S.
12
Chapter 3
Monads of Ultra lters and Their Algebras
De nition 3.1. Given a category C, a monad is a triple (T; ; ) where T is a functor
T : C!C,  is a natural transformation  : 1C ! T and  is a natural transformation
 : T2!T so that the following diagrams commute:
T3 T //
 T   
T2
 
  
T2  //T
T  T //
T   id
  
T2
 
  
T2  //T
WhereT is the natural transformation with component functionsT( X) : T3X!T2X
and  T is the natural transformation with component functions  TX : T3X!T2X for each
object X2ob(C).
This is the standard de nition of a monad, given, for instance, in [8]. In [1], Manes
gives the following equivalent de nition if C = Set:
De nition 3.2. A monad over Set is a triple (T; ;( )#) where T is a function T : Set!
Set,  X : X!TX for every set X, and for every pair of sets X;Y and function f : X!TY,
an extension f# : TX!TY so that:
1. f#  X = f
2. ( X)# = idTX
13
3. (g] f)] = g] f] for any g : Y !TZ
That the two de nitions of a monad are equivalent is given by the following:
Proposition 3.3. A monad (T; ; ) over Set is equivalent to a monad over Set (T; ;( )]).
Proof. First, suppose we are given a monad as (T; ;( )]). Extend T to a functor by de ning,
for f : X ! Y, Tf = ( Y  f)] : TX ! TY. T is a functor since if g : Y ! Z,
T(g f) = ( Z g f)] whereas Tg Tf = ( Z g)] ( Y f)] = (( Z g)]  Y f)] = ( Z g f)].
For any such f, Tf  X = ( Y f)]  X =  Y f so  is a natural transformation  : 1C!T.
For each set X let  X = id]TX : TTX!TX.  : T2 !T is a natural transformation
since for any f : X!Y,  Y TTf = id]TY ( TY Tf)] = (id]TY  TY Tf)] = (idTY Tf)] =
(Tf)] and Tf  X = ( Y f)]id]TX = (( Y f)] idTX)] = (Tf)]. The following computations
show that the diagrams in 3.1 commute and so (T; ; ) is a monad:
 X  TX = id]TX  TX = idTX
 X T( X) = id]TX ( TX  X)] = (id]TX  TX  X)] = (idTX  TX)] = ( TX)] = idTX
 X  TX = id]TX id]TTX = (id]TX idTTX)] = id]]TX and  X T( X) = id]TX T(id]TX) =
id]TX ( TX id]TX)] = (id]TX  TX id]TX)] = (idTX id]TX)] = id]]TX
Conversely, suppose we are given (T; ; ). If f : X!TY, de ne f] =  Y Tf : TX!
TY. Then we can check that for any f : X!TY and g : Y !TZ,
1. f]  X =  Y  Tf  X =  Y   TY  f = f
2. ( X)] =  TX T( X) = idTX
3. (g] f)] =  Z T(g] f) =  Z T(g]) Tf =  Z T( Z Tg) Tf =  Z T( Z) TTg Tf =
 Z  TZ TTg Tf =  Z Tg  Y  Tf = g] f]
Given ( )] and de ning  X = id]TX, note that for any f : X ! TY,  Y  Tf =
id]TY Tf = id]TY ( TY f)] = (id]TY  TY f)] = (idTY f)] = f] so we recover our original
14
notion of ( )]. Conversely, given  and de ning f] =  Y  Tf for f : X !TY, we have
id]TX =  X T(idTX) =  X idTTX =  X and we recover our original notion of  . Thus
these two de nitions are equivalent.
We will sometimes refer to T as a monad when  and  (or ( )]) are clear from context.
The prototypical example of a monad over Set is  , with  X : X! X by x7!prin(x)
for each set X and x2X and ( )] de ned by f](u) = fB Y jfx2X jB2f(x)g2ug
for each f : X! Y and u2 X.
Given a monad T, there is an important classical notion of an algebra of T:
De nition 3.4. Given a monad T on a categoryC, an algebra of T is an object X2C with
a morphism  : TX!X so that the following diagrams commute:
T2X T //
 X
  
TX
 
  
TX  //X
TX
 
!!X
 X
OO
id
//X
De nition 3.5. If A is an algebra of the monad T over Set, B A is a subalgebra if there
is a set map  0 which makes the following diagram commute:
TB
 0   
Ti //TA
   
B i //A
where i : B!A is the inclusion map.
Note that if B is a subalgebra of (A; ) as above then (B; 0) is itself an algebra.
15
3.1 Tp!
Throughout this section, p will denote an arbitrary free ultra lter on !. The following
de nitions are taken from [1].
De nition 3.6. GpX =fr2 Xjr = fp for some f : !!Xg
De nition 3.7. Tp is the smallest submonad of ( ; ;( )]) over Set such that GpX TpX
for each set X.
The importance of this de nition is established by the following:
Theorem 3.8. De ne a topology on Tp! by declaring A Tp! closed if and only if A is a
subalgebra of (Tp!;id]TX). With this topology, Tp! is a countably compact, countably tight,
separable, Urysohn, non-compact, non-sequential space.
In [1], Manes established this fact, giving the  rst ZFC construction of such a space.
We will prove this result later by giving a topological construction of the space Tp!.
16
Chapter 4
Topological Descriptions and Properties
4.1  pX
Given any topological space X and p2! , the set of p-compact subsets of X form the
closed sets of a new topology on X. We?ll examine some properties of this new topology and
then look at this construction on a particular space to get the example we seek.
Throughout this section, let (X; ) be an arbitrary completely regular, Hausdor space
and p2! a free ultra lter.
De nition 4.1.
 p =fU XjXnU is p-compact in (X; )g[f;g
Proposition 4.2.  p is a topology on X.
Proof. ;2 p by de nition and X2 p trivially. We show that the  nite union and arbitrary
intersection of p-compact subsets of X are p-compact.
First, let C1 and C2 be p-compact subsets of X and (xn) a sequence in C1[C2. Let
Bi = fnjxn 2Cig. Since p is an ultra lter and B1[B2 2p, by 2.11.3 we may assume
without loss of generality that B1 2 p. Fix an arbitrary z 2 C1 and let x0n = xn for all
n2B1 and x0n = z otherwise. Then (x0n) is a sequence in C1, so p-limx0n exists and is in
C1 by assumption and p-limxn = p-limx0n by 2.34. Thus p-limxn2C1[C2 and C1[C2 is
p-compact.
Next, let Ci be p-compact for each i in some index set. If (xn) is a sequence in\Ci, then
(xn) is a sequence in Ci for each i so p-limxn2Ci for each i and thus\Ci is p-compact.
17
As observed in the preceding proof,  p may de ned by declaring C  (X; p) closed if
and only if C (X; ) is p-compact.
Proposition 4.3. If X is p-compact and C X is closed, then C is p-compact.
Proof. If (xn) is a sequence inC thenx = p-limxn exists sinceX isp-compact andx2C = C
by 2.31. Thus C is p-compact in the subspace topology inherited from X.
Corollary 4.4. If X is p-compact, then  p is  ner than  .
Note that even though  p is usually  ner than  in the examples we?ll consider, p-limits
are still the same when considered in  or  p:
Proposition 4.5. Let (xn) be a sequence in X. If p-lim xn exists, then p-lim p xn = p-
lim xn.
Proof. Let x = p-lim xn. Suppose x6= p-lim p xn. Then there is some O 2 p such that
x2O butfnjxn2Og62p. Thusfnjxn62Og2p since p is a z-ultra lter. Fix z2XnO
and de ne a new sequence (x0n) by x0n = xn if xn62O and x0n = z otherwise. Then (x0n) is a
sequence in XnO which is p-compact in (X; ) and so p-lim x0n2XnO. But xn = x0n for
all n2fnjxn62Og, so by 2.34 p-lim xn = p-lim x0n62O, a contradiction.
Corollary 4.6. If (X; ) is p-compact, (X; p) is p-compact.
Proposition 4.7. If (X; ) is p-compact, then (X; p) is p-sequential.
Proof. Let A X be non-closed in  p. Then A X is not p-compact in  . So by de nition
there is a sequence (xn) in A so that x = p-limxn2XnA.
Corollary 4.8. If (X; ) is p-compact, cl p(Y) = A!1(fpg;Y)
Proof. By 2.47.
Corollary 4.9. If X is p-compact and Y  X, jcl p(Y)j jYj!.
18
Mimicking the proof of 2.47, we can show that the two corollaries above are true even
if (X; ) is not p-compact to begin with, but we will not need this fact in what follows.
Now let?s look at the speci c example of interest:
De nition 4.10. Let  be the standard topology on  X.  pX = cl p(X) with the subspace
topology inherited from ( X; p).
Note that  pX = \fY   X jX  Y and Y is p-compactg, so this notation is com-
mensurate with the notation found in [9], although there they consider  pX to have the
subspace topology inherited from  X.
We will show that  p! is a countably compact, countably tight, separable, Urysohn
space that is not compact or sequential. We will then show that TpX =  pX (regarding X
as a set and a discrete space, respectively), giving a topological proof of 3.8.
Proposition 4.11.  p! is p-compact and thus countably compact.
Proof.  ! is compact and so p-compact by 2.36. This follows from 4.6 and 2.41.
Proposition 4.12.  p! is p-sequential and thus countably tight.
Proof. By 4.7 and 2.49.
Proposition 4.13.  p! is separable.
Proof. !  p! is a countable dense set.
Proposition 4.14.  p! is Urysohn.
Proof. By 4.4, if  is the standard topology on  !,  p is  ner than  . Given x;y2 p!, since
 ! is Urysohn, there are Ux;Uy2 containing x and y respectively with cl (Ux)\cl (Uy) =;.
Since  p is  ner than  , Ux;Uy2 p and cl p(Ux)\cl p(Uy) cl (Ux)\cl (Uy) =;so ( !; p)
is Urysohn. Since  p! is a subspace of ( !; p), it is Urysohn as well.
Proposition 4.15.  p! is not compact.
19
Proof. Let i :  p! ! ! denote the inclusion map. i is continuous since the topology on
 p! is  ner than the standard topology on  ! by 4.4. If  p! were compact, then i( p!)
would be a compact set (by 2.4) with ! i( p!)   !, and since compact subsets of  !
are closed (by 2.18 and 2.3) and !  ! is dense (by 2.17), i( p!) =  !. Butj p!j 2! by
4.9 and j !j= 22! by 2.22, a contradiction.
Proposition 4.16.  p! contains no non-trivial convergent sequences.
Proof. Since the topology on  p! is  ner than the standard topology, the inclusion map
i :  p!! ! is continuous. Thus if xn!x in  p!, i(xn)!i(x) in  ! by 2.5 and thus (xn)
is eventually constant by 2.23.
Corollary 4.17.  p! is not sequential.
Proof. If F   p! and (xn) is a sequence in F with xn !x then x = xm for some m so
x2F. Thus every subset of  p! is sequentially closed. But  p! is not discrete (!  p! is
not closed), so  p! is not sequential.
We can now recover a proof of 3.8 by showing the following:
Theorem 4.18. Let X be a set. The topological spaces TpX and  pX (considering X as a
discrete space) are identical.
The proof will follow from the next several propositions. For the remainder of this
section, suppose that X is discrete. Recall the de nitions related to Tp from 3.1. TpX
denotes the functor Tp applied to the underlying set of X.
Proposition 4.19. For any A TpX and g : !!A, if i : A!TpX is the inclusion map,
id#TpX (Tpi)(gp) = p-lim Xg(n).
Proof. Let A, g and i be as in the hypothesis. By de nition,
id#TpX (Tpi)(gp) =fD Xjfx2TpXjD2xg2(Tpi)(gp)g
20
and
fx2TpXjD2xg2(Tpi)(gp) ()
9C2gp(C fx2TpXjD2xg) ()
9C2gp(8y2C(y2fx2TpXjD2xg)) ()
9C2gp(8y2C(D2y)) ()
9B2p^9C g(B)(8y2C(D2y)) ()
9B2p(8y2g(B)(D2y))
The last equivalence follows from taking C = g(B). Thus D2id#TpX (Tpi)(gp) () 9B2
p with D2\g(B), and so id#TpX (Tpi)(gp) =[B2p\g(B)
Given any basic open O  X containing [B2p\g(B), O2[B2p\g(B) so there is
some B2p so that O2x for every x2g(B). Thus B fnjO2g(n)g=fnjg(n)2Og
and so fnjg(n)2Og2p and [B2p\g(B) = p-limg(n) as required.
Corollary 4.20. For any g : !!TpX;id#TpX(gp) = p-limg(n).
Proof. Tp(idTpX) = idTpTpX.
Proposition 4.21. TpX =  pX as sets.
Proof. Since (Tp; ;( )]) is a monad and idTpX : TpX !TpX, id#TpX(TpTpX)  TpX. For
any sequence (xn) in TpX, let g : !!TpX by n7!xn, so gp2TpTpX and id#TpX(gp) = p-
limxn2TpX. Thus TpX is p-compact with X TpX  X and so TpX  pX.
On the other hand, recalling de nition 2.46, A0(fpg;X) = X  TpX trivially. If
A (fpg;X) TpX for all  < , then if  is a limit ordinal, A (fpg;X) TpX trivially. If
not, let  =  + 1 and x2A (fpg;X). Then there is a sequence (xn) in A (fpg;X) with
x = p-limxn. Let g : !!A (fpg;X) TpX by n7!xn. Then x = p-limxn = id#TpX(gp)2
TpX. Thus A (fpg;X) TpX for all   !1 and so  pX TpX.
21
Proposition 4.22. The topologies on TpX and  pX coincide.
Proof. Recall that A  TpX is closed provided there is a set map  0 : TpA ! A with
id#TpX  Tpi = i  0, where i is the inclusion map. Such a  0 will exist if and only if
id#Tp! (Tpi)(TpA)  A, or 8gp2TpA;id#Tp! (Tpi)(gp) = p-limg(n)2A. Thus A TpX 
 X is closed in TpX if and only if it is p-compact when considered as a subset of  X.
4.2 UX
For di erent choices of p, each  p! captures only a small part of the structure of  !
(sincej p!j= 2! andj !j= 22!). For this section,  x a completely regular, Hausdor space
X. In what follows, we?ll describe a general way to glue together the various  pX into a new
topology on the entire set  X.
By [9], if p C q then  pX  qX.
De nition 4.23. For p C q, let  pq :  pX ,! qX denote the inclusion map.
De nition 4.24. For any p2! , let  p :  pX ,! X denote the inclusion map.
Proposition 4.25. If p C q then  pq is continuous.
Proof. Suppose p C q. If C  qX is closed, then C is q-compact. Since p C q, C is also
p-compact. Thus   1pq (C) = C\ pX is p-compact and so closed in  pX.
De nition 4.26. let UX = lim !f pX; pqg where  pq :  pX! qX is the inclusion map.
As a set, UX =f[r] jr2 p! for some pg with [r] = [r0] if  pq(r) =  p0q(r0). Since each
 is an inclusion map, we will identify UX with  X as a set. By de nition, O UX is open
if and only if   1p (O) is open for every p.
Proposition 4.27. UX is ultracompact.
Proof. Let (xn) be a sequence in UX and p2! . For each n, xn2 pnX for some pn2! .
By [9], there is an r2! with p C r and pn  C r for each n. Also by [9],  pnX   rX
22
and so fxng  rX.  rX is r-compact and since p  C r,  rX is p-compact. Thus p-
limxn2 rX UX.
We can characterize the topology on UX in several ways:
Proposition 4.28. The following are equivalent:
1. C UX is closed.
2. C is ultracompact.
3. C\ pX is p-compact for every p2! .
4. UXnC is a weak P-set in  X.
Proof. (1) ) (2): Since UX is ultracompact and a closed subset of a p-compact space is
p-compact, any closed C UX is ultracompact.
(2) ) (3): Given p2! , if C is ultracompact, C is p-compact and thus C\ pX is
p-compact.
(3) ) (1): By de nition of the direct limit, C UX is closed if and only if   1p (C) =
C\ pX is closed for each p. But C\ pX is closed in  pX if and only if C\ pX is
p-compact.
(2) () (4): Suppose C UX is ultracompact. Let O = UXnC. If D UXnO = C
is countable and y is a limit point of D, then by 2.31, y = p-limxn for some sequence (xn)
in D C and p2! . Since C is ultracompact, y2C = UXnO. Thus O is a weak P-set.
Conversely, suppose O is a weak P-set. Let C = UXnO and let (xn) be a sequence in C.
For any p2! , p-limxn2fxng, so p-limxn62O by assumption. Thus p-limxn2C and C
is ultracompact.
U! gives us another (much larger) example of a countably compact, countably tight,
separable, Urysohn, non-compact, non-sequential space, as the next several propositions
demonstrate:
23
Proposition 4.29. U! is countably compact.
Proof. U! is ultracompact and thus p-compact for every p. Thus U! is countably compact
by 2.41.
Proposition 4.30. For any Y  U!, clU!(Y) = A!1(! ;Y).
Proof. For any sequence (xn) in A!1(! ;Y) and p2! , fxng A (! ;Y) for some  <
!1 and so, by construction, p-limxn 2 A +1(! ;Y)  A!1(! ;Y). Thus A!1(! ;Y) is
ultracompact and so closed. Thus clU!(Y) A!1(! ;Y)
On the other hand, A0(! ;Y) clU!(Y) trivially. Suppose A (! ;Y) clU!(Y) for all
 < . If  is a limit ordinal, A (! ;Y) clU!(Y) trivially. If  =  + 1 and x2A (! ;Y)
then x = p-limxn for some sequence (xn) in A (! ;Y) and p2! . Sincefxng clU!(Y) and
x2fxng by 2.31, x2clU!(Y). Thus, by trans nite induction, A!1(! ;Y) clU!(Y).
Corollary 4.31. U! is countably tight.
Proof. By 2.48, taking S = ! .
Proposition 4.32. The topology on U! is strictly  ner than that of  !.
Proof. If C  ! is closed in the usual topology, C is compact by 2.2 and thus p-compact
for every p2! by 2.36. Thus C is closed in U! as well. If p2! is a weak P-point in ! 
(which exist by 2.29) then ! nfpgis ultracompact in  !, sofpg[! is open in U! but not
open in  !.
Corollary 4.33. U! is Urysohn and not compact.
Proposition 4.34. U! is not p-sequential for any p.
Proof. By 4.9, since jU!j=j !j= 22!.
Unfortunately, U! does not satisfy any stronger separation axioms.
Proposition 4.35. U! is not regular.
24
Proof. If p2! is a weak P-point, then ![fpg is open in U!. If p2V  U with V open,
then V \! must be in nite and so V 6 U.
Though the topologies on U! and  ! di er, continuous functions on each space behave
very similarly.
Proposition 4.36. C(U!) = C( !)
Proof. Since the topology on U! is  ner, any continuous f :  ! ! R is also continuous
when considered as a map f : U!!R. Thus C(U!) C( !).
On the other hand, suppose f : U!!R is continuous. If im(f) were not compact, there
would be a sequence yn2im(f) with no limit point. Choose xn2U! so that f(xn) = yn.
Then since U! is ultracompact, for any p2! , p-limxn exists and f(p-limxn) = p-limyn is
a limit point offyng, a contradiction. Thus f : U!!im(f) is a map into a compact set. By
the universal property of  ! (2.19), fj! extends to a unique continuous F :  !!im(f) R.
F is still continuous when considered as a map from U! into R and F and f agree on the
dense set !. Since R is Hausdor , F = f by 2.6 and so f is a continuous as a map from  !
into R.
Proposition 4.37. C(U!;U!) = C( !; !).
Proof. If f :  !! ! is continuous and C is closed in U!, then C is ultracompact in  !.
Let (xn) be a sequence in f 1(C) and yn = f(xn)2C. Given p2! , let x = p-limxn. Then
by 2.33, f(x) = p-limf(xn) = p-limyn2C. Thus x2f 1(C) and so f 1(C) is ultracompact
and thus closed in U!.
Conversely, suppose f : U! !U! is continuous. Let i : U! ,! ! be the inclusion
map. Then i f is continuous. By the universal property of  !, i fj! extends to a unique
continuous function F :  ! ! !. As above, F is still continuous when considered as a
function from U! to  !. F and i f agree on the dense subset ! and since U! is Hausdor ,
F = i f, so f is continuous considered as a map from  ! to  !.
25
Proposition 4.38. C(U!n!)6= C(! ).
Proof. Let p2! be a weak P-point (2.29). Let  p : ! !R by p7! 0 and q7! 1 for all
q6= p.  p62C(! ) since p is not isolated in  !. But since p is a weak P-point, p is isolated
in U!n! and thus  p2C(U!n!).
26
Chapter 5
Applications of the Topological Viewpoint
Historically, the spaces Tp! were of interest primarily because they satis ed the strongest
separation properties among ZFC examples of countably compact, countably tight, non-
compact spaces. Tp! is Urysohn for every p2! so a natural question is: could Tp! be
regular for any p? While this is not known in general, we do know
Proposition 5.1. If p is a weak P-point in ! ,  p! is not regular.
Proof. Let p be a weak P-point in ! . First note that p-limn = p since if O  ! is a basic
open set about p then O =fnjn2Og2p. Thus p2 p!.
Since p is a weak P-point, p is not a limit point of any countable subset of ! nfpgand
so ! nfpg is p-compact. Thus ![fpg  p! is open.
Suppose p2U U ![fpgfor an open U  p!. fn2!jn2Ug2p and so !\U
must be in nite. But then it follows thatj!\Uj>!, sojUj>j![fpgj, a contradiction.
With what we have established so far about Tp!, we can answer a question posed by
Manes in [1].
De nition 5.2. p2! is an m-point if Gp! (Tp!.
Manes showed that there is an m-point if we assume the Continuum Hypothesis and
further conjectured that every p2! is an m-point in ZFC. We will prove Manes? conjecture
using some tools from [9]. The key idea lies in the following:
De nition 5.3. If  is a semigroup operation on !, we may extend  to a semigroup
operation on  !. The extension, also denoted  , may be de ned as follows: for m2! and
q2! , let m q = q-limm n as n ranges over !; for p;q2! , let p q = p-limm q as m
ranges over !.
27
Proposition 5.4. No p2! is an m-point.
Proof. Pick any p 2 ! . By [9, 2.1], Tp! =  p! = fq 2  ! j q  C pg is always a sub-
semigroup under , while Gp! =fq2 !jq = fp for some f : !!!g=fq2!jq RK pg
is never a sub-semigroup [9, 2.15].
We can also say a little bit more about the interplay between the order C, the semigroup
operation  and the topology on  p!. The next result says, in e ect, that for any p2! 
there is an r C p that is \in nitely right-divisible" by p:
Proposition 5.5. For any s2! and p2! , there is some q C p so that for every n2!,
q = sn r pn for some r C p.
Proof. Fix s2! and p2! . As observed in [9], the map f :  !! ! by r7!s r p is
continuous. By [9, 2.1],  p!  ! is a subsemigroup under  , so f restricts to a function
f :  p!! p!. If C  p! is closed and (xn) is a sequence in f 1(C) then f(p-limxn) = p-
limf(xn)2C since C is p-compact. Thus, since  p! is p-sequential, f 1(C) is closed and so
f is still continuous in the new topology on  p!. For each m2!, fm( p!) = sm  p! pm is
p-compact and thus closed in  p! since for each m2!, p-limsm rn pm = p-limfm(rn) =
fm(p-limrn)2fm( p!). Thus
 p! s  p! p s2  p! p2 :::
is a countable descending chain of closed subsets of a countably compact space. Thus by
2.42 there is some q2\sn  p! pn. Such a q has the property given in the statement.
We?ve seen that TpX and  pX de ne the same topological spaces when X is a discrete
space, but  pX can be de ned for any arbitrary space X. This gives us more  exibility and
allows us to construct examples that could not be constructed using the monads Tp alone.
Fix a free ultra lter p2! .
Proposition 5.6. For any set X, TpX contains no non-trivial convergent sequences.
28
Proof. Let X be a set. This proposition can be seen from the de nition of TpX, but in light
of 4.18, it is easier to see that  pX contains no non-trivial convergent sequences when X is
considered as a discrete space: since the topology on  pX is  ner than the standard subspace
topology, the inclusion map i :  pX ! X is continuous. If a sequence (xn) converges in
 pX, then (i(xn)) converges in  X by 2.5 and so i(xn) is eventually constant by 2.23.
Proposition 5.7. There is a space X so that UXnX contains a non-trivial convergent
sequence.
Proof. By [10], there is a completely regular space X so that X contains a non-trivial
convergent sequence. Let xn!x be that sequence. We claim that xn!x in UX as well.
For any p2! , p-limxn exists because UX is p-compact. If p-limxn = y6= x then let Ux
and Uy be disjoint open sets in  X around x and y respectively. Since xn!x,fnjxn2Uyg
is  nite and thus fnjxn2Uyg62p by 2.26 since p2! .
Suppose (xn) does not converge to x in UX. Then there is some open O UX so that
A =fnjxn62Ogis in nite. Fix an ultra lter p2! with A2p. Fix z62O and let x0n = xn
if xn62O and x0n = z otherwise. Then by 2.34, p-lim Xxn = p-lim Xx0n62O since  XnO is
ultracompact. In particular p-lim Xxn6= x, contradicting the fact that xn!x in  X.
Proposition 5.8. For any set X, TpX is extremally disconnected.
Proof. Let X be a discrete space. It su ces to show that  pX is extremally disconnected.
Let  be the standard topology on  X. Suppose B;C X with B\C =;and B[C = X.
We claim that cl p(B)\cl p(C) =; and cl p(B)[cl p(C) =  p!.
First, since  p is  ner than  , cl p(B) cl (B). Thus cl p(B)\cl p  cl (B)\cl (C) =;
by 2.20.
To prove the second assertion, recall that by 2.47  pX = AY!1(fpg;X) where Y is the
space ( X; p). So it su ces to show that A (fpg;X)  A (fpg;B)[A (fpg;C) for all
 < !1 (the reverse inclusion is clear). Note that A0(fpg;X) = A0(fpg;B)[A0(fpg;C).
If A (fpg;X)  A (fpg;X) [A (fpg;X) for all  <  and  is a limit ordinal, then
29
A (fpg;X) A (fpg;X)[A (fpg;X) trivially. On the other hand, if  =  + 1 for some
 and if a2A (fpg;X) then a = p-limxn for some sequence (xn) in A (fpg;X). By the
induction hypothesis, (xn) is a sequence in A (fpg;B)[A (fpg;C). Let B0 = fnjxn 2
A (fpg;B)g and C0 = fnjxn 2A (fpg;C)g. B0[C0 = X, so by 2.11.3 we may assume
without loss of generality that B02p . Fix z2A (fpg;B) and de ne x0n = xn if n2B0 and
x0n = z otherwise. Then (x0n) is a sequence in A (fpg;B) and so p-limx0n2A (fpg;B). But
since fnjxn = x0ng2p, by 2.34 a = p-limxn = p-limx0n2A (fpg;B) as required.
Now suppose U  pX is open. Then since X is dense in  pX, cl p(U) = cl p(U\X).
By the above, cl p(U\X) =  pXncl p(XnU), so cl p(U) is open and  pX is extremally
disconnected.
Proposition 5.9. For any connected, completely regular, Hausdor space X,  pX is con-
nected.
Proof. Suppose X satis es the hypotheses. Since X is connected, clY (X) is connected for
any space Y  X [5].  pX = clY (X) where Y is the space ( X; p) for  the standard
topology on  X.
Thus this new topological method allows us to describe a signi cantly larger class of
examples.
30
Bibliography
[1] E. Manes, Monads in topology, Topology Appl. 157 (2010), 961-989.
[2] P. Nyikos, Classic problems - 25 years later (part 2), Topology Proc. 27 (2003), 365-378.
[3] P. Nyikos and J. Vaughn, The Scarborough-Stone problem for Hausdor spaces, Topol-
ogy Appl. 44 (1992), 309-316.
[4] A. Dow, A countably compact, countably tight, non-sequential space, Proceedings of the
1988 Northeast Conference on General Topology and Applications (1990), 71-80.
[5] A. Munkres, Topology, Prentice Hall, 1975.
[6] A. Kombarov, Compactness and sequentiallity with respect to a set of ultra lters,
Moscow Univ. Math. Bull. 40 (1985), 15-18.
[7] J. van Mill, An introduction to  !, Handbook of Set-Theoretic Topology 1984, 503-567.
[8] S. Mac Lane, Categories for the working mathematician, Springer-Verlag, 1971.
[9] S. Garcia-Ferreira, N. Hindman, D. Strauss, Orderings of the Stone-  Cech remainder of
a discrete semigroup, Topology Appl. 97 (1999) 127-148.
[10] R. Fri c and P. Vojt a s, Convergent Sequences in  X, Proceedings of the 11th Winter
School on Abstract Analysis, Circolo Matematico di Palermo (1984), 133-137.
31