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 !, 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
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