Applications of Stationary Sets in Set Theoretic Topology
by
Steven Clontz
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
December 13, 2010
Keywords: topology, set-theoretic, stationary, linearly ordered topological spaces, ordinals
Copyright 2010 by Steven Clontz
Approved by
Gary Gruenhage, Chair, Professor of Mathematics
Michel Smith, Professor of Mathematics
Stewart Baldwin, Professor of Mathematics
Andras Bezdek, Professor of Mathematics
Abstract
The notion of a stationary subset of a regular cardinal, a set which intersects any
closed unbounded subset of that cardinal, is a useful tool in investigating certain properties
of topological spaces. In this paper we utilize stationary sets to achieve an interesting
characterization of paracompactness of a linearly ordered topological space. We also use
stationary sets to nd a pair of Baire spaces whose product is not Baire.
ii
Acknowledgments
Thanks rst must go to my adviser, Dr. Gary Gruenhage. I feel extremely lucky to
have an adviser who is an extremely talented adviser, teacher and mentor. His insight
and, perhaps more importantly, patience have been invaluable in guiding this thesis. In
addition, I want to thank the rest of my committee: Dr. Michel Smith, whose guidance
during my undergraduate career as part of the Mathematics Club encouraged me to study
mathematics further; Dr. Stewart Baldwin, whose introductory graduate course in topology
cemented my interest in the eld; and Dr. Andras Bezdek, whose mentorship towards the
Putnam Undergraduate Mathematics Competition rst inspired my love of higher math, and
whose guidance towards the writing of my undergraduate honors thesis was invaluable.
I would like to extend my heartfelt thanks towards the support of my parents, Craig
and Sherill Clontz. Their love and support for me over the years is what has kept me going
even in the most frustrating of times. I?d like to thank my siblings Phillip and Laura for
their support as well.
Finally, I?d like to thank my fellow colleagues, the \Math GTAs", and my girlfriend
Jessica Stuckey. Your collective support, advice, and needed distractions have all been
welcome and appreciated.
iii
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 !1, the First Uncountable Ordinal . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Generalizing to Any Regular Cardinal . . . . . . . . . . . . . . . . . . . . . . 17
5 Some Theorems on the Compactness of Linearly Ordered Topological Spaces . . 23
6 A Characterization of the Paracompactness of a Linearly Ordered Topological
Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
7 Stationary Sets and the Baire Property . . . . . . . . . . . . . . . . . . . . . . . 33
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
iv
Chapter 1
Introduction
A stationary subset of a regular cardinal is de ned to be any subset of that cardinal
which intersects every closed and unbounded subset of that cardinal. Stationary sets are
useful tools in investigating properties of linearly ordered topological spaces. An example of
this is the following theorem characterizing paracompactness in a linearly ordered topological
space (often abbreviated LOTS).
Theorem. A linearly ordered topological space X is paracompact i X does not contain a
closed subspace homeomorphic to a stationary subset of a regular uncountable cardinal.
This result was rst discovered by Engelking and Lutzer in 1977 [3], for a larger class
of \generalized ordered spaces", spaces which are a subspace of another linearly ordered
topological space. However, as we will see, the proof developed in his paper works in a way
that could easily be extended to cover these generalized spaces as well.
Of course, stationary sets can be used in scenarios beyond LOTS. In perhaps a more
surprising application, stationary subsets of !1 can also be used to construct a pair of non-
linearly ordered sets which are Baire, but whose product is not Baire.
Theorem. There are metrizable Baire spaces X and Y such that X Y is not Baire.
The rst discovery of this result is due to Fleissner and Kunen in 1978 [4].
The goal of this paper is to develop the tools necessary for the proof of both theorems.
In the next chapter, we will outline the basics of topology and relevant basic results in the
eld, in order to accommodate the reader and serve as a reference. Readers familiar with the
eld may begin at Chapter 3, wherein the basic de nitions and results concerning stationary
subsets of the rst uncountable ordinal !1. In Chapter 4, these results are generalized to
1
any regular cardinal by introducing some set-theoretic results. In Chapter 5, we investigate
the compactness of linearly ordered topological spaces, and in Chapter 6 we use the results
developed thus far to prove the above theorem on paracompactness. Chapter 7 changes pace
and quickly develops the tools to prove the second theorem concerning the Baire property.
It should be noted that all the theorems and lemmas in this paper were taken from
the class notes of Dr. Gary Gruenhage?s set-theoretic topology course held in Fall 2008 and
Spring 2009 at Auburn University, which the author regrettably was unable to take part in.
The proofs to these, however, are all due to the author, with no reference to the original
proofs of these results, but with the obvious assistance of Dr. Gruenhage. The basics covered
in Chapter 2 are based on the second edition of Topology by James R. Munkres [1] and Set
Theory: An Introduction to Independence Proofs by Kenneth Kunen [2].
2
Chapter 2
The Basics
To begin, let?s de ne the basic concept of a topological space.
De nition 2.1. A topology on a set X is a collection P(X) having the following
properties:
1. ;;X2 .
2. If U , then
[
U2 .
3. If U;V 2 , then U\V 2 .
(Or equivalently, any nite intersection of sets in is in .)
An ordered pair hX; i where is a topology on X is known as a topological space,
although we often refer to it as simply X when is known by context.
The basic concepts of topology are the notions of open and closed sets.
De nition 2.2. For a topological space hX; i, U X is said to be open in hX; i (or
simply open or open in X) if U2 .
K X is said to be closed in hX; i(or simply closed or closed in X) if XnK2 .
A set which is both closed and open is referred to as clopen.
Proposition 2.3. Any nite union of closed sets is closed, and any arbitrary intersection
of closed sets is closed.
The concept of a limit point is often used to identify closed sets.
De nition 2.4. x is a limit point of a set A in a topological space X if every open set
containing x intersects A at a point distinct from x.
3
Proposition 2.5. A set K in a topological space X is closed if and only if K contains all
its limit points.
Any set is contained within a minimal closed set, which we call its closure.
De nition 2.6. The closure A of a set A in a topological space X is the intersection of all
closed sets containing A.
Two basic topologies for any arbitrary set X are the discrete and indiscrete topologies.
De nition 2.7. The discrete topology on a set X is = P(X). The indiscrete (or
trivial) topology on a set X is =f;;Xg.
It is easily seen that these indeed satisfy the criteria in De nition 2.1.
If we have a topological space and wish to investigate a subset of that space, we may
easily apply a \subspace" topology.
De nition 2.8. Let hX; i be a topological space and Y X. Y = fU\Y : U 2 g is
known as the subspace topology on Y with respect to X.
Listing every open set in a topology can be tedious, so often topologies are described
using simpler collections called bases.
De nition 2.9. A basis on X is a collection B P(X) such that
1. For each x2X there is a basis element B2B with x2B.
2. If x2B1\B2 where B1;B2 2B, then there is a basis element B3 2B with x2B3
B1\B2.
The topology generated by B is
fU2P(X) : for all x2U there exists B2B such that x2B Ug:
4
It can be seen that any generated by a basis which satis es these requirements satis es
the requirements of a topology. A basis can be thought of as a collection of the basic elements
of a topology on X. Certainly, any basis element is an open set in the space. Indeed, it is a
fact that every open set in the generated topology is a union of basis elements.
Proposition 2.10. An open set of a topological space generated by the basis B is the union
of elements of B.
De nition 2.11. A local base at a point x in a topological space X is a collection of open
setsBx, each of which contains x, such that for every open set U containing x, there is some
B2Bx with x2B U.
Another example of a topological space is a product space of various topological spaces.
De nition 2.12. The Cartesian product
Y
i2I
Xi is the set of functions f : I !
[
i2I
Xi
where f(i)2Xi.
In the case that there is some X with Xi = X for all i2I, the cartesian product is
often written as XI.
De nition 2.13. The product topology on the cartesian product of topological spaces
Y
i2I
Xi is the topology generated by the basis of sets of the form
Y
i2I
Ui, where Ui is open in
Xi for all i2I, and Ui = Xi for all but nite i2I.
Certainly, the topological spaces X = f0;1;2g and Y = fa;b;cg with topologies X =
f;;f0g;Xgand Y =f;;fag;Xghave no interesting di erences other than the labels we give
the elements. We use the concept of a homeomorphism to link two topologically equivalent
spaces.
De nition 2.14. If f : X !Y is a function, S X, and T Y, then f00(S) = ff(s) 2
Y : s2Sg is the image of S under f, and f 1(T) = fs2X : f(s) 2Tg is the inverse
image of T under f.
5
De nition 2.15. For two topological spaces X;Y, the function h : X !Y is a homeo-
morphism if h is a bijection and for every set U open in X and V open in Y, f00(U) is open
in Y (making it an open map) and f 1(V) is open in X (making it continuous).
Proposition 2.16. A map with the property that for every point y in its range and open set
V containing y, there is an open set U in the domain with f00(U) V, is continuous.
De nition 2.17. Two topological spaces X;Y are said to be homeomorphic if there exists
a homeomorphism h : X!Y. We write X = Y.
Following are some properties of various topological spaces which will be referenced or
investigated in this paper.
De nition 2.18. A topological space X is said to be T3 if for every point x2X, fxg is
closed, and for every open set U containing x, there is another open set V with x V
V U.
De nition 2.19. An open cover U of a topological space X is a collection of open sets
such that SU = X.
De nition 2.20. A topological space X is said to be compact if for every open coverU of
X, there exists some nite subcover U U.
De nition 2.21. A re nement V of U is a set such that for every V 2V, there is some
U2U with V U.
De nition 2.22. A collection of setsAin a topological space X is said to be locally nite
if for every point x2X and there exists some open set U containing x such that U intersects
only nitely many members of A.
De nition 2.23. A topological space X is said to be paracompact if for every open cover
U of X there exists some re nementV ofU such thatV is an open cover of X and is locally
nite.
6
Proposition 2.24. Any closed subspace of a paracompact space is paracompact.
De nition 2.25. A topological space X is said to be connected if there does not exist a
nonempty proper clopen subset A X.
De nition 2.26. A metric on a space X is a function d : X X!R where:
1. d(x;y) = 0,x = y
2. d(x;y) 0 for all x;y2X
3. d(x;y) = d(y;x) for all x;y2X
4. d(x;z) d(x;y) +d(y;z) for all x;y;z2X
An open ball of radius r with respect to a metric d, written Br(x), is the setfy : d(x;y) <
rg.
De nition 2.27. A topological space X is said to be metrizable if there exists a metric d
such that fBr(x) : x2X and r> 0g forms a basis generating the topology on X.
Proposition 2.28. Any subspace of a metrizable space is metrizable.
De nition 2.29. A collection of sets A is said to be -locally nite if A=
[
n2N
An where
An is locally nite for all n2N.
The following two results are not trivial, but are basic results from introductory topology
needed in this paper.
Proposition 2.30. Every metrizable space is paracompact.
Proposition 2.31. Every space which is T3 and has a -locally nite basis is metrizable.
The following de nitions are needed in the nal chapter.
De nition 2.32. A subset A of a topological space X is said to be dense in the space if it
intersects every open set in the space.
7
Proposition 2.33. A subset A of a topological space X is dense in the space if and only if
A = X.
De nition 2.34. A topological space X is said to be Baire if every countable intersection
of dense open sets in the space is dense.
We now turn our attention to linearly ordered sets. If a set has a linear order on it,
there is a natural topology which arises from this order.
De nition 2.35. A relation < on X is called a linear order on X if it has the following
properties for all a;b;c2X:
Either a = b, a** in
TC.
Theorem 3.3. For any closed unbounded subset C of !1, there is a strictly increasing home-
omorphism from C to !1.
Proof. De ne f : !1 !C such that
f(0) = min(C)
12
f( ) = min(Cnf00( )).
We rst claim that f is an order isomorphism. It?s obviously order-preserving (and thus
injective). To see that it is onto, suppose by way of contradiction that there is some 2C
such that 62ran(f). By the de nition of f, f( ) < for all + 1. Inductively, it
is easily seen that as gn( + 1) 62C, gn+1( + 1) > gn( + 1). This is a strictly increasing
sequence, so it converges to a limit ordinal .
Consider any < . There is a minimum gn( + 1) such that , we know gf( ) maps onto . Thus our g is an onto function from a set
of cardinality to +, a contradiction.
(iii) Let be a limit ordinal, and = cf( ). There is a strictly increasing functionf : !
which is unbounded in .
Suppose by way of contradiction that there exists an unbounded strictly increasing
function g : ! for < . Composing f and g gives us an unbounded increasing
function f g : ! , a contradiction of the de nition of as the least cardinal which
has an unbounded map onto . Thus cf( ) = and cf( ) is regular.
As is shown in the following theorem, regular cardinals have many of the properties
commonly associated with \uncountable" versus \countable" sets.
Theorem 4.5. For an in nite cardinal , the following are equivalent:
(i) is regular.
(ii) For any A , if jAj< , then sup(A) < .
(iii) The union of < -many sets, each of cardinality < , has cardinality < .
Proof.
(i) ) (ii) (Shown by contrapositive.) Let A such that jAj= A < and sup(A) = . Then
i : A ! where i is the inclusion map (i( ) = ) has an unbounded range. Let
: A !A be a bijection. Then i : A ! has an unbounded range, showing
cf( ) A < and thus is not regular.
19
(ii) ) (i) Let < . Suppose by way of contradiction that there is a function f : ! whose
range was unbounded in , that is, sup(A) = for A =ff( ) : 2 g; contradiction.
Thus cf( ) = .
(i) ) (iii) Let < , and assume by induction that for < , the union of -many sets, each
of cardinality < , has cardinality < . Assume that for each < , U is a set with
jU j< . As is regular, it follows that the function f : ! where f( ) =
[
<
U
is bounded by some cardinal < . As
[
<
U =
[
<
[
<
U
!
, it follows that the
cardinality of the union of -many sets of cardinality must be of cardinality
max( ; ) < .
(iii) ) (i) Let < . Suppose by way of contradiction that there is a function f : ! whose
range is unbounded. Then
[
<
f( ) = , and the union of less than -many sets each
of cardinality less than has cardinality , a contradiction.
Now that we?ve established the rules for regular cardinals, we observe that they behave
in nice ways, that is, similar to the relationship between ! and !1. From this we can
generalize many of the theorems from the previous chapter by merely replacing !1 with any
regular cardinal , replacing ! with any cardinal < , assuming \uncountable" to mean
\of cardinality ", and assuming \countable" to mean \of cardinality < ".
Theorem 4.6. Let be an uncountable regular cardinal and < . Let C =fC : < g
be a collection of club sets in . Then TC is club.
Proof. See the proof of Theorem 3.2.
Theorem 4.7. Let be an uncountable regular cardinal. For any closed unbounded subset
C of , there is a strictly increasing homeomorphism from C to .
Proof. See the proof of Theorem 3.3.
20
Theorem 4.8. Let f : ! be a function and C =f : < )f( ) < g be a subset
of . Then C is closed and unbounded.
Proof. See the proof of Theorem 3.4.
Theorem 4.9 (Pressing Down Lemma Lite). Let S be a stationary set in . If for each
ordinal 2Snf0g, we choose an ordinal < , then there is some < such that =
for -many 2S.
Proof. See the proof of Theorem 3.6.
Theorem 4.10. If S is a stationary subset of a regular cardinal , then S is not paracompact.
Proof. See the proof of Theorem 3.7.
These nal results about regular cardinals, speci cally !1, will be needed in the nal
chapter.
Lemma 4.11. Let C;D be closed unbounded subsets of a regular cardinal , and : !D
be a strictly increasing homeomorphism. 00(C) is closed unbounded in .
Proof. Fix 2 . As D is unbounded, we may x 2 D such that > . There is a
2 such that ( ) = . As C is unbounded, we may x 2C such that > , yielding
( ) > ( ) = > . Thus 00(C) is unbounded in .
As is a homeomorphism, 00(C) is a closed subset of D. This means there is a closed
set E of such that E\D = 00(C). As 00(C) is the intersection of closed sets, 00(C) is
closed.
Lemma 4.12. There are two disjoint stationary subsets of !1.
Proof. Suppose not. Let Q be the set of rational numbers. Let f : !1 ! (0;1)nQ be
injective. (0; 12) and (12;1) cannot both contain the image of stationary sets in !1, so there is
a closed unbounded set C1 of !1 that maps by f into an open interval of irrational numbers
of length 12.
21
Now assume we have, by way of induction, a chain of club sets C1 ::: Cn such
that Ci maps by f into an open interval of irrational numbers with rational endpoints of
length 12i. Let h : !1 !Cn be a strictly increasing homeomorphism and g = f h. Assume
f00(Cn) (a;b)nQ where a;b 2 Q and b a = 12n. g : !1 ! (a;b)nQ is injective.
(a;a+b2 ) and (a+b2 ;b) cannot both contain the image of stationary sets in !1, so there is a
closed unbounded set C of !1 that maps by g to either (a;a+b2 ) and (a+b2 ;b), which are
both have rational endpoints and are of length 12n+1 . It then follows by Lemma 4.11 that
Cn+1 = h00(C ) Cn is also a club set, and f00(Cn+1) = f00(h00(C )) = g00(C ) is a subset of
an open interval of irrational numbers with rational endpoints of length 12n+1 .
We then observe that for all ny is), so it must not lie
in T. This mean that there is some sy 2S such that sy >y. But that contradicts the fact
that y was an upper bound for S, which proves that every subset of X must have a least
upper bound.
23
To see that every subset also must have a greatest lower bound, simply reverse all the
orders in the above argument. Alternately, note that we merely need to consider a \mirrored"
linear space XM =fxM : x2Xg with the order xM y. There is an obvious
homeomorphism between the two spaces, so XM is also compact. Any subset without a
greatest lower bound in X would yield a re ection in XM without a least upper bound, a
contradiction. This nishes the forward implication.
Now conversely, assume that every subset of X has a least upper bound and greatest
lower bound. This means that X has a greatest lower bound, and minumum element, a and
a least upper bound, and maximum element, b. Let U = f(a ;b ) : < g be an open
interval cover of X for some cardinal . Let S =fs2X : there is a nite subcover of U for
[a;s]g. S is nonempty as there is a nite subcover of U for [a;a] =fag, placing a2S. We
note sup(S)2S as there is an interval (a 0;b 0)2U containing sup(S), and a nite subcover
of U covering [a;a 0] (since a 0 < sup(S))a 0 2S), so by combining them we nd a nite
subcover of U covering [a;sup(S)].
We claim that sup(S) = b. To see this, assume by way of contradiction that sup(S) sup(S) in S and yields our
contradiction.
As sup(S) = b, there exists a nite subcover of U for [a;b] = X, demonstrating the
compactness of X.
24
Even if a linearly ordered set is not compact we can easily compactify it, that is, embed
it densely in a compact linearly ordered topological space.
Theorem 5.2. Every linearly ordered topological space X is a dense subset of a compact
linearly ordered topological space ^X .
Proof. We de ne ^X as such. ^X P(X) where A2 ^X i all of the following holds:
1. A is closed in X
2. a2A and b a. It then follows that as a < x, a2Ax. This
means A Ax, and as x2AxnA , we see that A Ax, which places Ax 2 (A ;A+).
This makes the closure of fAx : x2Xg to be ^X, and fAx : x2Xg is dense.
To conclude, we show that ^X is compact. Let ^S be a nonempty subset of ^X. We note
that \^S is a closed set in X and that a2\^S and b < a implies a2S for all S2 ^S, and
25
thus b2S for all S2 ^S yielding b2\^S, and thus \^S2 ^X. Also, if S2 ^S, then certainly
\^S S, so \^S is a lower bound of ^S. For any lower bound T of ^S, we note that t2T
implies t2S for any S2 ^S, and thus t2\^S, which shows T \^S. Thus\^S is the greatest
lower bound of ^S.
Now we note that [^S is a closed set in X and that a2[^S and b b in some S2 ^S, which then implies that
b2S [^S [^S.
So we have that [^S2 ^X.
Certainly [^S is a superset of all S 2 ^S, so [^S is an upper bound of ^S. And if T is
any upper bound of ^S, then it is closed in X and is a superset of [^S, and as [^S is the
intersection of all such sets, we see that [^S T and [^S is the least upper bound of ^S.
As any arbitrary ^S ^X has both a least upper bound and greatest lower bound, we
know that ^X is compact, nishing the proof.
It?s often a useful trick to compactify a space in order to gain some extra structure, as
we will see in a later proof.
26
Chapter 6
A Characterization of the Paracompactness of a Linearly Ordered Topological Space
We?re about ready to tackle the rst main result of this paper. First, we introduce
another sense of connectedness in the sense of a collection of subsets of a topological space.
De nition 6.1. For a topological space X, a collectionU P(X), and two points a;b2X,
the nite sequence hU0;:::;Un 1i of sets in U is called a nite linked chain joining a;b if
a2U0, b2Un 1, and for all 0 i!. Let ^Y be the compacti cation of Y
from Theorem 5.2. ^Y adds a greatest element p not in Y since Y had uncountable co nality.
Lemma 6.6 gives us that ^Y nfpg contains some closed co nal subset K homeomorphic to
the co nality 0 of ^Y nfpg.
We note that if f : 0! ^Ynfpgis a co nal map, then the density of Y in ^Ynfpggives
a point of Y in the open set (f( );f( + 1)) in ^Y, so the co nality of Y is 0. Of course,
if g : !Y is co nal, then its inclusion map ig : ! ^Y nfpg is co nal in ^Y, so 0 .
Thus 0 = .
31
Now observe that K\Y cannot be a stationary subset of K by our assumption that
Y does not contain a closed subset homeomorphic to a stationary subset of any regular
uncountable cardinal such as . This gives us a subset of KnY which is closed and unbounded
in K, call it ^K. We may assume without loss of generality that all elements of ^K are greater
than x0 as any nal interval of a closed unbounded set is also a closed unbounded set.
^K is homeomorphic to , so suppose : ! ^K is a homeomorphism. We may rst
construct a nite re nement of [U] on [x0; (0))\Y by using any progressive nite linked
chain connecting x0 to any point of Y greater than (0), with each element intersected with
the open set ( ; (0))\Y in the subspace Y of ^Y (dropping any empty sets). Similarly, for
each < , we can similarly construct a nite re nement of [U] for ( ( ); ( +1)) by using
any progressive nite linked chain connecting x0 to any point of Y greater than ( + 1),
with each element of the chain intersected with ( ( ); ( + 1))\Y. The union of these is a
locally nite re nement of [U] covering [x0;!)\Y. Of course we?ll call this re nementY!
as well.
We note that Y! is locally nite at x0, so only nite elements of it can extend left of
x0. We can then use similar arguments to generate a locally nite re nement Y which
covers ( ;x0]\Y and only has nitely many elements which extend right of x0. Lastly,
Y =Y [Y! is then a locally nite re nement of [U] covering Y, nishing the proof.
32
Chapter 7
Stationary Sets and the Baire Property
Slightly changing pace, we shall investigate how we may use stationary subsets of !1
to construct two Baire spaces whose product is not Baire. It should be noted that in this
chapter we assume !1 has the discrete topology.
De nition 7.1. For 2! 2nand _n 2n n; n is an initial restriction ofg of domain
> 2n. Thus f(n) = ( _n 2n n; n)(n) max( n; n) and g(n) = ( _n 2n n; n)(n) max( n; n).
This tells us that f = sup(fmax( n; n)jn < !g) = g , giving that hf;gi cannot be in
A B as A;B are disjoint.
We have thus observed a product which is not Baire, regardless of whether or not its
component spaces are Baire. We proceed to show that, indeed, we may nd two uncountable
subsets A;B of !1 such that A ;B are Baire, by utilizing stationary sets.
34
Lemma 7.6. Let U be dense open in !!1 . For each 2 !**