The Stone- Cech Compacti cation and the Number of Pairwise
Nonhomeomorphic Subcontinua of [0;1)n[0;1)
by
David Lipham
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
May 4, 2014
Keywords: Stone- Cech, Subcontinua, Ultrapower
Copyright 2014 by David Lipham
Approved by
Michel Smith, Chair, Professor of Mathematics
Stewart Baldwin, Professor of Mathematics
Gary Gruenhage, Professor of Mathematics
Piotr Minc, Professor of Mathematics
Abstract
An introduction to the Stone- Cech compacti cation X of a normal topological space X
is given. The method of invariantly embedding linear orders into ultrapowers is used to nd
2c pairwise nonhomeomorphic continua in R, under the assumption that the Continuum
Hypothesis fails.
ii
Acknowledgments
Thanks rst and foremost to my adviser, Dr. Michel Smith, for his invaluable guidance
and patience. Thanks to Dr. Stewart Baldwin and Dr. Piotr Minc for their inspiring
graduate courses in Set Theory and Topology, respectively, and to Dr. Gary Gruenhage for
letting me present some of this work in his seminar. I would also like to thank Dr. Baldwin
for his help clarifying many of the combinatorial arguments needed in the nal chapter.
iii
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Topology and Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 The Stone- Cech Compacti cation . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1 Filters and Normal Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Construction of X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Properties and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4 Additional Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.5.1 ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.5.2 R and H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5.3 !1 and L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Ultrapowers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1 De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 CH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5 The Continua Iu (u2! ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.1 De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.2 Decompositions and the Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . 30
6 Constructing Ultra lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
iv
6.1 Invariant Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6.2 The linear orders J ( < 2c) . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.3 A Quick Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.4 The Proof of Theorem 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
v
List of Figures
5.1 Theorem 5.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.1 IX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.2 Theorem 6.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.3 Jv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.4 Theorem 6.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.5 [L1 ;L2 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
vi
Chapter 1
Introduction
A continuum is a compact and connected topological space. The Stone- Cech remainder
HnH of the half-line H = [0;1) is a continuum, and every free ultra lter u on ! generates
a subcontinuum Iu of HnH. In [1], A. Dow nds a family of 2c free ultra lters on ! such
that the corresponding Iu?s are pairwise nonhomeomorphic. Thus, he proves the following.
Theorem 1.1 (: CH). There exist 2c pairwise nonhomeomorphic subcontinua of HnH.
The main result of [1] is achieved by rst noting that each Iu is closely related to
the linearly ordered ultrapower R!=u. The following theorem says that in order to nd 2c
pairwise nonhomeomorphic Iu?s, it su ces to nd 2c pairwise nonisomorphic completions
R!=u of ultrapowers R!=u.
Theorem 1.2. If u and v are free ultra lters on ! and Iu?Iv, then R!=u R!=v.
Finding 2c nonisomorphic R!=u?s is no trivial matter. It was rst established in [9] that
all ultrapowers R!=u are isomorphic if CH holds. Prior to [3], only c were known to exist
when CH fails. The authors of [3] indeed show that there are 2c nonisomorphic ultrapowers
when CH fails. A. Dow was able to modify some of their arguments to prove the following.
Theorem 1.3 (: CH). There exists a family fD : < 2cg of free ultra lters on ! such
that R!=D 6 R!=D for any < < 2c.
The goal of this paper is to develop the tools needed for proving the theorems stated
above. In the next chapter, we will state some relevant de nitions and theorems from
introductory topology and set theory. The Stone- Cech compacti cation is the subject of
Chapter 3, wherein we prove existence and uniqueness results and look at some examples.
1
In Chapter 4 we introduce the ultrapowers R!=u and prove the aforementioned CH result.
In Chapter 5 we examine the spaces Iu and prove Theorem 1.2. In Chapter 6 we work
through a series of results from [3] which lead to the proof of Theorem 1.3. As indicated
above, Theorems 1.2 and 1.3 yield a proof of Theorem 1.1. We o er an alternative proof of
Theorem 1.1 in Chapter 6 as well - one that does not require the full strength of Theorem
1.3.
2
Chapter 2
Topology and Set Theory
2.1 Topology
We refer the reader to Topology by J. Munkres [6] for the basics.
Theorem 2.1. Every closed subspace of a compact space is compact.
Theorem 2.2. Every compact subspace of a Hausdor space is closed.
Theorem 2.3. The continuous image of a compact space is compact.
Theorem 2.4. A continuous bijection f : X!Y is a homeomorphism if X is compact and
Y is Hausdor .
Suppose f : X !Y is a function. For A2P(X), let f(A) = ff(x) : x2Ag denote
the image of A under f. For B 2P(Y), let f 1(B) = fx 2 X : f(x) 2 Bg denote the
inverse image of B under f. For added clarity we will sometimes write f[A] instead of f(A)
(respectively, f 1[B] instead of f 1(B)). If f is invertible, f 1 may also denote the inverse
of f.
Theorem 2.5. f is continuous i f(clXA) clYf(A) for all A2P(X).
A collection C of subsets of X is said to have the nite intersection property if every
nite subcollection of C has nonempty intersection.
Theorem 2.6. X is compact i for every collection C of closed sets in X having the nite
intersection property, the intersection TC2CC of elements of C is nonempty.
Theorem 2.7 (Tychono ?s Theorem). A product of compact spaces is compact in the product
topology.
3
A compacti cation of X is a compact Hausdor space containing a dense copy of X.
Every locally compact Hausdor space has a one point compacti cation, de ned to be the
set X = X[f1g with the following topology: U X is open if
(i) U is open in X, or
(ii) 12U and XnU is compact in X.
Theorem 2.8. If A is a connected subspace of X then the closure of A in X is connected.
A continuum is a compact and connected topological space.
Theorem 2.9. The intersection of a family of continua with the nite intersection property
is a continuum. That is, if C is a collection of compact connected subspaces of X with the
nite intersection property, then TC2CC is compact and connected (as a subspace of X).
Theorem 2.10. Every metrizable space is normal.
Theorem 2.11. Every compact Hausdor space is normal.
Theorem 2.12. Every F subspace of a normal space is normal.
Proof. Suppose L is a countable union of closed sets in X, and A and B are relatively closed
disjoint subsets of L. We may write A and B as countable unions of closed subsets of X;
A = Si2!Ai and B = Si2!Bi.
Separate A0 and B0 with disjoint open sets U0 and V0 such that clXU0 \B = ? =
A\clXV0. (To see that the last condition may be met, note, for instance, that B = F\L
for some closed F X such that F\A0 = ?.)
Separate A1 and B1 with disjoint open sets U0 and V0 such that clXU0\B = ? =
A\clXV0. Let U00 = U0\(XnclXV0) and V00 = V0\(XnclXU1). These disjoint open
sets still contain A1 and B1, respectively. Now let U1 = U00[U0 and V1 = V00[V0. We
have disjoint open sets containing A1 and B1, clXU1\B = ? = A\clXV1, and U0 U1
and V0 V1. Continuing in this manner, Si2!Ui and Si2!Vi will be disjoint open sets in
X containing A and B, respectively.
4
Theorem 2.13 (Urysohn?s Lemma). If X is normal and A and B are disjoint closed subsets
of X, then there exists a continuous f : X![0;1] such that f(A) =f0gand f(B) =f1g
Theorem 2.14 (Tietze?s Extension Theorem). If X is normal and A is a closed subspace
of X, then any continuous map from A into R may be extended to a continuous map from
X into R.
Let C(X) be the ring of continuous real-valued functions on X. Let C (X) be the
subring consisting of the bounded members of C(X). Suppose A;B;Z X. A is C -
embedded if every function in C (A) can be extended to a function in C(X). E.g., Tietze?s
Extension Theorem says closed subsets of normal spaces are C -embedded. A and B are
completely separated if there exists f2C(X) such that f(A) =f0g and f(B) =f1g. E.g.,
Urysohn?s Lemma says disjoint closed sets in a normal space are completely separated. X
is completely regular if for each closed set A X and p2XnA there exists f2C(X) with
f(A) = f0g and f(p) = 1. If X is completely regular and T1, then distinct singletons are
completely separated. By Urysohn?s Lemma, every T1 normal space is completely regular. Z
is a zero-set if there is a continuous function f : X!R with Z = f 1(f0g). If f : X!R,
we let Z(f) = f 1(f0g) denote the zero set of f. Let Z(X) denote the collection of all zero
sets of X.
Theorem 2.15. Two subsets of X are completely separated i they are contained in disjoint
zero sets.
Proof. Suppose A and B are completely separated by f, with A f 1(f0g) and B
f 1(f1g). Then A Z(f), B Z(f 1), and Z(f)\Z(f 1) = ?. Conversely, suppose
A and B are contained in disjoint zero sets. Let f1;f2 2C(X) s.t. Z(f1) A, Z(f2) B,
and Z(f1)\Z(f2) = ?. De ne f = jf1jjf1j+jf2j. Then f is continuous, f 1(f0g) = Z(f1) A,
and f 1(f1g) = Z(f2) B.
Theorem 2.16. In a metric space, closed sets and zero sets are equivalent.
5
A linear order is a pair (L;<), where L is a set and < is a binary relation on L such
that for all a;b;c2L:
(i) Either a = b, a**!1, such that
supi!1. By Theorems 2.23 and 2.24,
Y
i q. Letting = supq2Q\[0;1) q and q2Q\[0;1). Since Q\[0;1) is dense in [0;1),
we have f(f g [0;1)) = f(f g [0;1)) for all ; > . Suppose for a contradiction that
f is not constant on S > f g [0;1). Since f g [0;1) is connected and f is continuous,
f(f g [0;1)) is connected in R. So there exists a nonempty interval (a;b) R such that
(a;b) f(f g [0;1)) for all > . Asj(a;b)j !1, there existsfr : < leth ;x i2f g [0;1) such that
f(h ;x i) = r . Then f is not eventually constant onfh ;x i: > g, a contradiction.
Corollary 3.38. L = L.
24
Chapter 4
Ultrapowers
4.1 De nition
Suppose X is a set and u is an ultra lter on !. De ne a relation on X! by
f g,fn2! : f(n) = g(n)g2u:
It is easily checked that is an equivalence relation: symmetry is obvious, re exivity
follows from the fact that X 2 u, and transitivity follows from the fact that u is closed
under nite intersections and supersets. The ultrapower X!=u is the set of corresponding
equivalence classes f=u. If X is a linearly ordered set then we may de ne a relation on X!=u
by
f=u
g(n)g is in u. By Theorem 3.3, fn2! : f(n) g(n)g2u, so
that f=uf=u.
25
We will be primarily interested in the case X = R. We may view R as a linearly ordered
subset of R!=u by identifying c 2R with the equivalence class f=u, where f is given by
f(n) = c for all n2!. Under this identi cation, R!=u = R when u is a principal ultra lter.
If u is free then R!=u properly contains R, and is sometimes called hyper-real. In Chapter
6 we will assume:CH and nd 2c free ultra lters u2! such that the corresponding R!=u
are pairwise nonisomorphic.
4.2 CH
A dense linear order L is countably saturated if for any countable subsets A and B with
A****! are regular cardinals. (A;B) is a ( ; )-cut of L if A****><
>>:
jik if B\Ai = Cik
0 otherwise
:
Claim ffB : B !g is as desired. Let fBk : k! and Sl2!Gl = !, there exist r2I1 and s2I2 such
that g 1(l) is supported by FIs(fft : t2In[r;s]g) mod D for all l2!. Thus Cg;r;s2u, so
either g=u
>>>
>>>>
>><
>>>>
>>>
>>>
:
1 if (i) and h (ft (k)) l h (ft (k))
2 if (i) and lh (ft (k))
4 if (ii)
:
De ne : !(T) !f1;2;3;4gf1;:::;bg by (h ) = (?1(h );:::;?b(h )). Then is an a-place
function on ! with nite range. By Ramsey?s Theorem there exists an in nite W !
such that W(T) is constant, i.e., such that ?k W(T) is constant for each k b. Let
h 2W(T). For any m we have
Ahm[h Ah
\
i l1 or l1 < h00(ft (k)). Again,
?k(h00) = 1. Let l2 2! satisfying (i), such that h00(ft (k)) l2 h00(ft (k)). We have l1 6= l2,
41
Ahm[h0 Tk g 1k (l1) mod D, Ahm[h00jTk g 1k (l2) mod D. But Ahm[h0 Tk = Ahm[h00 Tk. Con-
tradiction.
Case 2: ?k(h ) = 2. Let l2! such that Ahm[h Tk g 1k (l) mod D for m su ciently large
and l!1 is a regular cardinal, then there exists a set fI : < 2 g of linear
orders satisfying
(i) cf(I ) =jI j= .
(ii) If 6= and ? : I !L, ? : I !L0 are co nal invariant embeddings, then L
and L0 have no isomorphic nal segments.
Proof. There exists a partition fS : < g of the stationary set S =f < : cf( ) = !1g
into pairwise disjoint stationary subsets (Theorem 2.28). Fix X . For each < ,
de ne
X =
8
>><
>>:
!1 if 2S 2XS
!2 otherwise
.
De ne IX = f( ; ) : < ; < X g. Give IX the lexicographic ordering, with the order
reversed in the second factor.
( X0 ;0]( X1 ;0] ( X ;0]
Figure 6.1: IX
Suppose X6= Y and ?X : IX !L, ?Y : IY !L0 are co nal invariant embeddings
into linear orders L;L0, respectively. For a contradiction, suppose : M ! M0 is an
43
isomorphism between nal segments M, M0 of L, L0, resp. For each < , let
M =fm2M : m!1 then there exists a set fJ : < 2 g of linear orders satisfying
(i) jJ j=
(ii) coi(J ) = cf( ) +!2
(iii) if 6= and ? : I !L, ? : I !L0 are coinitial invariant embeddings, then L
and L0 have no isomorphic initial segments.
Proof. If is regular, this follows from the previous theorem. Suppose is singular. By
Theorem 2.25 there exists a setf i : i< cf( )gof regular cardinals, each i >!1, such that
supi?u( ;x) for some < and x2Ih( );u(h( ))g
M0 =fm02M0 : m0>?v( ;x) for some < and x2Ih( );v(h( ))g:
There exists 2 S s.t. v(h( )) 6= u(h( )) (h S maps onto cf( )). Choose 2
Sh( )\f < : [M ] = M0 g with M 6= ?. Note that h( ) = h( ). Consider the cuts in L
and L0 below M and M0 respectively.
Figure 6.4: Theorem 6.8
Recall cf(Ih( ); ) = h( ) and cf( ) = !1 are regular uncountable. By the invariant
property of ?X and ?Y , M nM and M0nM0 have Ih( );v(h( )) and Ih( );u(h( )) co nally
invariantly embedded. But [M ] = M0 , so MnM M0nM0 . This contradicts a property
of the linear orders fIh( ); : < 2 h( )g.
46
We conclude this section by making a slight modi cation to Theorems 6.7 and 6.8.
Suppose L and L0 are linear orders. An order preserving map L ! L0 is an invariant-
1 embedding if the image of each ( ; )-cut of L is lled by precisely one element of L0.
Theorems 6.7 and 6.8 hold if we replace \invariant" with \invariant-1." For instance, if in
the proof of Theorem 6.7 there are unique l2L and l02L0 such that M ! are regular cardinals, A;B I and x2I. Then
(A;x;B) is a ( ;1; )-cut of I if A c, there exist 1 6= 2 2 E such that
(A 1;x 1;B 1) = (A 2;x 2;B 2). In particular B 1 = B 2. But ? 1 J 1 and ? 2 J 2
49
are coinitial invariant-1 embeddings of J 1 and J 2 into B 1 and B 2, respectively. This
contradicts a property of the orders J .
We may now recursively de ne X 2 ,jXj= 2 , such that !1+J admits an invariant-1
embedding intoR!=D but notR!=D for any < 2X. Thus there is no order preserving
isomorphism between R!=D and R!=D , for < 2X. For a xed 2X, there is at
most one 2X such that there exists an order reversing isomorphism between R!=D and
R!=D , so there exists S X, jSj = 2c, such that there is no order preserving or order
reversing isomorphism between R!=D and R!=D , for < 2S.
By Theorem 5.13, we have the following.
Theorem 6.11 (: CH). There exists a family fD : < 2cg of free ultra lters on ! such
that ID 6?ID for any < < 2c.
6.5 Concluding Remarks
In [10], A. Dow proves there are also 2c subcontinua of HnHwhen CH holds. Combined
with Theorem 1.1, we have the following theorem of ZFC.
Theorem 6.12. There exist 2c pairwise nonhomeomorphic subcontinua of HnH.
Prior to this result approximately 20 subcontinua were found in the ZFC setting, many
by M. Smith in [7]. In [7] it is also shown that the layers of Iu are indecomposable continua
unlike Iu. Note that each ( ; )-cut of R!=u corresponds to a layer in Iu. A. Dow indicates
in [10] that the following question remains open: If CH fails, are there 2c pairwise nonhome-
omorphic indecomposable subcontinua of HnH? In particular, it is not known if one can
produce 2c pairwise nonhomeomorphic layers by the method of invariantly embedding linear
orders into ultrapowers.
50
Bibliography
[1] Dow, Alan, Some set-theory, Stone- Cech, and F-spaces. Topology Appl. 158
(2011), no. 14, 1749{1755.
[2] K.P. Hart, The Cech-Stone compacti cation of the real line, in: Recent
Progress in General Topology, Prague, 1991, North-Holland, Amsterdam,
1992, pp. 317{352.
[3] L. Kramer, S. Shelah, K. Tent, S. Thomas, Asymptotic cones of nitely
presented groups, Adv. Math. 193 (1) (2005) 142{173.
[4] S. Shelah, Classi cation Theory and the Number of Non-isomorphic Models,
2nd Edition, North-Holland, Amsterdam, 1990.
[5] T. Jech, Set Theory. The Third Millennium Edition, revised and expanded,
Springer Monogr. Math., 2003, 3rd rev. ed., corr. 4th printing, XIV, 772 pp.
[6] J. Munkres, Topology, Prentice-Hall, Englewood Cli s, NJ. 1975.
[7] M. Smith, The subcontinua of [0;1) [0;1), Topology Proc. 11 (1986),
385-413.
[8] Walker R.C., The Stone- Cech Compacti cation, Springer, New York-Berlin,
1974.
[9] P. Erd os, L. Gillman, M. Henriksen, An isomorphism theorem for real-closed
elds, Ann. of Math. (2) 61 (1955) 542{554.
[10] Dow, Alan, On subcontinua and continuous images of RnR, preprint.
51
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