applications of reflection topology · abstract applications of reflection to topology doctor of...
TRANSCRIPT
Applications of Reflection to Topology
Renata Grunberg Almeida Prado
A thesis submitted in conformity with the requirements
for the degree of Doctor of Philosophy
Graduate Department of Mathematics
University of Toronto
@ Copyright by Renata Grunberg Almeida Prado 1999
National t i brary of Canada
Bibliotheque nationale du Canada
Acquisitions and Acquisitions et Bibliograph~c Services services bibliographiques 395 Wellington Street 395, rue Wellington Ottawa ON K1 A ON4 OttawaON K I A ON4 Canada Canada
The author has granted a non- L'auteur a accorde me licence non exclusive licence allowing the exclusive pennettant a la National Library of Canada to Bibliotheque nationale du Canada de reproduce, loan, dis~bute or sell reproduire, preter, distribuer ou copies of this thesis in microfom, vendre des copies de cette these sous paper or electronic formats. la forme de rnicrofiche/film, de
reproduction sur papier ou sur format electronique .
The author retains ownership of the L'auteur conserve la propriete du copyright in this thesis. Neither the droit d'auteur qui protege cette these. thesis nor substantial extracts from it Ni la these ni des exbcaits substantiels may be printed or otherwise de celle-ci ne doivent 6 e imprimes reproduced without the author's ou autrement reproduits sans son permission. autorisation.
Abstract Applications of Reflection to Topology
Doctor of Philosophy - 1999
Renata Grunberg Almeida Prado Graduate Department of Mathematics
University of Toronto
This work looks at reflection properties for topological spaces both in the large
ca,rdinal/forcing context and in the elementary submodel context. We present a general
technique valid for an interesting class of spaces (e.g. locally compact, Cech-complete,
first-countable) in both contexts and which allows us to reproduce results, present new
results and also to obtain interesting upwards and downwards reflection results and
examples.
To my loving family.
iii
Acknowledgments I would like to thank:
CNPq-Brasil/Brasilia for supplying the major financial support for this work.
The Department of Mathematics at the University of Toronto for all the support,
including partial financial support.
My friends and colleagues in Brazil and in Toronto.
My supervisor, Prof. F'rFrann D.Td , for the support, patience and guidance during
the whole process of completing this work and also for the financial support.
Prof. Alan Dow for the many important comments and suggestions about this
work.
All the members of the Toronto/York joint Set-Theoretic topology seminar.
Ida Bulat for all the help during many years.
Nadia, Pat, Karin, Beverly, Marie and all the staff of the Department of Mathe-
matics for always being so helpful.
My mother Lidia, father Abraham, sisters Clarissa and Norma, Adolpho and my
nephew Gabriel and my grandparents who provided a lot of emotional support, friend-
ship and encouragement.
My husband, Eduardo, for all the love and encouragement. Without his support,
interest and friendship none of this would be possible.
Contents
Introduction
1 Reflection with large cardinals
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction
. . . . . . . . . . . . . . . . . . . . 1.2 The perfect projection using C*(X)
. . . . . . . . . . . . . . . . . 1.3 The perfect projection without functions
. . . . . . . . . . . . . . . . . . . . . 1.4 The equivalence of the projections
. . . . . . . . . . . . . . . 1.5 Some immediate consequences and examples
2 Some Applications
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction
2.2 Preserving hereditary normality for spaces of small pointwise type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Balogh's Theorem
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 kt-spaces
. . . . . . . . . . . . . . 2.5 Expandability in countably paracompact spaces
3 Reflection with elementary submodels
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction
. . . . . . . . . . . . . . . . . . . 3.2 Forcing over an elementary submodel
. . . . . . . . . . . . . . . . . . . . 3.3 Upwards reflection of ordinal spaces
3.4 Upwards reflection of countable tightness, sequentiality and R6chet . . 3.5 Some remarks on the large cardinal versus the elementary submodel
. . . . . . . . . . . . . . . . . . . . . . . approach and the space X(M)
APPENDIX
Bibliography
Introduction
This thesis consists of three chapters and an Appendix, the Appendix contains def-
initions used in this work and the reader may choose to start by reading it, even
before reading the rest of this introduction, if he is unfamiliar with concepts such as
elementary submodels, elementary embeddings, collectionwise normal spaces etc.. . . The reader that wishes to avoid, in a first moment, elementary embeddings, large
cardinals and forcing may wish also to start reading Chapter 3 (and even skip Sec-
tion 3.2), which uses elementary submodels and has a simpler approach to the tech-
nique, and later read the other chapters.
For topological definitions (that are not in the Appendix) and theorems we refer
the reader to [9]. For results about elementary embeddings and laxge cardinals we refer
the reader to [16] and [8]. For results and definitions on forcing we refer the reader to
[20] and [16].
The effect of set theoretic axioms, combinatorid properties and techniques in gen-
eral has, for many years, been a source of interesting results for topologists. A topo-
logical space has a simple definition, it is a set and a family of subsets of this set that
satisfies a few axioms. This structural simplicity allows us to use set theoretic tools
to built topological spaces, to obtain properties for known topological spaces and to
prove (consistent) results about topological spaces.
A simple example is the real line: if 2'0 = N1 then there is a family of wl first
category sets whose union is the real line ( take the singletons), however, under different
consistent axioms, the union of a family of wl first category sets is of first category
( [2ol I* Things get more complicated when one introduces forcing, large cardinals, elemen-
tary submodels and combinatorial principles, one can start by looking at the many
articles in [22] to appreciate the variety of results that can obtained using set theoretic
tools.
Reflection of a topological property is a technical term which means:
"If a topological space X does not have the property P, then there is a "small"
(e.g. of cardinality less than (XI, or Iess than some fixed cardinal) subspace of X which
does not have property P."
Loosely, it has come to mean results proved using supercompact cardinals or ele-
ment ary submodels.
First , in this work we explore the large cardinal/element ary embeddings technique.
In this context, at first the "small" subspace turns out to be j"(X) c j ( X ) , (where
j is the elementary embedding and X is the topological space). By elementarity, this
yields a "small" subspace of X. We are able, then, to extend the concept of reflection
by using " a small nice image of a subspace" instead of " a small subspace" in the
previous sentence, so as to be able to get away with weaker hypotheses on X than are
needed for full reflection.
We work with properties like (hereditary) normality, collectionwise normality, count-
able parracompactness, expandability. They all involve subfamilies of the topology with
special properties (discreteness, local finiteness.. .) that we want to expand to families
of open sets having the same property. To see how delicate this is, notice that if X is a
normal space, one would have some difficulties to prove that X remained normal after
forcing as, although the basis for the topological space does not change, new open and
thus new closed disjoint sets might appear.
The large cardinal/elementary embedding technique allows us to relate X "nicelyn
to j(X), (which has, by elernentarity, the same properties that X has that can be
stated in a formula - normality, for instance). For an interesting class of topological
spaces, X turns out to be a L'nice" image of a subspace of j(X). This, added to nice
technical forcing techniques available to us when adding Cohen or random reds, make
it possible to obtain results.
We then explore the elementary submodel technique. We first use elementary sub-
models as a tool to prove a result on the preservation of normality after Cohen forcing
in a non large cardinal context.
After that, we use elementary submodels to obtain new spaces related to the original
ones via the elementary submodel. This space is then considered to be a reflection of
the original space, although it is not always a subspace. We study how sequential
properties like tightness, sequent iality and F'rBchet behave and provide some results
and examples. We also study the case when this space obtained from a topological
space and an elementary submodel is a cardinal. It so happens that, in this context,
the topological space and its reflection via the elementary submodel are nicely related,
(again, a "nicen image of a subspace) for aa interesting class of spaces, to be more
precise, the same class that we have in the large cardinal/forcing context.
We finally present a discussion relating both contexts (that have some properties
and results in common) and present some examples.
In [8] , the authors obtained reflection results for topological spaces using forcing
techniques and elementary embeddings of the universe. The results were obtained
for spaces of small character (small meaning smaller than the critical point of the
elementary embedding). This condition was enough to identify the topological space
X with a subspace of j ( X ) , namely jt'(X) (which inherits its topology from j ( X )
making this identification not always valid). Then the authors proceeded with the
forcing and reflection results and proved for instance :
Theorem 0.0.1 [a] In the model obtained a f t e ~ the addition of tc supercompact many
Cohen or random reals normal spaces of character less than K, are collectionwise nonnal.
The following question arises :
Questionl: How is X related to j(X) if the character of the space is not "small"?
Can they be related in such a way that the same techniques can be applied and similar
results proved?
In this work, we show that for a bigger class of spaces which includes locally compact
spaces and Cech complete spaces, although X cannot be identified to a subspace of
j ( X ) it can be identified to a perfect image of a subspace of j ( X ) . This technique allows
us to produce both new results and, more straightforward proofs of known results.
Using this technique, we prove that in the model obtained after adding supercom-
pact many Cohen or rasdom reds:
(1) Normal spaces with E < 2 H ~ (in particular, locally compact spaces) are collec-
tionwise normal [2].
(2) Normal kt-spaces are collectionwise normal [4] ( see Definition 2.4.1 for the
definition of k' -spaces).
(3) Hereditarily normal spaces with X _< 2'0 (in particular, locally compact spaces)
remain hereditarily normal after the addition of my number of Cohen reals.
(4) Count ably paracompact spaces with X ( X ) = w are expandable [2].
We then change context and start exploring elementary submodels. First we use
elementary submodels to give a partial answer for the following question:
Question 2: For which spaces can we (consistently) preserve normality without
the use of large cardinals after adding Cohen reals?
Then we start working with spaces obtained via elementary submodels:
Question 3: If XM (see Definition 3.1.1) has a certain topological property P, is
it true that X also has P ? ( This kind of question was explored in [18].)
We present some results and examples concerning properties related to sequentiality.
Using the perfect mapping technique, replicated in the elementary submodel con-
text in [l8], we study the following question for cardinal spaces, providing results and
examples.
Question 4: If XM is known, can we describe X up to homeomorphism (see
Definition 3.1.1) ? ( This kind of question was explored in (291 .)
In the first chapter two proofs of the main theorem, characterizing X as a perfect
image of a subspace of j ( X ) for X of pointwise countable type are given, and some
examples are presented. The second chapter gives applications of this technique to
characterizations of normality and related properties. The third chapter deals with
element my submodels, in particular establishing the undecidability of the assertion
that if XM is homeomorphic to wl, so is X.
Chapter 1
Reflection with large cardinals
1.1 Introduction
This chapter deals with the proof of the perfect mapping result mentioned in the
Introduction of this Thesis. It presents two proofs of the same result. The one in
Section 1.2 appeared first, based on [7] which gives a proof of Theorem 2.3.2 in the
locally compact case, using the ring of continuous functions. The one in Section 1.3
appeared later but is considered to be simpler. Both are shown to be equivalent and
some examples are presented.
Recall that the definitions used in this chapter can be found in the APPENDIX.
Lemma 1.1.1 Let j : V -+ M be an elementary embedding such that MW C M and
let P be a cowttable chain condition partial order such that P E M and P M . Then
if G is P' -gene& over M , it is also P-generic over V .
Proof: Let A P be a maximal antichain. A M and Mw M so A E M and
M C A is a maximal antichain (since M V). So G n A # 0. It is worth mentioning that if P is the Cohen partial order then you don't need
MW C M as P n M is completely embedded in P.
In addition to the "normal implies collectionwise normal" result presented in the
Introduction of this Thesis (Theorem 0.0.1 ) the following Theorem, proved using
techniques presented in [S] is a good motivation for this chapter.
Lemma 1.1.2 [8] Let G be a generic subset of Fn(X, 2) o r of MA ( the partial order
for adding X random reals), where X is a cardinal. Suppose that i n V, < X , T > is a
topological space and y is a discrete collection of subsets of X and such that lyl 5 A.
If Y is normalized in V [ q , it is separated in V .
Theorem 1.1.3 In the model obtained by adding K m a n y Cohen reab, where K is
supercompact, let X be a hereditarily normal topological space such that for every x E
X , ~ ( x , X ) < 2'0 = K. Then X remains hereditarily normal after adding any number
of Cohen reals.
Proof of Theorem 1 .l.3. Suppose the theorem is false. Pick an a such that
V + "Ip, Il- X is a normal space and forcing with Pa destroys its normality."
Choose a "convenient" elementary embedding (for the technical details, see the
proof of Theorem 2.2.1 ) j : V t M with j(n) > a. Let Gj(,) be Pi(,)-generic over
M (and so over V by Lemma 1.1.1 - because "convenient" implies MW E M) and let
G, = Gj(,) n V,. Note that G, is P,-generic over both M and V.
Since P, has the countable chain condition, we can ( see e.g. [8]), extend j to:
From now on we will abuse notation and refer to " " as simply "j ". In M[Gj(3], j ( X ) is hereditarily normal, so since M[Gj(,)] C j W ( X ) is a subspace
of j(X), and M[Gj(,]] C j ( X ) is hereditarily normal, M[Gj(,)] C j"(X) is hereditarily
normal. So in M[Gj(Q], X is homeomorphic to a normal space. Now, the choice of j,
the fact that j(K) > a, the fact that forcing with Cohen reals preserves non-normality
(see Lemma 1.1.2), and that M[Gj(,)] C V[Gj(,)] give us the result.
Lemma 1.1.4 [8l Suppose that j is a n elementary embedding and that the critical
point of j is n. If X is a topological space such that for every x E X , x(z,X) < K then
X is homeomorphic to a subspace of j ( X ) , namely jU(X) .
By looking at the proof of Theorem 1.1.3, the reader will notice that the main
result used concerning the topological space X was that the fact that its having "small
character" made it possible to identify X with a subspace of j ( X ) . The hereditary
normality of j ( X ) together with forcing techniques then made it possible to obtain the
result.
The theorem that will be proved in this chapter allows us to present a somewhat
similar proof of the above result for a more general class of spaces, the ones with
pointwise type smaller than the critical point of the elementary embedding j. It should
be noted that such space need not be homeomorphically embeddable in j ( X ) .
1.2 The perfect projection using C*(X) .
In this section we shall prove the following theorem:
Theorem 1.2.1 Suppose j : V --+ M is a n elementary embedding, K is the critical
point of j , P E V is a partial order, G is P-generic over V , H is j(P)-generic over
M and j extends t o j : V[GI ---+ M[H] . Suppose also that V[q + X is a completely
regular space and X(X) 5 n. Suppose the elementary embedding j : V[q + M[H]
also satisfies the condition
(*) j ( ~ ) > X 2 I [0, and M[H] is closed under X -sequences in V [ H ]
whose elements have names in M .
Then M[H] C X is homeomorphic to a perfect image of a subspace of j ( X ) .
In particular, taking j ( ~ ) sufliciently large, this applies to the case of adding su-
percompact many Cohen or random reds. Also, the perfect mapping has the property
that it is precisely j-' when restricted to jl'(X).
Remark 1.2.2 Let P denote the partial order for adding supercompact many Cohen
or random reals. To see that the mapping j can be extended when you add Cohen
or random reals, see Proposition 2.1 and Proposition 2.2 in [8]. To see that you can
satisfk (*), jwt look at the definition of supercornpact cardinal (Definition 7 in the
APPENDIX). Let i be a P-name such that lr Il- 5 = [0, I ] ~ * ( ~ ) . Let A P be a
maximal antichain such that for every p E P there is a cardinal nP such that p Il- 1x1 =
itp. A s P has the c.c.c (see [go]), A is countable. Pick X > sup{n, : p E A). You can
choose j : V + M such that j(n) is bigger than A. When you extend j to V[q, the
extension will satisfy (*).
On our way toward finding the copy of X, having an elementary embedding with
the above conditions, we follow the technique of 171, and in V[Gl see X as its image in
[0, l]c'(X), using the canonical embedding
e : X t [0, I.]C*(X),
defined by:
In M[H], we will see j ( X ) as its image inside [0, l]c'(j(x)) under the embedding
j(e). Note that [O, 11 could now be larger !
Noticing that C* ( j ( X ) ) = j (C*(X)) and consequently, j" ( C * ( X ) ) C C*( j (X) ) .
Define f : j ( X ) = p( j ( X ) ) t [0, l]j"(c*(x)) to be the projection described as follows:
Let
We use:
Lemma 1.2.3 [9] I f f : X t Y is a pe~fect mapping and Z C Y , then f 1 f -'(Z) : fdl (2) t Z is aLso perfect.
We then have:
Lemma 1.2.4 n : Z -+ j(X)f i s a perfect mapping.
Proof: Observe that as j (X) is compact, f is perfect and apply Lemma 1.2.3 using
that j (X)t ftt(jO). In order to prove Theorem 1.2.1, the following alternate characterization of 7; will
be useful.
Lemma 1.2.5 [q For X completely regular, if Z(X) 5 r then PX \ X is the intersec-
tion of a family of F<, sets ( i .e . sets which are the anion of fewer than r closed sets)
in P X .
Proof of Theorem 1.2.1
We will prove that j(X)f includes a subspace that is homeomorphic to X. First of
all, note that condition (*) is needed to get that (X, C*(X), j"(C*(X))} M[H] and
that the homeomorphism that we will describe shortly is also in M[H]. Also, if r is a
real number, note that j ( r ) = r. Since j is an elementary embedding , observe that:
Defme now,
Thus: " c* X)) M[HJ C E x G [0,1]j( ( .
Ex contains the same sequences of r e d numbers as X, but instead of being indexed
by elements of C*(X) they are indexed by elements of j" (C*(X) ) , and so, using that
[0, 1IVM is first countable, we get that
[0, l]V[q = jtt([O, llV[q) is a subspace of [0, 1lM[W = j ( [ O , 1IV[q).
Consequently
(By [0, 1IV[q, we mean [O,1] in the sense of V[q, and similarly for [ O , l ] MIHI. )
We conclude that Ex is homeomorphic to X. Explicitly, h : X ---+ Ex is described
by h(x) = ( ( j ( f ))(j(x)))j(t)~jn(~*(x)).
To simplify our notation, if h E C * ( X ) , let's refer to j(h)(j(x)) as xh. If h E
C* (j(X)) (or, C* (X)) denote by IE its extension to P(j(X)) (PX, respectively ) , where
for any completely regular topological space Y, we consider PY as the closure of Y
when embedded in [0, l]C'(Y).
We will prove that
Assuming the claim, pick z in the above intersection. Pick a continuous function
on pX such that g(z) = 0 but ~ ( x ) = 1. Now g = g X E C*(X). Therefore z, # x, contradicting the fact that z is in the intersection. It remains to prove the claim.
Proof of Claim :
Suppose the claim is false. Then
Since X(X) 5 n, by Lemma 1.2.5 we may write
,OX \ X = n (~ ' : i E I) where each R' is a F,, in pX.
Then, since n(ffl(zh) : h E C * ( X ) ) consists of just one point, namely ( I ~ ) ~ ~ ~ * ( ~ ~ ,
we get
V[q C 3 E I such that n ( ~ - l ( x h ) : h E C*(X)) n = 0.
Now we can write R' = U{K, : u E rj)) fa some Tj < 6, where each Kc is closed
and thus compact in pX.
We have
Using that each KO is compact, pick for every 0 E 7. , fr , -....., fze E C * ( X ) such
that
-- 1 K, nz-'(Zfin) n ... n fg (zq) = 0.
We have
Apply j to the above formula, and get by the elementarity of j that
as it is a set of size T. < PC. Finally, we have
Thus,
contradicting (w). The claim is proved.
Thus we have
n : Z + (jX)f is a perfect mapping and Ex E (jX)r
Using Lemma 1.2.3, finally we have:
rr : n - ' ( ~ x ) C j ( X ) Ex is perfect .
13
Remark. In huge cardinal applications , it is useful to note that the condition that
X 2 I [0, l]C*(X) I can be replaced by "A 2 rna~(21~I, IC*(X) I)" if the application only
requires the homeomorphism between X and Ex to be in V [ H ] .
1.3 The perfect projection without functions
Now we obtain the same result as in Section 1.2 with a different technique. By avoiding
the trouble of identifying X with a subspace of [O, l]C*(x), we are able to produce a
seemingly simpler projection and the Theorem is valid for regulax spaces. Some results
of this Section appear in [Is]. For the sake of clarity, we restate the Theorem:
Theorem 1.3.1 Suppose j : V + M is a n elementary embedding, K is the critical
point of j , B E V is a partial order, G is P-generic over V , H is j(P)-generic over M
and j extends to j : V [ q + M[Ii]. Suppose aZso that V [ q t= (X,T) is a regular
space and E(x) 5 n. Suppose the elementary embedding j : V[q + M [ H ] also
satisfies the condition
(*) j ( n ) > X 2 2lT1 and M[H] is closed under X -sequences in V[H]
whose elements have names in M.
Then M[H] C X is a perfect image of a subspace of j(X).
Again, in paxticulas, this applies to the case of adding supercompact many Cohen or
random reds (see Remark 1.2.2) and the perfect mapping is exactly j-' when restricted
to j"(X).
From now on X will denote a topological space ( with no extra requirements) with
topology T and we will use the notation of Theorem 1.3.1.
We start by presenting some notation and necessary results to be able to prove the
Theorem:
Definition 1.3.2 1. Let K = (K X : K is compact and x ( K , X ) < K ) .
2. F o r x f X , let K , = { K EK: : x E K).
3. For x E X , let V , = {V : V w open and x E V ) .
4 . For K f X , let V K = {V, : CY < T ) , T < n, be a jxed outer base for K .
Notice that because of (*) , {j1'(E), j"(K,), jt'(VZ)} G MIH] and also, supposing
without loss of generality that X, as a set, is an ordinal, (X,?) E M[H].
We define in M[H]:
Definition 1.3.3 For x E X , let K, = n j U ( V = ) .
This idea of taking the intersection of "olds' open sets about x also appears in
Balogh's proof [2].
From now on, we assume X is Hausdorff.
Definition 1.3.4
by
n ( z ) = t if and only if z E Kt.
We need X to be a Hausdorff space as in this case, x # y implies that K, n K, = 0. This implies that the mapping T defined above is indeed well defined, one point in Z
belongs to one K, at most.
Theorem 1.3.5 if X is T3 and E(X) 5 n, then n is a perfect mapping.
We need the following results:
Proposition 1.3.6 [ I ] If X is a Tz space and K C F are compact subsets of X then
x ( K X) 5 x(K, F ) . x(F, XI*
In [I] a proof of the above result for the Tychonoff case is presented. We will present
a proof that holds for regular spaces:
Proof: Suppose x(K, F ) x(F,X) = K , and let {W,),<, be an outer base for F in
X and {Vp)B<c an outer base for K in F. As F aad K are compact, we get that for
every p < 6, we can find disjoint open sets (in X) Gp and Hp, such that K Gp and
F\Vp C Hp.
We claim that (Gp n W, : a,/? < n) is an outer base for K in X. Suppose that
K C V, where V is an open set in X. Then there is a P < K such that K C Vp E V n F.
Also if Go and Hp are as above, we have that F H p U V. But then there is an a < n such that F E W, Hp U V. Clearly, K Go fl W,. It remains to prove that
Gp n W, V . Suppose x E Gp n W,. Then x E Gp implies that x 4 Hp , and therefore
x E W, implies that x E V, and we are done.
0
Proposition 1.3.7 Suppose X is such that X(X) 5 T . If V is an open set and x E V
then there exists K compact, x(K, X) < r, x E K, K V.
Proof: Suppose V is open and x E V. Pick K compact such that x(K,X) < r. Using that K is regular, construct a sequence {Vn : n € ww) of open sets such that
Vo = V, x E Vn for all n, and Vn+l n K Vn n K. Observe that Kt = n,,, Vn n K is
a compact set, z E Kt, and x(Kt, X) 5 x(Kt, K) x(K, X) = No x(K, X) < T .
0
Lemma 1.3.8 If X is Hausdorff and X(X) I n then Kz = n jM(Kx).
Proof: Let z E n jU(Kx). We will prove that z E j(V), for all V E V,. Let
V E V,. By Proposition 1.3.7, there is a K E K, such that K E V. By definition,
2 E j(K) E j(V).
Here is precisely where we use that YX) 5 n. Suppose now z jU(lC,). There
is some K E K, such that z $ j(K). Now V[GI J= VK is an outer base for K, so by
elementarity, M[H] C j(VK) is an outer base for j ( K ) . Since lV~l < K, we have that
j(VK) = jO(VK) and consequently M[H] jf'(VK) is an outer base for j ( K ) . There
is, then, a V E VK such that z 4 j (V) .
We are able now prove the Theorem.
Proof of Theorem 1.3.5. Lemma 1.3.8 implies that inverse images of points are
compact, so it remains to show s is continuous and closed.
Since the topology 7 on X in V[G,] is a basis for the topology X gets in M[G,(,)],
to prove continuity it is enough to prove that rr-'(F) is closed, where X \ F E 7. Let
z E Z\a-'(F), then z E K, for some x $ F. By regularity of X, take disjoint open sets
V, W containing x and including F respectively. By the definition of K x , K, j ( V )
and n-' ( F ) C j ( W ) . Therefore j ( V ) n a-'(F) = 0 which implies Z \ r - ' ( F ) is open.
Notice that to prove that 7r was continuous the only thing we used about X was its
regularity.
To see that s is closed, let A be closed in Z and let x E X \ a(A) . We f i s t
show that there is a K E K, such that j ( K ) n A = 0. Assume otherwise asd set
F = ( j ( K ) n A : K E K,). Since EC, is closed under finite intersections, F is a
collection of compact sets with the finite intersection property (if fl C 3 is finite then
j ( 3 ) = j t f ( F ) ) , and so there is a z E n3. But then z E K, n A and so x E *(A) ,
contradiction.
Since j ( K ) n A = 0, there is a V E VK such that j (V) n A = 0 (same argument as
in the proof of Lemma 1.3.8- here we use X ( X ) < n). Since x E V, it remains to show
that V n r ( A ) = 0. If y E V n *(A) , then for some z E A, ~ ( z ) = y. This would imply
z E j ( V ) f l A, contradiction.
0
1.4 The equivalence of the projections
In this section we will prove that the two mappings that we have defined in Section 1.3
and Section 1.2 are equivalent. During this section, we will refer to the mapping defined
on Section 1.2 as fl : H + X and the one defined in Section 1.3 as f2 : Z + X. Other
than that we follow the notation of the previous sections.
By equivalent we mean that we will define h : Z + H a homeomorphism such
that f l ( ~ ) = f i ( h ( 4 ) .
From now on X will denote a Tychonoff topological space (not necessarily of small
pointwise type) with topology 7.
As before, we will denote the reds between 0 and 1 in the sense of the ground model
by [0, 1IVM and the ones in the sense of the extension by [0, 1JM[w
Theorem 1.4.1 Iff E C*(X) (in the sense ofV[G'l) then if z E K, we have:
Proof:
Let I,, = ( ( f ( z ) - l /n , f ( x ) + l/n))M[a and In' = ( ( f (3) - l/n, f ( x ) + l / n ) ) V M *
e l ( I ) = I Define On = f - ' ( I ~ ' ) . Using that z E K, C j (0 , ) we have
that :
Vn E E w t j(ff-'(I,')) = ( j f)- '( j(In')) (by elementarity of j) = (j f)-'(In).
So Vn E w we have that ( j f ) ( z ) E (f (I) - l/n, f (x) + l / n ) which implies ( j f ) ( z ) =
f (4 0
Definition 1.4.2 Define h : j ( X ) t [O, l ] j ( c * ( x f ) by h(z ) = (g(z)),Ej(~*(~))=~*(j(x)).
It is known that h is a homeomorphic embedding. We will prove that if we restrict
h to 2, it maps onto H.
Theorem 1.4.1 gives us that f l (z) = f2(h(z)) and also
If z E Z then h( z ) E H,
as when you project h(z ) onto [0, l]j"(c'(x)), via fl it goes to the point which is
identified to x, meaning that h(z) is in the preimage of X.
It remains to show that h is onto:
Theorem 1.4.3 h"(Z) = H.
Pro08
Pick w E H h f f ( j ( X ) ) (by the definition of H and f l ) . We have that 3t E j ( X )
such that h( t ) = w. Now, w = ( g ( t ) ) gEo*( j (x ) ) We will prove that t E 2.
As w E H we have that 3x E X such that f l (w) = ( g ( ~ ) ) , ~ c * ( x ) .
Claim : t E Kz.
Proof: If t C K, then 3V open set such that x E V but t $ j ( V ) . Using that X
is Tychonoff, we may assume that 39 E C*(X) such that V = 9-'(I) where I [O, I]
is open. Notice that j ( V ) = j(g)-' ( j ( I ) ) , so as t 4 j (V) we have that j (g)( t ) 4 j ( I ) .
Now, h( t ) = w which is projected into ( g ( ~ ) ) , , p ( ~ ) , so j ( g ) ( t ) = g(x), precisely
the gth coordinate of the projection (by the definition of h and fl) which implies
j (g)( t ) E I G j ( I ) . Contradiction.
1.5 Some immediate consequences and examples.
The following examples and results will try to better explain the nature of the mapping
defined in Section 1.3 by exploring questions like the ones below. We chose to proceed
with the mapping defined without the functions for its apparent simplicity. Unless
otherwise stated j , V, M, V [ q , M [ H ] , n are as in Theorem 1.3.1.
(1) When is a-I (X) = j (X)?
(2) Is n-I (X) closed in j (X)?
(3) Is the perfect mapping theorem valid for a broader class of spaces?
We first explore compactness.
Theorem 1.5.1 If V[q k x ( X ) < tc, then M [ H ] + n is a homeomorphism between
n-'(X) and X.
Proof:
Pick x E X and fix Vz = (Va : a < T ) a local basis for x where r < K . As T < n we have that jff(Vx) = j(V=). So n j" (V,) = n j(V,). To finish the proof, just notice
that M[H] C j(V,) is a. local base for j (x) at j ( X ) which implies n jt'(V,) = n j (V,) =
{ j ( x ) ) . So n is a 1-1 perfect mapping which is a homeomorphism.
0
Theorem 1.5.2 Suppose that M[H] + X is compact. Then M[H] + a-I ( X ) = j ( X ) .
Proof=
Suppose the conclusion is false and there is some z E j ( X ) \ T-'(X). This means
that M[H] b Vx E X there is E V+ such that z @ j(V,). Notice that W = {j(V,) :
x E X ) is an open cover of a-'(X) and that z @ UW. Also M[H] + n- ' (X) is
compact ( as M[H] k X is compact and a is perfect).
It has, consequently a finite subcover 7 = { j ( & ) , ...., j(&)) . Claim: V[q V = I & , ....., Vk) covers X(*).
Otherwise V[q C y fi! U V , so M[H] t= j ( y ) $ U T . But j ( y ) E ?r-'(X) and
M[H] k U 7 covers n-' ( X ) . Contradiction.
0
Now (*) implies M[H] C j ( 7 ) covers j ( X ) . This contradicts z 4 U W. Contradic-
tion.
0
Example 1.5.3 Suppose we are adding supercompact many Cohen or random reals.
In the previous Theorem it is not enough to have V[q X is compact . Jwt pick
X = [0, 1IVM. Being a first countable space, by Theorem 1.5.1, ?r-'(X) = X . But
j ( X ) = [0, 1IM[w which is much bigger.
One might ask the question whether the converse of Theorem 1.5.2 is true, that is
if r - ' (X) = X, can we conclude that X is compact?
The following example will show that the conjecture fails.
Example 1.5.4 Let X = n + w with the order topology. Notice that:
va < n -t m-'({a)) = (a}.
In this case X is locally compact but not compact and n - ' ( ~ ) = j ( X ) = j(n) + w.
Example 1.5.5 Take X to be the rationals. j ( X ) is also the rationals, so x- ' (X) = X
and X is not even locally compact.
Example 1.5.6 X = n. Clearly j (X) = j(x). By Theorem 1.5.1, rF1(n) = n which
is not even closed in j (X) .
The following example appears in [18], in the elementary submodef context:
Example 1.5.7 A Tychonoff space X , such that X is not a perfect image of a subspace
of d x ) .
Let X be n + 1 where for every a < n, {a) is open and {I@, n] : P < n} is a
base at K . j (X) is j(n + 1) = j(n) + 1 and for every u < j ( n ) , {a) is open and
{]/3,j(n)] : @ < j(n)) is a local base at j(tc).
Suppose that p : B j ( X ) t X is perfect. ~ - ' ( { r c ) ) cannot be open) for otherwise
F = B \ p-l((n)) is closed and then ptt(F) = X \ (n) would be closed. Contradiction.
So j ( ~ ) E B and p(j(n)) = n. Notice that
as otherwise there is 7 < j(n) such that V =]r,j(n)] n B satisfies V n r l ( n ) = 0. Then F = B \ V is a closed set and p M ( F ) = n. So n is closed in X and hence ( K ) is
open in X. Contradiction.
A s a < j(tc) and j(n) is regular, there is some ,O < K such that p-'((P}) is cofinal
in j ) by ( ) But then p-' (]P + 1, n]) cannot be open in j(X), contradicting the
continuity of p.
The simple examples above show that trying to explain the subspace rr-l(X) is not
an easy task, and in most cases, each space has to be studied separately.
Now we will present some simple extensions of the result:
Theorem 1.5.8 In M[H], suppose (Xi)isI is a family of topological spaces such that
for every i E I there is a perfect p; : Zi C j(Xi) ---+ Xi. Then M[H] C there is
Z C j(niEIXi) and a perfect p : Z + KEI X;.
By Theorem 3.7.9 in [9], since the projections are perfect, this mapping is perfect.
Notice now that :
V [ q C X = nierXi is a product space whose factors are indexed by I. So
M[H] + j ( X ) is a product space whose factors are indexed by j(I).
V [ q + Xi is the i-th factor in the product space X , for every i E I. So M[H] I= j(Xi) is the j(i)-th factor in the product space j(X).
Notice that 2 is not a subset of j(niEI Xi) , so from 2 we will define such 2.
Pick q E j(X) and call Q = (qm)rnEj(~)\jll(~)* Let T = niEI j(Xi) x @ UP to permu-
tation of indices (or up to homeomorphism) T is a subspace of j ( X ) homeomorphic to
X i Z = 2 x t j can be seen in the same fashion and clearly the mapping you
obtain after composing p and this homeomorphism induced by the permutation on the
indices remains perfect.
0
Notice that although the mapping is perfect it is not canonically nor naturally
defined by j as before.
Theorem 1.5.9 Suppose that M[H] C Y is a perfect image of a subspace of j(Y) and
that V[q + 3f : X + Y which is perfect and onto. Then M[H] C X is a perfect
image of a subspace of j(X).
Proof= We follow the notation of the diagram below:
First we will define, for every x X, a compact set Kz. For distinct points in X
the defined compact sets will be disjoint. Call kt(,) = rY-'(f (x)), for x E X. Define:
(3) nx : Zx + X mapping z E K, to x.
As the compact sets are clearly disjoint, nx is well defined. Also it is clearly onto.
Also notice that j( f ) maps Zx into Zy . We will prove now that it is a perfect mapping.
The above diagram commutes:
Proof=
Just notice that if z E KI then j( f ) (2) E Rfg1 by the definition. Now f (nx (2)) =
f (4 = ~ u M f )(z)).
0
Claim: ?rx is closed.
Proof: Pick F Zx closed and x 4 ?rjG ( F ) . We have then ax-' (x) n F = 0 which
implies j( f )(ax-' (x)) n j f ( F ) = 0 (by the elementarity of j( f )).
Now, j ( f ) (?rxhl(x)) = m u - ' ( f ( x ) ) . So f ( x ) n ? r Y ( j ( f ) ( F ) ) = 0. We know that
7ry is perfect so H = + ( j ( f ) ( F ) ) is closed in Y. There is, then a V open in Y such
that f ( x ) E V and V n H = 0. So f d l ( V ) n f- '(H) = 0 Now, by (*) we have that
H = P(r$ ( F ) ) which gives us that f -'(V) n f -l f I f (lrl;(F)) = 0. To finish the proof
just notice that 7rs(F) f - ' ( f"(r$(F))) .
0
Claim: ax is continuous.
To see that, let F X be closed and z E Zx \ ax-I(F) . As f is perfect f"(F) is
closed in Y. So that ny -' ( f " ( F ) ) is closed in Zy . Notice that by the definition of Zx we have that j ( f ) ( z ) $ xu- ' ( fN(F)) . So there
is an open set V in Zy about j ( f ) ( z ) such that V n a u - ' ( f f f ( F ) ) = 0. Notice now that
(j( f ))-I ( V ) n irx-I ( F ) = 0. 0
Definition 1.5.10 W e say that X has property P if M [ H ] + X is a perfect image of
a subspace of j ( X ) .
Remark 1.5.11 I n the model M[H], Theorem 1.5.8 and Theorem 1.5.9 say that P
is productive and a n inverse invariant under surjective perfect mappings. This implies
that a variety of spaces that can be constructed from a space X that has property P
also have property P. One such example is the Iliadis Absolute of a space X , E(X),
which is a subspace of the Gleason space of X (the space of ultra$lters o n the family of
regular closed subsets of X - they coincide when X is compact). In Chapter 6 of [24
it is proven that E ( X ) 4s a n extremally disconnected zero dimensional space and if X
is regular, it is a perfect image of E ( X ) . S o E ( X ) also has property P i f X does.
Chapter 2
Some Applications
2.1 Introduction
In this chapter we present applications of the general method that was proved in
Section 1.3 and we follow its notation.
The first section deals with the preservation of hereditary normality. The second one
presents a new pmof of Balogh's Theorem regarding the 'normal implies collectionwise
normal" problem for locally compact spaces.
The third one presents a new proof of Daniels' result that extends Balogh's to
kt-spaces and the last one gives another proof of Balogh's result that 'Lcountably
paracompact implies expandable" for spaces with pointwise countable type.
2.2 Preserving hereditary normality for spaces of
small pointwise type
We can now state and prove the preservation theorem for spaces that are not of small
character but of small pointwise type. We will present a proof dealing with the addition
of Cohen reals, but with slightly more complicated details the same machinery will work
for random reals.
By examining the proof, the reader will notice that except for the fact that we do not
have X identified to a subspace of j(X) the proof is similar to the one of Theorem 1.1.3.
The perfect mapping is "enoughn to obtain the result. Also, the same machinery works
for other topological properties that are invariant under perfect mappings although in
pazticular cases one may also need results depending on the particular problem being
investigated; here the particular result is Lemma 1.1.2.
Most of the results in this section and the next one appear in [15], and are joint
work with my supervisor Franklin D. Tall.
Theorem 2.2.1 Let rc be a supercornpact cardinal. Add K many Cohen reals. In the
~esulting model, hereditary normality is preserved for Hausdorflspaces with 76 5 2'0 = rc
by adding any number of Cohen reals.
Lemma 2.2.2 (see [9]) Suppose that there is a perfect mapping f from a topological
space X onto a topological space Y . If X is hereditarily normal then Y is hereditarily
normal (in fact we just need the mapping to be continuous and closed).
We are ready to prove Theorem 2.2.1.
Proof: As before we may assume that the base set for the topological space is an
ordinal.
We proceed by contradiction and repeat the same initial procedure as in Theo-
rem 1.1.3. Let P, denote the partial order for adding rc Cohen reals. There is a p E P,
forcing that there is a topological space (X, T), with E ( X ) 5 n, and an a such that X
has its hereditary normality destroyed after forcing with Pa. As X is an ordinal, we
may pick B (much bigger than the size of a given base for X), such that He contains
all the interesting information about open and closed sets (more precisely, nice names
for open and closed subsets of X).
Pick a cardinal X bigger than a, 1 He 1, 2l71 (notice that as 1 He 1 = 2<' which can be
bigger than 8, so X and 8 might be different), and pick j : V --+ M with j ( ~ ) > A
and M" M and ((Me = (V)e (here we mean the objects of rank < 0 ). Let GjU
be I",(,)-generic over M (and so over V) and such that p = j ( p ) E Gj(,). Again, let
G, = Gj(,) n V,.
Pick in V[Gn], Y , F and K, nice names for subsets of X, and 1 = lp, E P, such
that
V[G,] C "1 It- F and k are closed disjoint unseparated subsets of Y 2."
Observe that we have that (M[Gn])e = (V[G,])e (remember that B was chosen
to be large enough). So, for the properties that concern us, truth in one of them is
equivalent to truth in the other one. We mean,
M[GK] J== "1 Il- P and K are closed disjoint unseparated subsets of Y _C x."
This implies
M[G,] k "1 It- x is not hereditarily normal ."
Pick H IP, -generic over M[Gn] and I such that (see [8] and Chapter VIII in [20]) :
We have that
M[G,] [W] + X is not hereditarily normal.
Therefore since the addition of Cohen reds preserves non-normality (Lemma 1.1.2),
M[Gj(,]] k X is not hereditarily normal.
Now j ( X ) is hereditarily normal, so ael(X) is hereditarily normal and as ?r is
perfect Lemma 2.2.2 implies that:
M[Gj(,]] X is hereditarily normal,
Contradiction.
0
Corollary 2.2.3 Add n supercompact many Cohen reals. In the resulting model, i f X
is hereditarily normal and Z ( X ) 5 n then X is hereditarily collectionwise normal.
Proof: We know that X remains hereditarily normal after adding any number of
Cohen reals. Just apply Lemma 1.1.2 for every subspace Y of X and get the result.
We have been unable to answer the question of whether Theorem 2.2.1 remains valid
if "hereditay normality" is weakened just to "normality". It is also interesting to note
that it is unknown whether Corollary 2.2.3 is true if both instances of "hereditarily"
are removed. None of the proofs of Balogh [2], Fleissner [lo], or ourselves (see below)
work for the case K = 2".
2.3 Balogh's Theorem
In this section we present a different proof of Balogh's result that "in the model obtained
by adding supercompact many Cohen reals, normal spaces of pointwise countable type
are collectionwise normal" [2]. Dow's proof [7] of the locally compact case of Balogh's
theorem was our starting point but the one presented here avoids C*(X) and elementary
submodels and is considerably more general as it works for regular spaces of pointwise
type < 2 N ~ .
It is known that there are normal spaces whose normality can be destroyed by the
addition of one Cohen real [13], but these spaces are not of pointwise countable type.
As in the previous section, our proof will depend on the characterization of X as
a perfect image of a subspace of j(X) plus the usual forcing reflection argument for
normality (normalized in the extension implies separated in the ground model - see
Lemma 1.1.2 ). So this proof depends on the fact that normality is a topological
property invariant under perfect mappings (in fact we just need the mapping to be
closed), and the fact that normality has the forcing reflection property described above.
A similar procedure could be followed, as we do in Section 2.5, to obtain similar results
for other topological properties with similar reflection and invariance properties.
Lemma 2.3.1 Suppose X is a Hausdorff space, x ( X ) 5 n, and y is a disjoint collec-
tion of subsets of X such that V 2 E [y]c'C, Z is discrete. Then y is discrete.
Proof: Since y' = (Y : Y E Y ) also has subcollections of size < n discrete, we
may suppose that Y is a collection of closed sets. Also, as Y is a collection of pairwise
disjoint sets, to prove it is discrete it is enough to show that for every Z C y if -
x E U Z, then x is in some Z E Z. By contradiction assume there is 2 y and -
x E UZ \ UZ. Let KO be a compact set containing x , x(Ko,X) = X < K. Then KO
meets only a finite number of elements of 2 (otherwise it would include an infinite
closed discrete set.) If KO meets Zo, ....., Zn, notice that KO t I Ui<, - 2; is closed and
does not contain x, so V = X \ (KO n Us, Zi) is an open set containing x. Working in
KO, there is K, a closed Ga subset of KO, such that x E K C KO n V. We have that K is
compact in X and, by Lemma 1.3.6, x(K, X) 5 x(K, KO) x(Ko, X) 5 No A = A < K.
Also K n U Z = 0 .
Let {U,),<x be an outer base for K in X. Since x E U,, for each a we may
pick Y, E 2 such that U, n Y, # 0. Since subcollections of size < rc are discrete,
U = X\U,,x Y, is an open set including K which does not include any U, contradicting
the fact that (U,),<x was chosen to be an outer base for K.
0
Theorem 2.3.2 With the notation of Theorem 2.2.1,
V[GJ if X w a normal space with K(X) < 2'0, then X is collectionwise normal .
Proof: Suppose V[G,] + "1 It X is a normal space, X ( X ) < 2 X ~ , and C is a discrete
family of closed subsets of X" . Choose an elementary embedding as in Theorem 2.2.1.
Apply Theorem 1.3.1, and let T : Z z X be the perfect mapping. Observe that
(X, 7) and C are in M[G,] . We have
M[Gjw] j(C) is a discrete family of closed sets in j(X).
Claim: 2) = (n-'(C) : C E C) is discrete in j(X).
Pro0 f:
Since M[Gj(4 ] C z(j(X)) = j(E(x)) < n, by Lemma 2.3.1 we just need to prove
that every subfamily of 2, of size < n is discrete. In M[Gj(4], let E be a subset of V
of size < n. Notice that Pj(,) = PK * Pj(4\K and that Pj(,)\, has the countable chain
condition. Since C is in V[G,], there is an R E V[G,), < K , % C C, such that
1 Il-Pj(4,K £ {n-l(C) : C E 'fl) ( 3C is also in MIGjGl]), so without loss of generality,
we may assume that E = {a-'(C) : C E R , 31 C C, 13CI < n, 31 E V[GK].
In V[G,l, as X is normal and by the the discreteness of 'H, we can find open
sets {Uc : C E 3L) with Uc 2 C, such that C # C' implies n C' = 0. Again
by normality, we can find a (possibly empty) open U 2 X \ U(Uc : C E 311, with - U n U3t = 0. Let U = {U) U {Uc : C E 31). Observe that U covers X so j(U) = j"U
covers j ( X ) . Observe that j(U) "witnesses the discreteness" of E. First of all, by the
definition of x, x-'(C) j(Uc). Second, j (U . ) n ?r-'(C') = 0 if C # C'. For take
open Vet 3 C', Vc, n Uc = 0. Then a-'((7') C ~(VCI) and j(Uc) f~ j(Vct) = 0. Finally,
suppose z E j(U). Then z is in no nhl(C), for if z E Kc, c E C , then c E X \ u, so
z E j(X \ u ) = j (X) \ j (V), contradiction.
0
Remark 2.3.3 For the case of when the space is of pointwise countable type, the above
Claim can be proved just b y using that in this case X is a k-space [9] and so it is enough
to prove discreteness for countable subcollections, which follows since nomnality implies
countable collectionwise nomnality (see e.g. [28]).
Now, since $9 is discrete in j ( X ) and j(X) is normal, $9 is normalized in j(X) and
consequently in 2. To finally obtain that M[Gj(K)] + C is normalized in X, just use
that 7r is closed, as follows:
Let S C. Since 2) is normalized, pick U, V disjoint open in Z such that U(n-'(C) :
C E S) U and U{a-'(C) : C E C \ S) V. Notice that U(C : C E S)
x \ r ( z \ V ) , U { C : C € C \ S ) C X \ s ( Z \ V ) andthat X \ a ( Z \ U ) a n d X \ r ( Z \
V) are disjoint open sets in X. So M[Gj(.)] + C is normalized in X. Again, apply
Lemma 1.1.2 (collections normalized in a Cohen extension are separated in the ground
model), and get that M[G,] + C is separated. But then C is separated in V[GK] and
we are done. Cl
Definition 2.4.1 We say that a topological X is a kt-space if whenever x E X is such
that x E cl(F) where F E X, then there is K 2 X compact such that x E cl(F f~ K ) .
It is easy to see that kt-spaces are k-spaces but there are k-spaces which are not
kt-spaces (see [I]). Also there are kt-spaces which are not of pointwise countable type
(also in [I] ) and spaces of pointwise countable type which are not kt-spaces as the
following example will show.
We need the following proposition to be able to build the example. It is stated in
[l4]. The proof is simple and we present it here:
Proposition 2.4.2 Suppose that N C X E PN (where N represents the natural num-
bers) and X is a kt-space. Then X C PN is open and consequently locally compact.
Proof: We think of PN as the set of non-principal ultrafilters on N together with
N with the topology generated by N(B) = (U E ,ON : B E U) for all infmite B E N
and all points in N open.
Let Y = X\ N. Notice that for every y E Y we have that y E &(N), and as X is a
kt-space there is a compact set Ky X such that x E clx(K, n N). Call A = Ky n N.
NOW, N(A) is a clopen subset of PN and also y E N (A) = clplv(A) G Kg = clpN(K,) G
X. So X is an open subspace of a locally compact space which implies it is locally
compact.
The example is a clear consequence of the previous proposition:
Example 2.4.3 A space of pointwise countable type which is not a kt-space.
Let A PN \ N be infinite and countable and X = PN \ A. The space X is a Gs
subspace of PN which is compact and thus ~ e c h -complete and consequently it is ~ e c h
-complete which implies pointwise countable type (see [9]).
X , however cannot be a kt-space for otherwise it would be open, meaning that A
would be closed. This is a contradiction since A is countably infinite and IdaN(A)I = 2'
(see P?JJ
This definition and a more detailed study of those spaces appear in [I].
A natural next step after proving the consistency of the "normality implies collec-
tionwise normality" result for locally compact (pointwise countable type) spaces would
be to obtain the same result for k-spaces. Although this has not been shown, in [4],
Peg Daniels shows that in the model obtained after adding supercompact Cohen reals
normal k'-spaces are collectionwise normal. One would wish to be able to apply Theo-
rem 1.3.1 to obtain Daniels' Theorem. Our proof has some elements in common with
Fleissner's measure-theoretic proof of the kt-result in the random real model [ll].
Theorem 1.3.1 allowed us to use the defined projection in the above model to prove
collectionwise normality for those normal spaces for which the projection was perfect.
Unfortunately, although the mapping is continuous, it is not necessarily perfect or even
closed for k t -spaces. Example 7.11 in [18] provides a counterexample in the elementary
submodel context which can be translated into the large cardinal context:
Example 2.4.4 A kt-space X such that X is not a closed image of a subspace of
j(X). Our ground model is the model obtained after the addition of t~ supercompact
Cohen reab. Let X be w x w U { s ) , where w x w has poirats open and a neighborhood
o f s i s o f t h e f o r m V " f = ( < a , P > ~ w x w : / 3 > ~ ( c Y ) ) where f : w t w .
The proof is almost identical to the one in [18]; the result comes mainly from the
fact that x(s,X) = K, but ~ ( s , j(X)) = j(K) > K.
We are able, however to overcome the above difficulty and obtain the result below
by using Theorem 1.3.1 together with parts of the proof of the result for spaces of
small character (see [S]) and a seemingly technical but really simple idea which will be
explained later.
Theorem 2.4.5 In the model obtained by adding supercompact many Cohen (or ran-
dom) reals, normal k'- spaces are collectionwise normal.
To prove the Theorem we need the following Lemma due to ~rhan~el 'ski i [I]:
Definition 2.4.6 W e say that a mapping f : X + Y is a pseudo-open mapping if
whenever V X is open and f -l(C) V , then C int(f (V)) .
It is worth mentioning that pseudo-open mappings are precisely the hereditarily
quotient mappings [9] .
Lemma 2.4.7 [ I] Every k t topological space is a pseudo-open image of a locally com-
pact space.
The locally compact space above is the k-leader of the topological space, meaning
the disjoint sum of all compact subsets of the topological space, and the mapping
assigns each point in the k-leader to itself inside the topological space. This mapping
is continuous for every k-space but it is pseudo-open if and only if the topological
space is a k t -space.
Remark 2.4.8 Observe that the k-leader of a topological space Y , K ( Y ) is a locally
compact normal space ( as it is the disjoint sum of compact spaces). Its normality does
not depend on the normality of Y as closed disjoint subsets of K ( Y ) are disjoint sums
of closed disjoint subsets of compact spaces. They can be separated in each compact
set and so the disjoint sum of this separation is the desired separation for the original
disjoint closed sets.
Fkom now on, Y is a k' normal space, X is a locally compact space and f : X + Y is a pseudo-open continuous onto mapping. We follow the notation of the previous
sections. If P is the partial order for adding n (supercompact) many Cohen or random
reals then G will be a P-generic filter and H will be a j(P)-generic filter such that
the mapping j : V --+ M extends to j : V[G] t M [ H ] .
R e c d that the mapping defined in Section 1.3 was continuous provided the topo-
logical space was regular ( see the second paragraph of the proof of Theorem 1.3.5).
So we can define ny as described in Definition 1.3.3 and Definition 1.3.4, although we
must have in mind that rrry is not necessarily perfect, just continuous. We can also
define q, which will be perfect as X is locally compact.
With all the not ation as described above we are able to draw the following diagram:
The above diagram can help us to describe the proof of Theorem 2.4.5. The details
of this explanation are best described in the real proof of the Theorem.
If C = {C; : i E I) is a family of closed discrete subsets of Y, the first idea would
be to apply Theorem 2.3.2 directly to { f -' (Ci) : i E I} which is a discrete family
of subsets of X (see Remark 2.4.8) and obtain a disjoint open expansion of it. By
applying first f and later the interior operator to each element of its expansion one
would hope to get the desired expansion. It is clearly an open expansion, however it
might not be disjoint and this comes mainly from the fact that the open sets in X are
not necessarily of the form f -'(V) where V is open in Y.
To deal with that we "traveln through the above diagram as follows:
Notice that M[H] {nY-'(Ci) : i E I) is discrete in j(Y). Now, j(Y) is normal,
so every generic partition of the family generates two disjoint closed sets that can be
separated by 'generic" open sets. Apply j ( f )-' to those open sets, use ?rx which is
perfect to obtain "generic" open sets in X as in the proof of Theorem 2.3.2 , and then as in [8] (see Lemma 1.1.2), using endowments, obtain a disjoint separation of
{ f (C) : i I ) . The difference now, is that these open sets are obtained (after
applying a few operations ) from j(f)-' of the "generic" open sets. This, together
with the elementmity of j allows us to construct an open disjoint expansion of C =
{C; : i E I) from the open disjoint expansion of ( f - I (C;) : i E I) obtained through the
above described process, use the normality of Y and get an open discrete expansion.
Definition 2.4.9 Let I be a set and n a positive integer. An n-dowment for Fn(l ,2)
is a family Cn of finite subsets of Fn( I , 2) that satisfies the following conditions:
(1) For each maximal antichain A c Fn(l,2) there is L E L, such that L A.
(2) For any element p E Fn(I,2) with domain of size n and for any collection
{Ll, ..., L,) of elements of 13, there exists fl E Ll, ..., f, E L,, such that
{ p , fl, ..., fn) has a common lower bound.
In [8], i t is proved that there are n-dowments for Fn(l ,2) for every positive integer
n.
The following technical Lemmas are also needed:
Lemma 2.4.10 If A Y and A E V[q then M [ H ] C (j(f))-l(j(A)) = j(f-'(A)).
V[q C Vx,x E f - ' ( ~ ) iff 3y E A such that f(x) = y.
By the element arity of j, we have then :
M[H] + Vx, x E j(f-'(A)) iff 3y E j(A) such that j(f)jx) = y.
35
Lemma 2.4.11 I f z E ?rx-'(x) then j ( f ) ( + ) E r v - l ( f ( ~ ) ) .
Proof..
In V[q , let V be an open set such that f ( x ) E V . We need to prove that
M[H] j ( f ) ( z ) E j (V) . Now, f - '(V) is an open set containing x, so z E j( f -'(V)).
By Lemma 2.4.10 we then obtain that z E ( j ( f ) ) - ' ( j ( V ) ) and finally j ( f ) ( z ) E j (V) .
Lemma 2.4.12 [Ill Let Z be a compact space and U be a family of open subsets of Z
such that for every U, V E U 3W E U such that W U 17 V. Then U is an outer base
for nu.
Recall that outer base is defined in the APPENDIX, together with other concepts
mentioned in this chapter.
We can prove the Theorem now.
Proof=
Let C = {Ci : i E I) be a discrete family of closed sets in Y. Using the normality
of Y in V[q and that M[H] C j ( Y ) is a k-space, one uses the usual argument (see
Remark 2.3.3) to prove that:
M[H] t= {rY-'(ci) : i E I) is discrete in j ( Y ) .
Now M[H] C j ( Y ) is normal, so given h : I + 2 a generic function (h = U H)
there are I& open sets in j ( Y ) such that:
Define, open subsets of X in M [ H ] as follows:
As rrx is perfect both Wo and Wl are disjoint open subsets of X in M[H].
We also have that :
Proof: Suppose otherwise, then 32 E f -' (Ci) nrx (Zx \ ( j ( f ))-' (VhO) we have then
3y E Zx\(j( f))-'(Vh(i)) such that ?rx(y) = x. So y E rrx-' ( x ) and b y Lemma 2.4.11 we
havethat j ( f ) ( y ) E nv- ' ( f ( z ) ) . But f ( ~ ) E C; implies T Y - ' ( f ( + ) ) C TY-'(Ci) C &(i)-
Finally y E ( j ( f ))-'(TY-' ( f ( x ) ) ) c ( j ( f ) ) - I (h'hca) . Contradiction.
We have then:
Now we will define open sets about each f-'(Ci) in M [ q using the endowment
technique [8] :
Fix L2 a 2-dowment in P.
For y E f -' (Ci) , fix Ay a maxima3 antichain in P such that
V p E dy there is 6 open subset of X, y E V, and p ', Wh(i) .
Pick By Ay such that By E L2, and define Vy = npEB, 6. Define 0; = Uyej- l (Ci) Vy.
Remember that f is a pseudo-open mapping so we have that Ci i n t ( f ( 0 ; ) ) as
f - l (C i ) Oi.
Claim : Vi E I Ci n UjZi f (Oj ) = 0.
Proof: Suppose that x E Ci f~ U j f i f (Oj). As X is a kt-space there is K compact in
X such that:
Now 1 II- 3U, open set such that x E UI and j(U, fl K ) E Vh(i) , by Lemma 2.4.12.
To see that , consider U = ( j (U n K ) : U is open and x E CT) = j"((U n K : x E
U open )) and Z = j ( K ) and notice that nU = nx- ' (x) f7 j(K).
Pick A a maximal antichain in P such that Vp E A 3Up p IF j(U, tl K) G Vh(i).
Pick B C A such that B E La and define:
Now, by (**) , U n K n u j + i f ( O j ) n K # B
So
3 j # i such that U n K f l f ( O j ) # 0.
This implies
3y E f-'(Cj) such that U n K n f(V,) # 0.
Let p = (< i,O >,< j,l >) , and using that La is a Bdowment, find p, E
By and q E B such that p,,p, q have a common lower bound T .
Note that:
( 1 ) r Il- h(i) = O and h ( j ) = 1.
( 2 ) r Il- j ( U K) E Vo.
(3) r ll- V, E W l .
Pick z E U n K n f (V,). Then
3t E 1/, f (t) = z.
38
ives us Item(2) g*
r Il- j( f (t)) = j(z) G V,,
which implies
We then have that
As t E Vv , item(3) gives us that:
We have then:
(a) and (b) imply that (j( f ))-' (%) n ( j ( f))-' (W) # 0. Contradiction!
0
Having the claim , just defme fi = int( f (Oi) ) \ Ujgi f (Oj).
The claim above shows that Ci and the vs are obviously disjoint.
Alan Dow has very recently managed to simplify our machinery, leading to the hope
of applying it to more general classes of spaces.
2.5 Expandability in countably paracompact spaces
Balogh proves in [2] that:
Theorem 2.5.1 Suppose that in V , rc is a supercompact cardinal and P is either the
poset for adding K Cohen ~ea l s or K random ~eals . Let G be P-generic over V . Then
V[GJ C every countably paracompact space with E ( x ) = wl is expandable.
We will present a simpler proof of Balogh's result using the perfect mapping the-
orem. Using Theorem 1.3.1 we are able to do this by a simple modification of the
argument in [26] analogous to the modification we did for [8] and normality.
From now on X will denote a topological space (not necessarily regular or of small
pointwise type) with topology 7 and we will use the notation of the Theorem.
As usual, the proof has technical parts but the idea is very sirnilax to the one we
use to prove Theorem 2.3.2. First we note that if a topological space X is countably
paracompact after one adds "enough" Cohen reds (the statement of the lemma will
clarify the meaning of "enough" ), then X is expandable in the ground model (Theo-
rem 2.5.5). Recall that a similar result was used in the proof of Theorem 2.3.2 , that
is if the topological space is normal after the addition of "enough" Cohen reds, then it
is collectionwise normal in the ground model . So if X is countably paracompact and
C is a locally finite family of sets then j(X) is countably paracompact and contains a
locally finite family of sets V which is related to C via the perfect projection n. Count-
able "generic" partitions of D have open locally finite expansions, and a being perfect
gives us that countable "generic" partitions of C have open locally finite expansions (in
Theorem 2.3.2, a being closed assured us that the desired collection was normalized).
Theorem 2.5.5 finishes the proof.
We need the following results :
Definition 2.5.2 We call a space X 8-expandable if for every locally fitaite family
of subsets of X , L = (Li : i E I ) there is a sequence En = {Ein : i E I } of open
expansions ofC such that for all x E X there is an open set V containing x and n E w
such that V meets at most a jiraite number of elements of En. We require that for all
i E I , E~"+' C Ein.
Lemma 2.5.3 ('251) A countably paracompact 0-expandable space is expandable.
Lemma 2.5.4 ([23]) A space X is countably paracompact ifl every countable locally
finite family has a locally finite open expansion.
Proofs for the two previous Lemmas can be found in [lo]. The following Theorem is proved using a technique of Tall ([26] $71).
Theorem 2.5.5 Let Y be a locally finite collection of subsets of X . Add A Cohen reals
where card(X) 2 max(card(y) , card(X)) . If V [ q + X is countably paracompact, then
V j= Y is expandable.
Proof:
Throughout this proof the Cohen forcing will be represented by Fn(A, w, w). By a
result of Dow ([27]) , this partial order is n-dowed even if we make the extra restric-
tion that range(p) E n in the definition of n-dowed (Definition 2.4.9, item (2) and
Fn(X, w, w ) instead of F n ( l , 2 ) ) .
Let Y = {Y, : a < A) be a locally finite collection of subsets of X.
In V[q pick h = U G : X -t w the generic mapping and define
and
Clearly {Z, : n < w ) is a locally finite family, so that in V[q it has a locally finite
expansion {Wn : n E w}.
For every n < w pick an n-dowment L,. Also for every x E Y, Vy < X use that
1 Il- Y, Zh(,) Wh(-() and define:
A: = Ip E P : 3 4 open such that s E & and p Il- W h M )
Clearly AZ, P is dense. Pick B," E IS, such that Bz E A: and define:
Notice that all 0, are expansions of Y . Next we will prove the following claim which implies that the space is expandable
as it is countably paracompact.
Claim: {On : n E w ) is a Sexpansion for y. Proof:
Suppose that x E X is such that no 0, is locally finite for x. Pick p E P and U an
open set about x and m E w such that p IF u meets 4 m elements of {W, : n E w ) .
Let k = card(dom(p)) + sup(range(p)). Since we are assuming that U meets
infinitely many elements of (V,+k+l(7) : 7 < A) and since dom(p) is finite, we may
pick n, . . ., ym+l > max(m + k + 1 , lJ dom(p)) such that U n Vm+k+1 (y i ) # 0 aad choose
yi E Y, such that U hm+k+~ ( y i ) # 0. Define:
Clearly range(q) m + k + 1 , so we can use that for every 1 5 i 5 rn + 1 the set
B:+k+, E Lm+r+l and consequently pick pi E BE+,+, and s E P such that s 5 q and
' d , ' i l < i < m + l s ~ p ; .
Notice now that s IF h(7;) = k + i and that pi Ik hm+k+l ( y i ) E Wh(,) which
gives that s It- U fl Wk+i # 0 V i 1 5 i 5 m + 1. But as s 5 q 5 p we have that
s It- U meets m + 1 elements of { Wn : n E o). Contradiction!
0
As X is a @-expandable count ably paracompact space it is expandable and we have
the result.
0
We now need to prove Theorem 2.5.1. We will use the notation of Theorem 1.3.1
and its following paragraph.
Proof of Theorem 2.5.1:
Let C = CC, : 7 < A) be a locally finite family of subsets of X. Again, pick
h = U H : X + w a generic pastition of X and define Cn = {n-'(C,) : h(y) = n)
and Yn = UCn. Notice that by aa argument used in the discrete case, as X has
small type V = {a-'(C,) : 7 < A) is locally finite in j ( X ) . To see that, as in
Remark 2.3.3, one just needs to show that countable subcollections are locally finite.
If (C, : n E w } C C, as X is countably paracompact, there is a locally finite collection
of open sets O = {On : n E w ) such that for every n E w , Cn On. As in the discrete
case, rrB1(C,.,) & j(On) and j(O) = j"(O) is locally finite and so we have the result.
Consequently {Yn : n < w ) is also locally finite in j ( X ) and so it has an open locally
finite expansion {On : n E w).
Define now Vn = {C, : h(7) = n) and Dn = U Vn. Clearly T-'(D,) = Yn.
We can define now:
We obtain:
(1) Dn & W,: If x E Dn then T-'(x) r-'(D.) = Yn On. SO T-'(x)n j (X) \On =
0 which implies x @ a(j(X) \ On) and so x E Wn.
(2) Wn is open as T is closed.
(3) {W, : n E w ) is locally finite : Pick x E X, rr-'(x) is compact. So there is an
open set 0, n-' (3) 0 meeting just a finite number of elements of {On : n E w ) .
Define V = X \ n ( j ( X ) \ 0).
We will prove that V meets just a finite number of elements of {Wn : n E w):
Claim: If V n Wn # 0 then 0 n 0, # 8:
Pick y E V n Wn.
AS y E W, then y $ r ( j ( X ) \ On) which implies r-' ( y ) 0,.
As y E V then y 6 ?r(j(X) \ 0) which implies ?r-'(y) C 0.
So O n 0, # 0 and we have the result.
Repeat the proof of Theorem 2.5.5 for {Wn : n < w ) and we will have the result.
a
Notice that Theorem 2.5.1 is the only place in this thesis where we use the fact
that the mapping is perfect rather thas merely closed. It would be interesting to get
some class of topological spaces X for which we could obtain simply a closed mapping
between a subspace of j ( X ) and X, but we were unable to answer this question.
Theorem 2.5.1 is true for < 2'0 but there is a difficulty in proving n-' of a locally
finite collection is locally finite. We were unable to produce such a proof. Very recently,
A. Dow presented a proof that if h ( X ) < K = 2h and X is countably paracompact
then ?r-I of a locally finite collection is locally finite. The remaining steps of the proof
would be the same, making the result true for any countably paracompact space with - h < 2 N ~ . This provides a positive answer to Balogh's question on whether his result
could be extended to spaces with pointwise type < 2 N ~ .
Chapter 3
Reflection with elementary
submodels
3.1 Introduction
The definition of elementary submodel is in the APPENDIX. For a basic reference
about the subject see e. g. [19].
In the first section we will simply use elementary submodels as a tool to obtain a
forcing result concerning the preservation of normality after forcing.
Then we will consider the general following situation:
Definition 3.1.1 Let X , T be a topological space and M 4 He containing X and T .
We look at xM = x X M wdh the topology TM on XM generated by TM = (U n M :
U E T ~ M ) .
With this definition we are able to state the analogue of Theorem 1.3.1 in the
elementary submodel context:
Theorem 3.1.2 1181 Let (X,'T) be a regular space with h ( X ) 5 n and let M be a
elementary submodel of Ho such that ( X , 7) E M and such that n C M . Then there
is Y E X and rr : (Y ,V --+ XM such that n is perfect.
The mapping is defined as follows:
Let
K = {K C X : K is compact and x(K,X) 5 re ) ,
K , = { K E ~ ~ M : x E K ) , ~ o ~ x E X ~ ,
v , = { V ~ ~ n M : x ~ V ) , f o r x ~ X ~ .
For each x E XM define K, = n V,.
Note that, since X is HausdorfF, a simple elementary submodel argument shows
that if x, y E M and x # y, then K, n Kg = 0.
Define
Y = U{K, : s E X M ) ,
by
~ ( y ) = x if and only if y E Kz.
This Theorem is explored in the next section where we deal with the problem of
characterizing topological spaces X when we know that XM is a cardinal (ordinal).
We present the definition of T and not just the statement of the existence of a perfect
mapping since it will be useful later in the proofs.
The following is analogous to Lemma 1.1.4 and will be very useful:
Lemma 3.1.3 [I81 Suppose that M i s an elementary submodel, < X , T > E M is a
topological space and re C M . Iffor every x E X we have that x ( z , X ) 5 K , then TM is the subspace topology X n M gets fiom X . Moreover, following the notation of the
previous Theorem, Y = X n M and consequently T is the identity.
The third section presents the study of upwards reflection of topological properties
related to sequentiality . "Upwards reflection" means a property of XM transferring
to X.
Finally, we discuss the analogy between the large cardinal / forcing approach and
the elementary submodel approach.
3.2 Forcing over an elementary submodel
The question of preservation of normality in a non large cardinal context is explored in
this section. Here, the role of elementary submodels will be of a tool to prove the result
and not as arn essential part of the study as in the following sections. Some results in
this section appear in [15]. In this section we will prove:
Theorem 3.2.1 Let K be a regular cardinal. Adjoin tc Cohen reals, then for any a! < K,
normality of spaces of size < rc in which each point has character < K is preserved by
adjoining a! Cohen reals.
We could have stated Theorem 3.2.1 for spaces of pointwise type < rc. The two
formulations are equivalent as, for Hausdorff spaces of size < K, Z ( X ) 5 n implies that
every point has character < rc. To see this, note that for every compact Hausdorff
space K, w(K) 5 I K 1, so +(x, K) < rc which implies ~ ( x , K) < K as K is compact.
Now, apply Proposition 1.3.6 and get the result.
Although the proof has some technical details, the general idea is that we see the
addition of Cohen reals as an iteration. We will show that as the space is "small", it
appears in an earlier stage (the set and its topology). If the addition of a small number
of Cohen reds could kill its normality, the same would be true over this stage where
it appears. We finally use that non-normality is preserved by the addition of Cohen
reals to see that the space would not be normal in the final extension, which would
contradict the hypothesis.
We will follow the notation of and use the following result from [7]:
Proposition 3.2.2 Suppose P E M 4 He is a partial order and G is P-genem'c over
both V and M . Let a = rnin{r : T is a cardinal and T n M E 7).
Then
where
Notice that since M 4 He and P E M , G n M is P n M-generic over He. M n k, =
{ r : T is aP-narnein H,n M ) and ( ~ n k ) [ ~ n M ] = {ranM: T E ~ n & ) .
Proof of Theorem 3.2.1 :
Suppose that the theorem is false. Pick G P,-generic and, in V[q, pick (X, 23) a
counterexample (we work with a basis for the topology instead of a topology, as a basis
remains a basis after forcing even if it generates a new topology). We also suppose,
without loss of generality that X = E < n and B is a basis for 7 of size 5 E. We have a PKname B E V such that IB 15 6 and
V C "lrS Il-3 is a base for x and ( x , fi) is normal but forcing with Pa destroys its
normality. "
As IP, has the countable chain condition, we may suppose that 8 is of the form:
where each y is a nice P,-name for a subset of C. (We follow the notation of [20].)
Each ' r r ~ is of the form:
where each Af, is an antichain of P,, and consequently is countable.
Define
and let
Since K is a reguIar cardinal, ,O < K, and so we can pick y such that y < K , but
r > P,a.
Choose M 4 He (for B sufficiently large) such that I M I< n, M n n = p > 7, and
{n ,b ,P , , y )W c M.
As M 4 He we get that G is also P,-generic over M.
We get:
Applying Proposition 3.2.2 with Ipl+ _< n playing the role of a and n playing the
role of A we obtain:
By hypothesis we have that V[q C 3 ~ , K nice Pa-names for subsets of X (= t ) such that
Ip, It- F, k are closed disjoint non-separated subsets of X.
If H is Pa-generic over V [q , it is also over &[GI, and, since Pa, < X, 23 >, F, K E H, [q , we have that
HK[q C " lp, It- F, K axe closed disjoint unseparated sets."
Remember that 8 is a &-name and p > 7; consequently ( d ) ~ , = 23, and so
< X , B > E N. By elernentarity , we may take F, K E M n &[G,] and have
( M n a) [G,] I= "lp, Il- @, K axe closed disjoint unseparated sets."
Consequently there are P, and hence P,-names Q, R E M n k, such that (~)q = F
and (R )~ , ' = K.
4
Consequently
V C 30 < n such that Q, R are PU-names.
Since Q, fi E M and M 4 V, this o may be taken in M and so that o < p and
CT > a. Notice that if H is Pa-generic over ( M ~ & ) [ G , ] , it is as wellover (M~&)[G,],
where Gu = GpnPu. Using that G, = G n M = GnP, aad that Gu = G,,nP, = GnP,,
we can obtain H as above such that there is an I (see Chapter VIII in [20]) such that
As F is a nice name for a subset of c, we get that
k = ( ( 0 ) x A,: q < 6).
This implies
Notice that P, E M which implies (M n [G,] = (M n HK)[G,] 4 V[Gu], so L
can be defined in (M n H,)[G,] , and consequently can be taken in M. We can do the
same process for K and get a common L for both. Obviously, L < p. We may then take
a > ~ , p .
It follows that
This implies:
NI = (M n H.) [G,] [HI + (P)n, (k)n are not separated(**).
If they were, the separation would be in (M n H.)[G,][H], which contradicts what
we had above.
From (**) we obtain:
V[G,] [HI (k')H and( K ) ~ are closed disjoint nnsepaxated.
This is a contradiction to the fact that lP, Il- X is normal, since by Lemma 1.1.2 the
unsepasatedness is preserved after one adds Cohen reals, and by (*), V[q is a Cohen
real extension of V [G,] [HI.
0
One can actually remove the restriction a < IC in the previous Theorem.
Corollary 3.2.3 (A. Dow) Let n be a regular cardinal. Adjoin K Cohen reals, then
for any a, normality of spaces of size < rc in which each point has character < K is preserved by adjoining a Cohen reals.
Proof: We use the notation of the previous theorem. We work in V[q. Suppose
a, 2 R. We will prove that if there is an a 2 rc contradicting the result then there is
& < K that also contradicts the result.
Looking at the proof of the previous Theorem at some point we had:
V [ q C 3&', K nice Pa-names for subsets of X (= E ) such that
Ira IF P, K are closed disjoint non-separated subsets of X.
As 1x1 < tc by Lemma 2.2 in Chapter VIII of [20], there is an I rc with 111 < K
such that F and K are P = Fn(1, 2) names.
Now Fn(a, 2) is isomorphic to Fn(I , 2) x Fn(a \ I, 2). By absoluteness after forcing
with Fn(I , 2) the above sets are disjoint. By the fact that the basis of the topology
remains the same after forcing they are closed. If they were separated after forcing
with Fn(I,2), they would be separated after forcing with Fn(a,2), so in V [ q ,
1 ~ ~ ( ~ , ~ ) IF F , K are closed disjoint non-separated subsets of X.
Let & = 111. B < n and Fn(I,2) is isomorphic to Fn(b, 2). We obtain:
1Fn(6,2) I t - F , K are closed disjoint non-separated subsets of X.
3.3 Upwards reflection of ordinal spaces
In tbis section we research and provide answers to the question of what can we say
about X if we know that XM is homeomorphic (equal) to a cardinal (ordinal) in many
cases.
This is a particular step in the study of what caa we say about different XM7s for
a given topological space X, and if we know precisely how XM looks like what can we
say about X. We mean both questions not only in the topological sense but even with
regards to a mathematical structure related to the topology (algebraic , order...).
Previous works have addressed the questions above. In [21] the f i s t question is
explored in the case when X is the real line with its usual order and its field structure. In
[29] the second question is explored for a variety of topological spaces, e.g. uncountable
compact metric spaces, the reals , the rationals.
When appearing in this section, "n" will be the mapping described after Theo-
rem 3.1.2.
Theorem 3.3.1 Let K be an ordinal, < X , 'T > a topological space and let M 4 W ( 9 )
be an elementary submodel such that X , T, K E M. If XM = K. (meaning real equality
, not homeomorphism) then X = K .
Proof:
Notice that as XM = K:
(2) M Vx E X, x is an ordinal.
(3) M + Vx E X Vy E x we have that y E X.
(4) M + V A X A has an E -minimal element.
All of the above imply that M X is an ordinal, which implies that H(8) X
is an ordinal. Notice that this fact holds in the set sense, i.e. we still have to prove
that it is also true in the topological sense. Also notice that we used here that E is as
absolute relation, so the same arguments would not hold for "homeomorphic" instead
of 'Lequal''.
Claim 1 : For yl < 7.2 E K. we have that M + ( yl , .yz] is open in X.
Proof: Otherwise we would have M t= 3+y1, yz < K such that 32 E I =]yl, y2] \
W n , A)- (*I Pick 71,72, x E M satisfying the above sentence. We may say that x = y < K.
As I E M, x E I and XM = K , I is open in XM. So there is a V E TM such that
x E V n M I. Therefore M V is open, x E V and V 5 I. This contradicts (*)
and we have the result.
0
Claim 2: M + VV E 7 and V a E X with a E V there is /3 < a such that (P, a] E
v. Proof: Pick V E 7 n M. V f l M E Ti . Using that XM = M topologically we may
find ,8 < a such that (P, a] G V n M. As before (P, or] E M and so we have that
Ml=(P,a lG V. 0
Claim 1 and Claim 2 combined say that M + X is homeomorphic to an ordinal
and consequently H(8) X is homeomorphic to an ordinal. All we need to prove
now is that this ordinal is actually rcl.
Suppose not , then X = X > K . As tc E M and tc E M we would have X n M =
X n M which includes K + 1. This would contradict X n M = tc.
0
We will now state and prove a version of the Theorem when we have Uhomeomor-
phic" instead of "equal". As we do not have the topology on the space defined by an
absolute order , E, the technique is rather different.
First we need the following Lemmas:
Lemma 3.3.2 Let K be a cardinal , co f ( K ) 2 w l , with the topology induced by the
order and let Y K be such that with the subspace topology, Y is homeomorphic to K.
Then Y is a closed subset of K .
Proof: Fix h : Y + K a homeomorphism and notice that:
IY I = K as they are homeomorphic (*).
Suppose that Y is not closed. Pick a = min(8 : 9 E cl (Y) \ Y).
Claim :
ht'(Y n [0, a]) is bounded in K (w).
To see that, suppose otherwise. Then Cl = h"(Y n [O,d) is a closed unbounded
subset of K . In this case C2 = h"(Y \ [0, a]) is a closed subset of K which must be
bounded as cof ( K ) > wl and in this case two closed unbounded sets must meet and Cl
and C2 are obviously disjoint.
So C2 [O,P], which is compact and consequently Y \ [ O , a ] C h-'([O,P]) which is
also compact. So Y \ [0, or] C [0, 71 E K , for some 7 < K.
Notice now that this implies Y = ( Y n [0, a]) U (Y \ [0, a]) [0, mar (a, y)], which
implies IYI < K , contradicting (*).
cl
Now, let X = cof (a). Fix f : X + a a cofinal mapping.
By induction, we will construct a strictly increasing sequence a = {cl; : e < A) C Y
converging to a 4 Y with the following induction hypothesis:
(1) V/3 < A, {cu, : r < p) is strictly increasing.
(2) If p < A is a limit ordinal then crp = sup{& : E < P ) .
Choose a. E (f (0), a] n Y. If a. is chosen then:
a.+l is chosen in (a,, a] n (f ( E + I), a] n Y (* * *).
It remains to choose a@ for ,8 < X a limit ordinal. Notice that a@ = {a. : c < P ) is not closed in K . If it were closed, as it is bounded, it would have an upper bound in
9, and this would contradict the fact that it is strictly increasing. It has then a limit
point , this limit point has to be in Y by the minirnality of cu and as ,B < cof (a). We
choose cup to be this limit point. By (* * *), o converges to a.
Notice that by the way we constructed it, a U (a) is closed in n. So a is closed in
Y.
We have that h"(a) is closed in K and is bounded in n by (**). It is then a compact
subset of K . As h is a homeomorphism, a is compact in Y. As Y is a subspace of 6,
it is compact in n. This is a contradiction since it includes a sequence converging to a
point outside of it.
0
The condition that co f ( K ) 2 wl is a necessmy condition for Lemma 3.3.2:
Example 3.3.3 (A. Dow) Let Y = N,\(w). Observe that Y with the subspace topology
is homeomorphic to Nw but is clearly not closed in Nw.
We will define the homeomorphism f : H, ---+ Nw \ { w ) as follows:
3. If a = N n + l wheren EU, n > 1 then f(a) = n-1.
4. If a = N, + k where n , k E w, n 2 1, k 2 2 then ~ ( c Y ) = N, + k - 1.
5. Elsewhere, the function is the identity.
Lemma 3.3.4 (see [.7]) Suppose that M .is a n elementary submodel and X is a topo-
logical space such that < X , 7 > E M . If XM is compact then X is compact. Moreover
domain(.rr) = X.
The previous lemma is essentially proved earlier in this thesis in the supercompact
context (see Theorem 1.5.2).
Lemma 3.3.5 (see [29]) Suppose that M is a n elementary submodel and X is a topo-
logical space such that < X, 7- > E M. If XM is locally compact then X is locally
compact.
Lemma 3.3.6 (see [Zl]) Suppose that M is a n elementary submodel, X,Y E M and
Y E M . If 1x1 _< IYI then X M .
Theorem 3.3.7 Let M be a n elementary submodel, < X , 'J > a topological space and
n a cardinal with co f ( K ) 2 wl and such that a, X, T E M and also K M . If XM is
homeomorphic t o K, then X = XM and hence it is homeomorphic to K .
Proof: As XM is homeomorphic to K , it is locally compact. So by Lemma 3.3.5 X
is locally compact and we can use Theorem 3.1.2. There is , then, n : Z X + XM perfect. Fix f : K --+ XM a homeomorphism and call xu = f (a).
Then:
V a i rc 3V, E 7 n M such that x, E V, n M G ft'([xO, xu]).
Notice that:
Claim 1: V ]Val < tc. Otherwise M IV, I 3 n which implies M C 3B V, such that IBI = n.
Pick this B E M. As tc M y b y Lemma 3.3.6 we get that B C M. So B V, n M
has size tc which contradicts (*).
0
Claim 2 : V C Vx E X there is a V E 7 such that x E V and JVI < tc. It is enough to prove that M models the previous sentence.
Suppose otherwise, that :
M 3x E X such that VV E 7 such that x E V, IVI 2 n (**)
Pick this x E X tl M. There is some a such that x = x, and so the previously
defined V, contradicts (**).
0
Now,
V C Every point in X has a neighborhood of size < n. So V C Vx E X $(x, X) < n. Use now that X is locally compact and get that:
V + V X E X X ( X , X ) < K .
The previous sentence implies by Lemma 3.1.3 that XM = X n M has the subspace
topology inherited from X.
Claim 3 : X n M is open in X. To see that for x E X n M , use Claim 2 and get
V a neighborhood of x of size X < n. This V may be taken in M (as x E M) and so
by Lemma 3.3.6 we have that V M . Now, x E V C X t l M.
0
We will prove now that 1x1 5 n. Suppose otherwise that:
We obtain:
V 3Y open subset of X , which is homeomorphic to K and such that IX \ YI 2 n.
so,
V + 3Yopen subset of X, which is homeomorphic to n and such that 3T 2 X \ Y such that IT1 = K.
By elementarity,
M 3Yopen subset of X, which is homeomorphic to n and such that 3T X \ Y such that IT1 = K.
Pick those Y, T E M.
Again, by Lemma 3.3.6, T C X fl M and Y 2 X n M. In particular Y is open in
X n M .
Remember that X fl M = XM which is homeomorphic to K . So fix g : n ---+ X n M
a homeomorphism.
C = g-l (Y) is a subset of c homeomorphic to n. So it is closed in n by Lemma 3.3.2.
As Y is open, C is also open in K . So D = n \ C is a closed subset of n. Because
two clubs in K always meet (as cof ( K ) 3 wl), D is a bounded set in n having then
cardinality less than n. The contradiction comes from the fact that g-'(T) E D and
has size K. SO (* * *) is false which implies 1x1 = tc, and by Lemma 3.3.6 X E M , so
X = X n M and hence is homeomorphic to K .
0
The assumption that K M is a strong one; however the following two Theo-
rems will show that the condition that the cardinal be included in the model cannot
be eliminated from the hypothesis of Theorem 3.3.7 at least when tc is not weakly
inaccessible:
Theorem 3.3.8 Suppose that M is an elementary submodel, K is a cardinal and < X , 7 > is a topological space such that X , 7, n, n+ E M and n M but n+ M . If
XM is homeomolphic to n+ then X is not homeomorphic to rc+.
P~oof i
We have two cases. If 32 E X x(x,X) 2 R+ then trivially X is not homeomorphic
to s+. Suppose then that Vx E X , x (x ,X) 5 n. Now Vx E X , , X x , X ) 5 n implies
(by Lemma 3.1.3) that XM is a subspace of X and so Xn M and n+ are homeomorphic.
EX is homeomorphic to c+ , fix f : re+ + X = {x, : a < R+) a homeomorphism.
W e may pick f E M. Notice that Y = f-'(X n M) is a subset of n+ homeomorphic
to n+. By Lemma 3.3.2 Y is closed in rc+, which implies X n M closed in X .
As R+ M, X is not a subset of M (again just use that f E M). Pick now
7 = min{cr : x, 4 X n M ) . As f E M, we have that 7 is an infinite limit ordinal.
Now a = {x, : a < 7) X n M and x, E d(o) as 7 E &((a : CY < 7) and f is a
homeomorphism.
0
Theorem 3.3.9 Suppose that M is an elementary submodel, K is a cardinal with wl L
co f (n) = T < R and < X, 7 > is a topological space such that X , 7, n E M and T E M
but n M . If XM is homeomorphic to rc then X is not homeomorphic to n.
Proof: Again, we have two cases . If w ( X ) < n or w(X) > tc then clearly X is not
homeomorphic to c. We may suppose then that w ( X ) = n. Pick 23 = {Bp : < n) E
M a basis for X of size K . Fix f : T --+ K a strictly increasing cofinal map and define
f o r x E X n M :
VCY < T, CX,* = n { B P : P < f (a) and 5 E
Notice that as all variables are in M, C,,, E M.
Define also :
Notice that:
(2) Pz M ( as T M).
(3) n P' = {x) as if y # x then there is Bg such that y 6 Bg but z E Bg. Pick E
such that f (a) > ,O and notice that y 6 C,,,.
To finish the proof observe that:
So, again our usual a is a homeomorphism and the proof proceeds as in Theo-
rem 3.3.8. This is the one place in this Chapter where we actually use the particular
definition of a . 0
We next need
Theorem 3.3.10 [ . I ] ~f OX does not exist and IMI 2 tc then n C M .
Theorem 3.3.7 together with the previous theorem leads to the result:
Corollary 3.3.11 Suppose O# does not exist. Let M be an elementary submodel,
< X , 7 > a topological space and K cadinal with co f (6) 2 wl such that n , X , 7 E M .
Then i f X M is homeomorphic to n, then X is homeomorphic t o 6.
The reader may get information about O X in [16]. It is a special subset of w. Its
existence has many consequences, among them existence of large cardinals in inner
models. Also V = L implies O# does not exist, in fact the nonexistence of O# is
equivalent to Jensen's Covering Lemma for L [16], [5].
Any definable K such as wl, wz, etc.. is automatically in M. Also, the condition
that tc E M is sometimes a consequence of K M . This holds for instance for all
successor cardinals:
Proposition 3.3.12 Suppose that n = Np and P < n. If M is an elementary submodel
and K E M then n E M .
Proof: The proof is simple. We have that n = Np and P E M so Np is defmed in M
, so K; E M.
I7
We now present some examples:
Lemma 3.3.13 Let M be an elementary submodel and rc E M a cardinal with the
order topology 7. Then n~ is homeomorphic to ot(n n M ) , where by o t ( X ) we mean
the order type of X .
Proof: To see that, D = {[a, p] , [a, P ) : o c P < n ) is a basis for the topology of n
that lies in M. Because of that TM and & generate the same topology on KM.
Now [a, p ) E M if and only if a, p E M (the same for [a, PI). So if f : ot(n n M ) + nn M is strictly increasing and onto , it is a homeomorphism
between ot(n fl M ) and KM.
0
Example 3.3.14 A n elementary submodel M , and a topological space < X , 7 > E M
such that X is not homeomorphic to w but XM is homeomorphic to w.
Just pick X a discrete space of uncountable cardinality and M a countable elemen-
tary submodel such that X E M . In this case XM is a countable discrete space and
thus homeomorphic to u.
Notice that in the above example w E M, w M but co f (w) = w < w l .
If we are looking for an X such that XM is homeomorphic to wl but X # XM, by
Theorem 3.3.7 we need to find an M with wl g M. Such an M must have lwl MI countable and yet be uncountable since XM is uncountable. We thus assume Chang's
Conjecture in the following form:
There is an elementary submodel M such that IMI = WI, lwl n MI = No and
1wz n M I = N ~ .
Example 3.3.15 Assuming Chang5 Conjecture there 49 an elementary submodel M
, and a topological space < X , T > E M such that X is not homeomorphic to wl bat
XM is homeomorphic to wl.
Pick an elementary submodel M such that IMI = N 1 , lwl fl MI = No and 1 ~ 2 n MI =
Hl*
Observe that:
To see this, pick T < w2 E M . Then there is f E M a bijection between wl and T .
Becawe f E M , we have that IT n MI = Iwl n M 1 . Claim : The order type ofw2 n M is ~ 3 1 .
To see that suppose otherwise that $3 > u1 and g : ,6 + w2 n M strictly increasing
onto. Now f (wl ) < ~2 would contradict (*).
Pick X = wz with the order topology. By Lemma 3.9.19, XM is homeomorphic to
w1.
We now deal with ordinals:
Example 3.3.16 An elementary submodel M , an ordinal 7 and a topological space
< X , 7 > E M such that X is not homeomorphic to .y but XM is homeomorphic to y.
Pick M a countable elementary submodel of H ( 8 ) . Notice that in this case wl n M =
7 < wl (as w C M which implies that every countable element of M is a subset of M-
Lemma 3.3.6). Now by Lemma 3.3.13 ( ~ 1 ) ~ = y .
Notice that in the previous example +y 4 M.
We have two remaining questions:
I- What happens with Theorem 3.3.7 if K, is an uncountable cardinal with countable
cofinality? Because uncountable cofinality is necessary for Lemma 3.3.2 to hold (see
Example 3.3.3), we could not extend the result to this case and also we were not able
to provide a counterexample.
2- What can we say in the general case when y is an ordinal and not a cardinal?
Even if one could get a proof of Lemma 3.3.2 for ordinals, the serious difficulty of
producing "small" neighborhoods in the proof of the theorem (see Claim 1 in the
proof) would remain.
3.4 Upwards reflection of countable tightness, se-
quentiality and Fdchet
In (181 the authors provide an example of a topological space X, which is a Fk6chet
space and consequently a sequential space and a countably closed (needed in the proof)
elementary submodel M such that t ( X M ) > No. This example shows that countable
tightness and sequentiality do not necessarily reflect downwards.
In this section we will explore the other direction, upwards reflection.
Theorem 3.4.1 Suppose that M is an w-covering elementary submodel and that < X, 7 > E M is a topological space. Ift(x~) = No then t (X) = No.
Proof:
Suppose the conclusion is false. Then:
V C 3A E X and 32 E cl(A) but VB A, I BI < No, we have that x $ d(B) .
By element arity,
M 3A C X and 31 E d(A) but VB A, IBI 5 No, we have that x $ d(B).(*)
So we may take such A E M and x E M.
Claim: x E &,(A n M).
To see that, notice that if the claim is false then:
3 V c T n M suchthat x ~ V b u t V n A n M = 0 .
63
The above sentence is equivalent to:
which clearly contradicts (*).
0
As t ( X M ) = No , there is B A n M such that IBJ = No and x E cl,(B).
Using that M is w-covering, there is C E M, I CI = No and B E C.
Call D = C n A . Notice that D E M,D sA, IDI =No aad x ~ d , ( D n M).
To finish the proof note that V + a E cl(D). For otherwise
This implies,
Pick this V in 7 n M. The previous sentence tells us that V n D n M = 8. This
contradicts the fact that x E dTM(D f l M ) . So t(X) = No.
0
Theorem 3.4.2 Suppose that M is an w - closed elementary submodel and that <
X , T > E M is a topological space. If XM is sequential then X is sequential.
Proof:
Again, by contradiction, suppose that V + 3A sequentially closed but 3s E cl(A)\
A. By elementarity,
M + 3A sequentially closed but 32 E cl(A) \ A(*).
We again can pick such A and x in M.
Claim: A n M is sequentially closed in < XM, 7M >.
To see that, let
f : w --+ A n M be a convergent sequence to y E X n M(**).
(we mean convergence in the TM sense). Call B = { f (n) : n E w).
Notice that f w x B M and f is countable. So, using that M is w - dosed we
can conclude that f E M and B E M .
Notice now that
M + f : o ---+ B converges to y ( in the 7 sense ) .
To see that notice that otherwise M + 3V E 'T such that y E V and V n E w 3m > n such that f (m) c$ I/. Pick this V E T fl M. It contradicts (**).
So we have that
V + f : o + B C_ A converges to y ( in the 7 sense ) .
As V + A is sequentially closed then V + y E A. So y E A n M, which finishes
the proof of the Claim.
As XM is sequential , we have that A n M is closed in XM. Now M b x E d(A) \ A,
so M + W E 7, x E V implies V n A # 0. This, by the definition of TM means that
x E dTM(A n M) = A n M. So x E A n M A. This contradicts (*).
Theorem 3.4.3 Suppose that M is an w - closed elementary submodel and that < X,7 > E M is a topological space. If XM is fie'chet then X is fie'chet.
Proof=
Suppose that V + X is not a RBchet space. Then M + X is not a FkBchet space.
We get then,
M b 3A 2 X, 3s E X such that x E d ( A ) but no sequence contained in A
converges to x(*) .
Again, pick x aad A in M.
Claim 1: x E d, (A n M ) .
Otherwise 3V E T n M such that x E V asd V n A n M = 0. This is equivalent to
M + x E V and V f l A = 0, contradicting (*).
As XM is FEchet we cam pick f : w + A n M a sequence converging to x.
Now, M is w - closed, f S. M and f is countable so f E M.
As in the previous proof M + f converges to x ( in the 'T sense). We have then a
contradiction to (*).
0
It is important to notice that some restriction on the elementary submodel is nec-
essary for the above results to hold, one trivial example is:
Example 3.4.4 A topological space with uncountable tightness X and an elementary
submodel M such that XM is fie'chet.
Just pick any countable elementary submodel M and any < X,T > E M with
uncountable tightness. XM has a countable base so it is first countable and hence
fie'chet.
The condition that M is not only w-covering but w- closed required for Theo-
rem 3.4.2 and Theorem 3.4.3 is also (consistently) necessary. We thank Alan Dow for
providing the following example.
Definition 3.4.5 For two sets A and B, we say that A s* B i f A \ B is finite .
Definition 3.4.6 We say that a set A is a pseudo intersection of a family F if A G* F
for every F E F. We say that a family of countable sets has the strong finite inter-
section property - s.f.i.p. if every nonempty finite subfamily has infinite intersec-
tion.
Definition 3.4.7 p = rnin{lFl : .F is a subfamily of [w]" with s.8i.p. which has no
infinite pseudo intersection ).
The following is true about p:
Theorem 3.4.8 [3] For a cardinal n we have that n n p if and only if MAK for 0-
centered posets holds.
Lemma 3.4.9 (apparently folklore) If X is a countable topological space and w(X) =
K < p then X is a Fre'chet space.
Proof: Without loss of generality we may suppose that X = w.
Fix 8 = {B, : a < n} a basis for X and define for x E X:
a,= (B, E 8 : s E B,).
Suppose that A & X and x E cl(A). Notice that VB E &, IB n A1 = w. Call for
B, E a,:
and
= {A, : B, E a,).
Using that x E cl(A) and that B, is a local basis for x, we obtain that 3 is a
subfamily if [u]" with s.f.i.p. and as I&I 5 n, we have that 131 5 n.
Use that n < p and obtain B' s* A, for every A, E F. In particular B' c* A. c A,
which implies that there is as F finite such that B' \ F A. Define B = B' \ F and
notice that B C A and B C* A, for every A, E F.
It is easy to notice that B = (b , : n E w ) converges to 2, for just suppose that
B, f 23, is arbitrary, then B E* B, n A = A,, so there is m E w such that for every
n > m we have that b, E B, n A C B,.
0
Example 3.4.10 (p > wl ) A topological space X and an w-covering elementary
submodel M of size N1 containing X (you can get such a model fiom an elementary
chain of countable elementary submodels - see [6l) such that X is not sequential but
XM is Belchet,
Define X = w x w U { s ) , where points in w x w are open and neighborhoods of s are
of the form Vjk = {< m, n >: m > k and n > f (m)) U { s ) , where k E w and f E wW.
X is a countable space and IMI = N1 so w ( X M ) 5 wl. B y Lemma 3.4.9, XM is
a fie'chet space. X is not a sequential space as it does not have converging sequences
and it is not discrete.
To see that X does not have converging sequences, suppose that { x , : n E w ) is
a sequence in X , where xn =< a,, bn >. W e will prove that { x , : n E w) does not
converge to s.
W e have two cases:
Case I : Infinitely many an's are diferent.
Just pick f : w t w such that for those a , 3 , f (a,) = b + 1. V: does the job.
Case 2: There is a finite number of a, 5, say {anl, ..ank}
Pick k = max{anl , ..anr) + 1 and any f E wW, again Vt does the job.
One cannot get a ZFC example because L. Junqueira [17] has proved that, under
C H , all w -covering elementary submodels are w -closed.
3.5 Some remarks on the large cardinal versus the
elementary submodel approach and the space
Another way to reflect a topological space through an elementary submodel is with the
spaces of the form X ( M ) .
A general definition of X (M) appeass in [I81 :
Definition 3.5.1 Let < X , T > be a topological space and F be a (not necessarily
open) cover of X . Let M be an elementary submodel such that X , 7, 7 E M . For
every x, y E X , define x N y if and only i f x E V t-) y E V , for every V E Fn M .
Clearly - is an equivalence relation. We can then define the quotient
space
XF(M) = X/ -- .
Also let
V = ~ $ ~ : X + X ( M )
be the quotient map. We will denote by [ X I the equivalence class of x. We will suppress
the "F' in X(M) when it is clear from context.
The spaces X(M) and XM are related in some particular cases:
Theorem 3.5.2 [I81 Suppose X is a space with pointwise countable type. If x E
X n M and F = { K C X : Kis compact with countable character }, then y - x if and
only if y E K,. Thus XM is homeomorphic to a subspace of X( M ) .
In this section we will deal with a particular case that relates to the general frame-
work of this thesis in a way which will be explained after the definition.
Definition 3.5.3 Let X be a topological space and M a n elementary submodel such
that X, C*(X) E M. W e define the following equivalence relation o n X:
x -C.(X) y if and only if for every f E C * ( X ) n M we have that f (x) = f ( y ) .
W e will denote by [XI the equivalence class of x.
Definition 3.5.4 We define the topological space X(M)ce(x) as the quotient space of
X under the equivalence relation NC*(X) and ?r : X + X(M)c*(x) as the projection:
n (x ) = [XI .
Notice that if X is a Tychonoff space and F is a basis for X composed of functionally
open sets then both definitions coincide [18].
One can see the proof of the perfect mapping theorem using functions (Theo-
rem 1.2.1) as a X(M)cm(x) type construction and the proof of the perfect mapping
69
theorem in section 1.3 (Theorem 1.3.1 ) as a XM type construction where for a topo-
logicd space Y, j (Y) plays the role of X and Y plays the role of XM. Notice that in
the context of the perfect mapping theorem using functions, first a "kind of X(M)"
space is constructed and then it is proved that it contains a subspace, ("kind of XM9')
which is homeomorphic to X ( analogy to Theorem 3.5.2).
This analogy described above makes one wonder if everything that can be proved
using (or about) the perfect mapping theorem in the large cardinal context can be
proved in the elementary submodel context and vice-versa.
Although in some occasions the proofs of certain results in both contexts could not
be considered similar, we were not able to produce a statement that could be proved
in one context and refuted in the other context.
We give here an example of this difference between proofs. Remember that in
Theorem 1.5.8, we proved that in the large cardinal context the property of a topological
space X being a perfect image of a subspace of j (X) was productive. The same is true
in the elementary submodel context, however as it is not true that X,)M =
na<x(X,)M, an analogous proof would not be enough.
We need the following :
Lemma 3.5.5 Suppose that M is an elementary submodel, X E M and { X , : cr <
A) E M is a family of topological spaces (the family of the topologies is also in M ) .
Then (nu,* X,)M is homeomorphic to a subspace of n,EAnM(Xa)M
Corollary 3-5.6 With the above notation if each (X,)M (a E M ) is a perfect image
of a sabspace of X,, then (JJmCA Xcr)M is a perfect image of a subspace of naCx Xcr .
Pro0 f:
For a E X n M, each (Xa)M is EL perfect image of a subspace 2, of X,. As before
we have a perfect mapping (the product mapping) T : naExnM ZU --+ &EXnM(Xa)~.
Now (n,,, Xa)M is homeomorphic to a subspace B of naEAnM(Xa)~- To finish
the proof just notice that rr 1 n-'(B) is perfect and that naExnMZa is homeomorphic
to a subspace of X,.
0
Now we can prove the above stated lemma:
Proof of Lemma 3.5.5:
Define f : (&<A L)M -t I I , E A ~ M ( X ~ ) M as f ~ h w x
Clearly V a E M we have that x, E M as T E M so the mapping is well defined.
To see that i t is 1-1, pick 1 = ( X , ) ~ < ~ , Y = (yo)a,A E M . If f ( ~ ) = =(a) then
b'a E A fI M we have that x, = y,. So M + T = v, which implies that they are equal.
TO see that f is continuous pick I = (x,)aEAnu and V = n,EAnM\F(X, )~ x
naEF(K fl M ) an open set about Z where F is a finite subset of A f l M ( so F E M )
m d V , e T a n M .
Call W = n,ex\FX, x n,,,V,. W is an open set and also W E M. Notice that
f-'(V) = W, so f is continuous.
To see that it is open, call 7 the topology of n , , ~ X, and Y = f"((n,,, Xa)M).
Let V be a basic open set of TM.
V is of the form (&,A H,) n M where Ha) E 7 fl M, so:
Ha = Xu for a @ Iv,
and
Ha = V, for a E Iv and V, E Ta fl M,
where Iv E M is a finite subset of A.
Pick z E Y n f " ( V ) and define:
W = W, where for a E Iv we put W, = V, n M and otherwise W, =
X, n M.
Clearly -z; E W as f is a projection and Iv is a finite subset of M which is open in
n n E A n ~ ( ~ a ) ~ *
Claim: W n Y f"(V).
E W n Y implies 3 E Xc* and Vcr E X f l M, z, = y,. Clearly Z E V so
f (z) = g E Y ( V )
The claim finishes the proof. 0
Again, as in Remark 1.5.11 if we fix an elementary submodel M y and work with
topological spaces that are in M, the property of XM being a perfect image of a
subspace of X is both productive and an inverse invariant under perfect mappings (the
proof is analogous to the one in Section 1.5), so, again, a variety of spaces that cam be
constructed from spaces having this property also have this property (e.g. the Iliadis
absolute - see Remark 1.5.11).
To better explain the analogy between the construction in Section 1.2 and the
X ( M ) c * (x):
Proposition 3.5.7 If X is compact then X(M)o-(x) is compact and thus T y c h o n o ~ .
Proof: The proof is trivial: X ( M ) p is the continuous image of a compact space.
0
In Section 1.2, we start with j ( P X ) = P ( j ( X ) ) (here notice that C*(j(X)) is
isomorphic to C*(Pj(X)) ) and get (j(PX))(M), where the copy of X lies.
One has to be careful with the space X ( M ) as we shall see next. First a technical
lemma:
Lemma 3.5.8 if V X and V = f-'(I) and I E [O, 11, f E M, then x-l(lr(V))) = V.
Proof:
Clearly V E n-'(x(V))). If x E a-'(n(V))) then 3y E V such that x -c.(x) y. In
particular as f E M we have that f (x) = f (y) E I. So x E V.
0
Theorem 3.5.9 X(M)c*(x) is a Hawdor f f space.
Proof.-
Suppose [XI # [y]. Then 3f E C*(X) n M such that f (x) # f (y). Without loss of
generality, we may assume f (x) < f (y) and pick q, I rational numbers (consequently
elements of M) such that f (x) < q < I < f (y). Define V = f -' ( [0, q] ) and W =
f l ( , 1). By Lemma 3.5.8, and the definition of the topology on X(M)c*(x) both
r (V) and ?r(W) are open disjoint sets about [XI and [y].
0
In fact the following example gives us a regular non-nonnal topological space X
such that X ( M ) is compact.
Example 3.5.10 Let X be a regular topological space such that C*(X) contains only
constant functions (see Problem 2.7.1 7 in M). Clearly X cannot be normal or even
Fychonoff but any two points are equivalent, meaning that IX(M)c*(x) 1 = 1.
Notice that the behavior of X(M)c*(xl is pathological compared to that of X M ,
where all basic separation axioms (TI, Tz, T3) [18] reflect both upwards and downwards.
This comes from the fact that some of its equivalence classes may not contain a repre-
sentative that lies in M.
APPENDIX
I- Logical/Set theoretical definitions
Definition 1 M is a n elementary submodel of N (denoted by M 4 N ) if M N and
for every n < w and formula tp with at mos t n free variables and all .(al, ..., a n ) C M ,
Definition 2 Let 9 be a cardinal. Elo is the collection of all sets that have hereditary
cardinality less than 8 (informally, elements of He have cardinality less than 8, so do
elements of elements of He, elements of elements of elements of He, and so forth...).
For 8 big enough, He satisfies "enough" axioms of Set Theory, so it is a set that can
"represent" V, the collection of all sets. Using He rather than V avoids some technical
problems - see [lg], [21], [29] for amplification of this remark.
Definition 3 W e say that an elementary submodel M is w-closed or countably closed
if every countable subset of M is a n element of M .
Definition 4 W e say that a n elementary submodel M is w-cowering if for every count-
able A C M , there is a countable B E M such that A s B.
Definition 5 W e say that j : V --+ M is a n elementary embedding i f f f o r every n < w
and formula cp with at most n free variables and all { a l , ..., a,) V,
Definition 6 W e say that K is the critical point of a n elementary embedding j : V --+ M i f rc = min({a : a is a n ordinal and j (a) # a)).
If j is not the identity such rc exists and is a cardinal.
Definition 7 W e say that a cardinal rc is supercompact i f for every cardinal A there
exists an elementary embedding j : V ---+ M such that :
1. K is the critical point of j .
2. j ( n ) > A.
9. M" M (M is closed under A- sequences).
For all basic definitions on forcing we refer the reader to Chapter VII in [20].
Definition 8 If P is a partial order consisting o f f inc t ions and G i s a P-generic filter
over V , we say that h = U G is a generic mapping.
It is not difficult to see that if P is a partial order consisting of functions and G is
P-generic over V then U G is indeed a function.
11-Topological definitions
A more elaborate discussion on the concepts introduced can be found in [9].
Definition 9 Let X be a topological space and let x E X . x(z ,X) is the smallest
cardinal K such that there is a local base for x in X of size rc. W e define x(X) =
sup({x(x, X) : x E x)). If x(X) = o we say that X is first countable.
Definition 10 Let X be a topological space and Y C X . A family 23 of open sets all
including Y is called an outer base for Y if for every open V 2 Y, there is a B E B
such that B C V . The character of Y in X is x(Y, X) = rnin1.r : there is a n outer
base for Y of cardinality 7).
Definition 11 For X a topological space, the pointwise type of X , denoted by Yx) , is the least cardinal K such that X can be covered by compact sets, each of which has
character less than n. If E(X) = HI, we say that X is of pointwise countable type. W e
define h ( X ) as the least cardinal K such that X can be covered by compact sets, each
of which has character less than or equal t o K .
Some examples of spaces of pointwise countable type are locally compact spaces,
first countable spaces and ~ e c h complete spaces.
Definition 12 W e say that a topological space X is a k-space i f for every subset A
of X, A is closed in X if and only i f An K is closed in K, for every K compact subset
of x.
Definition 13 A topological space X is normal i f for every x E X , -(x} is closed in
X and for every pair of disjoint closed subsets of X , F, T , there are disjoint open sets
V, W (in X ) such that F 5 V and T W . W e say that X is hereditarily normal if
every subspace of X is normal.
Definition 14 W e say that a family F of subsets of a topological space X is normalized
if for every partition of .F into two subcollections 2 and 3.1, there exist two open disjoint
subsets of X , V and W , such that U 2 E V and U3C C W .
Definition 15 If .F = {Fa : a < A} is a family of pairwise disjoint subsets of X ,
we say that .F is separated if there is a family of open pairwise disjoint subsets of X ,
W = {W, : cr < A ) such that Fa W, for every cr < A. In this case we say that W is
a separation of F.
Definition 16 W e say that a family of subsets of a topological space X is discrete if
for every x E X there is an open set V such that x E V and V meets at most one
element of F.
Definition 17 W e say that a family of subsets of a topological space X is l ocdy finite
if for every x E X there is an open set V such that x E V and V meets at mos t a finite
number of elements of F.
Definition 18 W e say that a family of subsets of X , {W, : a < A) is a n expansion
of another family {Fa : a < A) if Fa W, for every a < A.
Definition 19 W e say that a topological space X is collectionwise normal if every
discrete family of subsets of X has a discrete expansion consisting of open sets (open
expansion).
If X is normal, separated discrete collections have discrete expansions.
Definition 20 W e say that a topological space X is expandable if every locally finite
family of subsets of X has a locally finite expansion consisting of open sets (open
expansion).
Definition 21 A topological space X is Frkchet if for every x E X and A X , x E &(A) is there is a sequence a C A such that cr converges to x.
Definition 22 A subset S of a topological space X is sequentially closed if every con-
verging sequence which is a subset of S converges t o a point in S. A topological space
X is sequential if every sequentially closed subset of X is closed.
Definition 23 If X is a topological space we define the tightness of X , denoted by t ( X )
to be the smallest cardinal X such that if x E X and A X are such that x E cl(A),
then there is B & A with IBI 5 A such that x E d(B).
Notice that if X is Fr6chet then X is sequential and t ( X ) = No.
Definition 24 W e say that p : X + Y is a perfect mapping if it is continuous,
closed, onto and inverse images of points are compact.
Definition 25 W e say that a topological space is countably paracompact if every count-
able open cover of X has a n open locally finite refinement.
Bibliography
[I] Arhangel'skii, A. Bicompact sets and the topology of spaces, ' h a s . Moscow Math.
SOC. 13 (1965) 1-62.
[2] Balogh, 2. O n the collectionwise nomnality of locally compact normal spaces,
Trans. Amer. Math. Soc. vol323 (1991) 389-411.
[3] Bell, M. On the combinatorial principle P(c), Fund. Math. 14 (1981) 149-157.
[4] Daniels, P. Normal k'-spaces are consistently collectionwise normal, Fund. Math.
138 (1991) 225-234.
[5] Devlin, K. ; Jensen, R. Marginalia to a theorem of Sdver, ISILC Logic Cod.
Lecture Notes in Math. 499, Springer Verlag, (1975) 115-142.
[6] Dow, A. A n introduction to applications of elementary submodels t o Topology,
Top. Proc. 13 (1988) 17-72.
[7] Dow, A. Set Theory in Topology, Recent progress in General Topology , North
Holland (1992) 167-197.
[8] Dow, A. ; Tall, F. ; Weiss, W. New proofs of the consistency of t h e Normal Moore
space conjecture I, Top. Appl. 37 (1990) 35-51.
[9] Engelking, R. General Topology, Heldermann Verlag 1989.
1101 Fleissner, W. Normal Measure Axiom and Balogh's theorems, Top. Appl. 39
(1991) 123-143.
[ll] Fleissner, W. Theorems f i o m Measure Axioms , Counterexamples from o++, The
work of Mary Ellen Rudin, Ed. Franklin D. Tall - Annals of the New York Academy
of Sciences vol 705 (1993) 66-77.
[12] Fleissner, W. If all normal Moore spaces are metrizable, then there is a n inner
model with a measurable cardinal, Thns. Amer. Math. Soc. 273 (1982) 365373.
[13] Fleissner, W. ; LaBerge, T.; Stanley, A. Killing normality with a Cohen real,
Top. Appl. 72 (1996) 173-181.
[14] RoloBk, 2. Topologically complete spaces, Comment. Mathematicae Univ. Caroli-
nae 1,3 (1960) 3-15.
[15] Grunberg, R. ; Junqueira, L. ; Tall, F. Forcing and Normality, Top. Appl. 84
(1998) 145-174.
[16] Jech, T. Set Theory, Academic Press 1978.
[17] Junqueira, L. Upwards preservation by elementary submodels, preprint.
[18] Junqueira, L. ; Tall, F. The topology of elementary submodels, Top. Appl. 82
(1998) 239-266.
[I91 Just, W. ; Weese, M. Discovering m o d e m set theory. 11, Graduate Studies in
Mathematics, 18 , American Mathematical Society (1997).
[20] Kunen, K. Set Theory - An Introduction t o Independence Proofs, North Holland
1980.
[21] Kunen, K. ; Tall, F . The real line in elementary submodels of Set Theory, Journal
of Symbolic Logic - to appear.
[22] Kunen, K. ; Vaughan, J. Handbook of set-theoretic topology, North Holland (1984).
[23] Mansfield, M. O n countably paracompact norrnal spaces, Canada J. Math. 9
(1 957) 443-449.
1241 Porter, J. ; Woods, R. Extensions and absolutes of Hausdorff spaces, Springer
Verlag (1987).
[25] Smith, J. O n B - expandable spaces, Glas. Math. 31 (1976) 335-346.
[26] Tall, F. Countably paracompact Moore spaces a n metrizable in the Cohen Model,
Top. Proc. 9 (1984) 145-148.
[27] Tall, F. Cowigendwn to "Countably paracompact Moore spaces are metrizable in
the Cohen Model", Top. Proc. 10 (1985) 175-176.
(281 Td, F. Normality versus collectionwise normality, Handbook of set-theoretic
topology (1984) 685-732.
[29] Tall, F. If it looks like and smells like the rea ls..., preprint.