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[1]\textbf\textitKeywords: #1
Countably compact groups and sequential order
Dmitri Shakhmatov, Alexander Shibakov
Abstract
We use ♢ to construct, for every α≤ω1 a
sequential countably compact topological group of sequential order
α. This establishes the independence of the existence of
sequential countably compact non Fréchet
groups from the usual axioms of ZFC and answers several questions of
D. Shakhmatov.
1 Introduction and notation.
The standard definition of a topological space contains no reference
to convergence, sequential or otherwise. In common mathematical practice,
however, convergence often appears as a valuable tool in many
application areas for topology, such as analysis. In an effort to
formalize the relationship between the sequential convergence and
topology, the class of Fréchet spaces was defined in which the
closure operator is directly described in terms of limits of
convergent sequences. Later, a more general class of sequential
spaces was introduced in [4] to encompass all the topological
spaces in which convergent sequences fully describe the topology (see
below for the definitions of these and other concepts used in the
introduction).
The addition of an algebraic structure appropriately coupled with the
topology (by requiring the operations to be continuous)
imposes a number of restrictions on both the topology and the
convergence. Thus separation axioms T1–T321 are
equivalent in topological groups, as are metrizability and the first
countability axiom (see [1]). The investigation into
sequential topological groups was started by V. Malykhin, P. Nyikos,
I. Protasov,
E. Zelenyuk, and others (see [23], [13] and the
references therein) in
the 1970s and 1980s. In the following decade, a number of questions about
sequential topological groups (more generally, about ‘sequential
phenomena’ in groups) had been stated, that have guided the future
development of this field.
The survey paper [14] (see also [15]) presents a
fairly complete
overview of the state of the art in the study of convergence in the
presence of an algebraic structure.
One of the problems posed in [14] (Question 7.5) is the existence of a
countably compact sequential group that is not Fréchet. This
question is stated for precompact and
pseudocompact sequential groups, as well. It was shown in [17]
that it is consistent with
the usual axioms of ZFC that there are no such countably compact
groups. The goal of this paper is to establish the consistency of the
existence of such groups thus showing that such existence is
independent of ZFC. Note that every compact sequential (or
even countably tight) topological group is metrizable (see [1]).
Another question in [14] deals with the measure of (sequential)
complexity of the closure operator in sequential groups, called the sequential order (see below for a definition). Namely,
Question 7.4 asks if countably compact (pseudocompact, precompact)
sequential groups of arbitrary sequential orders exist, consistently, or otherwise.
A similar question was asked by P. Nykos in [13] about the class
of all sequential groups. The existence of sequential groups with
nontrivial (∈{0,1,ω1}) sequential orders was shown to
be independent of ZFC in [17] and [18]. The example in
this paper (see Theorem 1) thus establishes that the answer
to Question 7.4(iii) from [14] (existence of countably compact
groups of arbitrary sequential orders)
is likewise independent of ZFC (for the classes of pseudocompact and
precompact groups it is still an open question whether it is consistent
that there are no such groups of sequential order ω1).
We use the standard set-theoretic notation, see [10]
and [1]. All spaces are assumed to be regular unless stated
otherwise.
Abusing the notation somewhat we write σ−1=σ′ where
σ≥1 is a successor ordinal and σ′+1=σ or σ=0
and σ′=−1 (we do not go as far as call −1 an ordinal though). For brevity,
we use the term increasing to mean non decreasing, and use
strictly increasing when a stronger condition is assumed.
Define α≤Lβ as α<β for a limit β and
α≤β otherwise. Note that if α is a successor
α≤β is equivalent to α≤Lβ.
Our notation for various ordinal invariants is more detailed than customary
as we must frequently keep track of several topologies on the same
space. Whenever the topology is clear from the context we omit it from
the notation as well. The following
definition is the starting point for most arguments about convergence.
Definition 1**.**
Let (X,τ) be a topological space, A⊆X.
Define the sequential closure of A,
[A]0τ=[A]τ={x∈X:∃S⊆A,S→x}.
Given an ordinal σ>0, define [A]στ=[[A]σ−1τ]τ if σ
is a successor and [A]στ=∪σ′<σ[A]σ′τ otherwise.
Sequential topological spaces may be defined as exactly those (X,τ) in which for every
A⊆X there exists a σ such that Aτ=[A]στ. It is a quick argument
to show that in such spaces σ≤ω1.
Definition 2**.**
Let (X,τ) be a sequential space, A⊆X, and x∈X.
Define so(x,A,τ)=inf{σ:x∈[A]στ}∪{ω1}.
As a quick observation, if x∈Aτ then
so(x,A,τ)=σ<ω1 is a successor
ordinal, and whenever σ>0 there are xn∈A such
that xn→x in τ and so(xn,A,τ)=σn is an
increasing (non decreasing) sequence of ordinals such that
σn<σ and σn→σ−1.
The central ordinal invariant in the study of convergence can now be defined as
follows.
Definition 3**.**
Let (X,τ) be a sequential space. Define the sequential order of X,
so(X,τ)=sup{so(x,A,τ):A⊆X,x∈Aτ}.
The construction below depends heavily on the algebraic properties of
the underlying group.
Recall that a group G is called boolean if a+a=0G for any a∈G.
All such groups are abelian and may be viewed as vector spaces over F2. One can thus
consider (linearly) independent subsets of G in the usual sense. If A⊆G
by ⟨A⟩ we denote the span of A in G. A convenient property of
boolean groups that is used without mentioning below is that a+b=a−b
for any a,b∈G for a boolean G.
The following construction will be used often.
Definition 4**.**
Let G be a boolean group, S⊆G be an independent
subset. Define the even subgroup of ⟨S⟩ (relative to S) as ⟨2S⟩=⟨S+S⟩.
Under most circumstances, the set S will be clear from the context and the ‘relative to S’
part will be omitted. We assume below that all groups are abelian
unless stated otherwise, although a number of statements hold under
more general assumptions.
Lemma 1**.**
Let G be a topological group, A,B⊆G. Let
a∈[A]α and b∈[B]β for some
α,β<ω1. Then a+b∈[A+B]max{α,β}.
Proof.
We may assume that both α and β are successor ordinals.
If max{α,β}=0 then a∈A and b∈B so
a+b∈A+B=[A+B]0.
Suppose the Lemma holds for all successor α′,β′ such that
max{α′,β′}<γ. Let so(a,A)=α and
so(b,B)=β, max{α,β}=γ. Pick bn→b and
an→a such that so(an,A)=αn, so(bn,B)=βn
where αn<γ, βn<γ are such that
αn→α−1 and βn→β−1. By the hypothesis
an+bn∈[A+B]max{αn,βn}=[A+B]γn where
γn<γ. Thus a+b∈[A+B]γ.
∎
Lemma 2**.**
Let G be a topological group and K⊆G be a sequentially
compact subspace. Let P⊆D+K. Then for any
α<ω1 [P]α⊆[D]α+K.
Proof.
Suppose the lemma holds for all α′<α and let so(x,P)=α. If α=0 the argument is trivial, otherwise, there
are xn=dn+gn such that xn→x, dn∈D, gn∈K and
so(xn,P)=αn<α. By the hypothesis, we may assume that
xn=dn′+gn′ where so(dn′,D)≤αn and gn′∈K. Using
the sequential compactness of K we may assume (after passing to a
subsequence if necessary) that gn′→g for some g∈K. Thus
dn′→d=x−g and so(d,D)=α′≤α.
∎
Corollary 1**.**
Let G be a topological group and K⊆G be a sequentially
compact subspace. Let P={pn:n∈ω} and
pn=an+dn+dn where an∈K,
dn→d, and g∈[{di:i∈ω}]σ.
Then there exists a p∈[P]max{σ,1} such that p∈g+d+K. If an→a one may assume that
p=a+g+d.
Proof.
Put D={dn+dn:n∈ω} and apply Lemma 1 to show that
g+d∈[D]max{σ,1}. Now note that D⊆P+(−K) and use Lemma 2 (with the roles of P and D
reversed) to find a p∈[P]max{σ,1}
and an a∈K such that p=g+d+a.
If an→a note that
p=a+g+d∈[P]max{σ,1} by
Lemma 1.
∎
The next concept is used to build approximations of the sequential
group topology.
Definition 5**.**
Let (X,τ) be a topological space. Then (X,τ) is called kω
if there exists a countable family K of subspaces of X such that
U∈τ if and only if U∩K is relatively open in K for
every K∈K. We say that τ is determined by K and
write τ=kω(K).
A rich source of kω topologies on X is provided by the following
well known construction. Let K be a countable family of subsets of X such
that each K∈K is endowed with a compact topology τK. Define
a new topology τ=kω(K)={U:∀K∈K(U∩K)∈τK}. It is easy to see
that such τ is automatically kω. The construction above makes
sense for uncountable K as well. We will use kω(K) to denote
the appropriate topology even though for an uncountable K, the
topology kω(K) is not necessarily kω.
Note that for an arbitrary K such τ is not guaranteed to be
Hausdorff (although it is always T1 provided each τK is), nor does
τK necessarily coinside with the topology inherited by K from
τ. Both of these properites are readily ensured by starting with
a countable family K of compact subspaces of X in some (not
necessarily kω) topology τ′ on X and taking τK to be
the appropriate subspace topology.
Group kω topologies have been well studied and are useful building
blocks for various examples of sequential groups (see, for
example [14] and the
references therein).
The following simple lemma demonstrates a straightforward way to build
kω group topologies (see [21], Lemma 4 for a proof of a more
general statement).
Lemma 3**.**
Let G be a boolean topological group and K be a countable family
of compact subspaces of G closed under finite sums. Then kω(K)
is a Hausdorff group topology on G. Moreover, kω(K) is the
finest group topology on G in which each K∈K remains compact.
2 Basic definitions.
Most known constructions of nontrivial sequential spaces use a
technique that separates the analysis of convergence from that of
the topology. Among the tools used to deal with convergence, the
standard test spaces (such as the sequential fan S(ω),
Arens’ space S2, Archangel’skii-Franklin space Sω, etc.) feature
prominently. Additional ordinal invariants, such as the
Cantor-Bendixson index, scatteredness rank
(see [24], [18]), etc. are often used to bound the sequential
order of ‘intermediate spaces’ in the construction.
To outline the reasons why such methods
are of limited utility for the problem in this paper consider the
following argument.
Lemma 5 in [21] states that given a kω boolean group G and a
closed discrete subset D⊆G one can find a coarser
kω-topology on G in which D has a limit point. This suggests
the following brute force strategy for building a countably compact
sequential group that is not Fréchet.
Consider a subgroup G of
2c (algebraically) generated by a subspace homeomorphic to a
compact sequential space K of sequential order ≥2 (for example, the
one point compactification of the well-known Mrowka’s space
ψ∗). Endow G with the natural kω topology determined by the
family K of all the iterated sums of K.
Recursively
(using an appropriate set-theoretic principle such as
♢, see [21] for details of similar constructions)
add new convergent sequences using the lemma from [21]
mentioned above to make G countably compact. The topology determined
by all the added compact subspaces together with the original family K
will be sequential and countably compact providing the desired example.
It is instructive to see why the naive approach above fails. If such a
group topology τ on G existed it would be easy to find a
quotient G′ of G such that the quotient topology on G′ has a
countable pseudocharacter. As was shown in [21] G′ will
then necessarily be countable (at least with the K chosen
above) thus yielding a contradiction, since G′ must be
countably compact. In fact, it is still an open question whether it is
consistent with the axioms of ZFC that a countably compact sequential
group may contain a compact subspace of sequential order ≥2.
A more detailed analysis of countably compact (boolean) groups helps
to reveal the main source of difficulties with the approach above.
Recall that a topological group G is called precompact (or
totally bounded) if it
can be embedded as a subgroup in some compact group. The following
lemma follows from Pontryagin’s duality for compact abelian groups and
the well-known characterization of precompact groups. Since this
result will never be used directly, its proof is omitted.
Lemma 4**.**
Every countably compact group is precompact. Every precompact (thus
every countably compact) boolean group has a linear topology,
i.e. has a base of neighborhoods of [math] consisting of (clopen) subgroups of
(necessarily) finite index.
The construction in [21] that ‘forces’ D to aquire a
limit point adds a single convergent sequence to the original topology
of G along with all of the iterated sums of such sequence.
Unless some precautions with the choice of the new
convergent sequence (as well as its limit) are taken, it may easily
destroy the
precompactness of any topology on G compatible with the
kω-topology.
The construction in this paper uses two separate topologies, one to
deal with the convergence, and the other to ensure that the limit space is a topological
group: a kω topology and a coarser precompact first countable
metrizable topology (see Definition 7 below), respectively. In order to
keep the topologies compatible throughout the construction, instead of
using the standard test spaces to estimate the sequential order of
points, estimates of the sequential order of some homogeneous
countable subspaces are used instead (the odd subspace, see
Definitions 4 and 11).
In the arguments below it will be convenient to consider kω groups
together with the countable family of compact subsets that determine the
topology. We therefore introduce the following shortcut.
Definition 6**.**
Call (G,K) a kω-pair (with respect to τ)
if (G,τ) is a boolean topological
group with the kω topology τ and K is a countable family of
compact subspaces of G closed under finite sums and intersections
such that τ=kω(K) and ∪K=G.
In almost every case, the existence of the Hausdorff topology τ
will be clear from the context and will not be discussed while
referring to a kω-pair.
As was noted above, any inductive construction of a countably compact
group must provide a mechanism for ensuring the precompactness of the
final topology. The following definition is used throughout
and forms one of the building blocks of the
construction.
Definition 7**.**
Call (G,K,U) a convenient triple if G is a boolean
group, U is a countable family of subgroups closed under finite
intersections that forms an open base of neighborhoods of [math] in some
Hausdorff precompact topology τ(U) on G, and K is a
countable family of compact (in τ(U)) subgroups of G closed under finite sums
and intersections such that ∪K=G.
Trivially, if (G,K,U) is a convenient
triple then (G,K) is a kω-pair (with respect to
kω(K)) and every K∈K is metrizable.
A number of arguments involve translating various subsets by compact
subspaces. The definition below lists a few ordinal invariants that
measure the effects of such shifts.
Definition 8**.**
Let (G,K) be a kω-pair, A⊆G.
Suppose D⊆G is countable and there exist an
α<ω1 and K∈K such that
D⊆[A]α+K. Call the smallest α with this
property the K-depth of D over A and write
hAK(D)=α. If no such α exists hAK(D) is
defined to be ω1.
Let C={Si:i∈ω} be a family of subsets of G. Call
m∈ω the K-depth of D over
C and write hCK(D)=m if m∈ω is the smallest
such that D⊆∑i≤m⟨Si⟩kω(K)+K for some
K∈K. If no such m exists hCK(D)=ω.
Let 0∈Pkω(K) for some P⊆G. Call
m∈ω the asymptotic K-depth of P over
C and write ωhCK(P)=m if m∈ω is the smallest
such that 0∈P∩(∑i≤m⟨Si⟩kω(K)+K) for some
K∈K. If no such m exists ωhCK(P)=ω.
The following property is an immediate corollary of the definition above.
Lemma 5**.**
Let (G,K) be a kω-pair, A⊆G, and
P,D⊆G be countable subsets. Let P⊆D+K for some
K∈K. Then hAK(P)≤hAK(D).
Note that adding or removing finitely many
points does not change the K-depth of a set. Combined with the
lemma above it follows that D⊆∗D′ implies
hAK(D)≤hAK(D′).
We now define some minimality properites of sets with respect to their K-depth.
Definition 9**.**
Let (G,K) be a kω-pair, A⊆G, A={Ai:i∈ω} be a family of subsets of G, and
D⊆G be a countable subset. Let
hAK(D)=δ≤ω1. Define
{labeling}M(A,K)
for any β<δ, K∈K there exists a
finite FK,β⊆D such that
[TABLE]
{labeling}
M(A,K)**
for any n∈ω, K∈K there exists a
finite FK,n⊆D such that
[TABLE]
By choosing FK,β=FK,n above for any K∈K and any
β<ω1 one shows that M(A,K) implies
M(⟨An⟩,K) and h⟨An⟩K(D)=ω1 for every
n∈ω. Also note that both properties imply that ⟨D⟩∩K
is finite for every K∈K so ⟨D⟩ is closed and discrete in kω(K).
The following lemma will be part of most ‘thinning out’ arguments
below.
Lemma 6**.**
Let (G,K) be a kω-pair, D′⊆G be
infinite, closed and discrete in kω(K), and let A={Ai:i∈ω} be a family of subsets of G.
There exists an
infinite independent D⊆D′ such that ⟨D⟩ is closed and
discrete in kω(K) and D satisfies M(Ai,K) for every
i∈ω. If hAK(D′)=ω then D can be
chosen to satisfy M(A,K).
Proof.
Passing to a subset if necessary, assume that D′ is countable.
Let hA0K(D′)=σ0≤ω1. Suppose
0<σ0≤ω1 and pick σn0→σ0 if
σ0<ω1 is a limit ordinal and σn0=σ0−1
otherwise. If σ0=ω1 let σn0=ω1.
Select points dn∈D′ so that
[TABLE]
If the recursion terminates at some n∈ω then
D′⊆[A0∪{0}]σn0+K for some K∈K. If
σn0<ω1 then hA0K(D′)≤σn0<σ0
contradicting the choice of σ0. If σn0=ω1 then
σ0=ω1. Since D′ is countable there is a
σ′<ω1 such that D′⊆[A0]σ′kω(K)+K
contradicting hA0K(D′)=ω1.
Let D0={dn:n∈ω} and D⊆∗D0. Let
K=Ki′∈K, β<σ0,
σm0≥β. By picking i≥i′ large
enough if necessary we may assume that ⟨D∖D0⟩+K⊆∑i′<iKi′. Let n>max{i,m}. Put FK={dj:j<n} and let d∈⟨D⟩∖⟨FK⟩. Then
d=dn′+d′+a where n′≥n, a∈⟨D∖D0⟩, and
d′∈⟨{dj:j<n′}⟩. Now
dn′∈∑i′<iKi′+d′+[A0∪{0}]βkω(K) so d∈K+[A0∪{0}]βkω(K). Thus ⟨D⟩∩K is finite for
every K∈K so ⟨D⟩ is closed in kω(K),
hA0K(D)=σ0, and M(A0,K) holds for every infinite
D⊆∗D0. The argument for σ0=0 is similar,
replacing [A0∪{0}]σn0kω(K) with {0}. If
hAK(D)=ω replace
[A0∪{0}]σn0kω(K) with ∑i≤n⟨Ai⟩kω(K).
Repeatedly using the construction above construct
D0⊇D1⊇⋯⊇Dn⊇⋯ such
that ⟨Di⟩ is closed and discrete in kω(K) for every
i∈ω and M(Ai,K) holds for every infinite D⊆∗Di.
Let D⊆∗Di for every i∈ω. If D0
satisfies M(A,K) put D=D0.
∎
The remark after Definition 9 now gives the following
corollary (after selecting a trivial A={0}).
Corollary 2**.**
Let (G,K) be a kω-pair, D⊆G be a countable closed and
discrete subspace of G. Then there exists an infinite D′⊆D such that ⟨D′⟩ is closed and discrete in G.
Lemma 7**.**
Let (G,K,U) be a convenient triple and let A={Ai:i∈ω} be a family of subsets of G. Let P⊆G be
a countable subset such that 0∈Pτ(U) and hAK(P∩U)=ω for every U∈U (in particular, if ωhAK(P)=ω).
Then there exists an S⊆P such that S→0 in τ(U)
and S satisfies M(A,K).
Proof.
Let K={Kn:n∈ω} and U={Un:n∈ω}. Select
points pn∈P∩∩i≤nUi so that
[TABLE]
Since hAK(P∩U)=ω for every U∈U the
choice of pn is always possible. Verifying that S={pn:n∈ω}→0 in τ(U) is routine.
The proof that S satisfies M(A,K) is is the same as
in Lemma 6.
To see that the condition of the lemma follows from ωhAK(P)=ω, let m∈ω, K∈K, and U∈U. Then ωhAK(P)=ω implies
[TABLE]
therefore P∩U⊆∑i≤m⟨Ai⟩kω(K)+K.
Thus hAK(P∩U)=ω.
∎
3 ‘Convexity’ and extensions in boolean groups
Given a convenient triple (G,K,U), a common operation is to extend G and K
by adding points from the compact completion of (G,τ(U′)) where
U′ extends U. The
convergence properties of the new convenient triple (G′,K′,U′) will
depend on the precompact topology whose extension produces G′. This
section lists a few results to help control these properties.
The following lemma may be viewed as a boolean
kω version of the classical Hahn-Banach theorem.
Lemma 8**.**
Let (G,K) be a kω-pair such that every K∈K is a (compact)
subgroup of G, let H be a subgroup of G closed in
kω(K), and g∈G∖H. Then there exists a subgroup
U⊆G of finite index open in kω(K) such that H⊆U
and g∈U.
Proof.
Let K={Ki:i∈ω}. Build by induction a sequence of closed
subgroups H⊆H0⊆⋯⊆Hn⊆⋯
such that Hn+1=Hn+Kn+1′ for some subgroup
Kn+1′⊆Kn+1 clopen in Kn+1, and g∈Hn. If Hn has been built,
g∈Hn so there exists a clopen subgroup Kn+1′⊆Kn+1 such that g∈Hn+1=Hn+Kn+1′. Then Hn+1 is
a closed subgroup of G and Hn+1∩Ki is relatively open in
Ki for every i≤n+1.
Let H′=∪n∈ωHn. Now the intersection H′∩K is relatively open in K for every
K∈K so H′ is an open subgroup of G in
kω(K). Let G′⊆G be a subgroup such that
G′∩⟨H′∪{g}⟩={0} and G′+⟨H′∪{g}⟩=G. Then
g∈U=H′+G′ is open in kω(K) and of finite index.
∎
In the statement of the next lemma we abuse the notation to use
τ(U0) for both the topology of the compact completion of G,
as well as the topology on G generated by U0.
Lemma 9**.**
Let (G,K,U) be a convenient triple, let H be a subgroup of G closed in
kω(K). Then there exists a countable family of open (in kω(K))
subgroups of finite index U0⊇U such that
Hτ(U0)∩G=∩{U∈U0:H⊆U}=H.
Proof.
Let K={Kn:n∈ω}. For each K∈K let {UnK⊆K:n∈ω} be a countable base of neighborhoods of
[math] in K consisting of clopen subgroups of finite index. The family
KnK={UnK+a:a∈K} is finite so
K+=∪m,n∈ωKnKm is countable. For each
K∈K+, let U(K)⊆G be an open in kω(K) subgroup of finite index
such that H⊆U(K) and H∩K=∅ if
such U(K) exists. Let U(K)=G otherwise. Let U0 be the closure
of U∪{U(K):K∈K+} under finite intersections.
Let g∈G∖H. By Lemma 8 there exists an open in
kω(K) subgroup U of finite index such that H⊆U and
g∈U. Let K′∈K be such that g∈K′ and let UnK′
be such that UnK′⊆K′∩U. Then K=g+UnK′∈K+ and
U∩K=∅ so
H⊆U(K), U(K)∩K=∅. Since U(K)∈U0 is clopen,
g∈Hτ(U0).
∎
Lemma 10**.**
Let (G,K,U) be a convenient triple, let S be a countable
independent subset of G, closed and
discrete in kω(K). Then there exist a countable family of open
subgroups of finite index U0⊇U such that
⟨S′⟩τ(U0)∩G=⟨S′⟩ for any S′⊆S and
⟨2S⟩τ(U0)∩⟨S⟩∖⟨2S⟩τ(U0)=∅.
Proof.
Note that ⟨S′⟩=∩{⟨S∖{s}⟩:s∈S∖S′}
and s∈⟨2S⟩ for any s∈S. Now apply Lemma 9
repeatedly to find U0⊇U such that
⟨S∖{s}⟩τ(U0)∩G=⟨S∖{s}⟩ for
every s∈S and ⟨2S⟩τ(U0)∩G=⟨2S⟩.
∎
We now define a basic extension operation used in the construction.
Definition 10**.**
Let (G,K,U) and (G′,K′,U′) be convenient triples and an
independent D⊆G be such that ⟨D⟩ is closed and discrete
in kω(K). Call (G′,K′,U′) a primitive sequential
extension (pse for short) of (G,K,U) over D if the
following conditions hold:
- (1)
G⊆G′, Uτ(U′)∈U′ for every U∈U, and
K′ is the closure of K∪{L} under finite sums where
L=⟨D⟩τ(U′) (thus L is compact in kω(K′));
2. (2)
⟨D∖F⟩τ(U′)∩G=⟨D∖F⟩* for any F;*
Abusing the notation we will also use τ(U′) to refer to the
topology on G induced by τ(U′).
If the new points added to G are ‘sufficiently far’ from some set
A⊆G, the convergence properties at the points ‘near A’
may not be affected.
Lemma 11**.**
Let (G′,K′,U′) be a pse of (G,K,U) over D⊆G. Let A⊆G, hAK(D)=δ≤ω1,
and let D satisfy M(A,K). If so(a,A,kω(K′))=σ≤Lδ then a∈G and so(a,A,kω(K))=σ.
Proof.
If a∈[A]0 then a∈A⊆G. Suppose the Lemma has been proved
for all σ′<σ. Pick an→a (in kω(K′)) such that
so(an,A,kω(K′))=σn for some increasing
σn→σ−1. Then by the inductive hypothesis an∈G and
so(an,A,kω(K))=σn.
Since an→a in kω(K′) there is a K′∈K such that
{an:n∈ω}⊆K′+L where
L=⟨D⟩τ(U′). By thinning out and reindexing
we may assume that an=an+dn where an∈K′, an→a′∈K′
and dn∈L, dn→d∈L. Since an,an∈G,
by (2) dn∈⟨D⟩.
Let β=σ if δ is limit or β=δ−1
otherwise. Then β<δ and σn≤β for every
n∈ω. Using M(A,K) pick a finite F⊆D such
that (d+K′)∩[A]β=∅ for every
d∈⟨F⟩. Then (d+K′)∩[A]σn=∅ for
every n∈ω and d∈⟨F⟩.
Now dn∈⟨F⟩ for every n∈ω. We may assume that dn=d
for every n∈ω so an=an+d∈K′′ for some K′′∈K and
an→a in kω(K). It follows that a∈G and so(a,A,kω(K))≤σ. Since kω(K)⊆kω(K′), so(a,A,kω(K))=σ.
∎
4 Parity and separation
While Lemma 11 provides one way for preserving the
sequential order at some points, it is not always possible to expect a
given set to be far from a fixed witness to the sequential order. A
different mechanism is needed, introduced in this section.
The next definition is a convenient way to set a lower bound on the
sequential order in a kω group. Note that it is not required that
0∈[S]σkω(K) (or 0∈Skω(K)).
Definition 11**.**
Let (G,K,U) be a convenient triple and S⊆G. Let
σ be a successor. Say that S(K,S,σ) holds if for every
K∈K and every σ′<σ−1 there exists a clopen
subgroup UK(K,S,σ′)⊆K such that UK(K,S,σ′)∩[⟨S⟩∖⟨2S⟩]σ′=∅.
Lemma 12**.**
Let (G,K,U) be a convenient triple and S⊆G be a
countable independent subset. Let σ<ω1 be a successor
ordinal. Then S(K,S,σ) holds if and only if
so(0,⟨S⟩∖⟨2S⟩,kω(K))≥σ.
Proof.
Suppose so(0,⟨S⟩∖⟨2S⟩,kω(K))=σ′<σ. Then there exist
sn→0 in kω(K) such that
so(sn,⟨S⟩∖⟨2S⟩,kω(K))=σn≤σ′−1. By taking a subsequence
if necessary we may assume that sn∈K for some K∈K. Since
σn≤σ′−1<σ′≤σ−1, by S(K,S,σ) there exists a
clopen subgroup U=UK(K,S,σ′−1)⊆K such that
U∩[⟨S⟩∖⟨2S⟩]σ′−1=∅. Since 0∈U and
sn∈[⟨S⟩∖⟨2S⟩]σ′−1, this contradicts sn→0. Hence S(K,S,σ) implies so(0,⟨S⟩∖⟨2S⟩,kω(K))≥σ.
The converse will not be used so its proof is
omitted. Its proof uses the property that each K∈K is first
countable.
∎
The algebraic tool used to control the convergence properties in the
extension is given by the following parity homomorphism.
Definition 12**.**
Let (G′,K′,U′) be a pse of (G,K,U) over D⊆G, and let
L=⟨D⟩τ(U′). Define
pS:[⟨S⟩]σkω(K)→2 and pL:L→2 by letting
pS(a)=0 if a∈[⟨2S⟩]σkω(K) and pS(a)=1 if
a∈[⟨S⟩∖⟨2S⟩]σkω(K). If d∈L put pL(d)=0 if
d∈⟨2D⟩τ(U′) and pL(d)=1 otherwise. If b=a+d for
some a∈[⟨S⟩]σkω(K) and d∈L put
p(b)=pS(a)+pL(d).
The next lemma shows that the parity homomorphism is well defined
under some conditions.
Lemma 13**.**
Let (G′,K′,U′) be a pse of (G,K,U) over
D⊆[⟨S⟩∖⟨2S⟩]σkω(K) where S⊆G is an
independent set, and let
L=⟨D⟩τ(U′). If
⟨2D⟩τ(U′)∩⟨D⟩∖⟨2D⟩τ(U′)=∅ and
S(K,S,σ+1) holds then p:[⟨S⟩]σkω(K)+L→2 is
a well defined homomorphism, continuous on L.
Proof.
Since ⟨S⟩=⟨2S⟩∪(⟨S⟩∖⟨2S⟩) if a∈[⟨S⟩]σkω(K) for
some σ<ω1 then either a∈[⟨S⟩∖⟨2S⟩]σkω(K) or
a∈[⟨2S⟩]σkω(K). If
a∈[⟨S⟩∖⟨2S⟩]σkω(K)∩[⟨2S⟩]σkω(K) then
a∈[⟨S⟩∖⟨2S⟩]σ′kω(K)∩[⟨2S⟩]σ′kω(K) for some
successor ordinal σ′≤σ. Thus there are
an0→a and an1→a such that an0∈[⟨2S⟩]σn0
and an1∈[⟨S⟩∖⟨2S⟩]σn1 for some
σn0,σn1≤σ′−1<σ. Now an0+an1→0 and
an0+an1∈[(⟨S⟩∖⟨2S⟩)+⟨2S⟩]σ′−1kω(K)=[⟨S⟩∖⟨2S⟩]σ′−1kω(K)
by Lemma 1. Pick a K∈K such that {an0+an1:n∈ω}⊆K. Then UK(K,S,σ′−1)∩{an0+an1:n∈ω}=∅
contradicting an0+an1→0.
Thus pS is well defined
on [⟨S⟩]σkω(K). A similar argument involving
Lemma 1 and (⟨S⟩∖⟨2S⟩)+(⟨S⟩∖⟨2S⟩)=⟨2S⟩ shows that
pS is a homomorphism.
It follows from the definition of pL and the choice of D that
pL is a continuous homomorphism on L.
If d∈L∩[⟨S⟩]σkω(K) then d∈⟨D⟩ by (2) so
pS(d)=pL(d) by Lemma 1 and
D⊆[⟨S⟩∖⟨2S⟩]σkω(K). Let
a+d=a′+d′∈[⟨S⟩]σkω(K)+L where a,a′∈[⟨S⟩]σkω(K) and
d,d′∈L. Then a+a′=d+d′ so d+d′∈G and thus
d+d′∈⟨D⟩⊆[⟨S⟩∖⟨2S⟩]σkω(K) by (2)
and Lemma 1. Now
pS(a)+pS(a′)=pS(a+a′)=pS(d+d′)=pL(d+d′)=pL(d)+pL(d′)
so pS(a)+pL(d)=pS(a′)+pL(d′) and p is well
defined.
∎
While the parity homomorphism is unlikely to be continuous on its
domain (even if an appropriate topology is agreed upon) it satisfies
the following weak continuity property.
Lemma 14**.**
Let (G′,K′,U′) be a pse of (G,K,U) over
D⊆G. Let D satisfy M(⟨S⟩∖⟨2S⟩,K) and
⟨2D⟩τ(U′)∩⟨D⟩∖⟨2D⟩τ(U′)=∅,
and let L=⟨D⟩τ(U′). Let
h⟨S⟩∖⟨2S⟩K(D)=δ≤ω1 and
D⊆[⟨S⟩∖⟨2S⟩]δkω(K) if δ<ω1. Let α
be a successor and S(K,S,α) hold. If δ<α, and
so(g,⟨S⟩∖⟨2S⟩,kω(K′))=σ<α then
g∈[⟨S⟩]σkω(K)+L, p(g)=1 where
p:[⟨S⟩]α−1kω(K)+L→2 is the homomorphism in
Definition 12.
Proof.
Since δ<α the homomorphism
p:[⟨S⟩]α−1kω(K)+L→2 is well defined by
Lemma 13.
Suppose the Lemma has been proved for all g′∈G, σ′<σ
such that so(g′,⟨S⟩∖⟨2S⟩,kω(K′))=σ′ and let
so(g,⟨S⟩∖⟨2S⟩,kω(K′))=σ.
If σ≤Lδ then so(g,⟨S⟩∖⟨2S⟩,kω(K))=σ by
Lemma 11. Since σ is a successor, assume below that
σ>δ and let gn→g in kω(K′) be such that
so(gn,⟨S⟩∖⟨2S⟩,K′)=σn where σn→σ−1 and
σn<σ.
Applying the inductive hypothesis each gn∈[⟨S⟩]σnkω(K)+L, and
p(gn)=1 so there exist an∈[⟨S⟩]σnkω(K) and dn∈L
such that gn=an+dn. Since gn→g in kω(K′) there
are (possibly after thinning out and reindexing) a K′∈K, an∈K′, and dn∈L such that an→a, dn→d and
gn=an+dn. Now an−an=dn−dn=dn′ so dn′∈⟨D⟩
by (2). Thus an=an+dn′ where
dn′∈[⟨S⟩]δkω(K). Therefore
an∈[⟨S⟩]σn′kω(K) for some σn′<σ by
δ<σ and Lemma 1.
After picking a subsequence and reindexing, assume that
p(an) and p(dn) are constant. Then p(a)=p(an),
p(d)=p(dn) by the definition of p and Lemma 13
so p(g)=p(a+d)=p(an+dn)=p(gn)=1 by Lemma 13.
∎
The main reason the parity homomorphism was defined is the proof of
the following lemma that states the conditions under which the
sequential order of some points is preserved across primitive
sequential extensions.
Lemma 15**.**
Let (G,K,U) be a convenient triple, S⊆G be such that S(K,S,α) holds for some successor ordinal α<ω1. Let
(G′,K′,U′) be a pse of (G,K,U) over D⊆G that has the following
properties.
- (1)
D* satisfies M(⟨S⟩∖⟨2S⟩,K), h⟨S⟩∖⟨2S⟩K(D)=δ and either
δ≥α−1 or D⊆[⟨S⟩∖⟨2S⟩]δkω(K);*
2. (2)
if δ<α−1 then
⟨D⟩∖⟨2D⟩τ(U′)∩⟨2D⟩τ(U′)=∅;
Then S(K′,S,α) holds so that UK(K′,S,σ)=UK(K,S,σ) for every K∈K and
σ<α−1.
Proof.
Let L=⟨D⟩τ(U′), K∈K, and σ<α−1.
Suppose σ<δ. Pick a finite FK⊆D using M(⟨S⟩∖⟨2S⟩,K) such
that (K+d)∩[⟨S⟩∖⟨2S⟩]σkω(K)=∅ for every
d∈⟨D⟩∖⟨FK⟩. Put UK+L(σ)=K+⟨D∖FK⟩τ(U′).
Suppose
UK+L(σ)∩[⟨S⟩∖⟨2S⟩]σkω(K′)=∅. Let
u∈K, d∈⟨D∖FK⟩τ(U′), and a∈G′ be such that
u+d=a where a∈[⟨S⟩∖⟨2S⟩]σkω(K′). By
Lemma 11 a∈[⟨S⟩∖⟨2S⟩]σkω(K) so d∈G. Then
d∈⟨D∖FK⟩ by (2) and
(K+d)∩[⟨S⟩∖⟨2S⟩]σkω(K)=∅ by the choice of FK
contradicting the choice of a.
Suppose σ≥δ. Then δ<α−1. Put
UK+L(σ)=UK(K,S,σ)+⟨2D⟩τ(U′). Then UK+L(σ) is a
compact subgroup of finite index in K+L and thus clopen in
K+L. Suppose
UK+L(σ)∩[⟨S⟩∖⟨2S⟩]σkω(K′)=∅. Then
there exist u∈UK(K,S,σ) and
d∈⟨2D⟩τ(U′) such that u+d=a for some
a∈[⟨S⟩∖⟨2S⟩]σkω(K′). By Lemma 13
and (2) p:[⟨S⟩]σkω(K)+L→2 is well defined
and a=a′+d′ where a′∈[⟨S⟩]σkω(K), d′∈L and
p(a′+d′)=1 by Lemma 14.
Now u=a+d thus a+d∈G. Since p(a′+d′)=1 by
Lemma 14 and p(d)=0 by Definition 12,
p(a′+d′+d)=1. Since a+d∈G and u=a+d=a′+d′+d, d′+d∈L∩G=⟨D⟩⊆[⟨S⟩]σkω(K) by (2) and
δ≤σ.
Since a′+d′+d∈[⟨S⟩]σkω(K) by Lemma 1 and
pS(a′+d′+d)=p(a′+d′+d)=1, a′+d′+d∈[⟨S⟩∖⟨2S⟩]σkω(K) by the
definition of pS contradicting the choice of UK(K,S,σ).
Now for every K∈K′ pick a p(K)∈K such that K⊆p(K)+L and p(K)=K if K∈K. Note that Up(K)(K,S,σ)⊆Up(K)+L(σ) for K∈K. Put UK(K′,S,σ)=Up(K)+L(σ)∩K if K∈K′∖K and UK(K′,S,σ)=UK(K,S,σ) otherwise.
∎
Let γ be an ordinal. Suppose for every σ<γ a
convenient triple (Hσ,Kσ,Uσ) is defined so that the following
conditions hold:
- (1)
Hσ′⊆Hσ, Kσ′⊆Kσ,
and Uσ′⊆{U∩Hσ′:U∈Uσ}
if σ′≤σ<γ;
2. (2)
Hσ′ is dense in Hσ in kω(Kσ) for every
σ′≤σ;
Define (H<γ,K<γ,U<γ) by taking
H<γ=∪σ<γHσ,
K<γ=∪σ<γKσ,
U<γ={Ukω(K<γ):U∈Uσ,σ<γ}.
Note that in the case of a successor γ, (H<γ,K<γ,U<γ)=(Hγ−1,Kγ−1,Uγ−1).
Lemma 16**.**
The family U<γ forms a base of clopen subgroups of finite
index for a precompact group
topology τ(U<γ) on H<γ and each Hσ, σ<γ is dense
in H<γ in kω(K<γ). If γ<ω1 then
(H<γ,K<γ,U<γ) is a convenient triple.
Proof.
Let U∈Uσ′ for some σ′<γ. Then U+F=Hσ′ for
some finite F⊆Hσ′. Thus
Ukω(Kσ)+F=Hσ for any
σ′≤σ<γ. Therefore,
Ukω(K<γ)+F=H<γ so Ukω(K<γ) is a
clopen subgroup of finite index in H<γ.
Furthermore, if g∈H<γ∖{0} then g∈Hσ′ for some σ′<γ and there exists a
U′∈Uσ′ such that g∈U′. If σ≥σ′ then
by (1) U′=U∩Hσ′ for some
U∈Uσ. Thus g∈U′kω(Kσ) for any
σ<γ. Now
g∈U′kω(K<γ)=∪σ<γU′kω(Kσ). Thus
τ(U<γ) forms a base of a precompact T1 group topology
on H<γ that consists of clopen subgroups. The rest of the
properties are routine.
∎
The lemma below shows that iterated extensions preserve the sequential
order some points have when they are added to the group.
Lemma 17**.**
Let (G,K,U) be a convenient triple, S⊆G be a countable
independent subset, and
δ<ω1 be a successor ordinal. Suppose the family
{(Hγ,Kγ,Uγ):γ<α} where α<ω1 has the
following properties.
- (1)
(Hγ,Kγ,Uγ)* is a primitive sequential extension
of (H<γ,K<γ,U<γ) over some Dγ⊆H<γ for
every 0≤γ<α and (H<0,K<0,U<0)=(G,K,U);*
2. (2)
if h⟨S⟩∖⟨2S⟩K<γ(Dγ)=δγ then either
δγ≥δ−1 or Dγ⊆[⟨S⟩∖⟨2S⟩]δγkω(K<γ),
Dγ satisfies M(⟨S⟩∖⟨2S⟩,K<γ), and
⟨Dγ⟩∖⟨2Dγ⟩τ(Uγ)∩⟨2Dγ⟩τ(Uγ)=∅;
3. (3)
S(Kγ,S,δ)* holds for every γ<α;*
Suppose g∈Hγ∖H<γ for some γ<α
and so(g,⟨S⟩∖⟨2S⟩,kω(K<α))=σ<δ. Then
so(g,⟨S⟩∖⟨2S⟩,kω(Kγ))=σ.
Proof.
Suppose the statement holds for all g′∈H<α such that
so(g′,⟨S⟩∖⟨2S⟩,kω(K<α))=σ′<σ for some
σ<δ, and let g∈Hγ∖H<γ be such that
so(g,⟨S⟩∖⟨2S⟩,kω(K<α))=σ. Let γ′ be the
smallest ordinal with the following property. There exist gn→g
in kω(Kγ′) such that
so(gn,⟨S⟩∖⟨2S⟩,kω(K<α))=σn<σ such that
σn→σ−1, gn∈Hγn∖H<γn,
and γn≤γ′ is increasing.
Note that gn as above exist by the definition of
so(g,⟨S⟩∖⟨2S⟩,kω(K<α)). By gn→g there exists a
K∈Kβ for some β<α such that gn∈K for
every n∈ω. Thus γn≤β and
γ′≤β<α is well defined.
Suppose γ′>γ. Then (Hγ′,Kγ′,Uγ′) is a primitive
sequential extension of (H<γ′,K<γ′,U<γ′) over some Dγ′⊆H<γ′ such that h⟨S⟩∖⟨2S⟩K<γ′(Dγ′)=δγ′.
If σ≤Lδγ′ then by Lemma 11
so(g,⟨S⟩∖⟨2S⟩,kω(K<γ′))=σ. Otherwise, since
S(Kγ′,S,δ) holds and δγ′<σ<δ,
p:[⟨S⟩]σkω(K<γ′)+⟨Dγ′⟩→2 is
defined by Lemma 13. By Lemma 14 g=a+s where
so(a,⟨S⟩,kω(K<γ′))≤σ,
s∈⟨Dγ′⟩τ(U<γ′), and p(a+s)=1. Since
g∈Hγ⊆H<γ′, s∈H<γ′ so by (2)
s∈⟨Dγ′⟩. By Lemma 1
so(a+s,⟨S⟩,kω(K<γ′))=max{σ,δγ′}=σ. Since
p(a+s)=1, so(g,⟨S⟩∖⟨2S⟩,kω(K<γ′))=σ.
Thus we may assume that gn are chosen so that gn→g in
kω(K<γ′). Let K∈Kβ for some β<γ′ be
such that gn∈K. Then gn→g in kω(Kβ) and
γn≤β contradicting the choice of γ′.
Thus γ′≤γ and gn∈Hγ for every n∈ω
so by the hypothesis so(gn,⟨S⟩∖⟨2S⟩,kω(Kγ))≤σn and
so(g,⟨Sm⟩∖⟨2Sm⟩,kω(Kγ))=σ.
∎
The next lemma shows that not only is the sequential order preserved,
it is preserved in a ‘uniform’ way if certain conditions are met.
Lemma 18**.**
Let (G,K,U) be a convenient triple and S⊆G be a
countable independent subset. Suppose
the family {(Hγ,Kγ,Uγ):γ<α}, α≤ω1
has the following properties:
- (1)
conditions (1) and (2) of Lemma 17
hold;
2. (2)
S(Kγ,S,δ)* holds so that UK(Kγ,S,σ)=UK(Kβ,S,σ) for any K∈Kβ,
σ<δ−1, and β≤γ;*
Then S(K<α,S,δ) holds by defining UK(K<α,S,σ)=UK(Kγ,S,σ) for any σ<δ−1 and
K∈Kγ.
Proof.
Let K∈Kγ, σ<δ−1, and g∈Hβ∖H<β where
β,γ<α. Suppose
so(g,⟨S⟩∖⟨2S⟩,kω(K<α))=σ′≤σ. Then by
Lemma 17 so(g,⟨S⟩∖⟨2S⟩,kω(Kβ))=σ′. If
β≤γ then g∈UK(Kγ,S,σ)=UK(K<α,S,σ). If β>γ then
g∈K.
∎
If the new points are sufficiently ‘far’ from a given set, the next
lemma shows that the sequential order of the points ‘near’ the set is
not affected.
Lemma 19**.**
Let α<ω1 and let (Hγ,Kγ,Uγ) be a primitive
sequential extension of (H<γ,K<γ,U<γ) over some Dγ for every
γ<α. Let A⊆G=H−1,
hAK<γ(Dγ)=ω1, and Dγ
satisfy M(A,K<γ).
Then so(g,A,kω(K<α))=so(g,A,kω(K<0)).
Proof.
Suppose the Lemma holds for all g′∈H<α such
that so(g′,A,kω(K<α))=σ′<σ for some
σ<ω1 and let so(g,A,kω(K<α))=σ. Let
gn→g in kω(K<α) so that so(gn,A,kω(K<α))<σ. By the hypothesis so(gn,A,kω(K<0))<σ. There exists a γ<α such that
{gn:n∈ω}⊆K∈Kγ so gn→g in
kω(Kγ) and so(g,A,kω(Kγ))=σ. Let γ
be the smallest such and suppose γ≥0.
Since hAK<γ(Dγ)=ω1 and Dγ
satisfies M(A,K<γ), the sequential order so(g,A,kω(K<γ))=σ by Lemma 11 so there are
gn→g in kω(Kγ′) for some γ′<γ and
so(gn,A,kω(K<α))=so(gn,A,kω(K<0))<σ
contradicting the choice of γ.
∎
The constructions above are intended for the case when the
sequential order is reflected by a single point in the group, i.e. when the sequential order desired is a successor ordinal. In the case
when the group to be constructed must have a limit sequential order,
some additional tools are required.
The following lemma presents a rough idea of how one might go about
handling the limit case. Note that the sequentiality of the
Σ-product of sequential spaces was proved in [8],
Corollary 2.5
(one can also use an argument similar to the one below).
Lemma 20**.**
Let G be a Σ-product of countably compact sequential spaces
Gα, α<γ for some ordinal γ. Then
[TABLE]
Here the min is taken over all the possible reorderings of I.
Sketch of proof.
Below we refer to the ‘center’ point of each Gα, as well as
the corresponding points in ΣGα and finite products of
Gα as [math].
Let A⊆G be such that A∋0. Introduce a finer
topology τ on G by making each Gα discrete. It is well
known that τ is Féchet. Consider a finite I⊆γ
and a basic
neighborhod UI={p∈G:p(α)=0\mboxifα∈I} of [math] in τ. Consider the natural
projection πI(A) of A into the finite
product ∏α∈IGα. Note that
πI(A)∋0 so by [12], Theorem 2.2
0∈[πI(A)]σI where σI≤min∑α∈Iso(Gα). Using
induction on σI (and the sequential compactness of G) one
shows that there exists an x∈UI such that
x∈[A]σI. Thus Xτ∋0 where X={x∈[A]σI:I⊆γ,∣I∣<ω}. Since τ is
Fréchet, this completes the proof.
∎
The lemma above exhibits two obstacles to obtaining a group of a
sequential order that is a limit ordinal. Consider the case of
ω+ω. If the product approach suggested by the lemma above
is to be followed, it is clear that infinitely many factors are
required, infinitely may of which must have the sequential order above
ω+1. The best estimate for the sequential order of a product of
n of such factors (obtained in [12]) will then exceed
nω>ω+ω. Thus one must be concerned with providing a
better growth rate for the sequential order of finite products of
groups.
The other obstacle is the +1 part of the uppper bound estimate. Even
if the sequential order of finite products can be made to grow slowly,
the points in the ‘last sequence’ converging to a given point in the
product might still have unbounded sequential orders thus leading to a
successor
sequential order for the group.
It is unclear to the authors whether the examples below can be given a
product structure, thus a different approach was taken. The examples
‘look like’ products of groups, although the countable product
structure is destoyed.
The lemma below is a parity neutral version of Lemma 14.
Lemma 21**.**
Let (G,K,U) be a convenient triple.
Let D⊆G be such that
h⟨A⟩K(D)=αK≤ω1. Suppose
D⊆⟨A⟩kω(K) if αK<ω1, and D
satisfies M(⟨A⟩,K). Let (G′,K′,U′) be a primitive sequential
extension of (G,K,U) over D.
If g∈⟨A⟩kω(K′) then g=a+s where
a∈⟨A⟩kω(K) and s∈L=⟨D⟩τ(U′).
Proof.
Let so(g,⟨A⟩,kω(K′))=σ. If σ≤LαK by
Lemma 11 so(g,⟨A⟩,kω(K))=σ.
Suppose the statement holds for all g′∈G such that
so(g′,⟨A⟩,kω(K′))=σ′<σ where
σ≤LαK.
Pick gn→g in kω(K′) such that
so(gn,⟨A⟩,kω(K′))=σn<σ. By the hypothesis
gn=gn+sn where gn∈⟨A⟩kω(K) and sn∈L. Since gn→g one can find a K∈K, an∈K, sn∈L
such that gn=an+sn, an→a, sn→s. Then
dn=sn+sn=gn+an∈G so by (2)
dn∈⟨D⟩. Thus an=gn+dn and
an∈⟨A⟩kω(K) by Lemma 1. Thus
g=a+s where a∈⟨A⟩kω(K) and s∈L.
∎
The next definition itroduces a product-like structure into the
group. The preservation of this structure is the subject of
several lemmas that follow.
Definition 13**.**
Let (G,K,U) be a convenient triple. Let C={Si:i∈ω} be a family of countable subsets of G. Call C discrete separated by K in G if for any infinite, closed and discrete
Di={dni:n∈ω}⊆⟨Si⟩kω(K), i∈I where I⊆ω is
finite the set {∑i∈Idni:n∈ω}⊆K for
any K∈K (equivalently, {∑i∈Idni:n∈ω} is
infinite, closed, and discrete).
The following lemma follows from the definition of discrete separation.
Lemma 22**.**
Let (G,K,U) be a convenient triple. Let C={Si:i∈ω} be a
family of countable subsets of G discrete separated by K in G.
Let I⊆ω be finite and let Ai⊆⟨Si⟩kω(K)
be closed (in kω(K)) subsets of G for i∈I. Then ∑i∈IAi is closed in kω(K).
Lemma 23**.**
Let (G,K,U) be a convenient triple. Let C={Si:i∈ω} be a
family of countable subsets of G discrete separated by K in G.
Let D⊆G be a countable closed discrete subspace such that D
satisfies property M(⟨Si⟩,K) for every i∈ω, and
D⊆⟨Sm⟩kω(K) for some m∈ω. Let (G′,K′,U′)
be a primitive sequential extension of (G′,K′,U′) over D. Then C is
discrete separated by K′ in G′.
Proof.
Suppose there exist infinite closed and discrete in kω(K′)
subspaces Di={dni:n∈ω}⊆⟨Si⟩kω(K′) where i∈I for
some finite I⊆ω, and a
K′∈K′ such that {∑i∈Idni:n∈ω}⊆K′.
Suppose i=m and h⟨Si⟩K(D)<ω1. Let D={pn:n∈ω} and pn=an′+pn′ where an′∈K′′ for some
K′′∈K and pn′∈⟨Si⟩kω(K). Passing to a subset, if
necessary, we may assume that pn′=pk′ if n=k, so
D′={pn′:n∈ω}⊆⟨Si⟩kω(K) is infinite,
closed, and discrete in kω(K). Then {pn+pn′:n∈ω}={an′:n∈ω}⊆K′′ contradicting
the property that C is discrete separated by K. Hence
h⟨Si⟩K(D)=ω1 for every i=m. By Lemma 11
dni∈⟨Si⟩kω(K)⊆G for i=m.
Let K′⊆K+L where L=⟨D⟩τ(U′) and
K∈K. Now ∑i∈Idni=an+dn where an∈K and
dn∈L. If m∈I then
dn∈G by the argument in the preceeding paragraph, so
by (2) and the choice of D,
dn∈⟨D⟩⊆⟨Sm⟩kω(K). By passing to a subset if
necessary, we may assume that the set Dm={dn=dnm:n∈ω} is
either infinite, closed, and discrete, or is a subset of some K′′∈K. In the first case
{∑i∈I∪{m}dni:n∈ω}⊆K, while in the
second case {∑i∈Idni:n∈ω}⊆K+K′′∈K,
contradicting the property that C is discrete separated. Thus
we may assume m∈I.
Since L⊆⟨Sm⟩kω(K′), the set {dnm+dn:n∈ω}⊆⟨Sm⟩kω(K′) is also closed and
discrete in kω(K′) so we may assume, after replacing Dm, if
necessary, that ∑i∈I∖{m}dni+dnm=an∈G. Then dnm∈G and by
Lemma 21 dnm=dn′+sn where
dn′∈⟨Sm⟩kω(K) and sn∈L. Since dnm∈G,
sn∈⟨D⟩ by (2) so Dm⊆⟨Sm⟩kω(K) by
Lemma 1. Now
Di⊆⟨Si⟩kω(K) are closed and discrete in kω(K)
contradicting the assumption that C is discrete separated in G.
∎
Lemma 24**.**
Let (G,K,U) be a convenient triple. Let C={Si:i∈ω} be a family of countable subsets of G discrete
separated by K in G. Let D⊆G be a countable closed
discrete subspace that satisfies M(C,K).
Let (G′,K′,U′) be a primitive sequential extension of (G,K,U) over
D. Then C is discrete separated by K′ in G′.
Proof.
Note that h⟨Si⟩K(D)=ω1 for every i∈ω.
Suppose there exist infinite closed and discrete subspaces Di={dni:n∈ω}⊆⟨Si⟩kω(K′) such that
{∑i∈Idni:n∈ω}⊆K′∈K′.
Let K′⊆K+L where L=⟨D⟩τ(U′) and
K∈K. Now ∑i∈Idni=an+dn where an∈K and dn∈L.
By Lemma 11 dni∈⟨Si⟩kω(K) for every
i∈I, n∈ω so dn∈G and by (2)
dn∈⟨D⟩. Use M(C,K) to find a finite F⊆D such
that
(d+K)∩(∑i∈I⟨Si⟩kω(K))=∅
if d∈⟨D⟩∖⟨F⟩. Thus {dn:n∈ω}⊆⟨F⟩ and
{∑i∈Idni:n∈ω}⊆K′′ for some K′′∈K
contradicting that C is discrete separated by K in G.
∎
It is not possible to preserve the sequential order of every new point
(even if a uniform bound is imposed) in the extension so the next
lemma deals with a much weaker property.
Lemma 25**.**
Let α<ω1 and let (Hγ,Kγ,Uγ) be a primitive
sequential extension of (H<γ,K<γ,U<γ) over some Dγ⊆H<γ for every γ<α. Let Sm⊆G=H<0, m∈ω
be countable subsets. Let
h⟨Sm⟩K<γ(Dγ)=δγm. Suppose for every
m∈ω the set Dγ satisfies M(⟨Sm⟩,K<γ) and has
the property that either δγm=ω1 or
Dγ⊆⟨Sm⟩kω(K<γ).
If g∈Hβ∖H<β for some β<α and
g∈⟨Sm⟩kω(K<α) then
g∈⟨Sm⟩kω(Kβ).
Proof.
Suppose the statement holds for all g′∈H<α′ such that
so(g′,⟨Sm⟩,kω(K<α′)=σ′ where
(α′,σ′)<(α,σ). Let
so(g,⟨Sm⟩,kω(K<α))=σ. If σ=0 then
g∈⟨Sm⟩, otherwise there are gn→g in kω(K<α)
such that so(gn,⟨Sm⟩,kω(K<α))=σn<σ. Since gn→g there exists a K∈Kγ for some γ<α such that
gn∈K.
Suppose α is a successor ordinal. If β=α−1
then K<α=Kβ so the statement holds. Otherwise β<α−1 and g∈H<α−1 so by Lemma 21 g=a+s where
a∈⟨Sm⟩kω(K<α−1) and
s∈⟨Dα−1⟩τ(Uα−1)⊆⟨Sm⟩kω(Kα−1)
if δα−1m<ω1. If δα−1m=ω1
then g∈⟨Sm⟩kω(K<α−1) by
Lemma 11. Since a,g∈H<α−1, it follows that
s∈H<α−1, so by (2) s∈⟨Dα−1⟩ and
g∈⟨Sm⟩kω(K<α−1). Since (α−1,σ)<(α,σ),
the statement holds by the hypothesis.
If α is a limit ordinal γ<α′ for some
α′<α. Since (α,σn)<(α,σ) and
gn∈Hγ, by the hypothesis
gn∈⟨Sm⟩kω(K<α′) so
g∈⟨Sm⟩kω(K<α′) and
g∈⟨Sm⟩kω(Kβ) by the hypothesis.
∎
Lemma 26**.**
Let α<ω1 and let (Hγ,Kγ,Uγ) be a primitive
sequential extension of (H<γ,K<γ,U<γ) over some Dγ for every
γ<α. Let Si⊆G=H<0 be countable subsets such
that C={Si:i∈ω} is discrete separated by Kγ
in Hγ for every γ<α. Let
h⟨Sm⟩K<γ(Dγ)=δγm. Suppose for every
m∈ω the set Dγ satisfies M(⟨Sm⟩,K<γ) and has
the property that
whenever δγm<ω1 the set
Dγ⊆⟨Sm⟩kω(K<γ).
Then C is discrete separated by K<α in H<α.
Proof.
Suppose the statement holds for all γ<α.
Let dni∈Hαni∖H<αni and
K∈Kβ, β<α be such that D={∑i∈Idni:n∈ω}⊆K, where Di={dni:n∈ω}⊆⟨Si⟩kω(K<α) are
closed discrete subspaces of H<α, i∈I, and I⊆ω is finite.
If γ=max{{supαni:i∈I}∪{β}}<α then
by Lemma 25
Di⊆⟨Si⟩kω(Kγ) so C is not
discrete separated in kω(Kγ).
Suppose, say, αn0 is unbounded in α. Pick n∈ω
so that αn0>β and let γ=maxi∈Iαni. By the inductive hypothesis, C is discrete
separated by K<γ in H<γ. Since ⟨Dγ⟩ is
closed and discrete, this implies that
δγm<ω1 for at most one m∈ω (note that Dγ
cannot be finite, otherwise Hγ∖H<γ=∅). Then γ>β
and
h⟨Si⟩K<γ(Dγ)=ω1 for all i∈I such that
i=m for some m∈ω. By Lemma 11
dni∈⟨Si⟩K<γ for every i=m so
m∈I (otheriwse maxi∈Iαni<γ). Thus ∑i∈Idni∈Hγ∖H<γ
contradicting ∑i∈Idni∈K⊆H<γ.
∎
Putting the concepts introduced above together, the next definition
introduces a basic ‘extension step’ of the construction.
Definition 14**.**
Let (G,K,U) and (G′,K′,U′) be convenient triples, C={Si:i∈ω} be a family of countable subsets of G, and
{σi:i∈ω}⊆ω1 be a set of successor
ordinals. Let D⊆G be a countable independent closed and
discrete subset of G in kω(K). Call (G′,K′,U′) a fine primitive
sequential extension (or fpse for short) of (G,K,U) over (D,C) (or
D if C is clear) if the following properties hold.
- (1)
(G′,K′,U′)* is a primitive sequential extension of (G,K,U) over D, which
satisfies M(K,⟨Si⟩) for every i∈ω, and
either D satisfies M(K,C) or D⊆⟨Sm⟩kω(K) for
some m∈ω;*
2. (2)
if h⟨Sm⟩∖⟨2Sm⟩K(D)=δDm<σm−1 then
D⊆[⟨Sm⟩∖⟨2Sm⟩]δDmkω(K), D satisfies M(K,⟨Sm⟩∖⟨2Sm⟩),
and
⟨D⟩∖⟨2D⟩τ(U′)∩⟨2D⟩τ(U′)=∅;
Definition 15**.**
Let (G,K,U) be a convenient triple. Let C={Si:i∈ω} be a
family of countable subsets of G. Call a family
C={(Hγ,Kγ,Uγ):γ0≤γ<α} of convenient triples an fpse-chain above (G,K,U) along {Dγ:γ<α} relative
to C if (Hγ,Kγ,Uγ) is an fpse of (H<γ,K<γ,U<γ) over
(Dγ,C) where Dγ⊆H<γ for γ>γ0
and (Hγ0,Kγ0,Uγ0) is an fpse of (G,K,U) over (Dγ0,C) where
Dγ0⊆G. Call each Dγ the free sequence at
γ.
Lemma 27**.**
Let α<ω1 and let (Hγ,Kγ,Uγ) be a primitive
sequential extension of (H<γ,K<γ,U<γ) over some
Dγ⊆H<γ for every
γ0<γ<α. Let Si⊆Hγ0, i∈ω be
such that for every γ<α there is at most one i∈ω such that
h⟨Si⟩K<γ(Dγ)<ω1, Dγ
satisfies M(⟨Si⟩,K<γ) for every γ0<γ<α
and every i∈ω, and
Dγ⊆⟨Si⟩kω(K<γ) whenever
h⟨Si⟩K<γ(Dγ)<ω1.
Let g=∑i∈Igi where
gi∈⟨Si⟩kω(K<α) for some finite
I⊆ω. If g∈Hγ∖H<γ and
gi∈Hγi∖H<γi then
gi∈⟨Si⟩kω(Kγi) and γ=maxi∈Iγi.
Proof.
By Lemma 25 gi∈⟨Si⟩kω(Kγi).
Let γ′=maxi∈Iγi. If γ′=γ0 then
γi=γ0 for every i∈I so g∈Hγ0. Otherwise
(Hγ′,Kγ′,Uγ′) is a primitive sequential extension of
(H<γ′,K<γ′,U<γ′) over Dγ′⊆H<γ′. If
h⟨Si⟩K<γ′(Dγ′)=ω1 then
gi∈⟨Si⟩kω(K<γ′) by Lemma 11 so
γi<γ′. Thus there is a unique j∈I such that
γj=γ′ and γi<γ′ for i=j. Now
g=∑i∈Igi so γ′=γ.
∎
Certain discrete separated families behave like direct sums
algebraically.
Lemma 28**.**
Let (G′,K′,U′) be a pse of (G,K,U) over
D⊆G. Let Si⊆G, i∈ω be such that
⟨Sm⟩kω(K)∩(∑i∈I⟨Si⟩kω(K))={0} for any
finite I⊆ω∖{m}. Suppose there is at
most one i∈ω such that h⟨Si⟩K(D)<ω1 and if
such i exists D⊆⟨Si⟩kω(K). Let D
satisfy M(K,⟨Si⟩) for every i∈ω.
Then ⟨Sm⟩kω(K′)∩(∑i∈I⟨Si⟩kω(K′))={0} for any finite
I⊆ω∖{m}.
Proof.
Let g=∑i∈Igi where I⊆ω is finite,
gi∈⟨Si⟩kω(K′) for i∈I, m∈I, and
g∈⟨Sm⟩kω(K′).
Suppose h⟨Sm⟩K(D)=ω1. Then g∈⟨Sm⟩kω(K)
by Lemma 11 and gi∈⟨Si⟩kω(K) by
Lemma 27 so g=0 by the property of C.
If h⟨Sm⟩K(D)<ω1 and g∈G′∖G then
h⟨Si⟩K(D)=ω1 and gi∈⟨Si⟩kω(K) for i∈I. Thus g=∑i∈Igi∈G contradicting the assumption on
g. Thus g∈G and by Lemma 21 g=a+s where
a∈⟨Sm⟩kω(K) and s∈Dτ(U′). Since g∈G,
s∈G and by (2) s∈⟨D⟩. Thus
g∈⟨Sm⟩kω(K) and g=0 by the condition on C.
∎
It is convenient to have an upper bound measure
for the sequential order of
the construction.
Definition 16**.**
Let (G,K,U) be a convenient triple. Let C={Si:i∈ω} be a
family of countable subsets of G and
ξ={σi:i∈ω}⊆ω1 be a set of successor
ordinals. Call (C,ξ) a sequential scale in (G,K,U) if
the following conditions are satisfied.
- (1)
C* is discrete separated by K in G;*
2. (2)
⟨Sm⟩kω(K)∩∑i∈I⟨Si⟩kω(K)={0}* for every
finite I⊆ω∖{m};*
3. (3)
S(K,Si,σi)* holds for every i∈ω;*
A trivial observation shows that a collection C={Si:i∈ω} of subsets will form a sequential scale for any
ξ⊆ω1 provided ∪i∈ωSi is independent,
and ⟨∪i∈ωSi⟩ is closed
and discrete in kω(K).
Lemma 29**.**
Let (G,K,U) be a convenient triple, C={Si:i∈ω} be such
that (C,ξ) is a sequential scale in (G,K,U) for some ξ⊆ω1. Let
P⊆G be an infinite subset closed and discrete in
kω(K). Suppose P⊆⟨Sm⟩kω(K) or hCK(P)=ω.
Then there exists an fpse (G′,K′,U′) of (G,K,U) such that S→s for
some infinite S⊆P and s∈G′∖G.
Proof.
Using the observation immediately following Lemma 5 and
passing to a subset if neccessary,
assume that h⟨Sm⟩∖⟨2Sm⟩K(P′)=σ for every infinite
P′⊆P. Let P={pn:n∈ω},
D={dn:n∈ω}⊆[⟨Sm⟩∖⟨2Sm⟩]σkω(K) if
σ<σm−1, K∈K, and an,a∈K be such that
pn=dn+an and an→a. If σ≥σm−1 put
D=P. Using Lemma 6 find an infinite
J⊆ω such that the set Q={dn:n∈J}
satisfies M(⟨Si⟩,K) for i∈ω and
satisfies M(⟨Sm⟩∖⟨2Sm⟩,K) if σ<σm−1. Note that
h⟨Sm⟩∖⟨2Sm⟩K(Q)=σ by the assumption on P and
Lemma 5.
Use Lemma 10 to find a family of clopen subgroups
U0⊇U of finite index in G such that
⟨Q′⟩τ(U0)∩G=⟨Q′⟩ for every infinite Q′⊆Q
and
⟨2Q⟩τ(U0)∩⟨Q⟩∖⟨2Q⟩τ(U0)=∅ if
σ<σm−1 (all closures with respect to τ(U0) are
assumed to be taken in the appropriate compact completion of G). By
passing to a subset if necessary assume Q→q in τ(U0) for some q in the compact completion of G. Put
L=⟨Q⟩τ(U0), let G′=G+L, and let K′ be the closure
of K∪{L} under finite sums and intersections. Let U′ be the
trace of U0 on
G′. Now (1), (2), (1),
and (2)
follow from the construction and S={dn+an:n∈J}→q+a=s in
τ(U′) so (G′,K′,U′) is the desired fpse.
If hCK(P)=ω use Lemma 6 to find an infinite
S⊆P that satisfies M(C,K) and use an argument similar
to the one above to construct (G′,K′,U′).
∎
Lemma 30**.**
Let (G,K,U) be a convenient triple, C={Si:i∈ω} be such
that (C,ξ) is a sequential scale in (G,K,U) for some ξ. Let
I⊆ω be finite, and let U∈U.
Then there exists a U′⊆U such that (∑i∈I⟨Si⟩kω(K))∩U′=∑i∈I(⟨Si⟩kω(K)∩U) and
U′ is a clopen subgroup of finite index in (G,kω(K)).
Proof.
The group H=∑i∈I(⟨Si⟩kω(K)∩U) is closed in kω(K)
by Lemma 22
and is of finite index in ∑i∈I⟨Si⟩kω(K). Let
g1,…,gk∈G∖H be such that H+{gi:i≤k}⊇∑i∈I⟨Si⟩kω(K). Use Lemma 8 to construct an open
subgroup of finite index Ui, i=1,…,k such that H⊆Ui and gi∈Ui. Put U′=∩i≤kUi∩U.
∎
To keep the sequential order low one must be able to construct
sequences converging to some points that are already present in the
group.
Lemma 31**.**
Let (G,K,U) be a convenient triple, C={Si:i∈ω} and
ξ={σi:i∈ω}⊆ω1 be such that
(C,ξ) is a sequential scale in (G,K,U). Let I⊆ω be
finite, Di={dni:n∈ω}⊆⟨Si⟩kω(K) be such
that Di→0 in τ(U) for every i∈I, and
h⟨Si′⟩∖⟨2Si′⟩K(Di′)≥σi′−1>0 for some i′∈I.
Then there exists an infinite J⊆ω and an fpse (G′,K′,U′) of
(G,K,U) over D={dni′:n∈J} such that D→0 in kω(K′)
and {dni:n∈J}→0 in τ(U′) for every i∈I.
Proof.
Pick an infinite J⊆ω such that either D(i)={dni:n∈J}→0 or ⟨D(i)⟩ is closed and discrete in
kω(K) and D(i′) satisfies M(⟨Si⟩,K) for every
i∈ω. Since h⟨Si′⟩∖⟨2Si′⟩K(Di′)>0 the set
⟨D(i′)⟩ is infinite, closed and discrete in kω(K). By taking a subset of
J if necessary assume that D(i′) satisfies M(⟨Si⟩∖⟨2Si⟩,K) and the
set ∪i∈ID(i) is independent (using (2)). Since
C is discrete separated and D(i)⊆⟨Si⟩kω(K), the
group ⟨∪i∈I′D(i)⟩ is closed and discrete in kω(K) where
I′={i∈I:⟨D(i)⟩ is closed discrete in kω(K)}.
Use Lemma 8 and the argument in Lemma 9 to find
a U′⊇U such that for every U∈U′ and every i∈I
the set D(i)∖U is finite and ⟨∪i∈I′D(i)∖Fi⟩τ(U′)∩G=⟨∪i∈I′D(i)∖Fi⟩ for any finite Fi⊆G. Then
D(i)→0 in τ(U′).
Put L=Dτ(U′) and let K′ be the closure of
K∪{L} under finite intersections and
sums. Now D(i′)→0 in kω(K′), (1)
and (2) hold, and, since
h⟨Si⟩∖⟨2Si⟩K(D)≥σi′−1,
properties (1)–(2) hold so (G′,K′,U′)
is an fpse of (G,K,U) over D.
∎
The next lemma presents the basic structure of the example.
Lemma 32**.**
Let (G,K,U) be a convenient triple, C={Si:i∈ω} and
ξ={σi:i∈ω} be such that (C,ξ) is a
sequential scale in (G,K,U). Let {(Hγ,Kγ,Uγ):γ<α} be
an fpse-chain over (G,K,U) along {Dγ:γ<α}
relative to C.
Then (C,ξ) is a sequential scale in (H<α,K<α,U<α) and the
following properties hold.
- (1)
UK(K<α,Si,σ)=UK(Kγ′,Si,σ)=UK(Kγ,Si,σ)* for every K∈Kγ,
γ≤γ′<α, and σ<σi−1;*
2. (2)
if g∈Hγ∖H<γ for γ<α and
g∈⟨Si⟩kω(K<α) then
g∈⟨Si⟩kω(Kγ);
3. (3)
if g∈Hγ∖H<γ for some γ<α and
so(g,⟨Si⟩∖⟨2Si⟩,kω(K<α))=σ<σi then
so(g,⟨Si⟩∖⟨2Si⟩,kω(Kγ))=σ;
Proof.
Suppose the statement has been proved for all
α′<α. Suppose α is a successor ordinal such that
α′=α−1. Then by the hypothesis (C,ξ) is a sequential
scale in (H<α′,K<α′,U<α′) and (H<α,K<α,U<α)=(Hα′,Kα′,Uα′) is an fpse of
(H<α′,K<α′,U<α′) over some Dα′⊆H<α′. Now (1) and (2) hold by
Lemmas 23, 24, and 25,
and (1). Property (2) follows from
Lemma 28
and (1). Property (3) follows from
Lemma 15 applied to every Si and
property (2). Moreover, by Lemma 15 one may
choose UK(K<α,Si,σ)=UK(Kα′,Si,σ)=UK(Kγ,Si,σ) for any K∈Kγ,
σ<σi−1, i∈ω, γ<α so (1)
holds. Now (3) follows from
Lemma 17, (1), and (2).
Let α be a limit ordinal. Then (1) holds by
Lemma 26, the hypothesis,
and (1). Property (2) holds by
Lemma 25 and (1).
Let I⊆ω be finite and
gi∈⟨Si⟩kω(K<α) for i∈I, m∈I and
gm∈⟨Sm⟩kω(K<α). Then
gi∈⟨Si⟩kω(Kγ) where γ=maxi∈I∪{m}γi and gi∈Hγi∖H<γi
for i∈I∪{m}. Then γ<α and
property (2) holds by the
hypothesis. Properties (3) and (1) hold by
Lemma 18, (2), and the
hypothesis. Property (3) holds by Lemma 17.
∎
5 Stability
A number of arguments below require that new sequences are added in a
strict order (both to make the recursion work, as well as to
keep the convergence structure intact) depending on the sequential
order of the points involved.
The sequential order may change as the new sequences are added, however, so a
mechanism to keep the order fixed is needed. One such mechanism is
described below.
While the results below hold for any type of fpse-chains, the proofs
are more involved and the additional generality is not required in the
constructions that follow. We thus introduce a narrow subclass of
fpse-chains.
Definition 17**.**
Let (G,K,U) be a convenient triple and C={Si:i∈ω} be
such that (C,ξ) is a sequential scale in (G,K,U) for some
ξ. Call an fpse-chain
C={(Hγ,Kγ,Uγ):γ0≤γ≤γ1} above (G,K,U) along
{Dγ:γ0≤γ≤γ1} relative to
(C,ξ) of finite type if
Dγ⊆⟨Si(γ)⟩kω(K<γ) for
every γ.
If i(γ)=m for every γ say that C is close to
⟨Sm⟩. Otherwise, if i(γ)=m for every γ say that
C is away from ⟨Sm⟩.
The constructions below add new sequences to several ‘summands’ of the
‘direct sum’. The next construction presents a way to rearrange the
order in which new sequences are added.
Definition 18**.**
Let (G,K,U) be a convenient triple and C={Si:i∈ω} be
such that (C,ξ) is a sequential scale in (G,K,U) for some
ξ. Let C={(Hγ,Kγ,Uγ):γ0≤γ≤γ1} be an
fpse-chain of finite type above (G,K,U) along {Dγ:γ0≤γ≤γ1} relative to (C,ξ).
Let m∈ω and
γ(m)={γ0≤γ≤γ1:Dγ⊆⟨Sm⟩kω(K<γ)}.
Let μ<ω1 be such that h:μ→γ(m) is the unique
1-to-1 monotone map. Define a chain
C(m)(μ′)={(H(m)λ,K(m)λ,U(m)λ):λ<μ′≤μ} above (G,K,U) along {Dh(λ):λ<μ′} recursively as follows.
Let C(m)(∅)=∅. Suppose C(m)(μ′′) has
been defined for all μ′′<μ′. If μ′ is a limit ordinal, put
C(m)(μ′)=∪μ′′<μ′C(m)(μ′′). Otherwise
μ′=μ′′+1 for some μ′′<ω1. Put
Lμ′=Dh(μ′)τ(Uh(μ′)). Let
H(m)μ′=H(m)μ′′+Lμ′ and let K(m)μ′ be
the closure of K(m)μ′′∪{Lμ′} under finite sums
and intersections. Let U(m)μ′={U∩H(m)μ′:U∈Uh(μ′)}. In the definition above,
H(m)λ⊆Hγ1 and
K(m)λ⊆Kγ1. It is an easy argument to see that
each (H(m)λ,K(m)λ,U(m)λ) is a convenient triple.
Call C(m)=C(m)(μ) the m-reduction of C.
It is not immediately clear that the m-reduction is an
fpse-chain. This is the subject of Lemma 38.
The lemma below holds for any pse-chains, thus the sequential scale is
only needed to narrow its statement to fpse-chains and plays no other
role in the proof.
Lemma 33**.**
Let (G,K,U) be a convenient triple and C={Si:i∈ω} be
such that (C,ξ) is a sequential scale in (G,K,U) for some
ξ. Let {(Hγ,Kγ,Uγ):γ≤γ0} be an
fpse-chain above (G,K,U) along {Dγ:γ≤γ0} relative to (C,ξ). Let
Lγ=Dγτ(Uγ).
Let K⊆Hγ be compact in kω(Kγ). Then
K⊆K′+∑λ∈FLλ where K′∈K and
F⊆γ+1 is finite.
Proof.
Induction on γ0.
∎
Further similarity with the direct sum is provided by the following
‘projection’ result.
Lemma 34**.**
Let (G,K,U) be a convenient triple and C={Si:i∈ω} be
such that (C,ξ) is a sequential scale in (G,K,U) for some
ξ. Let K⊆G be compact in kω(K), I⊆ω
be finite and m∈ω. Then the set
πI,m(K)={d∈⟨Sm⟩kω(K):d+d′∈K for some d′∈∑i∈I∖{m}⟨Si⟩kω(K)} is compact in kω(K).
Proof.
Suppose πI,m(K) is not compact. Then there exists an infinite
closed and discrete subset Dm={dnm:n∈ω} such that for
some dni∈⟨Si⟩kω(K) and finite I⊆ω, I∋m, and D={∑i∈Idni:n∈ω}⊆K. After thinning out and reindexing we
may assume that each Di={dni:n∈ω} is either closed and
discrete in kω(K) or Di→di for some di∈G. Let
I′⊆I be the set of all i∈I such that Di is closed
and discrete in kω(K). Then m∈I′ and {∑i∈I′dni:n∈ω}⊆K′ for some K′∈K contradicting the
assumption that C is discrete separated.
∎
Lemma 35**.**
Let (G,K,U) be a convenient triple and C={Si:i∈ω} be
such that (C,ξ) is a sequential scale in (G,K,U) for some
ξ. Let C={(Hγ,Kγ,Uγ):γ0≤γ≤γ1} be an
fpse-chain of finite type above (G,K,U) along {Dγ:γ0≤γ≤γ1} relative to (C,ξ).
Let m∈ω and
γ(m)={γ0≤γ≤γ1:Dγ⊆⟨Sm⟩kω(K<γ)}.
Let μ<ω1 be such that h:μ→γ(m) is the unique
1-to-1 monotone map. Let
C(m)={(H(m)λ,K(m)λ,U(m)λ):λ<μ} be the
m-reduction of C and let so(g,D,kω(Kγ))=σ<ω1 where D⊆⟨Sm⟩.
Then g∈H(m)λ and so(g,D,kω(K(m)λ))=σ where
λ=sup{λ′:h(λ′)≤γ}. If λ is a
limit ordinal and γ=h(λ), λ may be replaced with
with <λ.
Proof.
Suppose the satement holds for all g′∈Gγ such that so(g′,D,kω(Kγ))=σ′<σ and let so(g,D,kω(Kγ))=σ. If σ=0 then the proof is immediate
so assume that σ>0 and there are gn→g in kω(Kγ)
such that so(gn,D,kω(Kγ))=σn<σ.
By the hypothesis so(gn,D,kω(K(m)λ))=σn where
λ is defined as in the statement of the Lemma. Since gn→g in kω(Kγ) there exists a compact K∈Kγ such
that gn∈K and by Lemma 33 K∈K′+∑ν∈FLν for some finite ν⊆γ+1. Thus
gn=gn+∑ν∈Fgnν where gn∈K′⊆G, gnν∈Lν, and
Lν=Dντ(Uν)⊆⟨Si(ν)⟩kω(Kν).
Let I={i(ν):ν∈F}∪{m}. By Lemma 1
gn∈∑i∈I⟨Si⟩kω(Kγ) so gn=∑i∈Ign(i) where gn(i)∈⟨Si⟩kω(Kγ). Since gn∈G by
Lemma 27 gn(i)∈⟨Si⟩kω(K) for every i∈I.
Since (C,ξ) is a sequential scale
gn(i)+∑i(ν)=ignν=0 for every i=m by (2) so
gn=gn(m)+gnν where gn(m)∈πI,m(K′) and
gnν∈∑i(ν)=mLν. If i(ν)=m then
Lν∈K(m)ν where ν≤γ so ν=h(λ′) for
some λ′≤λ. Thus gn∈K′′∈K(m)λ and
so(g,D,kω(K(m)λ))=σ.
∎
Lemma 36**.**
Let (G,K,U) be a convenient triple and C={Si:i∈ω} be
such that (C,ξ) is a sequential scale in (G,K,U) for some
ξ. Let C={(Hγ,Kγ,Uγ):γ0≤γ≤γ1} be an
fpse-chain of finite type above (G,K,U) along {Dγ:γ0≤γ≤γ1} relative to (C,ξ).
Let m∈ω and
γ(m)={γ0≤γ≤γ1:Dγ⊆⟨Sm⟩kω(K<γ)}.
Let μ<ω1 be such that h:μ→γ(m) is the unique
1-to-1 monotone map. Let
C(m)={(H(m)λ,K(m)λ,U(m)λ):λ<μ} be the
m-reduction of C and let D⊆⟨Sm⟩,
P0⊆⟨Sm⟩kω(Kγ) .
Then hDKγ(P0)=hDK(m)λ(P0) where
λ=sup{λ′:h(λ′)≤γ}. If λ is a
limit ordinal and γ=h(λ), λ may be replaced
with <λ.
Proof.
Let P1⊆[D]δkω(Kγ) and K∈Kγ be
such that P0⊆P1+K. By Lemma 35
P1⊆[D]δkω(K(m)λ) and by
Lemma 33 K=K′+∑ν∈FLν where
F⊆γ+1 is finite, K′∈K, and
Lν=Dντ(Uν)⊆⟨Si(ν)⟩kω(Kν). Let
g=p1−p0∈K for some p1∈P1 and p0∈P0. Then
g∈⟨Sm⟩kω(Kγ) by Lemma 1 and
g=g′+∑ν∈Fgν where gν∈Lν⊆⟨Si(ν)⟩kω(Kγ) and g′∈K′.
As in the proof of Lemma 35 g=g′′+gm where g′′∈πI,m(K′)
and gm∈∑i(ν)=mLν where I={i(ν):ν∈F}∪{m}. Thus
p1−p0∈πI,m(K′)+∑i(ν)=mLν⊆K′′∈K(m)λ so P0⊆P1+K′′.
∎
Lemma 37**.**
Let (G,K,U) be a convenient triple and C={Si:i∈ω} be
such that (C,ξ) is a sequential scale in (G,K,U) for some
ξ. Let m∈ω and
C={(Hγ,Kγ,Uγ):γ0≤γ≤γ1} be an fpse-chain
of finite type above (G,K,U) away from ⟨Sm⟩.
Let D⊆⟨Sm⟩ and P⊆⟨Sm⟩kω(Kγ).
Then P⊆G and hDKγ(P)=hDK(P).
If P is closed and discrete in kω(K) then P is closed
and discrete in kω(Kγ).
Proof.
Note that C(m)=∅. Applying Lemma 35 to
each p∈P one shows that
P⊆G.
Let hDKγ(P)=δ. It follows from
Lemma 36 and C(m)=∅ that
hDK(P)≤δ. To see that P is closed and
discrete note that otherwise for some infinite P′⊆P there
exists a compact K∈Kγ1 such that P′⊆K. Just
as in the proof of Lemma 36 we may assume that K∈K
contradicting the assumption that P is closed and discrete in kω(K).
∎
Lemma 38**.**
Let (G,K,U) be a convenient triple and C={Si:i∈ω} be
such that (C,ξ) is a sequential scale in (G,K,U) for some
ξ. Let C={(Hγ,Kγ,Uγ):γ0≤γ≤γ1} be an
fpse-chain of finite type above (G,K,U) along {Dγ:γ0≤γ≤γ1} relative to (C,ξ).
Then the m-reduction
C(m)={(H(m)λ,K(m)λ,U(m)λ):λ<μ} is an
fpse-chain above (G,K,U) along {Dh(λ):λ<μ}
relative to (C,ξ) close to ⟨Sm⟩.
Proof.
Let m∈ω and
γ(m)={γ0≤γ≤γ1:Dγ⊆⟨Sm⟩kω(K<γ)}.
Let μ<ω1 be such that h:μ→γ(m) is the unique
1-to-1 monotone map.
Each (H(m)λ,K(m)λ,U(m)λ) is a convenient triple. To
show (1) and (2) note that each
Dh(λ)⊆⟨Sm⟩kω(K(m)<λ) by
Lemma 35 and
hPK(m)<λ(Dh(λ))=hPK<h(λ)(Dh(λ))
for P∈{⟨Sm⟩,⟨Sm⟩∖⟨2Sm⟩} by Lemma 36 and
K(m)<λ⊆K<h(λ).
∎
The next definition and lemma introduce the concept of stability as
well as a way to pass to a stable subset in some cases.
Definition 19**.**
Let (G,K,U) be a convenient triple, C={Si:i∈ω} and
ξ⊆ω1 be such that (C,ξ) is a sequential scale
in (G,K,U). Let P⊆⟨Sm⟩kω(K) be closed and discrete in
kω(K). Call P (C,ξ)-stable (or simply stable
if (C,ξ) is clear from the context) in (G,K,U) if for any infinite
P′⊆P and any fpse-chain
{(Hγ,Kγ,Uγ):γ0≤γ≤γ1} of finite type
above (G,K,U) relative to (C,ξ)
h⟨Sm⟩∖⟨2Sm⟩Kγ1(P′)=h⟨Sm⟩∖⟨2Sm⟩K(P) provided P′ is
closed and discrete in kω(Kγ1).
Lemma 39**.**
Let (G,K,U) be a convenient triple, C={Si:i∈ω} and
ξ⊆ω1 be such that (C,ξ) is a sequential scale
in (G,K,U). Let P⊆⟨Sm⟩kω(K) be closed and discrete in
kω(K). Then there exists an infinite P′⊆P and an
fpse-chain {(Hγ,Kγ,Uγ):γ0≤γ<γ1} above
(G,K,U) close to ⟨Sm⟩ such that P′ is (C,ξ)-stable in
(Hγ1,Kγ1,Uγ1).
Proof.
Let δ<ω1 be the smallest ordinal such that there exists
an infinite P′⊆P and an fpse-chain
{(Hγ,Kγ,Uγ):γ0≤γ<γ1} above (G,K,U) close
to ⟨Sm⟩ such that P′ is closed and discrete in
kω(Kγ1) and h⟨Sm⟩∖⟨2Sm⟩Kγ1(P′)=δ.
If P′′⊆P′ is not stable in (Hγ1,Kγ1,Uγ1) for some
infinite P′′, there exists an
fpse-chain C={(Hγ,Kγ,Uγ):γ1≤γ≤γ2} of
finite type above (Hγ1,Kγ1,Uγ1) relative to (C,ξ) such
that
h⟨Sm⟩∖⟨2Sm⟩Kγ2(P′′′)<h⟨Sm⟩∖⟨2Sm⟩Kγ1(P′)=δ
for some infinite P′′′⊆P′′ closed and discrete in
kω(Kγ2).
Using the m-reduction C(m) of C instead of C and
Lemmas 36 and 38 leads to a
contradiction with the minimality of δ.
∎
Call the δ in the proof above the (C,ξ)-stable
K-height of P above ⟨Sm⟩∖⟨2Sm⟩.
6 Graded boolean group extensions
This section introduces an algebraic mechanism for keeping track and
altering of the sequential order of points.
Definition 20**.**
Let (G,K,U) be a convenient triple, C={Si:i∈ω} and
ξ={σi:i∈ω} be such that (C,ξ) is a sequential
scale in (G,K,U). Let P⊆[⟨Sm⟩∖⟨2Sm⟩]Σkω(K) for some m∈ω
and Σ<σm−1.
Call W={(Hα,Kα,Uα,Sα,sα):α0≤α≤Ω} a well-graded group stack
above (P,G,K,U,Σ) or wggs for short if it
satisfies the following properties:
- (1)
{(Hα,Kα,Uα):α0≤α≤Ω}* is an fpse-chain of
finite type above (G,K,U) along
{Sα:α≤Ω} relative to C close to ⟨Sm⟩;*
2. (2)
Sα→sα∈Hα∖H<α* in kω(Kα);*
3. (3)
if α is a limit ordinal there exist increasing αn<α such
that αn→α,
so(sαn,⟨Sm⟩∖⟨2Sm⟩,kω(Kαn)) is increasing, and
Sα={sαn:n∈ω};
4. (4)
if α is a successor ordinal there exist
pn∈P such that Sα={pn:n∈ω};
5. (5)
if g∈Hα∖H<α, α≤Ω then
so(g,⟨Sm⟩∖⟨2Sm⟩,kω(Kα))≥so(sα,⟨Sm⟩∖⟨2Sm⟩,kω(Kα));
6. (6)
⊤(W)=so(sΩ,⟨Sm⟩∖⟨2Sm⟩,kω(KΩ))>so(sα,⟨Sm⟩∖⟨2Sm⟩,kω(Kα))*
for any α<Ω;*
Lemma 40**.**
Let (G,K,U) be a convenient triple. Let C={Si:i∈ω} and ξ={σi:i∈ω} be such that
(C,ξ) is a sequential scale in (G,K,U).
Let {(Hγ,Kγ,Uγ):γ<α} be a an fpse-chain above (G,K,U)
relative to (C,ξ) where α is a limit ordinal.
Let m∈ω and
S={sn:n∈ω}⊆⟨Sm⟩∖⟨2Sm⟩kω(K<α) be
such that sn∈Hγn∖H<γn where
γn→α is strictly increasing and cofinal in
α. Suppose
so(g,⟨Sm⟩∖⟨2Sm⟩,kω(Kγn))≥σn<σm
for any g∈Hγn∖H<γn. Then S is
independent,
h⟨Sm⟩∖⟨2Sm⟩K<α(S)≥σ=supσn and ⟨S⟩ is
closed and discrete in kω(K<α). In particular, if
so(sn,⟨Sm⟩∖⟨2Sm⟩,kω(Kγn))=σn then
h⟨Sm⟩∖⟨2Sm⟩K<α(S)=σ.
Proof.
Let K∈K<α and β<σ. Then K∈Kγ for some
γ<α. Let γi>γ be such that
so(si,⟨Sm⟩∖⟨2Sm⟩,kω(Kγi))=σi>β. If
a+g=si for some a∈K and g∈H<α then g∈Hγi∖H<γi. Then either
so(g,⟨Sm⟩∖⟨2Sm⟩,kω(K<α))≥σm>σi>β or
so(g,⟨Sm⟩∖⟨2Sm⟩,kω(K<α))=so(g,⟨Sm⟩∖⟨2Sm⟩,kω(Kγi))≥so(si,⟨Sm⟩∖⟨2Sm⟩,kω(Kγi))=σi>β by the assumption
and Lemma 32.
Thus for any such K the set
S∖(K+[⟨Sm⟩∖⟨2Sm⟩]βkω(K<α))=∅ for
every β<σ so h⟨Sm⟩∖⟨2Sm⟩K<α(S)≥σ.
If s∈⟨S⟩ then s=∑n∈Isn for some finite
I⊆ω and s∈Hγ′∖H<γ′ where
γ′=maxn∈Iγn since γn are strictly
increasing. Thus if s∈⟨F⟩ where F={sn:γn≤γ} then s∈K. Therefore ⟨S⟩∩K is
finite and ⟨S⟩ is closed and discrete in kω(K<α).
∎
Lemma 41**.**
Let (G,K,U) be a convenient triple. Let C={Si:i∈ω} and ξ={σi:i∈ω} be such that
(C,ξ) is a sequential scale in (G,K,U). Let m∈ω and
P={pn:n∈ω}⊆[⟨Sm⟩∖⟨2Sm⟩]Σkω(K) be an
independent subset of G such that ⟨P⟩ is closed and discrete in
kω(K), h⟨Sm⟩∖⟨2Sm⟩K(P)=Σ<σm−1, and P is
(C,ξ)-stable in (G,K,U). Let D={dn:n∈ω}⊆G and d∈[D]μkω(K) where μ≥1 is
a successor ordinal, Σ+μ≤σ≤σm−1 for some
successor ordinal σ.
Then there exists a wggs W={(Hα,Kα,Uα,Sα,sα):α≤Ω} above (P,G,K,U,Σ)
such that ⊤(W)=σ and the following property holds.
- (1)
d+sΩ∈[{dn+pn:n∈ω}]ηkω(KΩ)* where*
[TABLE]
In addition, if P′⊆G is such that P′∩P=∅,
P∪P′ is independent, and ⟨P′∪P⟩ is closed and
discrete in kω(K) then one may assume the following property holds.
- (1)
⟨P′⟩* is closed and discrete in kω(KΩ);*
Proof.
Let σ=Σ+1<σm. Then μ=1 so by passing to subsets
and reindexing we may assume that dn→d in kω(K). Use
Lemma 10 to find an S0⊆P, and U0⊇U
such that ⟨S′⟩τ(U0)∩G=⟨S′⟩ for every infinite
S′⊆S0 and
⟨2S0⟩τ(U0)∩⟨S0⟩∖⟨2S0⟩τ(U0)=∅. By passing
to subsets and applying Lemma 6 we may assume that
S0→s0 in τ(U0) and S0 satisfies M(⟨Sm⟩∖⟨2Sm⟩,K)
and M(⟨Si⟩,K) for every i∈ω. Let (H0,K0,U0) be the fpse
of (G,K,U) over S0. Put (H<0,K<0,U<0)=(G,K,U) and
W={(Hi,Ki,Ui,Si,si):i<1}. Properties (1)–(4) are
immediate, (6) holds vacuously,
and (1) holds by the construction.
Let g∈H0∖G and let
so(g,⟨Sm⟩∖⟨2Sm⟩,kω(K0))=σ′. If σ′≤Σ
then so(g,⟨Sm⟩∖⟨2Sm⟩,kω(K))=σ′ by Lemma 11, in
particular g∈G. Thus σ′>Σ so
σ′≥Σ+1≥so(s0,⟨Sm⟩∖⟨2Sm⟩,kω(K0))
and (5) holds.
Suppose K∈K is such that (K+L)∩⟨P′⟩ is infinite where
L=⟨S0⟩τ(U0). Let an+dn=pn∈⟨P′⟩ be
distinct points in ⟨P′⟩ such that an∈K and dn∈L. Then
dn∈G so by (2) dn∈⟨S0⟩⊆⟨P⟩. If
the set {dn:n∈ω} is infinite then K∩⟨P′∪P⟩
is infinite since P′ and P are disjoint and P′∪P is
independent, contradicting ⟨P′∪P⟩ is closed and discrete in
kω(K). Thus {dn:n∈ω} is finite so K′∩⟨P′⟩ is
infinite for some K′∈K contradicting ⟨P′∪P⟩ is closed
and discrete in kω(K), so (1) holds.
Suppose W that satisfy (1) and (1)
can be constructed for every σ′<σ where
Σ+1<σ≤σm−1.
If μ>1 let μn→μ−1 be increasing such that dn′→d in
kω(K) where so(dn′,D,kω(K))=μn. Using regularity, construct disjoint
In⊆ω such that so(dn′,Dn,kω(K))=μn where
Dn={di:i∈In}.
If μ=1 assume, by
possibly thinning out and reindexing that dn→d, put
μn=1 and let In⊆ω be arbitrary infinite disjoint
subsets.
Let σi→σ−1>Σ be successor ordinals such that
σi≥Σ+μi. Let Pn={pi:i∈In}. Note that
every subset of P (including Pn, and their arbitrary unions) is
(C,ξ)-stable in (G,K,U) and
h⟨Sm⟩∖⟨2Sm⟩K(Pn)=Σ.
Build, by induction, a sequence of
Wi={(Hα,Kα,Uα,Sα,sα):γi−1<α≤γi} such that W0 is a wggs above
(P0,G,K,U,Σ), Wi+1 is a wggs above
(Pi+1,Hγi,Kγi,Uγi,Σ) for
i∈ω, and the following properties hold.
- (1)
the subgroup ⟨P′∪∪j>iPj⟩ is closed and discrete in
kω(Kγi);
2. (2)
for each i∈ω
so(sγi,⟨Sm⟩∖⟨2Sm⟩,kω(Kγi))=σi;
3. (3)
dn′+sγn∈[{di+pi:i∈In}]ηnkω(Kγi) where
[TABLE]
At the n-th step the set ∪j>nPj is closed, discrete, and
(C,ξ)-stable in (Hγn,Kγn,Uγn),
h⟨Sm⟩∖⟨2Sm⟩Kγn(∪j>nPj)=Σ, and σi<σ so Wn+1 exists
by the hypothesis.
Let Ω=limγi and consider the conventient triple (H<Ω,K<Ω,U<Ω).
Put SΩ′={sγi:i∈ω} and let g∈H<Ω be such that g∈Hα∖H<α and
so(g,⟨Sm⟩∖⟨2Sm⟩,kω(K<Ω))=σ′ for some σ′<σm
and α<Ω. Then so(g,⟨Sm⟩∖⟨2Sm⟩,kω(Kα))=σ′
by (1), Lemma 32, and the construction of
Wi, in particular,
so(sγi,⟨Sm⟩∖⟨2Sm⟩,kω(K<Ω))=σi.
The sequence of ordinals γi is cofinal in
Ω and so(g,⟨Sm⟩∖⟨2Sm⟩,kω(Kγi))≥σi for
every g∈Hγi∖H<γi
by (5) and the hypothesis so
h⟨Sm⟩∖⟨2Sm⟩K<Ω(SΩ)=σ−1 for any infinite subset
SΩ⊆SΩ′ and ⟨SΩ′⟩ is closed and
discrete in kω(K<Ω) by Lemma 40.
Construct an fpse (HΩ,KΩ,UΩ) of (H<Ω,K<Ω,U<Ω) over some
infinite SΩ⊆SΩ′ such that SΩ→sΩ in kω(KΩ) using Lemma 29 and let
L=⟨SΩ⟩τ(UΩ).
Suppose so(g′,⟨Sm⟩∖⟨2Sm⟩,kω(KΩ))<σ for some g′∈HΩ. Then so(g′,⟨Sm⟩∖⟨2Sm⟩,kω(KΩ))≤Lσ−1 so by
Lemma 11
so(g′,⟨Sm⟩∖⟨2Sm⟩,kω(K<Ω))≤Lσ−1<ω1 and g′∈H<Ω. Thus so(g,⟨Sm⟩∖⟨2Sm⟩,kω(KΩ))≥σ for any
g∈HΩ∖H<Ω, in particular,
so(sΩ,⟨Sm⟩∖⟨2Sm⟩,kω(KΩ))=σ.
Now (1)–(6) hold by
construction, (1) holds by the choice of dn′.
Let K∈K<Ω be such that K+L∩⟨P′⟩ is
infinite. Suppose pn=an+sn are distinct points in ⟨P′⟩ such
that an∈K and sn∈L. Then sn∈H<Ω so
sn∈⟨SΩ⟩ by (2). Let K∈Kβ where
β<Ω. If the set {sn:n∈ω} is infinite there is
an n∈ω such that sn∈Hα∖H<α for
some α≥β so pn=an+sn∈G contradicting pn∈P′. Thus {sn:n∈ω} is finite and pn∈K′∈Kγi for some γi<Ω. Then ⟨P′⟩ is not
closed and discrete in kω(Kγi) contradicting the
properties of Wi. Hence (1) holds.
∎
Lemma 42**.**
Let C={Si:i∈ω} be discrete separated in some kω-pair
(G,K). Let g∈Pkω(K) for some g∈G and P={pj:j∈ω}⊆G. Let D={dj:j∈ω} where
dj=∑i∈Idji for some finite I⊆ω, and
dji∈⟨Si⟩kω(K). Suppose pj−dj∈K for some K∈K
and every j∈ω.
Then there are di∈⟨Si⟩kω(K), i∈I and a∈K such
that
- (1)
g=a+∑i∈Idi;
2. (2)
for any U, UK, U(i), i∈I open in kω(K) such that g∈U, a∈UK, and di∈U(i) there is an n∈ω such that
pn∈U, pn−dn∈UK, and dni∈U(i);
Proof.
Suppose the Lemma holds for all g′∈G such that so(g′,P,kω(K))=σ′<σ and let so(g,P,kω(K))=σ. If
σ=0 the proof is immediate so assume σ>0. Then there are
gn∈[P]σn where σn<σ and gn→g. By the
hypothesis there are an∈K and di,n∈⟨Si⟩kω(K) for
i∈I such that gn=an+∑i∈Idi,n and for any
m∈ω and any open in kω(K) subsets U∋gm, UK∋am,
U(i)∋di,m where i∈I there is an n∈ω such that
pn∈U, dni∈U(i), pn−dn∈UK.
Since {∑i∈Idi,n:n∈ω}⊆K′ for
some K′∈K and C is discrete separated by K in G,
there is an i∈I such that the set {di,n:n∈ω} is not
closed and discrete. By thinning out and reindexing we then may assume
that di,n→di for some di∈⟨Si⟩kω(K). Replacing K′
with some K′′⊇K′+{di,n:n∈ω} and repeating this
argument we may find di such that di,n→di for each i∈I. By thinning out and reindexing we may assume that an→a for some
a∈K.
Then g=a+∑i∈Idi.
Let U, UK, U(i), i∈I be open in kω(K) such that g∈U,
a∈UK, and di∈U(i) for every i∈I. Pick an m∈ω
such that gm∈U, am∈UK, and di,m∈U(i) for every i∈I. By the
hypothesis there is an n∈ω such that pn∈U, pn−dn∈UK, and dni∈U(i), i∈I.
∎
Lemma 43**.**
Let ξ={σm:m∈ω}⊆ω1∖2, m∈ω be a
family of successor ordinals. Then there exists a convenient triple
(G,K,U) and a family C={Si:i∈ω} of discrete
separated subsets of G such that so(0,⟨Sm⟩∖⟨2Sm⟩,kω(K))=σm for
every m∈ω and (C,ξ) is a sequential scale in (G,K,U).
Proof.
For each m∈ω let (G(m),K(m),U(m)) be a convenient triple such that
kω(K(m)) is neither discrete nor compact.
Let Cm={S(n):n∈ω} be a family of countable disjoint
subsets of G(m)∖{0} such that ∪n∈ωS(n) is an
independent set and ⟨∪n∈ωS(n)⟩ is closed and
discrete in kω(K(m)). Then (Cm,ξ) is a trivial sequential scale in
(G(m),K(m),U(m)).
Let σnm→σm−1.
Let S(m)=∪n∈ωS(m,n) where S(m,n) are infinite and
disjoint. Each S(m,n) is trivially (Cm,ξ)-stable in (G(m),K(m),U(m))
(its K(m)-height is [math]). Put (G−1,K−1,U−1)=(G(m),K(m),U(m)),
γ−1(m)=−1, and use
Lemma 41 to construct wggs
[TABLE]
above (S(m,n),Gγn−1(m),Kγn−1(m),Uγn−1(m),0)
relative to (Cm,ξ) such that
⊤(Wn(m))=σnm for every n∈ω. Let
α(m)=limγn(m).
Just as in the proof of Lemma 41, the set S={sγ(n):n∈ω} is closed and discrete and
h⟨S(m)⟩∖⟨2S(m)⟩K<α(m)(S)=σm−1. Use
Lemma 29 to find an fpse (Gα(m),Kα(m),Uα(m)) of
(G<α(m),K<α(m),U<α(m)) such that S→s∈G<α(m) in
kω(Kα(m)). Then so(s,⟨S(m)⟩∖⟨2S(m)⟩,kω(Kα(m)))=σm−1,
the set s+S(m) is independent,
⟨s+S(m)⟩∖⟨2(s+S(m))⟩=s+⟨S(m)⟩∖⟨2S(m)⟩ so S(Kα(m),s+S(m),σm) holds. Put Sm=s+S(m), (Gm,Km,Um)=(Gα(m),Kα(m),Uα(m)).
Let G be the direct sum of Gm, U be the appropriate basis of
the topology inherited from the product of (Gm,τ(Um)), and
K be the closure of ∪m∈ωKm under finite sums.
Put C={Sm:m∈ω}. Then the desired properties hold by
the construction.
Note that in the proof above, the full sequential scale Cm was
only required to formally satisfy the conditions of
Lemma 41.
∎
Let σm be an increasing sequence of successor ordinals in
ω1. Let σki→σi−1 be an increasing sequence of
successor
ordinals for every i∈ω such that
supσki=σi−1.
Since all groups Gα in the construction have cardinality
2ω we will assume that every Gα is a subgroup
(algebraically) of 2ω. Let {Cα:α<ω1}⊆[2ω]ω and {Pα:α<ω1 is a limit ordinal}⊆[2ω]ω.
Let (G0,K0,U0)=(G,K,U) where the convenient triple (G,K,U) and the
sequential scale (C,ξ) have been constructed in Lemma 43.
Below we use the notation
cCα for the closure of Cm(α) in kω(K<m(α)).
Lemma 44**.**
There exists an fpse-chain {(Gα,Kα,Uα):α<ω1} relative to
(C,ξ) and an increasing continuous m:ω1→ω1 such that
- (1)
if α is a limit ordinal and Pα⊆Gm(α+1) is an infinite subset that is closed and
discrete in kω(Km(α+1)) then
there exists an infinite Sα⊆Pα such that
Sα→sα in kω(Km(α+2));
2. (2)
if α is a limit ordinal and
0∈Pαkω(Km(α+1)) then so(0,Pα,kω(Km(α+2)))≤σm for some m∈ω;
3. (3)
if α is a limit ordinal then one of the following conditions holds
- a.
if hCK<m(α)(cCα∩U)=ω for every
U∈U<m(α) then S→0 in kω(Km(α)) for some
S⊆cCα;
2. b.
suppose cCα⊆∑i∈I⟨Si⟩kω(K<m(α)) for some finite
I⊂ω (so (3.a) does not hold), and cCα∩∑i∈I′⟨Si⟩kω(K<m(α))=∅ for any I′⊂I, I′=I; if there exists an fpse-chain
C={(Gγ,Kγ,Uγ):m(α)≤γ≤γ0} such that
cCαkω(Kγ0)∩∑i∈I′⟨Si⟩kω(Kγ0)=0 for some I′⊂I,
I′=I; then cCαkω(Km(α+1))∩∑i∈I′′⟨Si⟩kω(Km(α+1))=0 for some I′′⊂I, I′′=I;
3. c.
suppose neither (3.a) nor (3.b) holds,
cCα⊆∑i∈I⟨Si⟩kω(K<m(α)) for some finite I⊂ω, and
cCα∩∑i∈I′⟨Si⟩kω(K<m(α))=∅ for any
I′⊂I, I′=I; suppose there exists an fpse-chain
[TABLE]
and
for some γn, n∈ω, strictly increasing and cofinal in
γ0 the following properties hold:
there are d(n)∈cCαkω(Kγ0) such that
d(n)=∑i∈Id(i,n) for every n∈ω where
d(i,n)∈⟨Si⟩kω(Kγ0)∩⋂{Uτ(Uγ0):U∈U<m(α)},
and
so(g,⟨Si(n)⟩,kω(Kγ0))≥σni(n)*
for any g∈Gγn∖G<γn where d(i(n),n)∈Gγn∖G<γn for some choice of i(n)∈I
and d(i,n)∈G<m(α) for i∈I;
then the condition above holds if
γ0 is replaced by m(α+1).*
4. (4)
if α is a limit ordinal then for every finite
I⊂ω
[TABLE]
where H(I)=∑i∈I⟨Si⟩kω(Km(α+3))
and for any U∈Um(α+3) there is a U′⊆U such
that U′∈Um(α+3) and
[TABLE]
Proof.
Put m(0)=0. If α is a
successor such that α=β+n where n∈ω and β is
a limit and n>3 or β=0 and n>0 let
m(α)=m(α−1)+1 and let (Gm(α),Km(α),Um(α))
be an arbitrary fpse of (Gm(α−1),Km(α−1),Um(α−1)).
Let α be a limit ordinal and Gα′ etc., m(α′)
have been constructed for all α′<α. Then m(α) is
uniquely defined by continuity.
Let K<m(α)={Ki:i∈ω} and U<m(α)={Ui:i∈ω}.
Suppose hCK<m(α)(cCα∩U)=ω for every
U∈Um(α). Find an infinite S⊆cCα such that
S→0 in τ(U<m(α)) and S
satisfies M(C,K<m(α)) using
Lemma 6. Use Lemma 31 to find an
fpse (Gm(α),Km(α),Um(α)) of (G<m(α),K<m(α),U<m(α)) over S such that
S→0 in Km(α). Let (Gm(α+1),Km(α+1),Um(α+1)) be
an arbitrary fpse of (Gm(α),Km(α),Um(α)) and put
m(α+1)=m(α)+1 to satisfy (3.a).
If the wggs from the statement of (3.b)
or (3.c) exists put
m(α+1)=γ0.
Suppose Pα is closed and discrete in
Gm(α+1).
If
hCKm(α+1)(Pα)=ω use
Lemma 29 to find an fpse (Gm(α+2),Km(α+2),Um(α+2))
of (Gm(α+1),Km(α+1),Um(α+1)) over Sα⊆Pα such
that Sα→s∈Gm(α+2) in
kω(Km(α+2)) where m(α+2)=m(α+1)+1.
Otherwise Pα⊆∑i≤m⟨Si⟩kω(Km(α+1))+K for some
K∈Km(α+1). Pick an infinite D={dn=∑i≤mdni:n∈ω} where
dni∈⟨Si⟩kω(Km(α+1)) and pn=dn+an
for some distinct pn∈Pα and an∈K. By passing to
subsequences and reindexing we may assume that an→a for some
a∈K and either dni→di or Di={dni:n∈ω} is
closed and discrete in K<m(α). Repeatedly using
Lemma 29 find a finite fpse-chain
[TABLE]
for some k≤m such that {dni:n∈J}→di in
kω(Km(α+2)) for some infinite J⊆ω.
Let 0∈Pαkω(Km(α+1)) and suppose
ωhCKm(α+1)(Pα)<ω. Let Pα={pj:j∈ω} and
pj−dj∈K for some K∈Km(α+1) where dj=∑i∈Idji for some finite I⊂ω, and
dji∈⟨Si⟩kω(Km(α+1)).
Using Lemma 42 find
di∈⟨Si⟩kω(Km(α+1)) and a∈K such that
0=a+∑i∈Idi and for any open in kω(Km(α+1))
subsets U∋0, UK∋a, U(i)∋di, i∈I there is an n∈ω such
that pn∈U, dni∈U(i), and pn−dn∈UK, i∈I.
Replacing each dni with dni+di, each dn with dn+a and picking
K∈Km(α+1) so that di∈K, i∈I, we may assume
that the following properties hold.
- (1)
dj=∑i∈Idji,
dji∈⟨Si⟩kω(Km(α+1)), and pj−dj∈K for
every j∈ω;
2. (2)
for any U∋0 open in kω(Km(α+1)) there exists an
n∈ω such that pn,dni,pn−dn∈U
for every i∈I;
Indeed, let U be open in kω(Km(α+1)). Find a U′∋0,
open in kω(Km(α+1)) such that U′+⋯+U′⊆U where the sum has ∣I∣ terms. Put UK=U′+a, U(i)=U′+di and
let n∈ω be such that pn∈U′, dni∈U(i), and
pn−dn∈UK. Then dni+di∈U′ and pn−(dn+a)=pn−dn+a∈U′.
Let (G(0),K(0),U(0))=(Gm(α+1),Km(α+1),Um(α+1)). By induction on k∈ω
build convenient triples (G(k),K(k),U(k)) and points
d(i,k)∈⟨Si⟩kω(K(k)) where i∈I, such that the
following properties
hold for k>0:
- (1)
(G(k),K(k),U(k))=(Gγ(k),Kγ(k),Uγ(k)) for some γ(k)<ω1 where
[TABLE]
is an fpse-chain above
(G(k−1),K(k−1),U(k−1)) relative to (C,ξ);
2. (2)
either d(i,k)=0 or d(i,k)∈Gγ(i,k)∖G<γ(i,k) where γ(k−1)<γ(i,k)≤γ(k);
3. (3)
if g∈Gγ(i,k)∖G<γ(i,k) then
so(g,⟨Si⟩∖⟨2Si⟩,Kγ(i,k))≥σki;
4. (4)
d(i,k)∈Uτ(U(k)) and p(k)−(∑i∈Id(i,k))∈K∩Uτ(U(k)) for every U∈U(k−1) where
p(k)∈[Pα]η(k)kω(K(k)), and
η(k)≤maxi∈Iσi−1;
Suppose (G(k′),K(k′),U(k′)) etc. that satisfy (1)–(4)
have been built for all k′<k. Let K(k−1)={Kn:n∈ω} and U(k−1)={Un:n∈ω}. Pick
n(j)∈ω by induction using (1) and (2) so
that
[TABLE]
In the rest of the construction below an infinite set J⊆{n(j):j∈ω} will be selected so that D(J,i)={dji:j∈J}
are independent subsets that satisfy some additional properties. Note that
D(J′,i)→0 in τ(U(k−1)) for every i∈I and any infinite
J′⊆J.
Let i∈I. If D(J,i) is not closed and discrete in kω(K(k−1))
then by (2) there exists an infinite Ji⊆J such that
D(Ji,i)→0 in kω(K(k−1)). Otherwise one can find an infinite
Ji⊆J such that for any infinite J′⊆Ji
h⟨Si⟩∖⟨2Si⟩K(k−1)(D(Ji,i))=h⟨Si⟩∖⟨2Si⟩K(k−1)(D(J′,i)).
Repeating this argument for every i∈I in the natural order one
can build Ji such that Ji′⊆Ji if i′≤i and one of
the two alternatives above holds. After replacing J with J=∩i∈IJi we may assume that either D(J,i)→0 in kω(K(k−1)) or D(J,i) is closed and
discrete in kω(K(k−1)) and
h⟨Si⟩∖⟨2Si⟩K(k−1)(D(J,i))=h⟨Si⟩∖⟨2Si⟩K(k−1)(D(J′,i)) for any
infinite J′⊆J.
Let I=Iu∪Iz∪Ih where D(J,i)→0 in kω(K(k−1)) for
every i∈Iu, D(J,i) is closed
and discrete in kω(K(k−1)) for i∈Iz∪Ih,
h⟨Si⟩∖⟨2Si⟩K(k−1)(D(J′,i))≥σi−1 for
each infinite J′⊆J and i∈Iz, and
h⟨Si⟩∖⟨2Si⟩K(k−1)(D(J,i))<σi−1 for every i∈Ih.
Passing to a subset of J if necessary we may assume that
⟨D(J,i)⟩ is closed and discrete in kω(K(k−1)) for every i∈Iz∪Ih. Since (C,ξ) is a sequential scale the set ∪i∈Iz∪IhD(J,i) is independent and the group
⟨∪i∈Iz∪IhD(J,i)⟩ is closed and discrete in
kω(K(k−1)). If i∈Iz use D(J,i)→0 in τ(U(k−1)) and
Lemma 31 to find, after thinning out J if necessary, an fpse
(Gγ(k−1)+1,Kγ(k−1)+1,Uγ(k−1)+1) of (G(k−1),K(k−1),U(k−1)) such that D(J,i)→0 in
kω(Kγ(k−1)+1), D(J,i′)→0 in τ(Uγ(k−1)+1) for every
i′∈Iz∖{i} and D(J,i′′) is closed and discrete in
kω(Kγ(k−1)+1) for every i′′∈Ih∪Iz∖{i}. Note that while formally Lemma 31 does not
guarantee that each D(J,i′′) remains closed and discrete in
kω(Kγ(k−1)+1) for i′′∈Ih∪Iz∖{i}, it
follows from D(J,i′′)→0 in
τ(Uγ(k−1)+1) that if D(J,i′′) is not closed and discrete
in kω(Kγ(k−1)+1), after passing to a smaller J if
necessary we may assume that D(J,i′′)→0 in
kω(Kγ(k−1)+1). We may thus move such i′′ to Iu and
proceed with the argument. Alternatively, one may note that the fpse
constructed above is of finite type and use Lemma 37.
Repeating this argument for every i∈Iz in the natural order and
possibly passing to a subset of J if necessary one can build a
finite fpse-chain
{(Gγ,Kγ,Uγ):γ(k−1)<γ≤γz=γ(k−1)+∣Iz∣}
above (G(k−1),K(k−1),U(k−1)) such that D(J,i)→0 in kω(Kγz) for
every i∈Iu∪Iz and D(J,i) is closed and discrete in
kω(Kγz) for every i∈Ih.
Let i∈Ih. Use Lemma 39 to find an fpse-chain
{(Gγ,Kγ,Uγ):γz<γ<γs(i)} above
(Gγz,Kγz,Uγz) close to ⟨Si⟩ and an infinite Ji⊆J
such that D(Ji,i) is closed, discrete, and stable in
(Gγs(i),Kγs(i),Uγs(i)). Note that D(J,i′) remains closed
discrete in (Gγs(i),Kγs(i),Uγs(i)) for every i′∈Ih by
Lemma 37.
Repeating this argument for every i∈Ih in the natural order and
possibly passing to a subset of J if necessary one may build an
fpse-chain {(Gγ,Kγ,Uγ):γz<γ<γs} of finite
type over (Gγz,Kγz,Uγz) such that each D(J,i), i∈Ih is
closed, discrete, and stable in (G<γs,K<γs,U<γs). Put
δi=h⟨Si⟩∖⟨2Si⟩K<γs(D(J,i)) and note that
δi<σi−1 for every i∈Ih.
For every i∈Ih pick a K(i)∈K<γs, a D+(J,i)={dj(i):j∈J}⊆[⟨Si⟩∖⟨2Si⟩]δikω(K<γs) and
dji=dj(i)+aj(i) for every j∈J where aj(i)∈K(i). By
thinning out J we may assume that aj(i)→a(i) for some
a(i)∈K(i).
Let ν=∣Ih∣ and Ih={i1,…,iν} where the order of
ij is chosen so that the following assumption holds. Let successor
ηi(k)<ω1 for i∈Ih be chosen so that
σi−1≥δi+ηi(k)≥σki and assume that
ηij(k) are increasing for each k∈ω. Put dn′=∑i∈Iu∪Izdn(i). Then dn′→0 in
kω(K<γs). Use Lemma 41 to
construct a wggs
[TABLE]
above
[TABLE]
so that
[TABLE]
and
[TABLE]
Recursively applying Lemma 41 and using
δij+ηij−1≤δij+ηij for j>1 construct
wggs
[TABLE]
above
[TABLE]
for every j≤ν so that
[TABLE]
and
[TABLE]
Put γ(k)=γ(iν,k), (G(k),K(k),U(k))=(Gγ(k),Kγ(k),Uγ(k)) and
d(i,k)=a(i)+sγ(i,k) if i∈Ih, otherwise put
d(i,k)=0.
If U∈U(k−1) then D(J,i)⊆∗U and {aj(i):j∈J}⊆∗a(i)+U so D+(J,i)⊆∗a(i)+U. Now
d(i,k)∈a(i)+D+(J,i)kω(K(k))⊆Uτ(U(k)).
By the construction, ∑i∈Ihsγ(i,k)∈[{∑i∈Idn(i):n∈J}]ηiν(k)kω(K(k)) and ∑i∈Ian(i)→∑i∈Ia(i), so ∑i∈Id(i,k)∈[{∑i∈Idni:n∈J}]η(k)kω(K(k)) by
Lemma 1,
where η(k)=ηiν(k)≤max{σi−1:i∈I}.
Since pn−∑i∈Idni∈K for all but finitely many n∈J
by the choice of pn(j), there exists a
p(k)∈[Pα]η(k)kω(K(k))
such that p(k)−∑i∈Id(i,k)∈K by Lemma 2. Since
dni,pn,pn−∑i∈Idni∈Uτ(U(k)) for all but
finitely many n∈J, p(k)∈Uτ(U(k))
so (4) holds.
Let g∈Gγ(i,k)∖G<γ(i,k) and
so(g,⟨Si⟩∖⟨2Si⟩,kω(K(k)))<σki≤σi−1.
Then i=ij∈Ih for some j≤ν by (1),
Lemma 37 and the
construction of Wj.
Then by
Lemma 32 so(g,⟨Si⟩∖⟨2Si⟩,kω(Kγ(i,k)))<σki
contradicting so(g,⟨Si⟩∖⟨2Si⟩,kω(Kγ(i,k)))≥so(sγ(i,k),⟨Si⟩∖⟨2Si⟩,kω(Kγ(i,k)))≥σki
by (5) and the construction of Wj.
Thus (3) holds.
Let γ(ω)=limk∈ωγ(k) and
(G(ω),K(ω),U(ω))=(G<γ(ω),K<γ(ω),U<γ(ω)).
If U′∈U(ω) then U′=Uτ(U(ω)) where
U∈U(k−1) for some k∈ω. By (4) d(i,n),p(n),p(n)−∑i∈Id(i,n)∈Uτ(U(n))⊆U′ for
all n≥k. Since b(n)=p(n)−∑i∈Id(i,n)∈K
by (4), it follows that b(n)→0 in
kω(K(ω)).
By thinning out and reindexing we may assume that d(i,n)=0 if
i∈I0 and d(i,n)=0 if i∈I0 for some I0⊆I and every n∈ω. If g∈Gγ(i,n)∖G<γ(i,n) then
so(g,⟨Si⟩∖⟨2Si⟩,kω(Kγ(i,n)))≥σni
by (3) and γ(i,n) is cofinal in
limk∈ωγ(k). Thus by (2) and Lemma 40
h⟨Si⟩∖⟨2Si⟩K(ω)(Di)=σi−1 where Di={d(i,n):n∈ω} for every i∈I0 and ⟨∪i∈I0Di⟩
is closed and discrete in kω(K(ω)).
Using Lemma 31 we can build a
finite fpse-chain
[TABLE]
above (G(ω),K(ω),U(ω)) relative to (C,ξ) such that D0i→0
in kω(Kγt) for each i∈I0 where D0i={d(i,n):n∈J0} for some infinite J0⊆ω. Then {p(n):n∈J0}→0 and
p(n)∈[Pα]η′kω(Kγt) where
η′=supk∈ωη(k)≤max{σi−1:i∈I} so
0∈[Pα]ηkω(Kγt) where
η≤max{σi:i∈I}. Put m(α+2)=γt.
Put m(α+3)=m(α+2)+1. Let (G′,K′,U′) be an arbitrary
fpse of (Gm(α+2),Km(α+2),Um(α+2)). Using
Lemmas 22, 9, and 30
extend U′ to a countable family of open (in
kω(Km(α+2))) subgroups of Gm(α+2) of
finite index U′′⊇U′
so that (4) holds after replacing Um(α+3)
with U′′ and let
(Gm(α+3),Km(α+3),Um(α+3))=(G′,K′,U′′).
∎
Suppose ♢ holds and let {Cα:α<ω1} be a
♢-sequence. Identifying ω1 and 2ω we may assume that
each Cα⊆2ω.
Let {Pα:α<ω1 is a limit ordinal} list all infinite countable subsets of
2ω so that each Pα is listed ω1 times.
Lemma 45**.**
Let (Gω1,Kω1,Uω1)=(G<ω1,K<ω1,U<ω1) where (Gα,Kα,Uα),
α<ω1 have been constructed in Lemma 44. Then kω(Kω1)=τ(Uω1),
Gω1 is countably compact, and
so(kω(Kω1))=sup{σm:m∈ω}.
Proof.
Suppose A⊆Gω1 is such that 0∈A, A∩K is closed
for every K∈Kω1 and A∩U=∅ for every
U∈Uω1.
Let θ be a large enough cardinal. Consider the sets of the form A∩M where M is a countable elementary submodel of H(θ)
and X∈M is a countable set containing the details of the construction
of Gω1. The set
[TABLE]
is a club in ω1. Thus Cγ=A∩M for some
γ<ω1 where M∩ω1=γ. Note that
γ=m(α) for some limit α<ω1 and
cCα=A∩G<m(α).
Let K<γ={Ki:i∈ω}, Uω1∩M={Ui:i∈ω}. Note that for any U∈U<γ there is a
Ui such that Ui∩G<ω=U (since
UKω1∈Uω1). Pick points dn∈A∩(∩i≤nUi)∩M
so that
[TABLE]
If the recursion does not terminate by (3.a) there exists an
S⊆D={dn:n∈ω}⊆A such that S→0 in
kω(Km(α+1)) contradicting the choice of A.
If the recursion stops at some n∈ω let K′∈K<γ∩M be such that ∑i<nKi+⟨{di:i<n}⟩⊆K′. Then
for every d∈A∩(∩i≤nUi)∩M there is an
s∈∑m<n⟨Sm⟩kω(K<γ)⊆∑m<n⟨Sm⟩kω(Kω1)
such that d∈s+K′.
By elementarity A∩U⊆∑i≤m⟨Si⟩kω(Kω1)+K for some m∈ω, U∈U<ω1,
and K∈Kω1. We may assume that A=A∩U by picking a subset if
necessary.
Pick points dn∈A∩(∩i≤nUi)∩M so that
dn=an+∑i≤mdni where
dni∈⟨Si⟩kω(K<γ) and an∈K.
By passing to a subsequence and reindexing if necessary, assume
an→a. Now dn→0 so ∑i≤mdni→a in
τ(U<γ). Let K∈Kβ for some β<γ.
If a∈∑i≤m⟨Si⟩kω(Kβ′) for any
β′<γ let α′∈ω1 be a limit such that
β<m(α′+3)<γ. Such α′ exists since
β,m∈M. Then a∈∑i≤m⟨Si⟩kω(Km(α′+3)) and by (4)
there is a clopen U∈Um(α′+3) such that a∈U and ∑i≤m⟨Si⟩⊆U. Note that U=Ui∩Gm(α′+3) for some i∈ω so a∈Ui and ∑i≤mdni∈Ui for every n∈ω contradicting ∑i≤mdni→a in τ(U<γ).
Thus
a∈∑i≤m⟨Si⟩kω(K<γ). By (1) there
exists a γ0<ω1 such that γ0>γ and Di→di in
kω(Kγ0) for each i≤m where Di⊆{dni:n∈ω} is infinite. By thinning out and reindexing, we may
assume that dni→di in kω(Kγ0). Then dn→d=a+∑i≤mdi in kω(Kγ0) so d∈A∩∑i≤m⟨Si⟩kω(Kγ0) by
Lemma 1.
Let U∈Uω1∩M then dn∈U for all but
finitely many n∈ω so d∈U. Thus
for every U∈Uω1∩M there is a γ′<ω1 and a d∈A∩(∑i≤m⟨Si⟩kω(Kγ′))∩Uτ(Uγ′). Then
the set A′=A∩(∑i≤m⟨Si⟩kω(Kω1)) is closed in
kω(Kω1) and by elementarity 0∈A′τ(Uω1) so assume
that A⊆∑i≤m⟨Si⟩kω(Kω1).
Let I⊆m be such that 0∈A′τ(Uω1) and
0∈A′′τ(Uω1) for any A′′=A∩(∑i∈I′⟨Si⟩kω(Kω1)) where A′=A∩(∑i∈I⟨Si⟩kω(Kω1)), I′⊂I, I′=I. Using this
property of I and picking a closed (in kω(Kω1))
subset of A′ if necessary we may assume that
every d∈A has the property that d=∑i∈Idi∈A where di=0 and
di∈⟨Si⟩kω(Kω1).
Pick points dn=∑i∈Idni∈A∩(∩k≤nUk)∩M so
that dni∈⟨Si⟩kω(K<γ)∩(∩k≤nUk)
for every i∈I.
If the recursion terminates at some n∈ω then there exists a
U∈Uω1∩M such that for every d∈A∩M d∈∑i∈I(⟨Si⟩kω(Kγ′)∩U) for
any γ′∈M. Let γ′<γ be such that U∩Gγ′∈Uγ′. Then U∩Gγ′′∈Uγ′′
for any γ′′≥γ′. Let α′<γ be a limit such
that γ′<m(α′+3)<γ. By (4) there
exists a U′∈Uω1∩M such that U′⊆U, U′∩Gm(α′+3)∈Um(α′+3), and U′∩∑i∈I⟨Si⟩kω(Km(α′+3))=∑i∈I(⟨Si⟩kω(Km(α′+3))∩U′). Let
γ0≥m(α′+3) by such that γ0<γ and
d=∑i∈Idi∈A∩U′∩M for some
di∈⟨Si⟩kω(Kγ0). Such d and γ0 exist
by elementarity since A,U′,Kω1,C∈M. Note that for every i∈I
the set ⟨Si⟩kω(Kγ0)∩U′ is closed in
kω(Kγ0) so by Lemma 22 the set
∑i∈I(⟨Si⟩kω(Kγ0)∩U′) is closed in
kω(Kγ0). Since U′∩∑i∈I⟨Si⟩ is dense in
U′∩∑i∈I⟨Si⟩kω(Kγ0) this shows that
U′∩∑i∈I⟨Si⟩kω(Kγ0)=∑i∈I(⟨Si⟩kω(Kγ0)∩U′). Then
di∈⟨Si⟩kω(Kγ0)∩U′⊆⟨Si⟩kω(Kγ0)∩U (note that d
uniquely determines di by (2)) contradicting the
choice of U.
Thus Di={dni:n∈ω}→0 in τ(U<γ) for each
i∈I. If Di is not closed and discrete in kω(K<γ)
for some i∈I then for some infinite J⊆ω {dni:i∈J}→0 in kω(Kω1).
Using (1) and possibly thinning out J we may assume
that {dnj:n∈J}→d(j) for every j∈I in kω(Kω1). Then d=∑i′∈Id(i′)∈∑i′∈I∖{i}⟨Si⟩kω(Kω1) and d∈A contradicting the
minimality of I.
Thus each Di is closed and discrete in
kω(K<γ). If h⟨Si′⟩K<γ(Di′)≥σi′−1
for some i′∈I then by Lemma 31 there exists an fpse
(Gγ,Kγ,Uγ) of (G<γ,K<γ,U<γ) such that {dni′:n∈J}→0 in
kω(Kγ) and a finite fpse-chain
{(Gλ,Kλ,Uλ):γ<λ<γ+k} for some k∈ω
such that {dni:n∈J}→d(i) in kω(K<γ+k) for some d(i)∈G<γ+k and some infinite J⊆ω. As above d=∑i∈Id(i)∈∑i∈I∖{i′}⟨Si⟩kω(Kγ) and d∈AKγ. Thus by (3.b) A∩∑i∈I′⟨Si⟩kω(Km(α+1))=∅ for some
I′⊂I, I′=I contradicting the minimality of I.
Now h⟨Si⟩K<γ(Di)<σi−1 for every i∈I.
Pick an i′∈I and use Lemma 10 to build an fpse-chain
{(Gγ,Kγ,Uγ):m(α)≤γ≤m(α)+k} of finite
type above (G<m(α),K<m(α),U<m(α)) away from ⟨Si′⟩ for some
k∈ω so that for some infinite J⊆ω {dni:n∈J}→di∈G<m(α) in kω(Km(α)+k) for i∈I∖{i′}. Then P={dni′:n∈J} is closed and
discrete in kω(Km(α)+k) by Lemma 37.
Use Lemma 19 to construct an fpse-chain
{(Gγ,Kγ,Uγ):m(α)+k<γ≤γ0} such that, after
possibly thinning out J, P is stable in (Gγ0,Kγ0,Uγ0) and
⟨P⟩ is closed and discrete in kωKγ0.
Let J=∪j∈ωJj be such that Jj are infinite and
disjoint and Pj={dni′:n∈Jj}.
Repeatedly apply Lemma 41 to build wggs
[TABLE]
over (Dj,Gγj,Kγj,Uγj,δ)
where
δ=h⟨Si′⟩∖⟨2Si′⟩Kγj(Pj)=h⟨Si′⟩∖⟨2Si′⟩Kγ0(P)
and
Dj⊆[⟨Si⟩∖⟨2Si⟩]δkω(K<γj) is such that
Pj⊆Dj+K(j) and Dj⊆Pj+K(j) for some
K(j)∈Kγj (due to the
stability of P we may assume that K(j)=K∈Kγ0 but this
stronger property is not needed) such that
[TABLE]
and ⟨∪k>jPj⟩ is closed and discrete in
kω(Kγj+1). Let
γ0=limγj.
Use
Corollary 1 to find a
d(i′,n)∈Pnkω(Kγn+1) such that
d(i′,n)=a(n)+sγn+1 for some a(n)∈K(n). Note that d(i′,n)∈Gγn+1∖G<γn+1 since a(n)∈K(n)∈K<γn+1 and by (5)
[TABLE]
for any g∈Gγn+1∖G<γn+1. Let d(i,n)=di for i∈I∖{i′}. Put d(n)=∑i∈Id(i,n). Then
d(n)∈Akω(Kγ0) by Corollary 1. Since Di→0 in
τ(U<γ),
d(i,n)∈∩{Uτ(U<γ0):U∈U<γ}.
Now by (3.c) for every n∈ω there exists a
d(n)∈Gm(α+1)∖G<γ such that
d(n)∈Akω(Km(α+1))
and d(n)=∑i∈Id(i,n) where
[TABLE]
and
so(d(i′,n),⟨Si⟩∖⟨2Si⟩,kω(Km(α+1)))≥σni′
for some i′∈I. Note that d(i,n)∈M.
Thus for any U∈Uω1∩M, any γ′∈ω1∩M, and any n∈ω there exists a
d(n)=∑i∈Id(i,n)∈A where
d(i,n)∈⟨Si⟩kω(Km(α+1))∩U, d(i,n)∈Gγ′, and
so(d(i′,n),⟨Si⟩∖⟨2Si⟩,kω(Km(α+1)))≥σni′
for some i′∈I, α<ω1.
By elementarity such d(n) and d(i,n) exist in every
(∩j≤nUj∖G<γn)∩M
for some γn cofinal in γ. Now
h⟨Si⟩∖⟨2Si⟩K<γ({d(i′,n):n∈ω})≥σi′−1 by
Lemma 40 and d(i′,n)→0 in U<m(α)
so by an argument similar to the one above using (3.b) one obtains a contradiction
with the minimality of I.
The upper bound estimate for so(Gω1) follows from (2), while
the lower bound estimate follows from Lemma 32. Countable
compactness follows from (1).
∎
Theorem 1** (♢).**
Let σ≤ω1. There exists a countably compact sequential
group G such that so(G)=σ.
Proof.
The existence of such G for σ<ω1 follows from
Lemma 45. For σ=ω1 consider a Σ product of
ω1 countably compact sequential groups Gα such that
so(Gα)=α.
Note that the group of sequential order ω1 constructed above
is not separable unlike the examples of smaller sequential order. A
modified construction similar to that of
Lemma 45 may be used to construct a separable example of
sequential order ω1, as well, although we omit the details.
∎
Corollary 3**.**
The existence of a countably compact sequential non Fréchet group is
independent of the axioms of ZFC.
Proof.
See Theorem 1 and [17] Theorem 2.
∎
We conclude by listing some open questions.
Question 1**.**
Does Theorem 1 follow from
CH alone?
Question 2**.**
Is the existence of a countably compact sequential group G such that
K⊆G for some compact subspace K, so(K)≥2
consistent with ZFC?
Question 3**.**
Can the construction in Theorem 1 be made
Cohen-indestructible (see, for example, [6] for the relevant
definitions)?
The last question requires some clarification. Since the group G
will contain compact subspaces homeomorphic to the Cantor cube
2ω, the addition of any reals will destroy
the sequentiality of G. Each such subspace, as well as G itself, will inherit
a precompact topology
from the ground model, however, so the group G⊇G in the extension may
be defined by taking the Raikov completion of each compact subspace in
Kω1 and letting the topology of G be determined by the new family
of compact subspaces.
7 Acknowledgements.
The research in this paper was started when one of the authors
(Alexander Shibakov) was visiting the Department of Mathematics at
Ehime University in the Summer of 2017. He would like to thank the
Department and Prof. Shakhmatov for the support that made this visit
possible
and productive.