Countably compact groups without non-trivial convergent sequences
Michael Hrušák, Jan van Mill, Ulises Ariet Ramos-García, Saharon Shelah
aa r X i v : . [ m a t h . GN ] J un COUNTABLY COMPACT GROUPS WITHOUT NON-TRIVIALCONVERGENT SEQUENCES
M. HRUˇS ´AK, J. VAN MILL, U. A. RAMOS-GARC´IA, AND S. SHELAH
Abstract.
We construct, in
ZFC , a countably compact subgroup of 2 c withoutnon-trivial convergent sequences, answering an old problem of van Douwen. Asa consequence we also prove the existence of two countably compact groups G and G such that the product G × G is not countably compact, thusanswering a classical problem of Comfort. Introduction
The celebrated Comfort-Ross theorem [11, 7] states that any product of pseudo-compact topological groups is pseudo-compact, in stark contrast with the examplesdue to Nov´ak [28] and Terasaka [34] who constructed pairs of countably compactspaces whose product is not even pseudo-compact. This motivated Comfort [9](repeated in [8]) to ask:
Question 1.1 (Comfort [8]) . Are there countably compact groups G , G such that G × G is not countably compact? The first consistent positive answer was given by van Douwen [45] under MA ,followed by Hart-van Mill [21] under MA ctble . In his paper van Douwen showed thatevery Boolean countably compact group without non-trivial convergent sequencescontains two countably compact subgroups whose product is not countably com-pact, and asked: Question 1.2 (van Douwen [45]) . Is there a countably compact group without non-trivial convergent sequences?
In fact, the first example of such a group was constructed by Hajnal and Juh´asz[20] a few years before van Douwen’s [45] assuming CH . Recall, that every com-pact topological group contains a non-trivial convergent sequence, as an easy con-sequence of the classical and highly non-trivial theorem of Ivanovski˘ı-Vilenkin-Kuz’minov (see [25]) that every compact topological group is dyadic , i.e., a contin-uous image of 2 κ for some cardinal number κ .Both questions have been studied extensively in recent decades, providing a largevariety of sufficient conditions for the existence of examples to these questions, much Date : June 24, 2020.2010
Mathematics Subject Classification.
Primary 22A05, 03C20; Secondary 03E05, 54H11.
Key words and phrases.
Products of countably compact groups, p -compact groups, ultrapow-ers, countably compact groups without convergent sequences.The research of the first author was supported by a PAPIIT grant IN100317 and CONA-CyT grant A1-S-16164. The third named author was partially supported by the PAPIIT grantsIA100517 and IN104419. Research of the fourth author was partially supported by EuropeanResearch Council grant 338821. Paper 1173 on the fourth author’s list. work being done by Tomita and collaborators [17, 18, 23, 30, 33, 40, 41, 42, 37, 43,44, 38], but also others [10, 13, 14, 27, 35]. The questions are considered central inthe theory of topological groups [1, 2, 7, 8, 15, 32, 36].Here we settle both problems by constructing in ZFC a countably compact sub-group of 2 c without non-trivial convergent sequences.The paper is organized as follows: In Section 2 we fix notation and review basicfacts concerning ultrapowers, Fubini products of ultrafilters and Bohr topology. InSection 3 we study van Douwen’s problem in the realm of p -compact groups. Weshow how iterated ultrapowers can be used to give interesting partial solutions tothe problem. In particular, we show that an iterated ultrapower of the countableBoolean group endowed with the Bohr topology via a selective ultrafilter p producesa p -compact subgroup of 2 c without non-trivial convergent sequences. This on theone hand raises interesting questions about ultrafilters, and on the other handserves as a warm up for Section 4, where the main result of the paper is proved byconstructing a countably compact subgroup of 2 c without non-trivial convergentsequences using not a single ultrafilter, but rather a carefully constructed c -sizedfamily of ultrafilters. 2. Notation and terminology
Recall that an infinite topological space X is countably compact if every infinitesubset of X has an accumulation point. Given p a nonprincipal ultrafilter on ω (for short, p ∈ ω ∗ ), a point x ∈ X and a sequence { x n : n ∈ ω } ⊆ X we say(following [5]) that x = p -lim n ∈ ω x n if for every open U ⊆ X containing x theset { n ∈ ω : x n ∈ U } ∈ p . It follows that a space X is countably compact if andonly if every sequence { x n : n ∈ ω } ⊆ X has a p -limit in X for some ultrafilter p ∈ ω ∗ . Given an ultrafilter p ∈ ω ∗ , a space X is p -compact if for every sequence { x n : n ∈ ω } ⊆ X there is an x ∈ X such that x = p -lim n ∈ ω x n .For introducing the following definition, we fix a bijection ϕ : ω → ω × ω , andfor a limit ordinal α < ω , we pick an increasing sequence { α n : n ∈ ω } of smallerordinals with supremum α . Given an ultrafilter p ∈ ω ∗ , the iterated Fubini powers or Frol´ık sums [16] of p are defined recursively as follows: p = pp α +1 = { A ⊆ ω : { n : { m : ( n, m ) ∈ ϕ ( A ) } ∈ p α } ∈ p } and p α = { A ⊆ ω : { n : { m : ( n, m ) ∈ ϕ ( A ) } ∈ p α n } ∈ p } for α limit.The choice of the ultrafilter p α depends on (the arbitrary) choice of ϕ and the choiceof the sequence { α n : n ∈ ω } , however, the type of p α does not (see e.g. , [16, 19]).For our purposes we give an alternative definition of the iterated Fubini powersof p : given α < ω we fix a well-founded tree T α ⊂ ω <ω such that(i) ρ T α ( ∅ ) = α , where ρ T α denotes the rank function on h T α , ⊆i ;(ii) For every t ∈ T α , if ρ T α ( t ) > t ⌢ n ∈ T α for all n ∈ ω .For β α , let Ω β ( T α ) = { t ∈ T α : ρ T α ( t ) = β } and T + α = { t ∈ T α : ρ T α ( t ) > } .If p ∈ ω ∗ , then L p ( T α ) will be used to denote the collection of all trees T ⊆ T α such that for every t ∈ T ∩ T + α the set succ T ( t ) = { n ∈ ω : t ⌢ n ∈ T } belongs to p .Notice that each T ∈ L p ( T α ) is also a well-founded tree with ρ T ( ∅ ) = α . Moreover,the family { Ω ( T ) : T ∈ L p ( T α ) } forms a base of an ultrafilter on Ω ( T α ) which has OUNTABLY COMPACT GROUPS 3 the same type of p α . If T ∈ L p ( T α ) and U ∈ p , T ↾ U denotes the tree in L p ( T α )for which succ T ↾ U ( t ) = succ T ( t ) ∩ U for all t ∈ ( T ↾ U ) + .Next we recall the ultrapower construction from model theory and algebra. Givena group G and an ultrafilter p ∈ ω ∗ , denote by ult p ( G ) = G ω / ≡ , where f ≡ g iff { n : f ( n ) = g ( n ) } ∈ p. The
Theorem of L´os [26] states that for any formula φ with parameters [ f ] , [ f ] , . . . [ f n ], ult p ( G ) | = φ ([ f ] , [ f ] , . . . [ f n ]) if and only if { k : G | = φ ( f ( k ) , f ( k ) , . . .f n ( k )) } ∈ p . In particular, ult p ( G ) is a group with the same first order propertiesas G .There is a natural embedding of G into ult p ( G ) sending each g ∈ G to theequivalence class of the constant function with value g . We shall therefore consider G as a subgroup of ult p ( G ). Also, without loss of generality, we can assume thatdom( f ) ∈ p for every [ f ] ∈ ult p ( G ).Recall that the Bohr topology on a group G is the weakest group topology makingevery homomorphism Φ ∈ Hom( G , T ) continuous, where the circle group T carriesthe usual compact topology. We let ( G , τ Bohr ) denote G equipped with the Bohrtopology.Finally, our set-theoretic notation is mostly standard and follows [24]. In par-ticular, recall that an ultrafilter p ∈ ω ∗ is a P -point if every function on ω isfinite-to-one or constant when restricted to some set in the ultrafilter and, an ul-trafilter p ∈ ω ∗ is a Q -point if every finite-to-one function on ω becomes one-to-onewhen restricted to a suitable set in the ultrafilter. The ultrafilters p ∈ ω ∗ whichare P-point and Q-point are called selective ultrafilters. For more background onset-theoretic aspects of ultrafilters see [6].3. Iterated ultrapowers as p -compact groups In this section we shall give a canonical construction of a p -compact group forevery ultrafilter p ∈ ω ∗ . This will be done by studying the iterated ultrapowerconstruction.Fix a group G and put ult p ( G ) = G . Given an ordinal α with α >
0, let ult αp ( G ) = ult p lim −→ β<α ult βp ( G )) ! , where lim −→ β<α ult βp ( G ) denotes the direct limit of the direct system h ult βp ( G ) , ϕ δβ : δ β < α i with the following properties:(1) ϕ δδ is the identity function on ult δp ( G ), and(2) ϕ δβ : ult δp ( G ) → ult βp ( G ) is the canonical embedding of ult δp ( G ) into ult βp ( G ),defined recursively by ϕ δ,α +1 ([ f ]) = the constant function with value [ f ],and ϕ δ,α ([ f ]) = the direct limit of ϕ δ,β ([ f ]) , β < α for a limit ordinal α .In what follows, we will abbreviate ult α − p ( G ) for lim −→ β<α ult βp ( G ). Moreover, wewill treat ult α − p ( G ) as S β<α ult βp ( G ) and, in such case, we put ht( a ) = min { β <α : a ∈ ult βp ( G ) } for every a ∈ ult α − p ( G ). This is, of course, formally wrong, but is HRUˇS ´AK, VAN MILL, RAMOS-GARC´IA, AND SHELAH facilitated by our identification of G with a subgroup of ult p ( G ). In this way wecan avoid talking about direct limit constructions.We now consider ( G , τ Bohr ). Having fixed an ultrafilter p ∈ ω ∗ , this topologynaturally lifts to a topology on ult p ( G ) as follows: Every Φ ∈ Hom( G , T ) naturallyextends to a homomorphism Φ ∈ Hom( ult p ( G ) , T ) by letting(3.1) Φ([ f ]) = p - lim n ∈ ω Φ( f ( n )) . By L´os’s theorem, Φ is indeed a homomorphism from ult p ( G ) to T and hencethe weakest topology making every Φ continuous, where Φ ∈ Hom( G , T ), is a grouptopology on ult p ( G ). This topology will be denoted by τ Bohr .The following is a trivial, yet fundamental fact:
Lemma 3.1.
For every f : ω → G , [ f ] = p - lim n ∈ ω f ( n ) in τ Bohr .Proof.
This follows directly from the definition of Φ and the identification of G witha subgroup of ult p ( G ). (cid:3) The group that will be relevant for us is the group ult ω p ( G ), endowed with thetopology τ Bohr induced by the homomorphisms in Hom( G , T ) extended recursivelyall the way to ult ω p ( G ) by the same formula (3.1).The (iterated) ultrapower with this topology is usually not Hausdorff (see [12,3]), so we identify the inseparable functions and denote by ( Ult ω p ( G ) , τ Bohr ) thisquotient. More explicitly,
Ult ω p ( G ) = ult ω p ( G ) /K, where K = T Φ ∈ Hom( G , T ) Ker(Φ). The natural projection will be denoted by π : ult ω p ( G ) → ult ω p ( G ) /K. The main reason for considering the iterated Fubini powers here is the followingsimple and crucial fact:
Proposition 3.2.
Let p ∈ ω ∗ be an ultrafilter. (1) ult αp ( G ) ≃ ult p α ( G ) for α < ω , and (2) ( Ult ω p ( G ) , τ Bohr ) is a Hausdorff p -compact topological group.Proof. To prove (1), fix an α < ω . For given [ f ] ∈ ult αp ( G ), recursively define atree T f ∈ L p ( T α ) and a function ˆ f : T f → ult αp ( G ) so that • succ T f ( ∅ ) = dom( f ∅ ) and ˆ f ( ∅ ) = [ f ∅ ], where f ∅ = f ; • if ˆ f ( t ) is defined say ˆ f ( t ) = [ f t ], then succ T f ( t ) = dom( f t ) and ˆ f ( t ⌢ n ) = f t ( n ) for every n ∈ succ T f ( t ).We define ϕ : ult αp ( G ) → ult p α ( G ) given by ϕ ([ f ]) = [ ˆ f ↾ Ω ( T f )] . Claim 3.3. ϕ is an isomorphism. OUNTABLY COMPACT GROUPS 5
Proof of the claim.
To see that ϕ is a surjection, let [ f ] ∈ ult p α ( G ) be such thatdom( f ) = Ω ( T f ) for some T f ∈ L p ( T α ). Consider the function ˇ f : T f → ult αp ( G )defined recursively by • ˇ f ↾ Ω ( T f ) = f and, • if t ∈ T + α , then ˇ f ( t ) = [ h ˇ f ( t ⌢ n ) : n ∈ succ T f ( t ) i ].Notice that the function ˇ f satisfies that ˇ f ( t ) ∈ ult ρ Tf ( t ) p ( G ) for every t ∈ T f . Inparticular, ˇ f ( ∅ ) ∈ ult αp ( G ) and, a routine calculation shows that ϕ ( ˇ f ( ∅ )) = [ f ].To see that ϕ is injective, suppose that ϕ ([ f ]) = ϕ ([ g ]). Then there exists a tree T ∈ L p ( T α ) such that ˆ f ↾ Ω ( T ) = ˆ g ↾ Ω ( T ) . If set h := ˆ f ↾ Ω ( T ), then we can verify recursively that ˇ h ( ∅ ) = [ f ] = [ g ].Therefore, ϕ is a one-to-one function.Finally, using again a recursive argument, one can check that ϕ preserves thegroup structure.To prove (2) note that by definition Ult ω p ( G ) is a Hausdorff topological group. Tosee that Ult ω p ( G ) is p -compact, since Ult ω p ( G ) is a continuous image of ult ω p ( G ), itsuffices to check that ult ω p ( G ) is p -compact. Let f : ω → ult ω p ( G ) be a sequence andlet n ∈ ω . So f ( n ) ∈ ult p ( ult ω − p ( G )), that is, there exists f n : ω → S α<ω ult αp ( G )such that f ( n ) = [ f n ]. Thus, for every n ∈ ω there exists α n < ω such that f ( n ) ∈ ult α n p ( G ) and hence [ f ] ∈ ult αp ( G ) for α = sup { α n : n ∈ ω } < ω . Then[ f ] = p -lim n ∈ ω f ( n ) in τ Bohr as by the construction Φ([ f ]) = p -lim Φ( f ( n )) for everyΦ ∈ Hom( G , T ). This gives us the p -compactness of ult ω p ( G ). (cid:3) The plan for our construction is as follows: fix an ultrafilter p ∈ ω ∗ , find a suitabletopological group G without convergent sequences and consider ( Ult ω p ( G ) , τ Bohr ).The remaining issue is: Does (
Ult ω p ( G ) , τ Bohr ) have non-trivial convergent se-quences?While our approach is applicable to an arbitrary group G , in the remainder ofthis paper we will be dealing exclusively with Boolean groups, i.e., groups whereeach element is its own inverse. These groups are, in every infinite cardinality κ , isomorphic to the group [ κ ] <ω with the symmetric difference △ as the groupoperation and ∅ as the neutral element. Every Boolean group is a vector spaceover the trivial 2-element field which we identify with 2 = { , } . Hence, we cantalk, e.g., about linearly independent subsets of a Boolean group. Also, since everyhomomorphism from a Boolean group into the torus T takes at most two values (inthe unique subgroup of T of size 2) we may and will identify Hom([ ω ] <ω , T ) withHom([ ω ] <ω ,
2) to highlight the fact that there are only two possible values. Hencealso Hom([ ω ] <ω ,
2) is a Boolean group and a vector space over the same field.The following theorem is the main result of this section.
Theorem 3.4.
Let p ∈ ω ∗ be a selective ultrafilter. Then ( Ult ω p ([ ω ] <ω ) , τ Bohr ) is a Hausdorff p -compact topological Boolean group without non-trivial convergentsequences. In order to prove this theorem, we apply the first step of our plan. The general case will be dealt with in a separate paper.
HRUˇS ´AK, VAN MILL, RAMOS-GARC´IA, AND SHELAH
Proposition 3.5.
The group [ ω ] <ω endowed with the topology τ Bohr is a non-discrete Hausdorff topological group without non-trivial convergent sequences.Proof.
It is well-known and easy to see that τ Bohr is a non-discrete Hausdorff grouptopology ( e.g., see [2] Section 9.9). To see that τ Bohr has no non-trivial convergentsequences, assume that f : ω → [ ω ] <ω is a non-trivial sequence. Then rng( f ) is aninfinite set. Find an infinite linearly independent set A ⊆ rng( f ) and split it intotwo infinite pieces A and A , and take Φ ∈ Hom([ ω ] <ω ,
2) such that A i ⊆ Φ − ( i )for every i <
2. Therefore, Φ is a witness that the sequence f does not converge. (cid:3) We say that a sequence h [ f n ] : n ∈ ω i ⊂ ult p ([ ω ] <ω ) is p -separated if for every n = m ∈ ω there is a Φ ∈ Hom([ ω ] <ω ,
2) such that Φ([ f n ]) = Φ([ f m ]). In otherwords, a sequence h [ f n ] : n ∈ ω i ⊂ ult p ([ ω ] <ω ) is p -separated if and only if itselements represent distinct elements of Ult p ([ ω ] <ω ) = ult p ([ ω ] <ω ) /K where K = T Φ ∈ Hom([ ω ] <ω , Ker(Φ) and π : ult p ([ ω ] <ω ) → Ult p ([ ω ] <ω ) is the corre-sponding projection.We next show that, in general, the plan does not work for all p ∈ ω ∗ . Lemma 3.6.
The following are equivalent: (1)
There exists a p ∈ ω ∗ such that ( Ult p ([ ω ] <ω ) , τ Bohr ) has non-trivial convergentsequences. (2) There exist a sequence h Φ n : n ∈ ω i ⊂ Hom ([ ω ] <ω , and a mapping H : Hom ([ ω ] <ω , → ω such that for every n ∈ ω the family { [ ω ] <ω \ Ker (Φ n ) } ∪ { Ker (Φ) : H (Φ) n } is centered.Proof. Let us prove (1) implies (2). Let ˜ f : ω → Ult p ([ ω ] <ω ) be a non-trivial se-quence, say ˜ f ( n ) = π ( f ( n )) ( n ∈ ω ) where f : ω → Ult p ([ ω ] <ω ). Without loss ofgenerality we can assume that ˜ f is a one-to-one function converging to π ([ h ∅ i ]),here h ∅ i denotes the constant sequence where each term is ∅ . So h [ f n ] : n ∈ ω i is a p -separated sequence τ Bohr -converging to [ h ∅ i ], where [ f n ] = f ( n ) for n ∈ ω .By taking a subsequence if necessary, we may assume that for every n ∈ ω thereis a Φ n ∈ Hom([ ω ] <ω ,
2) such that Φ n ([ f n ]) = 1. Now, by τ Bohr -convergenceof h [ f n ] : n ∈ ω i , there is a mapping H : Hom([ ω ] <ω , → ω such that for eachΦ ∈ Hom([ ω ] <ω ,
2) and each n > H (Φ) it follows that Φ([ f n ]) = 0. Now we willcheck that for every n ∈ ω the family { Ker(Φ n ) c } ∪ { Ker(Φ) : H (Φ) n } is cen-tered. For this, since ([ ω ] <ω , τ Bohr ) is without non-trivial convergent sequences and[ f n ] τ Bohr −−−→ [ h ∅ i ], we may assume that [ f n ] = [ h a i ] for every h n, a i ∈ ω × [ ω ] <ω , thatis, f n [ U ] is infinite for all h n, U i ∈ ω × p . Now, fix n ∈ ω and let F ⊂ Hom([ ω ] <ω , H (Φ) n for every Φ ∈ F . Then Φ([ f n ]) = 0 forevery Φ ∈ F and hence there exists U F ∈ p such that Φ( f n ( k )) = 0 for every h k, Φ i ∈ U F × F . Since Φ n ([ f n ]) = 1, there exists U n ∈ p such that Φ n ( f n ( k )) = 1for every k ∈ U n . Put U = U F ∩ U n ∈ p . Then f n [ U ] ⊂ Ker(Φ n ) c ∩ T Φ ∈ F Ker(Φ),so we are done. For a subset A of the group [ ω ] <ω , A c = [ ω ] <ω \ A . OUNTABLY COMPACT GROUPS 7
To prove (2) implies (1), first we observe that there is a sequence h f n : n ∈ ω i ⊂ ([ ω ] <ω ) ω such that for each F ∈ [ ω ] <ω and every σ : F → [ ω ] <ω there exists k ∈ ω such that f i ( k ) = σ ( i ) for all i ∈ F . Now, define A ,n = { k ∈ ω : Φ( f n ( k )) = 0 } and A ,n = { k ∈ ω : Φ( f n ( k )) = 1 } for all (Φ , n ) ∈ Hom([ ω ] <ω , × ω .Fix h Φ n : n ∈ ω i ⊂ Hom([ ω ] <ω ,
2) and H : Hom([ ω ] <ω , → ω as in (2). Claim 3.7.
The collection S n ∈ ω { A n ,n } ∪ { A ,n : H (Φ) n } forms a centeredfamily which generates a free filter F . Proof of the claim.
To show that such family is centered, let m > i < m fix a finite set { Φ j : j < m i } ⊂ H − [ i + 1]. Then, considering all choicefunctions σ : n → [ i 2) such that Φ( f n ( k )) = 1 for infinitely many n .Then pick one of such n with H (Φ) n and, k / ∈ A ,n ∈ F .Let p ∈ ω ∗ extend F . By Claim 3.7, it follows that(i) Φ n ([ f n ]) = 1, for every n ∈ ω .(ii) The sequence h Φ([ f n ]) : n ∈ ω i converges to 0, for every Φ ∈ Hom([ ω ] <ω , i.e., h [ f n ] : n ∈ ω i is a τ Bohr -convergent sequence to [ h ∅ i ].Finally, taking a subsequence if necessary, we can assume that h [ f n ] : n ∈ ω i is p -separated and, hence h π ([ f n ]) : n ∈ ω i is a non-trivial convergent sequence in( Ult p ([ ω ] <ω ) , τ Bohr ). (cid:3) Remark . Note that the filter F is actually an F σ -filter. Theorem 3.9. There exists a p ∈ ω ∗ such that ( Ult p ([ ω ] <ω ) , τ Bohr ) has non-trivialconvergent sequences.Proof. We will show that the second clause of the Lemma 3.6 holds. To see this,choose any countable linearly independent set { Φ n : n ∈ ω } ⊂ Hom([ ω ] <ω , W be a vector subspace of Hom([ ω ] <ω , 2) such that Hom([ ω ] <ω , 2) = span { Φ n : n ∈ ω } ⊕ W . We define the mapping H : Hom([ ω ] <ω , → ω as follows: H (Φ) = min { n : Φ ∈ span { Φ i : i < n } ⊕ W } . Now, let n ∈ ω and fix a finite set { Φ j : j < m } ⊂ H − [ n + 1]. In order to showthat Ker(Φ n ) c ∩ \ j Fact 3.10 ([31], p. 124) . Let V be a vector space and Φ , Φ , . . . , Φ m − linearfunctionals on V . Then the following statements are equivalent:(1) T j Ult p ([ ω ] <ω ) , τ Bohr ) is a topological subgroup of ( Ult ω p ([ ω ] <ω ) , τ Bohr ) thereare ultrafilters (even P-points assuming CH ) such that Ult ω p ([ ω ] <ω ) has a non-trivialconvergent sequence.Selective ultrafilters and Q-points, have immediate combinatorial reformulationsrelevant in our context. Given a non-empty set I and G a Boolean group, we shallcall a set { f i : i ∈ I } of functions f i : ω → G p -independent if ( n : a + X i ∈ E f i ( n ) = ∅ ) / ∈ p for every non-empty finite set E ⊂ I and every a ∈ G . Note that, in particular,a function f : ω → G is not constant on an element of p if and only if { f } is p -independent. Now, we will say that a function f : I → G is linearly independent if f is one-to-one and { f ( i ) : i ∈ I } is a linearly independent set and, a function f : I → ult p ( G ) is p -independent if f is one-to-one and { f i : i ∈ I } is a p -independentset, where f ( i ) = [ f i ] for i ∈ I . Proposition 3.12. Let p ∈ ω ∗ be an ultrafilter. Then: (1) p is a Q-point if and only if for every finite-to-one function f : ω → [ ω ] <ω thereis a set U ∈ p such that f ↾ U is linearly independent. (2) The following are equivalent (a) p is selective; (b) for every function f : ω → [ ω ] <ω which is not constant on an element of p there is a set U ∈ p such that f ↾ U is linearly independent; OUNTABLY COMPACT GROUPS 9 (c) for every p -independent set { f n : n ∈ ω } of functions f n : ω → [ ω ] <ω , thereis a set U ∈ p and a function g : ω → ω so that f n ↾ U \ g ( n ) is one-to-onefor n ∈ ω , f n [ U \ g ( n )] ∩ f m [ U \ g ( m )] = ∅ if n = m , and G n ∈ ω f n [ U \ g ( n )] is linearly independent. Proof. Let us prove (1). Suppose first that p is a Q-point. Let f : ω → [ ω ] <ω be afinite-to-one function. Recursively define a strictly increasing sequence h n k : k ∈ ω i of elements of ω and a strictly increasing sequence of finite subgroups h H n : n ∈ ω i of [ ω ] <ω so that(i) H n ∩ rng( f ) = ∅ for all n ∈ ω , and(ii) n k = max f − [ H k ] & f ′′ [0 , n k ] ⊂ H k +1 , for all k ∈ ω .Then partitioning ω into the union of even intervals, and the union of odd inter-vals, one of them is in p , say A = [ i ∈ ω [ n i , n i +1 ) ∈ p. Applying Q-pointness we can assume that there exists a U ∈ p such that | [ n i , n i +1 ) ∩ U | = 1 for every i ∈ ω, and U ⊆ A . By item (ii) and since h H n : n ∈ ω i is a strictly increasing sequence, itfollows that f ↾ U is one-to-one and { f ( n ) : n ∈ U } is linearly independent.Suppose now that for every finite-to-one function f : ω → [ ω ] <ω there is a U ∈ p such that f ↾ U is one-to-one and { f ( n ) : n ∈ U } is linearly independent. Let h I n : n ∈ ω i be a partition of ω into finite sets. Define a finite-to-one function f : ω → [ ω ] <ω by putting f ( k ) = { n } for each k ∈ I n . Then there is an U ∈ p suchthat f ↾ U is one-to-one and { f ( n ) : n ∈ U } is linearly independent. Note thatnecessarily | I n ∩ U | n ∈ ω and therefore p is a Q-point.(2) To see (a) implies (b), let f : ω → [ ω ] <ω be a function which is not constanton an element of p . Using P-pointness, we may assume without loss of generalitythat f is a finite-to-one function. So, by item (1), there is an U ∈ p such that f ↾ U is one-to-one and { f ( n ) : n ∈ U } is linearly independent.To see (b) implies (a), let f : ω → [ ω ] <ω be a function which is not constant onan element of p . By item (b), there is an U ∈ p such that f ↾ U is one-to-one and { f ( n ) : n ∈ U } is linearly independent, and hence p is a P-point. To verify that p is a Q-point, notice that every finite-to-one function f : ω → [ ω ] <ω is not constanton an element of p . Thus, by clause (1) we get the desired conclusion.To prove (a) implies (c), assume that { f n : n ∈ ω } is a p -independent set offunctions f n : ω → [ ω ] <ω . Fact 3.13. Given a finite p -independent set { f i : i < n } , and a finite linearlyindependent set A ⊂ [ ω ] <ω , the set of all m ∈ ω such that A ⊔ { f i ( m ) : i < n } islinearly independent, belongs to p . (cid:3) Here ⊔ denotes the disjoint union. Using Fact 3.13, we can recursively construct a p -branching tree T ⊂ ω <ω suchthat for every t ∈ T , it follows thatsucc T ( t ) = { m : A t ⊔ { f i ( m ) : i | t |} is linearly independent } , where A t = { f i ( t ( j )) : i < | t | & j ∈ [ i, | t | ) } .By Galvin-Shelah’s theorem ([4, Theorem 4.5.3]), let x ∈ [ T ] be a branch suchthat rng( x ) ∈ p . Thus, if we put U = rng( x ) and g ( n ) = max( x ↾ n ) for n ∈ ω , weget the properties as in (c).Finally, notice that (b) is a particular instance of (c) when { f n : n ∈ ω } = { f } .Therefore, (c) implies (b). (cid:3) Remark . In the previous theorem, it is possible to change the group [ ω ] <ω toany arbitrary Boolean group and, the conclusions of the theorem remain true.For technical reasons, it will be necessary to reformulate the notion of p -indepen-dence. Lemma 3.15. Let G be a Boolean group and < α < ω . Then: (1) A set { f i : i ∈ I } of functions f i : ω → G is p -independent if and only if thefunction ˜ f : I → ult p ( G ) / ult p ( G ) defined by ˜ f ( i ) = π ([ f i ]) for i ∈ I is linearly independent, where π : ult p ( G ) → ult p ( G ) / ult p ( G ) denotes the natural projection. (2) A set { f i : i ∈ I } of functions f i : ω → ult αp ( G ) is p -independent if and only ifthe set { ˜ f i : i ∈ I } of functions ˜ f i : ω → ult αp ( G ) / ult α − p ( G ) is a p -independentset, where each ˜ f i is defined by ˜ f i ( n ) = π αα − ( f i ( n )) for n ∈ ω and π αα − : ult αp ( G ) → ult αp ( G ) / ult α − p ( G ) denotes the natural projection.Proof. To see (1), note that X i ∈ E [ f i ] = [ h a i ]iff ( n : a + X i ∈ E f i ( n ) = ∅ ) ∈ p, for every non-empty finite set E ⊂ I and every a ∈ G .To see (2). Let E ⊆ I be a non-empty finite set and a ∈ ult αp ( G ) and, notice that ( n : X i ∈ E ˜ f i ( n ) = π αα − ( a ) ) ∈ p iff ( n : a + X i ∈ E f i ( n ) ∈ ult α − p ( G ) ) ∈ p OUNTABLY COMPACT GROUPS 11 iff ( n : ( a + [ f ]) + X i ∈ E f i ( n ) = ∅ ) ∈ p, where for some U ∈ p we have that f ( n ) = a + P i ∈ E f i ( n ) ∈ ult α − p ( G ) for n ∈ U . (cid:3) Note also that if ht([ f ]) = α for α > 0, then f is not constant on an element of p (equivalently, { f } is p -independent). Lemma 3.16. Let < α < ω , [ f ] ∈ ult αp ([ ω ] <ω ) and p a selective ultrafilter. If f is not constant on an element of p , then there is a tree T ∈ L p ( T α ) with T ⊆ T f such that ˆ f ↾ Ω ( T ) is linearly independent. Proof. First, if α = 1, then the conclusion of the lemma follows from Proposition3.12 (2) (b). Thus, we may assume that α > T ∈ L p ( T α ) with T ⊆ T f , so that the following holdfor any β α : • if β > 0, then h ˆ f ( t ) : t ∈ Ω β ( T ) i forms a p -independence sequence; • if β = 0, then h ˆ f ( t ) : t ∈ Ω ( T ) i forms a linearly independent sequence.In order to do this, first, we recursively construct a tree T ∗ ∈ L p ( T α ) with T ∗ ⊆ T f , so that the following hold for any t ∈ T ∗ with ρ T ∗ ( t ) > • if ht( ˆ f ( t )) = 1, then h ˆ f ( t ⌢ n ) : n ∈ succ T ∗ ( t ) i ⊂ [ ω ] <ω forms a linearlyindependent sequence; • if ht( ˆ f ( t )) = β + 1 with β > 1, then h ˆ f ( t ⌢ n ) : n ∈ succ T ∗ ( t ) i ⊂ ult βp ([ ω ] <ω )forms a p -independent sequence; • if ht( ˆ f ( t )) is a limit ordinal, then h ht( ˆ f ( t ⌢ n )) : n ∈ succ T ∗ ( t ) i is a strictlyincreasing sequence of non-zero ordinals.At step t . If ht( ˆ f ( t )) = 1 and h ˆ f ( t ⌢ n ) : n ∈ succ T f ( t ) i is not constant on anelement of p , then ρ T f ( t ) = 1 and applying Proposition 3.12 (2) (b) there exists U ∈ p with U ⊆ succ T f ( t ) such that h ˆ f ( t ⌢ n ) : n ∈ U i is linearly independent.Therefore, in this case we put succ T ∗ ( t ) = U .If ht( ˆ f ( t )) = β + 1 with β > h ˆ f ( t ⌢ n ) : n ∈ succ T f ( t ) i is not constant onan element of p , then consider the sequence˜ f t : succ T f ( t ) → ult βp ([ ω ] <ω ) / ult − βp ([ ω ] <ω )defined by ˜ f t ( n ) = π ββ − ( ˆ f ( t ⌢ n )) for n ∈ succ T f ( t ). Since h ˆ f ( t ⌢ n ) : n ∈ succ T f ( t ) i isnot constant on an element of p , by Lemma 3.15 (2), the sequence ˜ f t is not constanton an element of p . Therefore, applying Proposition 3.12 (2) (b) and Remark 3.14,we can find an element U ∈ p with U ⊆ succ T f ( t ) such that ˜ f t ↾ U is linearlyindependent. Thus, by Lemma 3.15 (1), putting succ T ∗ ( t ) = U we can concludethat h ˆ f ( t ⌢ n ) : n ∈ succ T ∗ ( t ) i forms a p -independent sequence. Here, we are using the notation from the proof of Proposition 3.2 (1). If ht( ˆ f ( t )) = β is a limit ordinal, then for every δ < β we set U δ = { n ∈ succ T f ( t ) : ht( ˆ f ( t ⌢ n )) = δ } . Then G δ<β U δ = succ T f ( t ) , where each U δ / ∈ p . The selectiveness of p implies that there is an U ∈ p such that | U ∩ U δ | δ < β . Thus, in this case put succ T ∗ ( t ) = U \ U . Thisconcludes recursive construction of T ∗ .Notice that ρ T ∗ ( t ) = ht( ˆ f ( t )) for every t ∈ T ∗ . Now given a tree T ′ ∈ L p ( T α )with T ′ ⊆ T ∗ , we can canonically list its members t ′ ∈ T ′ as { t T ′ k : k < ω } so that • t T ′ k ⊂ t T ′ l entails k < l ; • t T ′ k = t ⌢ n , t T ′ l = t ⌢ m , ht( ˆ f ( t )) is a limit ordinal, and ht( ˆ f ( t ⌢ n )) < ht( ˆ f ( t ⌢ m )) entails k < l ; • t T ′ k = t ⌢ n , t T ′ l = t ⌢ m , ht( ˆ f ( t )) is a successor ordinal, and n < m entails k < l .Choose a sufficiently large regular cardinal θ and a countable elementary sub-model M of h H ( θ ) , ∈i containing all the relevant objects such as p and T ∗ . Fix U ∈ p so that U is a pseudo-intersection of p ∩ M . Put T ∗∗ = T ∗ ↾ U and V t = succ T ∗∗ ( t ) for t ∈ ( T ∗∗ ) + .We unfix t , and construct by recursion on k the required condition T = { t Tk : k ∈ ω } ∈ L p ( T α ) with T ⊆ T ∗∗ , as well as an auxiliary function g : T + → ω and sets W t ⊆ V t for t ∈ T + such that the following are satisfied:(a) W t = V t \ g ( t ) = succ T ( t ) for all t ∈ T + (by definition).(b) For all k , • if ρ T ( t Tk ) = 1, then D ˆ f ( t Tl ⌢ n ) : ∃ l k (cid:16) n ∈ W t Tl & ρ T ( t Tl ⌢ n ) = 0 (cid:17)E ⊆ [ ω ] <ω forms a linearly independent sequence; • if ρ T ( t Tk ) = β + 1 with β > 1, then D ˆ f ( t Tl ⌢ n ) : ∃ l k (cid:16) n ∈ W t Tl & ρ T ( t Tl ⌢ n ) = β (cid:17)E ⊂ ult βp ([ ω ] <ω )forms a p -independence sequence; • if ρ T ( t Tk ) = β is a limit ordinal, then D ht( ˆ f ( t Tl ⌢ n )) : ∃ l k (cid:16) n ∈ W t Tl & ρ T ( t Tl ) = β (cid:17)E forms an one-to-one sequence, andsup n ht( ˆ f ( t Tl ⌢ n )) : ∃ l < k (cid:16) ρ T ( t Tl ) = β & n ∈ W t Tl & ρ T ( t Tl ⌢ n ) < β (cid:17)o < min n ht( ˆ f ( t Tk ⌢ n )) : n ∈ W t Tk o . Before describing the construction let us recall a simple fact from linear algebra: Fact 3.17. Let A and B be linearly independent sets in a Boolean group with A a finite set. Then there is A ′ ⊆ B such that | A ′ | ≤ | A | and A ⊔ ( B \ A ′ ) is linearlyindependent. OUNTABLY COMPACT GROUPS 13 Proof of the fact. Let V A = span( A ) and V B = span( B ). Then dim( V A ∩ V B ) | A | , so there exists a set A ′ ⊆ B such that span( A ′ ) = V A ∩ V B . Therefore, | A ′ | | A | and A ⊔ ( B \ A ′ ) is linearly independent. Basic step k = 0. So t T = ∅ . We put g ( t T ) = 0 and hence W t T = V t T . Theconditions (a) and (b) are immediate. Recursion step k > 0. Assume W t Tl (for l < k ) as well as g ↾ k have been definedso as to satisfy (a) and (b). In particular, we know already t Tk , for it is of the form t Tl ⌢ n for some n ∈ W t Tl where l < k . Put ρ T ( t Tk ) = γ and assume γ > 1. Notethat, since (b) is satisfied for l , we must have ρ T ( t Tl ) = γ + 1 and D ˆ f ( t Tj ⌢ m ) : ∃ j l (cid:16) m ∈ W t Tj & ρ T ( t Tj ⌢ m ) = γ (cid:17)E ⊂ ult γp ([ ω ] <ω )is a p -independent sequence. Put A l = { t Tl ′ : l ′ k & ρ T ( t Tl ′ ) = γ }⊂ n t Tj ⌢ m : ∃ j l (cid:16) m ∈ W t Tj & ρ T ( t Tj ⌢ m ) = γ (cid:17)o and A − l = A l \ { t Tk } .If γ = 1, then applying Proposition 3.12 (2) (c) there exists V ∈ p and a function g l : A l → ω such that D ˆ f ( t ⌢ m ) : t ∈ A l & m ∈ V \ g l ( t ) E ⊆ [ ω ] <ω is a linearly independent sequence. Using the elementarity of M and our assumptionabout U we conclude that there exists a function g l,U : A l → ω such that D ˆ f ( t ⌢ m ) : t ∈ A l & m ∈ U \ g l,U ( t ) E ⊆ [ ω ] <ω is a linearly independent sequence. Note that V t Tk \ g l,U ( t Tk ) ⊆ U \ g l,U ( t Tk ) and W t \ g l,U ( t ) ⊆ U \ g l,U ( t ) for t ∈ A − l . Since A l is a finite set, using Fact 3.17, wecan find a natural number g ( t Tk ) > g l,U ( t Tk ) so that D ˆ f ( t ⌢ m ) : t ∈ A − l & m ∈ W t E ∪ D ˆ f ( t Tk ⌢ m ) : m ∈ V t Tk \ g ( t Tk ) E forms a linearly independent sequence, as required.For the case γ = β + 1 with β > 1, we will proceed in a similar way as theprevious case. Given t ∈ A l , let˜ f t : V t → ult βp ([ ω ] <ω ) / ult β − p ([ ω ] <ω )be defined by ˜ f t ( m ) = π β − β ( ˆ f ( t ⌢ m )) for m ∈ V t . By Lemma 3.15 (2), { ˜ f t : t ∈ A l } is a p -independent set. Thus, applying Proposition 3.12 (2) (c) and Remark 3.14,we can find an element V ∈ p and a function g l : A l → ω such that D ˜ f t ( m ) : t ∈ A l & m ∈ V \ g l ( t ) E ⊆ ult βp ([ ω ] <ω ) / ult β − p ([ ω ] <ω )is a linearly independent sequence. By elementarity of M and the property of U we have that there exists a function g l,U : A l → ω such that D ˜ f t ( m ) : t ∈ A l & m ∈ U \ g l,U ( t ) E is a linearly independent sequence. Since A l is a finite set, V t Tk \ g l,U ( t Tk ) ⊆ U \ g l,U ( t Tk ) and W t \ g l,U ( t ) ⊆ U \ g l,U ( t ) for t ∈ A − l , using Fact 3.17, we can find anatural number g ( t Tk ) > g l,U ( t Tk ) so that D ˜ f t ( m ) : t ∈ A − l & m ∈ W t E ∪ D ˜ f t Tk ( m ) : m ∈ V t Tk \ g ( t Tk ) E forms a linearly independent sequence and, by Lemma 3.15 (1), this means that D ˆ f ( t ⌢ m ) : t ∈ A − l & m ∈ W t E ∪ D ˆ f ( t Tk ⌢ m ) : m ∈ V t Tk \ g ( t Tk ) E ⊂ ult βp ([ ω ] <ω )forms a p -independent sequence, as required.If γ is a limit ordinal, then applying Proposition 3.12 (2) (c) there exists V ∈ p and a function g l : A l → ω such that D ˆ f ( t ⌢ m ) : t ∈ A l & m ∈ V \ g l ( t ) E ⊂ ult γ − p ([ ω ] <ω )is a linearly independent sequence. Thus, proceeding as previous cases, it is possibleto find a function g l,U : A l → ω and a natural number g ( t Tk ) > g l,U ( t Tk ) so that D ˆ f ( t ⌢ m ) : t ∈ A − l & m ∈ W t E ∪ D ˆ f ( t Tk ⌢ m ) : m ∈ V t Tk \ g ( t Tk ) E forms a linearly independent sequence. In particular, D ht( ˆ f ( t ⌢ m )) : t ∈ A − l & m ∈ W t E ∪ D ht( ˆ f ( t Tk ⌢ m )) : m ∈ V t Tk \ g ( t Tk ) E forms an one-to-one sequence and, since γ is a limit ordinal, one sees that withoutloss of generality, we may assume thatsup n ht( ˆ f ( t Tl ⌢ m )) : ∃ l < k (cid:16) ρ T ( t Tl ) = γ & m ∈ W t Tl & ρ T ( t Tl ⌢ m ) < γ (cid:17)o < min n ht( ˆ f ( t Tk ⌢ m )) : m ∈ V t Tk \ g ( t Tk ) o , as required. (cid:3) Now we are ready to prove the main theorem of this section. Proof of the Theorem 3.4. According to Proposition 3.2, Ult ω p ([ ω ] <ω ) isa Hausdorff p -compact topological group. It remains therefore only to show that Ult ω p ([ ω ] <ω ) contains no non-trivial convergent sequences to π ([ h ∅ i ]). To see this,let ˜ f : ω → Ult ω p ([ ω ] <ω ) be a non-trivial sequence, say ˜ f ( n ) = π ( f ( n )) ( n ∈ ω )where f : ω → ult ω p ([ ω ] <ω ). Without loss of generality we can assume that ˜ f is aone-to-one function. Thus, since ult ω p ([ ω ] <ω ) = ult p [ α<ω ult αp ([ ω ] <ω ) ! , there exists 0 < α < ω so that [ f ] ∈ ult αp ([ ω ] <ω ) and f is not constant on an elementof p . By Lemma 3.16, there is a tree T ∈ L p ( T α ) with T ⊆ T f such that ˆ f ↾ Ω ( T )is linearly independent. Note that ˆ f [Ω ( T )] ⊆ [ ω ] <ω . Take Φ ∈ Hom([ ω ] <ω , 2) sothat ˆ f [Ω ( T )] ⊆ Φ − (1). So Φ([ ˆ f ]) = 1 and hence Φ([ f ]) = 1. Thus, Φ is a witnessthat the sequence f does not τ Bohr -converge to [ h ∅ i ] and, since ˜ f is one-to-one, infact ˜ f does not converge to π ([ h ∅ i ]). OUNTABLY COMPACT GROUPS 15 Countably compact group without convergent sequences In this section we develop the ideas introduced in the previous section into a ZFC construction of a countably compact subgroup of 2 c without non-trivial convergentsequences. Recall that any boolean group of size c (in particular Ult ω p ([ ω ] <ω )) isisomorphic to [ c ] <ω . In fact, the extension of homomorphisms produces a (topo-logical and algebraic) embedding h of ( Ult ω p ([ ω ] <ω ) , τ Bohr ) into 2 c ≃ Hom([ ω ] <ω , defined by h ([ f ])(Φ) = Φ([ f ]) . Similarly to the ultrapower construction, we shall extend the Bohr topology τ Bohr on [ ω ] <ω to a group topology τ Bohr on [ c ] <ω to obtain the result. The differenceis that rather than using a single ultrafilter, we shall use a carefully constructed c -sized family of ultrafilters. Theorem 4.1. There is a Hausdorff countably compact topological Boolean groupwithout non-trivial convergent sequences.Proof. We shall construct a countably compact topology on [ c ] <ω starting from([ ω ] <ω , τ Bohr ) as follows:Fix an indexed family { f α : α ∈ [ ω, c ) } ⊂ ([ c ] <ω ) ω of one-to-one sequences suchthat(1) for every infinite X ⊆ [ c ] <ω there is an α ∈ [ ω, c ) with rng( f α ) ⊆ X ,(2) each f α is a sequence of linearly independent elements, and(3) rng( f α ) ⊂ [ α ] <ω for every α ∈ [ ω, c ).Given a sequence { p α : α ∈ [ ω, c ) } ⊂ ω ∗ define for every Φ ∈ Hom([ ω ] <ω , 2) itsextension Φ ∈ Hom([ c ] <ω , 2) recursively by puttingΦ( { α } ) = p α - lim n ∈ ω Φ( f α ( n )) . Note that [ ω ] <ω together with the independent set {{ α } : α ∈ [ ω, c ) } generatethe group [ c ] <ω so the above definition uniquely extends Φ to a homomorphismΦ : [ c ] <ω → τ Bohr induced by { Φ : Φ ∈ Hom([ ω ] <ω , } on [ c ] <ω as the weakest topology making all Φ continuous (for Φ ∈ Hom([ ω ] <ω , { Ker(Φ) : Φ ∈ Hom([ ω ] <ω , } as asubbasis of the filter of neighbourhoods of the neutral element ∅ . It followsdirectly from the above observation that independently of the choice of the ul-trafilters the topology is a countably compact group topology on [ c ] <ω . Indeed, { α } ∈ { f α ( n ) : n ∈ ω } τ Bohr for every α ∈ [ ω, c ), in fact { α } = p α - lim n ∈ ω f α ( n ).Call a set D ∈ [ c ] ω suitably closed if ω ⊆ D and S n ∈ ω f α ( n ) ⊆ D for every α ∈ D . The following claim shows that the construction is locally countable. Claim 4.2. The topology τ Bohr contains no non-trivial convergent sequences if andonly if ∀ D ∈ [ c ] ω suitably closed ∃ Ψ ∈ Hom([ D ] <ω , 2) such that(1) ∀ α ∈ D \ ω Ψ( { α } ) = p α - lim n ∈ ω Ψ( f α ( n ));(2) ∀ i ∈ |{ n : Ψ( f α ( n )) = i }| = ω . Proof of the claim. Given an infinite X ⊆ [ c ] <ω there is an α ∈ [ ω, c ) suchthat rng( f α ) ⊆ X . Let D be suitably closed with α ∈ D , and let Ψ be the given homomorphism. It follows directly from the definition, and property (1) of Ψ, that,if Φ = Ψ ↾ [ ω ] <ω then in turn Ψ = Φ ↾ [ D ] <ω , which implies that h f α ( n ) : n ∈ ω i (and hence also X ) is not a convergent sequence as Φ takes both values 0 and 1infinitely often on the set { f α ( n ) : n ∈ ω } .The reverse implication is even more trivial (and not really necessary for theproof).Note that if this happens then, in particular, K = \ Φ ∈ Hom([ ω ] <ω , Ker(Φ)is finite, and [ c ] <ω /K with the quotient topology is the Hausdorff countably com-pact group without non-trivial convergent sequences we want.Hence to finish the proof it suffices to produce a suitable family of ultrafilters. Claim 4.3. There is a family { p α : α < c } of free ultrafilters on ω such that forevery D ∈ [ c ] ω and { f α : α ∈ D } such that each f α is an one-to-one enumeration oflinearly independent elements of [ c ] <ω there is a sequence h U α : α ∈ D i such that(1) { U α : α ∈ D } is a family of pairwise disjoint subsets of ω ,(2) U α ∈ p α for every α ∈ D , and(3) { f α ( n ) : α ∈ D & n ∈ U α } is a linearly independent subset of [ c ] <ω . Proof of the claim. Fix { I n : n ∈ ω } a partition of ω into finite sets such that | I n | > n · X m 0, and { f α n ( m ) : m ∈ B ∩ I l , l ∈ B n and n ∈ ω } is linearly independent . In order to obtain the set B we recursively use Fact 3.17 to construct a sequence { C l : l ∈ ω } of finite sets such that: • C = ∅ ; • If l > 0, then(i) C l ⊆ I l ,(ii) | C l | P i Put B = S l ∈ ω I l \ C l . By (ii), it follows that B ∈ B ⊆ F . Since B n = ∗ A n and B ∈ F , is clear that U n := B ∩ S l ∈ B n I l ∈ p α n . By (iii), it follows that { f α n ( m ) : m ∈ U n and n ∈ ω } is linearly independent. Therefore, the sequence h U n : n ∈ ω i is as required.Now, use this family of ultrafilters as the parameter in the construction of thetopology described above. By Claim 4.2 it suffices to show that given a suitablyclosed D ⊆ c and α ∈ D \ ω there is a homomorphism Ψ : [ D ] <ω → ∀ α ∈ D \ ω Ψ( { α } ) = p α - lim n ∈ ω Ψ( f α ( n ))(2) ∀ i ∈ |{ n : Ψ( f α ( n )) = i }| = ω .By Claim 4.3, there is a sequence h U α : α ∈ D \ ω i such that(1) { U α : α ∈ D \ ω } is a family of pairwise disjoint subsets of ω ,(2) U α ∈ p α for every α ∈ D \ ω , and(3) { f α ( n ) : α ∈ D \ ω & n ∈ U α } is a linearly independent subset of [ c ] <ω .Enumerate D \ ω as { α n : n ∈ ω } so that α = α . Recursively define a function h : { f α ( n ) : α ∈ D \ ω & n ∈ U α } → h takes both values 0 and 1 infinitely often on { f α ( n ) : n ∈ U α \ { α }} ,(2) Ψ ( { α } ) = p α -lim k ∈ U α Ψ ( f α ( k )), and(3) if { α n } is in the subgroup generated by { f α m ( n ) : m < n & n ∈ U α m } thenΨ n ( { α n } ) = p α n - lim k ∈ U αn Ψ n ( f α n ( k )), and making sure that(4) Ψ n ( { α n } ) = p α n - lim k ∈ U αn Ψ n ( f α ( k )) . Where Ψ n is a homomorphism defined on the subgroup generated by { f α m ( n ) : m < n & n ∈ U α m } ∪ {{ α m } : m < n } extending h ↾ { f α m ( n ) : m < n & n ∈ U α m } . Then let Ψ be any homomorphismextending S m ∈ ω Ψ m . Doing this is straightforward given that the set { f α ( n ) : α ∈ D \ ω & n ∈ U α } is linearly independent.Finally, note that if we, for a ∈ [ c ] <ω , let H ( a )(Φ) = Φ( a )then H is a continuous homomorphism from [ c ] <ω to 2 Hom([ ω ] <ω , whose kernel isthe same group K = T Φ ∈ Hom([ ω ] <ω ) Ker(Φ), which defines a homeomorphism (andisomorphism) of [ c ] <ω /K onto a subgroup of 2 Hom([ ω ] <ω ) ≃ c . (cid:3) Concluding remarks and questions Even though the results of the paper solve longstanding open problems, they alsoopen up very interesting new research possibilities. In Theorem 3.4 we showed thatif p is a selective ultrafilter then Ult ω p ([ ω ] <ω ) is a p -compact group without non-trivial convergent sequences. This raises the following two interesting questions,the first of which is the equivalent of van Douwen’s problem for p -compact groups. Question 5.1. Is there in ZFC a Hausdorff p -compact topological group without anon-trivial convergent sequence? A closely related problem asks how much can the property of being selectivebe weakened in Theorem 3.4. Recall that by Corollary 3.11 it is consistent thatthere is a P-point p for which Ult ω p ([ ω ] <ω ) does contain a non-trivial convergentsequence. On the other hand, Ult ω p ([ ω ] <ω ) ≃ Ult ω p α ([ ω ] <ω ) for every α < ω , sothere are consistently non-P-points for which ( Ult ω p ([ ω ] <ω ) contains no non-trivialconvergent sequences. Question 5.2. Is the existence of an ultrafilter p such that Ult ω p ([ ω ] <ω ) contains nonon-trivial convergent sequences equivalent to the existence of a selective ultrafilter? Question 5.3. Is it consistent with ZFC that Ult ω p ([ ω ] <ω ) contains a non-trivialconvergent sequence for every ultrafilter p ∈ ω ∗ ? Assuming Ult ω p ([ ω ] <ω ) contains no non-trivial convergent sequences, it is easy toconstruct for every n ∈ ω a subgroup H of Ult ω p ([ ω ] <ω ), such that H n is countablycompact while H n +1 is not. It should be possible to modify the construction inTheorem 4.1 to construct such groups in ZFC . These issues will be dealt with in aseparate paper.Another interesting question is: Question 5.4. Is it consistent with ZFC that there is a Hausdorff countably compacttopological group without non-trivial convergent sequences of weight < c ? Finally, let us recall a 1955 problem of Wallace: Question 5.5 (Wallace [46]) . Is every both-sided cancellative countably compacttopological semigroup necessarily a group? It is well known that a counterexample can be recursively constructed inside ofany non-torsion countably compact topological group without non-trivial conver-gent sequences [29, 39]. The fact that we do not know how to modify (in ZFC ) theconstruction in Theorem 4.1 to get a non-torsion example of a countably compactgroup seems surprising. Also the proof of Theorem 3.4 does not seem to easilygeneralize to non-torsion groups. Hence: Question 5.6. Is there, in ZFC , a non-torsion countably compact topological groupwithout non-trivial convergent sequences? Question 5.7. Assume p ∈ ω ∗ is a selective ultrafilter. Does ( Ult ω p ( Z ) , τ Bohr ) contain no non-trivial convergent sequence? Here the τ Bohr is defined as before as the weakest topology on ult ω p ( Z ) whichmakes all extensions of homomorphisms from Z to T continuous, and the group Ult ω p ( Z ) = ult ω p ( Z ) /K with K being the intersection of all kernels of the extendedhomomorphisms. Acknowledments. The authors would like to thank Alan Dow and OsvaldoGuzm´an for stimulating conversations. The authors also wish to thank the anony-mous referee for a thorough reading of the text and for helpful suggestions. References 1. A. V. Arhangel’skii, Selected old open problems in general topology , Bul. Acad. S¸tiint¸e Repub.Mold. Mat. (2013), no. 2-3, 37–46. MR 3241473 OUNTABLY COMPACT GROUPS 19 2. Alexander Arhangel ′ skii and Mikhail Tkachenko, Topological groups and related structures ,Atlantis Studies in Mathematics, vol. 1, Atlantis Press, Paris, 2008. MR 2433295 (2010i:22001)3. Paul Bankston, Coarse topologies in nonstandard extensions via separative ultrafilters , IllinoisJ. Math. (1983), no. 3, 459–466. MR 6983084. Tomek Bartoszy´nski and Haim Judah, Set theory , A K Peters, Ltd., Wellesley, MA, 1995, Onthe structure of the real line. MR 1350295 (96k:03002)5. Allen R. Bernstein, A new kind of compactness for topological spaces , Fund. Math. (1969/1970), 185–193. MR 02516976. Andreas Blass, Ultrafilters and set theory , Ultrafilters across mathematics, Contemp. Math.,vol. 530, Amer. Math. Soc., Providence, RI, 2010, pp. 49–71. MR 27575337. W. W. Comfort, Topological groups , Handbook of set-theoretic topology, North-Holland, Am-sterdam, 1984, pp. 1143–1263. MR 7766438. , Problems on topological groups and other homogeneous spaces , Open problems intopology, North-Holland, Amsterdam, 1990, pp. 313–347. MR 10786579. , Letter to K. A. Ross , (August 1, 1966).10. W. W. Comfort, S. U. Raczkowski, and F. J. Trigos-Arrieta, Making group topologies with,and without, convergent sequences , Appl. Gen. Topol. (2006), no. 1, 109–124. MR 228493911. W. W. Comfort and Kenneth A. Ross, Pseudocompactness and uniform continuity in topo-logical groups , Pacific J. Math. (1966), 483–496. MR 20788612. Mauro Di Nasso and Marco Forti, Hausdorff ultrafilters , Proc. Amer. Math. Soc. (2006),no. 6, 1809–1818. MR 220749713. Dikran Dikranjan, Countably compact groups satisfying the open mapping theorem , TopologyAppl. (1999), no. 1-3, 81–129, II Iberoamerican Conference on Topology and its Applica-tions (Morelia, 1997). MR 171999614. Dikran Dikranjan and Dmitri Shakhmatov, Forcing hereditarily separable compact-like grouptopologies on abelian groups , Topology Appl. (2005), no. 1-3, 2–54. MR 213974015. , Selected topics from the structure theory of topological groups , Open problems intopology II, Elsevier, 2007, pp. 389–406.16. Zdenˇek Frol´ık, Sums of ultrafilters , Bull. Amer. Math. Soc. (1967), 87–91. MR 020367617. S. Garcia-Ferreira and A. H. Tomita, Countably compact groups and p -limits , Bol. Soc. Mat.Mexicana (3) (2003), no. 2, 309–321. MR 202927918. S. Garcia-Ferreira, A. H. Tomita, and S. Watson, Countably compact groups from a selectiveultrafilter , Proc. Amer. Math. Soc. (2005), no. 3, 937–943. MR 211394719. Salvador Garc´ıa-Ferreira, Three orderings on β ( ω ) \ ω , Topology Appl. (1993), no. 3,199–216. MR 122755020. A. Hajnal and I. Juh´asz, A separable normal topological group need not be Lindel¨of , GeneralTopology and Appl. (1976), no. 2, 199–205. MR 043108621. Klaas Pieter Hart and Jan van Mill, A countably compact topological group H such that H × H is not countably compact , Trans. Amer. Math. Soc. (1991), no. 2, 811–821. MR 98223622. M. Hruˇs´ak, D. Meza-Alc´antara, E. Th¨ummel, and C. Uzc´ategui, Ramsey type properties ofideals , Ann. Pure Appl. Logic (2017), no. 11, 2022–2049. MR 369223323. Piotr B. Koszmider, Artur H. Tomita, and S. Watson, Forcing countably compact group topolo-gies on a larger free abelian group , Proceedings of the 15th Summer Conference on GeneralTopology and its Applications/1st Turkish International Conference on Topology and its Ap-plications (Oxford, OH/Istanbul, 2000), vol. 25, 2000, pp. 563–574 (2002). MR 192570724. Kenneth Kunen, Set theory , Studies in Logic and the Foundations of Mathematics, vol. 102,North-Holland Publishing Co., Amsterdam-New York, 1980, An introduction to independenceproofs. MR 59734225. V. Kuz ′ minov, Alexandrov’s hypothesis in the theory of topological groups , Dokl. Akad. NaukSSSR (1959), 727–729. MR 010475326. Jerzy Lo´s, Quelques remarques, th´eor`emes et probl`emes sur les classes d´efinissables d’alg`ebres ,Mathematical interpretation of formal systems, North-Holland Publishing Co., Amsterdam,1955, pp. 98–113. MR 007515627. V. I. Malykhin and L. B. Shapiro, Pseudocompact groups without convergent sequences , Mat.Zametki (1985), no. 1, 103–109, 139. MR 79223928. J. Nov´ak, On the Cartesian product of two compact spaces , Fund. Math. (1953), 106–112.MR 0060212 29. Desmond Robbie and Sergey Svetlichny, An answer to A. D. Wallace’s question about count-ably compact cancellative semigroups , Proc. Amer. Math. Soc. (1996), no. 1, 325–330.MR 132837330. Manuel Sanchis and Artur Hideyuki Tomita, Almost p -compact groups , Topology Appl. (2012), no. 9, 2513–2527. MR 292184131. H. H. Schaefer and M. P. Wolff, Topological vector spaces , second ed., Graduate Texts inMathematics, vol. 3, Springer-Verlag, New York, 1999. MR 174141932. Dmitri Shakhmatov, A comparative survey of selected results and open problems concern-ing topological groups, fields and vector spaces , Topology Appl. (1999), no. 1, 51–63.MR 166679533. Paul J. Szeptycki and Artur H. Tomita, HFD groups in the Solovay model , Topology Appl. (2009), no. 10, 1807–1810. MR 251921634. Hidetaka Terasaka, On Cartesian product of compact spaces , Osaka Math. J. (1952), 11–15.MR 005150035. M. G. Tkachenko, Countably compact and pseudocompact topologies on free abelian groups ,Izv. Vyssh. Uchebn. Zaved. Mat. (1990), no. 5, 68–75. MR 108331236. Michael G. Tkachenko, Topological features of topological groups , Handbook of the history ofgeneral topology, Vol. 3, Hist. Topol., vol. 3, Kluwer Acad. Publ., Dordrecht, 2001, pp. 1027–1144. MR 190026937. A. H. Tomita, Countable compactness and finite powers of topological groups without conver-gent sequences , Topology Appl. (2005), 527–538. MR 210716938. A. H. Tomita and S. Watson, Ultraproducts, p -limits and antichains on the Comfort grouporder , Topology Appl. (2004), no. 1-3, 147–157. MR 208028739. Artur H. Tomita, The Wallace problem: a counterexample from MA countable and p -compactness , Canad. Math. Bull. (1996), no. 4, 486–498. MR 142669440. Artur Hideyuki Tomita, On finite powers of countably compact groups , Comment. Math. Univ.Carolin. (1996), no. 3, 617–626. MR 142692641. , A group under MA countable whose square is countably compact but whose cube isnot , Topology Appl. (1999), no. 2, 91–104. MR 166451642. , Two countably compact topological groups: one of size ℵ ω and the other of weight ℵ ω without non-trivial convergent sequences , Proc. Amer. Math. Soc. (2003), no. 8, 2617–2622. MR 197466343. , A solution to Comfort’s question on the countable compactness of powers of a topo-logical group , Fund. Math. (2005), no. 1, 1–24. MR 216309944. , The weight of a countably compact group whose cardinality has countable confinality ,Topology Appl. (2005), no. 1-3, 197–205. MR 213367845. Eric K. van Douwen, The product of two countably compact topological groups , Trans. Amer.Math. Soc. (1980), no. 2, 417–427. MR 58672546. A. D. Wallace, The structure of topological semigroups , Bull. Amer. Math. Soc. (1955),95–112. MR 67907 OUNTABLY COMPACT GROUPS 21 Centro de Ciencias Matem´aticas, Universidad Nacional Aut´onoma de M´exico, CampusMorelia, Morelia, Michoac´an, M´exico 58089 E-mail address : [email protected] URL : KdV Institute for Mathematics, University of Amsterdam, Science Park 105-107,P.O. Box 94248, 1090 GE Amsterdam, The Netherlands E-mail address : [email protected] Centro de Ciencias Matem´aticas, Universidad Nacional Aut´onoma de M´exico, CampusMorelia, Morelia, Michoac´an, M´exico 58089 E-mail address : [email protected] Einstein Institute of Mathematics, Edmond J. 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