Algebraic structure of countably compact non-torsion Abelian groups of size continuum from selective ultrafilters
aa r X i v : . [ m a t h . GN ] J un ALGEBRAIC STRUCTURE OF COUNTABLY COMPACTNON-TORSION ABELIAN GROUPS OF SIZE CONTINUUMFROM SELECTIVE ULTRAFILTERS
MATHEUS KOVEROFF BELLINI, ANA CAROLINA BOERO,VINICIUS DE OLIVEIRA RODRIGUES, AND ARTUR HIDEYUKI TOMITA
Abstract.
Assuming the existence of c incomparable selective ultrafilters, weclassify the non-torsion Abelian groups of cardinality c that admit a countablycompact group topology. We show that for each κ ∈ [ c , c ] each of these groups hasa countably compact group topology of weight κ without non-trivial convergentsequences and another that has convergent sequences.Assuming the existence of 2 c selective ultrafilters, there are at least 2 c nonhomeomorphic such topologies in each case and we also show that every Abeliangroup of cardinality at most 2 c is algebraically countably compact. We also showthat it is consistent that every Abelian group of cardinality c that admits a count-ably compact group topology admits a countably compact group topology withoutnon-trivial convergent sequences whose weight has countable cofinality. Introduction
Some history.
Under Martin’s Axiom, Dikranjan and Tkachenko [9] showedthat if G is a non-torsion Abelian group of size continuum, then the following con-ditions are equivalent:a) G admits a countably compact Hausdorff group topology without non-trivialconvergent sequences;b) G admits a countably compact Hausdorff group topology;c) the free rank of G is equal to c and, for all d, n ∈ N with d | n , the group dG [ n ] is either finite or has cardinality c .The implications a) → b) and b) → c) hold in ZFC [9].The classification of all torsion groups of arbitrary cardinality using a single selec-tive ultrafilter and a mild cardinal arithmetic hypothesis appears in Castro-Pereiraand Tomita [7]. The example for torsion groups can be p -compact for the selective Key words and phrases. countable compactness, convergent sequences, topological group.MSC: primary 54H11, 22A05; secondary 54A35, 54G20. ultrafilter p and this cannot be expected for non-torsion Abelian groups. Indeed,there are no p -compact group topologies on free Abelian groups [17].Boero and Tomita [5] showed that the almost torsion-free groups of cardinality c admit a countably compact group topology using c selective ultrafilters.In [4], it was shown that the free Abelian group of cardinality c admits a countablycompact group with a convergent sequence using c selective ultrafilters. In [1] it wasshown from p = c that every Abelian group of cardinality c that admits a countablycompact group topology also admits a countably compact group topology with aconvergent sequence.Malykhin and Shapiro [16] showed that every pseudocompact Abelian group whoseweight has countable cofinality has a convergent sequence and showed a forcingexample of a pseudocompact group topology without non-trivial convergent sequecewhose weight is ℵ < ( ℵ ) ω for the group ( Z ) ( c ) . Tomita [19] showed that it isconsistent that Z ( c ) admits a countably compact group topology without non-trivialconvergent sequences whose weight is ℵ ω .In this paper, we improve the main results of both [1] and [8]. Indeed, we showfrom the existence of c selective ultrafilters that condition c) implies condition a), d)and e) where d) and e) are as follows:d) G admits a countably compact Hausdorff group topology with a convergentsequence.e) for each κ ∈ [ c , c ], G admits a countably compact Hausdorff group topologywithout convergent sequences whose weight is κ .Assuming 2 c selective ultrafilters, there are 2 c non-homeomorphic such group topolo-gies satisfying a), b), d) and e). We note that Martin’s Axiom implies the existenceof 2 c selective ultrafilters; on the other hand the existence of 2 c selective ultrafiltersis consistent with the negation of Martin’s Axiom.Also using 2 c selective ultrafilters, we show that every Abelian group of cardinalityat most 2 c is algebraically countably compact.1.2. Basic results, notation and terminology.
We recall that a T topologicalspace is countably compact iff every infinite subspace has an accumulation point inthe space.The following definition was introduced in [2] and is closely related to countablecompactness. OUNTABLY COMPACT GROUP TOPOLOGIES WITH CONVERGENT SEQUENCES 3
Definition 1.1.
Let p be a free ultrafilter on ω and let s : ω → X be a sequencein a topological space X . We say that x ∈ X is a p -limit point of s if, for everyneighborhood U of s , { n ∈ ω : s ( n ) ∈ U } ∈ p . (cid:3) If X is a Hausdorff space a sequence s has at most one p -limit point x and wewrite x = p -lim s .The set of all free ultrafilters on ω will be denoted by ω ∗ . It is not difficult to showthat a T topological space X is countably compact iff, each sequence in X thereexists p ∈ ω ∗ such that s has a p -limit point. Similar to what happens to the regularconvergence of sequences, we have the following proposition with a very similar andwell known proof: Proposition 1.2. If p ∈ ω ∗ and ( X i : i ∈ I ) is a family of topological spaces, then( y i ) i ∈ I ∈ Q i ∈ I X i is a p -limit point of a sequence (( x ni ) i ∈ I : n ∈ ω ) in Q i ∈ I X i if, andonly if, y i = p − lim( x ni : n ∈ ω ) for every i ∈ I . (cid:3) The following proposition is also very easy to prove. A proof can be found in [12](Theorem 3.54).
Proposition 1.3.
Let X , Y be topological spaces and f : X → Y be a continuousfunction, s : ω → X be a sequence in X and p ∈ ω ∗ . It follows that if x = p -lim( s n : n ∈ ω ), then f ( x ) = p -lim( f ( s n ) : n ∈ ω ). (cid:3) Since + and − are continuous functions in topological groups, it follows from thetwo previous propositions that: Proposition 1.4.
Let G be a topological group and p ∈ ω ∗ .(1) If ( x n : n ∈ ω ) and ( y n : n ∈ ω ) are sequences in G and x, y ∈ G aresuch that x = p − lim( x n : n ∈ ω ) and y = p − lim( y n : n ∈ ω ), then x + y = p − lim( x n + y n : n ∈ ω );(2) If ( x n : n ∈ ω ) is a sequence in G and x ∈ G is such that x = p − lim( x n : n ∈ ω ), then − x = p − lim( − x n : n ∈ ω ). (cid:3) The unit circle group T will be identified with the metric group ( R / Z , δ ) where δ is given by δ ( x + Z , y + Z ) = min {| x − y + a | : a ∈ Z } , for every x, y ∈ R . Given asubset A of T , we will denote by δ ( A ) the diameter of A with respect to the metric δ . The set of all non-empty open arcs of T will be denoted by B .We also fix U a basis of T c of cardinality c consisting of basic open sets which areproducts of elements of B . As such, for each U ∈ U and µ ∈ c we define U µ ∈ B tobe the µ -th coordinate of U and let supp U = { µ ∈ c : U µ = T } .Let X be a set and G be a group. We denote by G X the product Q x ∈ X G x where G x = G for every x ∈ X . The support of g ∈ G X is the set { x ∈ X : g ( x ) = 0 } ,which will be designated as supp g . The set { g ∈ G X : | supp g | < ω } will be denotedby G ( X ) . If f : ω → G ( X ) is a sequence, then supp f = S n ∈ ω supp f ( n ). M. K. BELLINI, A. C. BOERO, V. O. RODRIGUES, AND A. H. TOMITA
The torsion part T ( G ) of an Abelian group G is the set { x ∈ G : ∃ n ∈ N ( nx = 0) } .Clearly, T ( G ) is a subgroup of G . For every n ∈ N , we put G [ n ] = { x ∈ G : nx = 0 } .In the case G = G [ n ], we say that G is of exponent n provided that n is the minimalpositive integer with this property. The order of an element x ∈ G will be denotedby o( x ).A non-empty subset S of an Abelian group G is said to be independent if 0 S and, given distinct elements s , . . . , s n of S and integers m , . . . , m n , the relation m s + . . . + m n s n = 0 implies that m i s i = 0, for all i . The free rank r ( G ) of G is thecardinality of a maximal independent subset of G such that all of its elements haveinfinite order. It is easy to verify that r ( G ) = | G/T ( G ) | if r ( G ) is infinite.An Abelian group G is called divisible if, for each g ∈ G and each n ∈ N \ { } , theequation nx = g has a solution x ∈ G .We end this section by presenting some of the notation that will be used throughoutthis article (with the exception of the example of cardinality 2 c , where the notationis adapted to 2 c ).We fix a two element partition { P , P } of c such that | P | = | P | = c and ω + 1 ⊆ P . We add the following notation: Definition 1.5.
We define the groups W = ( Q / Z ) ( P × ω ) , W = Q ( P ) , W = W ⊕ W , X = Q ( P × ω ) , X = Q ( P ) , X = X ⊕ X .Given w ∈ W (or w ∈ X ), we denote by w and w the respective projectionsto W and W ( X and X ); that is, the unique elements of W (or X ) such thatsupp w ⊆ P × ω , supp w ⊆ P and w = w + w .Similarly, given g a sequence from ω into W (or X ), we define g and g . So g = g + g , supp g ⊆ P × ω and supp g ⊆ P . (cid:3) It will be useful to be able to easily transform an element of X into an element of W , so we use the following definition: Definition 1.6.
We denote by [ · ] the homomorphism from X = X ⊕ X onto W = W ⊕ W so that X → W is the quotient map coordinatewise and X → W = X is the identity.Given a function g : ω → X , we also define [ g ] : ω → W be given by [ g ]( n ) = [ g ( n )]for every n ∈ ω . (cid:3) Now we introduce the notion of a nice subgroup of W . These will be the maingroups we will be working with throughout the article. Associated to each nice sub-group W of W we have a subgroup Q of X of the elements which generate theelements of W and the group Z of elements of integer coordinates of Q . All thesegroups will be useful in the construction. Definition 1.7.
Given a family of positive integers ~n = ( n ξ : ξ ∈ P ), we define ~P ,~n = S ξ ∈ P { ξ } × n ξ . OUNTABLY COMPACT GROUP TOPOLOGIES WITH CONVERGENT SEQUENCES 5
We say that W is a nice subgroup of W if there exists a family of positive integers ~n = ( n ξ : ξ ∈ P ) such that W = ( Q / Z ) ( ~P ,~n ) ⊕ Q ( P ) . In this case, we define W = ( Q / Z ) ( ~P ,~n ) and W = Q ( P ) .In this case, we define the following subgroups of X : Q W = Q ( ~P ,~n ) , Q W = Q ( P ) and Q W = Q W ⊕ Q W , Z W = Z ( ~P ,~n ) , Z W = Z ( P ) and Z W = Z W ⊕ Z W . (cid:3) Note that, given a nice subgroup W , the restriction of [ · ] to Q W is a homomor-phism onto W .Throughout this article we will never deal with more than one nice subgroup of W at the same time. Consequently, given a nice subgroup W of W , we just write ~P = ~P ,~n , Q = Q W , Z = Z W , and Q j = Q j W , Z j = Z j W for j = 0 or 1.The following notation is also useful: Definition 1.8.
Given E ⊆ c , we define E = E ∩ P and E = E ∩ P . (cid:3) Definition 1.9.
Given ξ ∈ P , the function χ ξ : P → Q is given by χ ξ ( µ ) = (cid:26) µ = ξ µ = ξ. (cid:3) Definition 1.10. if r ∈ Q / Z , then p ( r ) and q ( r ) are the unique integers p, q suchthat q >
0, gcd( p, q ) = 1, 0 ≤ p < q and r = pq + Z . Likewise, if r ∈ Q , p ( r ) and q ( r )are the unique integers p, q such that q >
0, gcd( p, q ) = 1 and r = pq .We also define, for w ∈ W , p ( w ) = max {| p ( w ( z )) | : z ∈ supp w } and q ( w ) =max { q ( w ( z )) : z ∈ supp w } if w = 0. We define p (0) = 0 and q (0) = 0. (cid:3) Structure of the article.
In Section 2, we recall the definition of selectiveultrafilters and some properties which are necessary to construct homomorphismsrestricted to countable subgroups.In Section 3, we recall the notion of arc homomorphism that was defined in [1]which helped simplify the construction of the homomorphisms.In Section 4, we define the types and state the theorem in [1] that shows that every1-1 sequence can be associated to one of the types.In Section 5, we improve the result in [1] to have an estimate of the size of theoutput arc that depends only on the size of the input arc.In Section 6, we prepare the arc homomorphism to make the sequences have thepre-assigned p -limits and we produce the homomorphisms on countable subgroups.Here lie the main differences between constructing homomorphisms using p = c andselective ultrafilters. For selective ultrafilters, we have to know the size of the outputarcs in advance to guarantee the assigned points will be the p -limits. M. K. BELLINI, A. C. BOERO, V. O. RODRIGUES, AND A. H. TOMITA
In Section 7, we present the algebraic immersion, extend the homomorphism oncountable subgroups and define group topologies with a non-trivial convergent se-quence and without non-trivial convergent sequences.In Section 8 we show that the main construction can be used to obtain some moreexamples which were already mentioned in the introduction.2.
Selective Ultrafilters
In this section we review some basic facts about selective ultrafilters, the Rudin-Keisler order and some lemmas we will use in the next sections.
Definition 2.1.
A selective ultrafilter (on ω ), also called Ramsey ultrafilter, is a freeultrafilter p on ω such that for every partition ( A n : n ∈ ω ) of ω , either there exists n such that A n ∈ p or there exists B ∈ p such that | B ∩ A n | = 1 for every n ∈ ω . (cid:3) The following proposition is well known. We provide [13] as a reference.
Proposition 2.2.
Let p be a free ultrafilter on ω . Then the following are equivalent:a) p is a selective ultrafilter,b) for every f ∈ ω ω , there exists A ∈ p such that f | A is either constant orone-to-one,c) for every function f : [ ω ] → A ∈ p such that f | [ A ] is constant.The Rudin-Keisler order is defined as follows: Definition 2.3.
Let U be a filter on ω and f : ω → ω . We define f ∗ ( U ) = { A ⊆ ω : f − [ A ] ∈ U } . (cid:3) It is easy to verify that: f ∗ ( U ) is a filter; if U is an ultrafilter, so is f ∗ ( U ); if f, g : ω → ω , then ( f ◦ g ) ∗ = f ∗ ◦ g ∗ ; and (id ω ) ∗ is the identity over the set of allfilters. This implies that if f is bijective, then ( f − ) ∗ = ( f ∗ ) − . Definition 2.4.
Let U , V be filters. We say that U ≤ V (or
U ≤ RK V , if we need toavoid ambiguity) iff there exists f ∈ ω such that f ∗ ( U ) = V .The Rudin-Keisler order is the set of all free ultrafilters over ω ordered by ≤ RK .We say that two ultrafilters p, q are equivalent iff p ≤ q and q ≤ p . (cid:3) It is easy to verify that ≤ is a preorder and that the equivalence defined above isin fact an equivalence relation. Moreover, the equivalence class of a fixed ultrafilteris the set of all fixed ultrafilters, so the relations restricts to ω ∗ without modifyingthe equivalence classes. We refer to [13] for the following proposition: Proposition 2.5.
The following are true:(1) If p, q are ultrafilters, then p ≤ q and q ≤ p is equivalent to the existence ofa bijection f : ω → ω such that f ∗ ( p ) = q . OUNTABLY COMPACT GROUP TOPOLOGIES WITH CONVERGENT SEQUENCES 7 (2) The selective ultrafilters are exactly the minimal elements of the Rudin-Keisler order.This implies that if f : ω → ω and p is a selective ultrafilter, then f ∗ ( p ) is eithera fixed ultrafilter or a selective ultrafilter. If f ∗ ( p ) is the ultrafilter generated by n ,then f − [ { n } ] ∈ p , so, in particular, if f is finite to one and p is selective, then f ∗ ( p )is a selective ultrafilter equivalent to p .The existence of selective ultrafilters is independent from ZFC. Martin’s Axiom forcountable orders implies the existence of 2 c pairwise incompatible selective ultrafiltersin the Rudin-Keisler order.The lemma below appears in [20] and will be used for the construction of homo-morphisms on countable subgroups. Lemma 2.6.
Let ( p k : k ∈ ω ) be a family of pairwise incomparable selectiveultrafilters. For each k , let ( a k,i : i ∈ ω ) be a strictly increasing sequence in ω such that { a k,i : i ∈ ω } ∈ p k and i < a k,i for each k, i ∈ ω . Then there exists { I k : k ∈ ω } ⊂ [ ω ] ω such that:a) { a k,i : i ∈ I k } ∈ p k , for each k ∈ ω .b) I j ∩ I j = ∅ whenever i, j ∈ ω and i = j , andc) ([ i, a k,i ] : k ∈ ω and i ∈ I k ) is a pairwise disjoint family.3. Arc homomorphisms
Throughout this section we fix a nice subgroup W of W .In [1], we introduced the concept of arc homomorphism to construct homomor-phisms from countable subgroups of W to T . This technique replaces the arc func-tions that were used in [22] to treat free Abelian groups.In this section, we recall this concept and some of the useful results. More detailscan be seen in [1].First, we define some subgroups of W . A countable subset of c generates a count-able subset of W . We may restrict this subgroup to the elements which are annihi-lated by an integer K . Formally: Definition 3.1.
Given E a subset of c , denote by W E = { w ∈ W : supp w ⊆ ( E × ω ) ∪ E } . Given positive integer K , we define W ( K ) = { w ∈ W : ∃ u ∈ Q , w = [ u ] and Ku ∈ Z } . Finally denote W E,K = W E ∩ W ( K ) . (cid:3) M. K. BELLINI, A. C. BOERO, V. O. RODRIGUES, AND A. H. TOMITA
In particular, if E ⊆ P , then W E,K is the subgroup of ( Q / Z ) ( ~P ∩ ( E × ω )) of theelements which are annihilated by K , and if E ⊆ P , then W E,K is the subgroup of Q ( E ) of the elements that become an element of Z ( E ) when multiplied by K . Also,notice that W E,K = W E ,K ⊕ W E ,K is a finitely generated group whenever E isfinite.Now we recall the definition of a ( E, K, ǫ )-arc homomorphism.
Definition 3.2.
Given a positive real ǫ , a subset E of c and a positive integer K ,an ( E, K, ǫ )-arc homomorphism is a pair φ = ( φ , φ ) where:a) φ : W E ,K → T is a homomorphism,b) φ : { K χ ξ : ξ ∈ E } → B ,c) φ ( K χ ξ ) is an arc of length ǫ , for every ξ ∈ E . (cid:3) φ can naturally be extended to W E ,K as follows: Definition 3.3.
Let E ⊆ c , K be a positive integer, ǫ > φ = ( φ , φ ) be an( E, K, ǫ )-arc homomorphism and w ∈ W E,K . Then we define:ˆ φ ( w ) = φ ( w ) + X ξ ∈ supp w Kw ( ξ ) φ (cid:16) χ ξ K (cid:17) We define the empty sum of arcs to be the set { Z } . (cid:3) Notice that if supp w ⊆ ~P then ˆ φ ( w ) = { φ ( w ) } .As in [1], the following notion of extension will be very useful: Definition 3.4.
Given ǫ ′ , ǫ >
0, positive integers
K, K ′ , subsets E, E ′ of c , φ =( φ , φ ) an ( E, K, ǫ )-arc homomorphism and ψ = ( ψ , ψ ) an ( E ′ , K ′ , ǫ ′ )-arc homo-morphism, we say that ψ < φ ( ψ extends φ ) iff:a) E ⊆ E ′ , K | K ′ and ǫ ′ ≤ ǫ ,b) φ ⊆ ψ ,c) cl (cid:0) K ′ K ψ ( K ′ χ ξ ) (cid:1) ⊆ φ ( K χ ξ ), for each ξ ∈ E .d) m ! K ′ K ψ ( χ m K ′ ) is contained in the arc ( − ǫ K , ǫ K ) + Z , for each m ∈ ( E ′ ∩ ω ) \ E . (cid:3) The following proposition appears in [1] and its proof is straightforward.
Proposition 3.5.
Suppose ( φ n : n ∈ ω ) is a sequence of ( E n , K n , ǫ n )-arc homomor-phisms such that: φ n +1 < φ n for every n ∈ ω ; ( ǫ n : n ∈ ω ) →
0; and each positiveinteger divides cofinitely many elements of ( K n : n ∈ ω ). Let E = S n ∈ ω E n . Then: OUNTABLY COMPACT GROUP TOPOLOGIES WITH CONVERGENT SEQUENCES 9 (1) For every w ∈ W E , T { ˆ φ n ( w ) : w ∈ W E n ,K n } is a singleton.(2) The function ψ : W E → T so that ψ ( w ) ∈ T { ˆ φ n ( w ) : w ∈ W E n ,K n } is agroup homomorphism.(3) If ω ⊆ E and E n is finite for every n , then for every positive integer S ,( ψ (cid:0) m ! S χ m (cid:1) : m ∈ ω ) converges to 0 ∈ T . (cid:3) Types of sequences
Introduction.
Throughout this section we fix a nice subgroup W of W . Wewill recall, from [1], the definition of the 11 types of sequences in the next subsectionand the theorem that states that every sequence is related to one of them. The typesare defined with respect to a subgroup G of W .We call H the class of all the sequences that are of one of the 11 types.The theorem that we are going to need, proved in [1], is the following: Theorem 4.1.
Let f : ω → Q be given by f ( n ) = (0 , n ! χ n ) for every n ∈ ω . Let G be a subgroup of W containing (0 , χ n ) for every n ∈ ω . Let g : ω → Q with [ g ] ∈ G ω .Then there exists h : ω → Q such that h ∈ H or [ h ] is a constant sequence in G ω , c ∈ Q with [ c ] ∈ G , F ∈ [ ω ] <ω , p i , q i ∈ Z with q i = 0 for every i ∈ F , ( j i : i ∈ F )increasing enumerations of subsets of ω and j : ω → ω strictly increasing such that g ◦ j = h + c + X i ∈ F p i q i f ◦ j i with q i ≤ j i ( n ) for each n ∈ ω and i ∈ F (which implies [ p i q i f ◦ j i ] ∈ G ω since q i | (( j i ( n ))!) for each i ∈ F and n ∈ ω ). (cid:3) A discussion on the types can be seen in [1].To state the definition of the 11 types, the following notation is useful:Given w ∈ Q , we call w , and w , the natural projections of w into Q ( ω ) and Q ( P \ ω ) .Similarly, given g : ω → W (or Q ), we define g , and g , so that g = g , + g , ,with g , ∈ ( Q ω ) ω and g , ∈ ( Q P \ ω ) ω .4.2. The types.
The types are grouped by where the main element of the supportis.
The types related to P \ ω Definition 4.2.
Let G be a subgroup of W . We define the first three types of se-quence (with respect to G ) as follows: Let g : ω → Q be such that [ g ] ∈ G ω . We say that g is of type 1 if supp g , ( n ) \ S m
Let G ⊆ W be a subgroup. Let g : ω → Q be such that [ g ] ∈ G ω .Then we define types 4 to 9 (with respect to G ) as follows:We say that g is of type 4 if q ( g ( n )) > n , for every n ∈ ω .We say that g is of type 5 if there exists M ∈ T n ∈ ω supp g , ( n ) such that { q ( g ( n )) : n ∈ ω } is bounded and | p ( g ( n )( M )) | > n , for every n ∈ ω .To define types 6, 7 and 8, suppose g is such that for each n ∈ ω , there exists M n ∈ supp g , ( n ) \ ∪ m
We define types 10 and 11 (with respect to G ) as follows: Let g : ω → Q be such that [ g ] ∈ ( W ) ω ∩ G ω .We say that g is of type 10 if q ( g ( n )) > n , for every n ∈ ω .We say that g is of type 11 of order k if the family { [ g ( n )] : n ∈ ω } is an independentfamily whose elements have a fixed order k , for some positive integer k . (cid:3) OUNTABLY COMPACT GROUP TOPOLOGIES WITH CONVERGENT SEQUENCES 11 Extensions of Arc homomorphisms
Throughout this section we fix a nice subgroup W of W .We will state theorems on extensions of arc homomorphisms related to the typeswhich will be important to build homomorphisms in countable subgroups of W . Wehave to make an estimate for the output arc compared to the input arc. In [1], all thatwas needed was to find an output arc to continue the induction, and this dependedsolely on having a homomorphism that solved a certain arc equation.The construction of the arc homomorphisms were divided in two theorems (type 1-10) and type 11 because of the preliminaries. Since we are not proving the theoremshere we will rewrite those theorems from [1] in a unique statement: Theorem 5.1.
Let the following be given:i) E ∈ [ c ] <ω ,ii) K a positive integer,iii) ǫ ∗ , ǫ positive reals with ǫ < ,iv) φ an ( E, K, ǫ )-arc homomorphism,v) h a sequence of type 1 to 11,vi) γ > h is of type 1 to 10,vii) k as the order of h , if h is of type 11.Then there exists a cofinite set A ⊆ ω such that for every n ∈ A and for everyfinite E ′ ⊃ E , and positive integer K ′ such that K | K ′ and (cid:2) K h ( n ) (cid:3) ∈ W E ′ ,K ′ forevery arc U of length at least γ (if h is of type 1-10) or r ∈ T of order k (if h is oftype 11) there exists ǫ ′ ≤ ǫ, ǫ ∗ K ′ and an ( E ′ , K ′ , ǫ ′ )-arc homomorphism φ ′ < φ suchthat: • If h is of one of the types 1 to 10, ˆ φ ′ (cid:0)(cid:2) K h ( n ) (cid:3)(cid:1) ⊆ U . • If h is of type 11, ( φ ′ ) ([ h ( n )]) = r . (cid:3) We will improve Theorem 5.1 so that ǫ ′ will no longer depend on U and φ , butonly in the size of U . First, we prove a lemma. Lemma 5.2.
Given E ∈ [ c ] <ω , K a positive integer, ǫ a positive real, there exists afinite set S of ( E, K, ǫ )-homomorphisms such that for every ( E, K, ǫ )-homomorphism φ there exists ψ ∈ S such that ψ < φ . Proof.
Let T ⊆ T be a finite ǫ -dense subset. Let J be the set of homomorphismsfrom W E ,K to T , which is finite, and let J be the set of all functions from E to T , which is also finite. Let J = J × J . For each j = ( f j , z j ) ∈ J , let φ j be the( E, K, ǫ )-arc homomorphism such that φ j = f j and φ j ( K χ ξ ) be the arc centered in z j ( ξ ) of length ǫ . Let S = { φ j : j ∈ J } . (cid:3) Now we prove a new version Theorem 5.1 that we need later.
Theorem 5.3.
Let the following be given:i) E ∈ [ c ] <ω ,ii) K a positive integer,iii) ǫ ∗ , ǫ positive reals with ǫ < ,iv) h a sequence of type 1 to 11,v) γ > h is of one of the first 10 types,vi) if h is of type 11, let k be the order of h .Then there exists a cofinite set A ⊆ ω such that for every n ∈ A , for every finite E ′ ⊇ E and positive integer K ′ satisfying K | K ′ and (cid:2) K h ( n ) (cid:3) ∈ W E ′ ,K ′ , there existsa positive real ǫ ′ ≤ ǫ, ǫ ∗ K ′ such that for every ( E, K, ǫ )-arc homomorphism φ : • if h is of one of the first ten types, then for every arc U of length ≥ γ thereexists an ( E ′ , K ′ , ǫ ′ )-arc homomorphism φ ′ < φ such that ˆ φ ′ (cid:0)(cid:2) K h ( n ) (cid:3)(cid:1) ⊆ U . • if h is of type 11, then for every r ∈ T [ k ] there exists an ( E ′ , K ′ , ǫ ′ )-archomomorphism φ ′ < φ such that ( φ ′ ) ([ h ( n )]) = r . (cid:3) Proof.
Suppose h is of one of the first 10 types. The other case is similar since T hasonly a finite number of elements of order k . We obtain A by applying Theorem 5.1using ǫ instead of ǫ and γ instead of γ . Now fix E ′ , K ′ . It follows that:(*) for every open arc V of length ≥ γ and every ( E, K, ǫ )-arc homomorphism ψ there exist ǫ ′ n,V,ψ ≤ ǫ , ǫ ∗ K ′ and an ( E ′ , K ′ , ǫ ′ n,V,ψ )-arc homomorphism φ ′ < ψ such that (cid:2) K h ( n ) (cid:3) ∈ W E ′ ,K ′ and ˆ φ ′ (cid:0)(cid:2) K h ( n ) (cid:3)(cid:1) ⊆ V .We apply Lemma 5.2 to obtain a finite set S of ( E, K, ǫ )-homomorphisms suchthat for every ( E, K, ǫ ) homomorphism φ there exists ψ ∈ S such that ψ < φ .Let T be a finite γ -dense subset of T . Let V be the (finite) set of the arcs of length γ centered in a point of T .Fix n ∈ A . We apply (*) for each V ∈ V and φ ∈ S to obtain ǫ n ′ ,V,ψ for which thereexists an ( E ′ , K ′ , ǫ ′ n,V,ψ )-arc homomorphism φ ′ n,V,ψ < ψ such that (cid:2) K h ( n ) (cid:3) ∈ W E ′ ,K ′ and ˆ φ ′ n,V,ψ (cid:0)(cid:2) K h ( n ) (cid:3)(cid:1) ⊆ V .Let ǫ ′ n = min n ǫ ′ n,V,ψ : V ∈ V , ψ ∈ S o .Now given φ and U as in the statement of this Theorem, there exists ψ ∈ S suchthat ψ < φ , and V ∈ V such that V ⊆ U . We shrink the size of the arc homomorphism φ ′ n,V,ψ to ǫ ′ n the proof is complete. (cid:3) We will fix the objects that are obtained through the application of Lemma 5.3and give them a notation.
OUNTABLY COMPACT GROUP TOPOLOGIES WITH CONVERGENT SEQUENCES 13
Definition 5.4.
For each E ∈ [ c ] <ω , K positive integer, ǫ, ǫ ∗ positive reals less than1 / h sequence of type 1-11 and γ >
0, let A ( E, K, ǫ, ǫ ∗ , h, γ ) be a fixed set A ob-tained by applying the previous theorem. Moreover, for each n ∈ A ( E, K, ǫ, ǫ ∗ , h, γ ),for every E ′ ⊇ E and for every positive integer K ′ satisfying K | K ′ and (cid:2) K h ( n ) (cid:3) ∈ W E ′ ,K ′ , we let E ( E, K, ǫ, ǫ ∗ , h, γ, n, E ′ , K ′ ) be a fixed positive real ǫ ′ such that ǫ ′ ≤ ǫ, ǫ ∗ /K ′ and such that for every ( E, K, ǫ )-homomorphism φ : • if h is of one of the first ten types, then for every arc U of length ≥ γ thereexists an ( E ′ , K ′ , ǫ ′ )-arc homomorphism φ ′ < φ such that ˆ φ ′ (cid:0)(cid:2) K h ( n ) (cid:3)(cid:1) ⊆ U . • if h is of type 11, then for every r ∈ T [ k ] there exists an ( E ′ , K ′ , ǫ ′ )-archomomorphism φ ′ < φ such that ( φ ′ ) ([ h ( n )]) = r . (cid:3) Homomorphisms on a countable subgroup
In this section we show how to construct homomorphisms on countable subgroupsof W . For the next proposition, given a ∈ Q , we define: k a k = X ξ ∈ supp a | a ( ξ ) | . Proposition 6.1.
Let E be a countable subset of c with ω ⊆ E , e ∈ W E with e = 0,a countable G ⊆ H (where H is defined with respect to W E ) and ( p g : g ∈ G ) afamily of pairwise incomparable selective ultrafilters.Fix a family ( c g : g ∈ G ) of elements of Q E such that [ c g ] ∈ W E , c g is a non torsionelement if g is of one of types from 1 to 10, and [ c g ] has the same order as [ g ] if g isof type 11.Then there exists a homomorphism ρ : W E → T such that:(1) ρ ( e ) = 0,(2) p g -lim( ρ ([ g ( n )]) : n ∈ ω ) = ρ ([ c g ]), for each g ∈ G , and(3) (cid:0) ρ (cid:0) n ! S χ n (cid:1) : n ∈ ω (cid:1) converges to 0 + Z , for every integer S > (cid:3) Before we prove the proposition it is easy to produce the objects below satisfyingcondition (i)-(xvi):Enumerate G = { g m : m ∈ ω } faithfully.Let E be a finite subset of E , K be a positive integer, ǫ > A be a cofinitesubset of ω such that(i) [ e ] , [ c g ] , [ g (0)] ∈ W E ,K ;(ii) ǫ k c g k < ;(iii) A = ω \ { } ; The three items above comprise a sufficient condition for the existence of an ( E , K , ǫ )-arc homomorphism σ such that 0 / ∈ ˆ σ ([ e ]), which will be used to prove condition (1)for ρ .Let ( E t : t ∈ ω ) be an arbitrary sequence of finite subsets of E such that(iv) S t ∈ ω E t = E ;(v) E t +1 ⊇ E t , for every t ∈ ω ;(vi) { [ c g t ] } ∪ S { [ g m ( m ′ )] : m, m ′ ≤ t } ⊂ W E t , for every t ∈ ω ;Fix ( K t : t ∈ ω ) an arbitrary sequence of integers so that:(vii) K t | K t +1 and t ! | K t , for every t ∈ ω ;(viii) { [ c g t ] } ∪ S { h K t − g m ( m ′ ) i : m, m ′ ≤ t } ⊆ W E t ,K t , for every t ∈ ω \ { } ;Recursively, we define sequences ( ǫ t : t ∈ ω \ { } ), ( ǫ ∗ t : t ∈ ω \ { } ) of positivereal numbers less than , a sequence of real numbers ( γ t : t ∈ ω ) and a sequence ofcofinite sets ( A t : t ∈ ω ) satisfying:(ix) γ t = ǫ t − K t − min {k [ c g l ] k : l ≤ t } , for every t ∈ ω ;(x) ǫ ∗ t = ǫ t − t K t max {k c gl k : l ≤ t } (xi) ǫ t ≤ min {E ( E i , K i , ǫ i , ǫ ∗ i , g j , γ i , n, E t , K t ) : j ≤ i ≤ t − n ∈ A i ∩ t } ,(xii) A t = ∩ i ≤ t A ( E t , K t , ǫ t , ǫ ∗ t , g i , γ t ) \ ( t + 2).Denote the ultrafilter p g k as p k . The family { A t : t ∈ ω } ⊆ p k , for each k ∈ ω . Bythe selectivity of p k there exists an increasing ( a k,i ) i ∈ ω such that(xiii) a k,i ∈ A i and i < a k,i , for each i ∈ ω and(xiv) { a k,i : i ∈ ω } ∈ p k , for each k ∈ ω .Applying Lemma 2.6, we have a family ( I k : k ∈ ω ) of pairwise disjoint subsets of ω such that:(xv) { a k,i : i ∈ I k } ∈ p k and(xvi) ([ i, a k,i ] : k ∈ ω and i ∈ I k ) is a family of pairwise disjoint sets.Finally, we may also suppose, my removing finitely many elements, that:(xvii) I k ∩ k = ∅ for every k ∈ ω . Proof. (of Proposition 6.1)We will proceed by induction on I = S k ∈ ω I k . Let ( i n : i ∈ ω ) be an increasingenumeration of I and let k n ∈ ω be the unique k such that i n ∈ I k and set a n = a k n ,i n .Fix φ ′ any ( E , K , ǫ )-arc homomorphism such that 0 / ∈ cl ˆ φ ′ ([ e ]). The existenceof such arc homomorphism follows from conditions (i) and (ii). Since i ≥
0, we canfix an ( E i , K i , ǫ i )-arc homomorphism φ i such that φ i < φ ′ . From the inequality,it follows that:(a) 0 / ∈ cl ˆ φ i ([ e ]). OUNTABLY COMPACT GROUP TOPOLOGIES WITH CONVERGENT SEQUENCES 15