DiGiMiAGT.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 7, No. 1, 2006 pp. 1-39 Weakly metrizable pseudocompact groups Dikran Dikranjan, Anna Giordano Bruno and Chiara Milan∗ Abstract. We study various weaker versions of metrizability for pseudocompact abelian groups G: singularity (G possesses a compact metrizable subgroup of the form mG, m > 0), almost connectedness (G is metrizable modulo the connected component) and various versions of extremality in the sense of Comfort and co-authors (s-extremal, if G has no proper dense pseudocompact subgroups, r-extremal, if G ad- mits no proper pseudocompact refinement). We introduce also weakly extremal pseudocompact groups (weakening simultaneously s-extremal and r-extremal). It turns out that this “symmetric” version of ex- tremality has nice properties that restore the symmetry, to a certain extent, in the theory of extremal pseudocompact groups obtaining sim- pler uniform proofs of most of the known results. We characterize doubly extremal pseudocompact groups within the class of s-extremal pseudocompact groups. We give also a criterion for r-extremality for connected pseudocompact groups. 2000 AMS Classification: Primary 22B05, 22C05, 40A05; Secondary 43A70, 54A20. Keywords: pseudocompact group, Gδ-dense subgroup, extremal pseudo- compact group, dense graph. ∗The first and the third author were partially supported by Research Grant of the Italian MIUR in the framework of the project “Nuove prospettive nella teoria degli anelli, dei moduli e dei gruppi abeliani” 2002. The third author was partially supported by Research Grant of the University of Udine in the framework of the project “Torsione topologica e applicazioni in algebra, analisi e teoria dei numeri”. 2 D. Dikranjan, A. Giordano Bruno and C. Milan 1. Introduction The metric spaces and their generalizations are of major interest in Gen- eral Topology. We consider here several weak versions of metrizability that work particularly well for pseudocompact groups. The class of pseudocom- pact groups, introduced by Hewitt [25] was proved to be an important class of topological groups in the last forty years. 1.1. Extremal pseudocompact groups. In the framework of pseudocom- pact groups, extremality is a generalization of metrizability discovered by Com- fort and co-authors. The starting point was a theorem of Comfort and Soun- dararajan [15] who proved that for a compact, metrizable and totally discon- nected group there exists no strictly finer pseudocompact group topology. In 1982 Comfort and Robertson [9] extended this result to all compact, metrizable groups and proved also that a compact abelian group is metrizable if and only if it has no proper dense pseudocompact subgroup (see also Corollary 4.14 here for a short simultaneous proof of both theorems). This result motivated the study of the extremal pseudocompact groups, which can be considered as a generalization of the metrizable (compact) ones. In 1988 Comfort and Robertson [11] defined extremal pseudocompact groups; however, they did not distinguish among different kinds of extremality. This was done explicitly somewhat later (see [1]). A pseudocompact group G is • s-extremal if G has no proper dense pseudocompact subgroups; • r-extremal if there exists no strictly finer pseudocompact group topol- ogy on G; • doubly extremal if it is both s-extremal and r-extremal. Since the pseudocompact groups of countable pseudocharacter are compact, they cannot contain proper dense subgroups and every continuous bijection between such groups is a homeomorphism; hence it follows that every metriz- able pseudocompact group is doubly extremal. Note that this result holds for arbitrary pseudocompact groups (not necessarily abelian) and introduces an important class of doubly extremal pseudocompact groups. Extremality of zero-dimensional pseudocompact abelian groups was consid- ered by Comfort and Robertson [11]. It was shown that a zero-dimensional pseudocompact abelian group that is either s- or r-extremal is metrizable [11, Theorem 7.3]. An important step in the proof was the case of abelian el- ementary p-groups, for which s- and r-extremality coincide and both imply metrizability (cf. [11, Theorem 5.19]). Hence it is worth asking whether there exist non-metrizable pseudocompact groups that are s-, r- or doubly extremal and what is the relation between s- and r-extremality. In the following list we collect questions posed in [11], [6], [4], [23] and [1]: (A) Is every s-extremal pseudocompact group metrizable? (B) Is every r-extremal pseudocompact group metrizable? (C) Is every doubly extremal pseudocompact group metrizable? Weakly metrizable pseudocompact groups 3 (D) Is every s-extremal pseudocompact group also r-extremal? (E) Is every r-extremal pseudocompact group also s-extremal? Clearly, (A) implies (C) and (D), while (B) implies (C) and (E). Moreover, if the conjunction of (D) and (E) is true, then (A), (B) and (C) are equivalent. Nevertheless, none of the above questions has a complete answer, even when the attention is restricted to the context of abelian groups, hence it is worth studying extremality in the class of pseudocompact abelian groups. In many particular cases it has been proved that some forms of extremality are equivalent to metrizability (e.g., for totally disconnected pseudocompact abelian groups, countably compact abelian groups, pseudocompact abelian groups of weight at most c, where c denotes the cardinality of the continuum). As far as the relation between s- and r-extremality is concerned, in some cases these two notions turn out to be equivalent. 1.2. Further levels of extremality. An essential tool in the study of pseu- docompact groups is the notion of Gδ-density (a subgroup H ≤ G is Gδ-dense if non-trivially meets every non-empty Gδ-set of G). The first very important theorem on pseudocompact group is due to Comfort and Ross (cf. Theorem 2.1); it shows the relation between a pseudocompact group G, its completion G̃ and Gδ-density. In this paper we intend to face the problem of a better description of the relations among the three levels of extremality. To this end we introduce some weaker forms of extremality (in what follows r0(G) will denote the free rank of G). Definition 1.1. A pseudocompact abelian group is (a) d-extremal if G/H is divisible for every Gδ-dense subgroup H of G; (b) c-extremal if r0(G/H) < c for every Gδ-dense subgroup H of G; (c) weakly extremal if it is both d-extremal and c-extremal. While there exist plenty of non-metrizable c-extremal (e.g., torsion) or d- extremal (e.g., divisible) pseudocompact abelian groups, we are not aware whether every weakly extremal pseudocompact abelian group is metrizable. This property as well as the one given in the following theorem make the weak extremality the most relevant extremality property: Theorem A. (Theorem 3.12) If a pseudocompact abelian group G is either s-extremal or r-extremal, then G is weakly extremal. The same conclusion of the above theorem can be deduced from [3, Theorem 4.4]; the proof is given in §3.2 and is based on the dense graph theorem (cf. Theorem 3.8), proved in 1988 by Comfort and Robertson [11]. In particular, Theorem 3.12 implies that if the problem of extremality for a certain class of pseudocompact abelian groups admits a solution in the case of weak extremality, then this problem is solved also for s-extremality, r- extremality and double extremality, which are the main forms of extremality. The notion of weakly extremal group introduces the following relevant question: 4 D. Dikranjan, A. Giordano Bruno and C. Milan (F) Is every weakly extremal pseudocompact group metrizable? Let us note that a positive answer to this question will lead to a simultaneous positive answer to all questions (A)-(E), since r-extremal or s-extremal implies weakly extremal by Theorem A. On the other hand, one may anticipate that in all cases where a positive answer to one of questions (A)-(E) is given for both s- and r-extremality, it is possible to extend these results to the case of weakly extremal groups. Moreover, the notion of weak extremality introduces a new approach to the study of extremal groups and in most cases allows us to extend to r-extremal groups results proved only for s-extremal ones (cf. Theorem 4.15). 1.3. Singularity and almost connectedness of pseudocompact groups and their connection to extremality. The original aim of this paper (hav- ing large intersection with [24]), was to be a comprehensive survey on extremal pseudocompact groups. Gradually some original ideas and results appeared. In June 2004, shortly before the submission of this paper, we received a copy of [3], kindly sent by the authors before the publication. Although most of the results were announced earlier [22, 23], the proofs became accessible to us only at that point. Let us describe in detail the content of the present paper. Following [10], we denote by Λ(G) the family of all closed (normal) Gδ- subgroups of a pseudocompact topological group G and by m(G) the minimum cardinality of a dense pseudocompact subgroup of G. Then every Gδ-set con- taining 0 contains also a subgroup N ∈ Λ(G). In §2 we recall some important facts concerning pseudocompact groups G and the family Λ(G), that will be essential in the sequel. Most of the proofs are omitted but hints or due references are given in all cases. In §3 we consider some stability properties of the various classes of extremal groups, with particular attention to the behavior of extremal groups under tak- ing closed (pseudocompact) subgroups and quotients with respect to subgroups N ∈ Λ(G). §3.2 is opened by the dense graph theorem (Theorem 3.8), which is at the basis of the proofs of other results of §§3.2 and 6.1, concerning in particular the relations among various kinds of extremality. Comfort, Gladdines and van Mill showed in [4, Corollary 4.6] that s-extremal groups have free rank at most c. The same conclusion was proved for r-extremal groups by Comfort and Galindo in [3, Theorem 5.10 (b)] (announced in [23, Theorem 7.3]). These results are simultaneously extended to c-extremal groups in Theorem 3.6 (although c-extremal pseudocompact abelian groups G with r0(G) = c need not be metrizable, Example 4.4). On the other hand, d-extremal pseudocompact abelian groups need not have bounded free rank (take any large pseudocompact divisible abelian group). In §4 we introduce the concept of singular group generalizing simultaneously metrizability and torsion (cf. Theorem 2.14). If m is a positive integer and G is a group, let G[m] = {x ∈ G : mx = 0}. Weakly metrizable pseudocompact groups 5 Definition 1.2. A topological group G is singular if there exists m ∈ N+ such that G[m] ∈ Λ(G). The concept of singular abelian group implicitly appeared already in [20] in the case of compact groups. We prove that for singular pseudocompact abelian groups d-extremality is equivalent to metrizability (cf. Theorem 4.6). Note that d-extremality and c- extremality behave in a completely different way w.r.t. singularity: d-extrema- lity complements singularity to metrizability, whereas c-extremality is a weaker version of singularity (Proposition 4.7). On the other hand, Theorem 4.13 shows that a c-extremal pseudocompact abelian group G is singular whenever w(G) ≤ c or G is compact. In particular, Theorems 4.6 and 4.13 imply Theorem 4.15, imposing the restraint w(G) > c for non-metrizable weakly extremal pseudocompact abelian groups G. So this theorem simultaneously strengthens the known results of Comfort, Gladdines and van Mill [4, Theorem 4.11] and Comfort and Galindo [3, Theorem 5.10] where the same conclusion is obtained for s-extremal (resp., r-extremal) pseudocompact abelian groups. It is not possible to replace in Theorem 4.15 weak extremality neither by c-extremality nor by d-extremality (just take a non-metrizable pseudocompact abelian group of weight ≤ c that is either torsion or divisible). In §5 we define almost connected groups as follows (here c(G) denotes the connected component of G): Definition 1.3. A topological group G is almost connected if c(G) ∈ Λ(G). If G is an almost connected pseudocompact group, then the quotient G/c(G) is compact. Let us recall here, that some authors [5, p.82] call a locally com- pact group G almost connected if only compactness is required for the quotient G/c(G) (whereas, in our setting this quotient is compact metrizable). Through- out this paper almost connected will be understood always in the sense of Defi- nition 1.3. Since the locally compact pseudocompact groups are compact, there is no interference at all except in the case of compact groups. It follows from the definition that both connected groups and metrizable groups are almost connected. Moreover, totally disconnected pseudocompact abelian groups are almost connected if and only if they are metrizable (cf. Proposition 5.6). As a matter of fact, almost connectedness and singularity form a well balanced pair of generalizations of metrizability for pseudocompact abelian groups: G is singular if mG is metrizable for some m > 0 (Lemma 4.1), whereas G is almost connected if G/mG is metrizable for every m > 0 (Theorem 5.8). Hence G is metrizable whenever it is simultaneously singular and almost connected. The almost connectedness is stable under taking Gδ-dense subgroups and closed Gδ-subgroups (cf. Theorem 5.8). Moreover, if a topological group G admits a dense almost connected pseudocompact subgroup, then G itself is pseudocompact and almost connected (cf. Lemma 5.2). 6 D. Dikranjan, A. Giordano Bruno and C. Milan For non-metrizable almost connected pseudocompact abelian groups some cardinal invariants are preserved by subgroups that are in Λ(G). In particular, the connected component c(G) of a non-metrizable almost connected pseudo- compact abelian group satisfies m(G) = m(c(G)) ≤ r0(G) and w(G) ≤ 2 r0(G) (cf. Theorem 5.13). The properties of almost connected groups are used in §5 to generalize some results concerning s-extremality of connected pseudocompact abelian groups proved in [6]. The main tool is given by Theorem 5.14 which shows that a d-extremal pseudocompact abelian group is almost connected. Note that c- extremal pseudocompact abelian groups need not be almost connected (take any torsion, non-metrizable pseudocompact abelian group). The proof of the following result, due to Comfort and Robertson, is not covered by the proofs given here. For the rest this paper is self-contained. Theorem 1.1. ([11, Corollary 7.5]) Let G be a torsion pseudocompact abelian group that is s- or r-extremal. Then G is metrizable. This theorem can easily be generalized to d-extremal groups: Theorem B. (Corollary 5.17) Let G be a pseudocompact abelian group that is either totally disconnected or singular. Then G is d-extremal if and only if it is metrizable. Since torsion pseudocompact abelian groups are singular, this is a general- ization of Theorem 1.1. Moreover, since a zero-dimensional topological group is totally disconnected, this corollary generalizes also [11, Theorem 7.3], limited to the case of zero-dimensional groups. Comfort and van Mill [6, Theorem 7.1] proved that every connected s- extremal group is divisible. An analogous result for r-extremal groups was announced in [1, Corollary 5.11] and proved in [3, Corollary 5.11]. These re- sults are simultaneously extended to d-extremal groups in Corollary 5.22 where it is proved that a d-extremal pseudocompact abelian group is divisible if and only if it is connected. This follows by the more general fact (relying on proper- ties of almost connectedness) that the connected component of an almost con- nected pseudocompact abelian group is divisible if and only if it is d-extremal (Theorem 5.20). The following theorem, due to Comfort and Galindo (announced in [23, Theorem 8.2], [1, Theorem 6.1]), establishes some relevant necessary condi- tions satisfied by a non-metrizable r-extremal (resp. s-extremal) group. It introduces a certain asymmetry between s-extremality and r-extremality for pseudocompact abelian groups. Theorem 1.2. ([3, Theorem 7.1]) Let G be a pseudocompact abelian group. Then: (a) if G is s-extremal and non-metrizable, then there exists p ∈ P such that G[p] is not Gδ-dense in G̃[p]; (b) if G is r-extremal, then G[p] is dense in G̃[p] for every p ∈ P. Weakly metrizable pseudocompact groups 7 As an immediate corollary one can see that for a non-metrizable doubly extremal pseudocompact abelian group G, there exists p ∈ P such that G[p] is dense but not Gδ-dense in G̃[p], i.e., G[p] is not pseudocompact ([3, Theorem 7.2(ii)]). According to [30]: an abelian group G is almost torsion-free if G[p] is finite for every prime p. This generalization of the notion “torsion-free group” immediately gives: if G is a doubly extremal pseudocompact almost torsion-free abelian group, then G is metrizable. For torsion-free groups this result was obtained in [3]. The problem in attacking question (D) is the current lack of sufficient con- ditions for r-extremality. Theorem 3.8 and its consequences allow us to obtain a sufficient condition for a s-extremal group G to be doubly extremal (hence, can be considered as a partial solution to (D)). Theorem C. (Theorem 6.10) Let G be a s-extremal pseudocompact abelian group. Then G is doubly extremal if and only if G[p] = G̃[p] for every p ∈ P. This result strengthens Theorem 4.4 (b) of [3] where it is shown that every divisible s-extremal pseudocompact abelian group G such that G[m] = G̃[m] for every m ∈ N is r-extremal. In §6.2 we give a criterion for r-extremality of connected pseudocompact groups making no recourse to “external” issues such as characters or even different topologies on the group: a connected pseudocompact abelian group G is r-extremal if and only if G is weakly extremal and every N ∈ Λ(G) with G/N ∼= T is d-extremal (Theorem 6.11). According to [4, Theorem 4.8], every s-extremal pseudocompact abelian group has cardinality at most c. In §7 we generalize this result as follows: an infinite weakly extremal pseudocompact abelian group G has |G| = c if and only if m(G) = |G| (cf. Theorem 7.1 and Corollary 7.2). It turns out that c-extremality is stable with respect to taking Gδ-subgroups: Theorem D. (Theorem 4.11) Every closed Gδ-subgroup of a c-extremal pseu- docompact abelian group G is c-extremal. Nevertheless, such a stability need not be available for all extremality prop- erties. Indeed, it was proved by Comfort and Galindo [3, Theorem 6.1] (an- nounced in [23, Lemma 8.1] and [1, Theorem 5.16 (b)]) that a pseudocompact abelian group G is metrizable whenever every N ∈ Λ(G) is s-extremal. This motivates the necessity to define stronger versions of extremality as follows. Definition 1.4. A pseudocompact group G is said to be: (a) strongly extremal if every N ∈ Λ(G) is r-extremal; (b) strongly d-extremal if every N ∈ Λ(G) is d-extremal. The property in item (a) (without a specific term) was introduced and used in [3]. We will prove in Theorem 6.7 that s-extremal pseudocompact abelian groups that are also strongly d-extremal must be metrizable. Moreover, in Theorem 6.8 we will see that if every N ∈ Λ(G) is weakly extremal, then G is 8 D. Dikranjan, A. Giordano Bruno and C. Milan strongly extremal (i.e., every c-extremal and strongly d-extremal pseudocom- pact abelian group is strongly extremal). The next diagram shows the relations among all forms of weak metrizability that appear in the paper: metrizable ↓ strongly extremal ←− doubly extremal ւ ց ւ ց str. d-extr. r-extremal s-extremal ց ւ ց weakly extremal singular ւ ց ւ almost conn. ←− d-extremal c-extremal In the compact case only three distinct classes remain: metrizable groups (as weakly extremal compact groups are metrizable by Corollary 4.14), singu- lar groups (as c-extremal compact groups are singular by Theorem 4.13) and almost connected groups (as almost connected compact groups are strongly d-extremal according to Theorem 5.15). The intersection of that last two is the class of metrizable groups. Since every non-metrizable compact connected abelian group is strongly d-extremal, but not singular (hence not c-extremal), strongly d-extremal does not imply c-extremal. 1.4. Notation and terminology. The symbols Z, P, N and N+ are used for the set of integers, the set of primes, the set of natural numbers and the set of positive integers, respectively. The circle group T is identified with the quotient group R/Z of the reals R and carries its usual compact topology. Let G be an abelian group. The subgroup of torsion (p-torsion) elements of G is denoted by t(G) (resp., tp(G)). The group G is said to be bounded torsion if there exists n ∈ N+ such that nG = 0. If m is a positive integer, Z(m) is the cyclic group of order m. We denote by r0(G) the free rank of G (if G is free abelian this is simply its rank, otherwise this is the maximum rank of a free subgroup of G). For n ∈ N let ϕn : G → G be defined by ϕn(x) = nx for every x ∈ G. Then kerϕn = G[n]. If H is a group and h : G → H is a homomorphism, then we denote by Γh := {(x,h(x)), x ∈ G} the graph of h. For a topological space X, we denote by w(X) the weight of X (i.e., the minimum cardinality of a base for the topology on X). A space X is said to be zero-dimensional if X has a base consisting of clopen sets. A space X is pseudocompact if every continuous real-valued function on X is bounded, ω-bounded if the closure of every countable subset of X is compact. Throughout this paper all topological groups are Hausdorff and completeness is intended with respect to the two-sided uniformity, so that every topological group has a completion which we denote by G̃. A group G is precompact if G̃ is compact. With c(G) we indicate the connected component of 0 in G; a group G is totally disconnected if c(G) is trivial. Weakly metrizable pseudocompact groups 9 For a topological group G, we will denote by VG(0) the filter of 0-neighbor- hoods in G, by χ(G) the character of G (that is, the minimal cardinality of a basis of VG(0)) and by ψ(G) the pseudocharacter of G (i.e., the minimum size of a family B of 0-neighborhoods of G such that ⋂ U∈B U = {0}). If M is a subset of a topological group G, then 〈M〉 is the smallest sub- group of G containing M and M is the closure of M. For any abelian group G let Hom(G,T) be the group of all homomorphisms from G to the circle group T. When (G,τ) is an abelian topological group, the set of τ-continuous homomorphisms χ : G → T (characters) is a subgroup of Hom(G,T) and is denoted by Ĝ. For a subset H of G the annihilator of H in Ĝ is the subgroup A(H) = {χ ∈ Ĝ : χ(H) = {0}} of Ĝ. For undefined terms see [19, 26]. 2. Background on pseudocompact groups The following theorem guarantees precompactness of the pseudocompact groups and characterizes the pseudocompact groups among the precompact ones. Theorem 2.1. ([13, Theorems 1.2 and 4.1]) (a) Every pseudocompact group is precompact. (b) Let G be a precompact group. Then the following conditions are equiv- alent: (b1) G is pseudocompact; (b2) G is Gδ-dense in G̃; (b3) G̃ is the Stone-Cech compactification of G. For the sake of easier reference we isolate the following lemma from Theorem 2.1. Lemma 2.2. Let G be a topological group and let H be a dense, pseudocompact subgroup of G. Then G is pseudocompact and H is Gδ-dense in G. Lemma 2.2 immediately yields that a dense subgroup H of a pseudocompact group G is pseudocompact if and only if H is Gδ-dense in G. Corollary 2.3. Let G be a pseudocompact topological group. (a) If G is metrizable, then G is compact; (b) G is connected if and only if G̃ is connected; (c) G is zero-dimensional if and only if G̃ is zero-dimensional. Lemma 2.4. ([11, Theorem 3.2]) Let G be a pseudocompact group such that {0} is a Gδ-set. Then G is metrizable and compact. A useful pseudocompact criterion via quotients follows. Theorem 2.5. ([11, Lemma 6.1]) Let G be a precompact group. Then G is pseudocompact if and only if G/H is compact metric for every H ∈ Λ(G). 10 D. Dikranjan, A. Giordano Bruno and C. Milan Corollary 2.6. Let G be a pseudocompact abelian group. Then: (a) N is pseudocompact for every N ∈ Λ(G); (b) if w(G) > ω, then w(G) = w(N) for every N ∈ Λ(G); (c) if N ∈ Λ(G) and H is a closed subgroup of G such that N ⊆ H, then H ∈ Λ(G); (d) if N ∈ Λ(G) and L ∈ Λ(N), then L ∈ Λ(G). Now we see that Gδ-density in G is completely controlled by the subgroups N ∈ Λ(G). Lemma 2.7. Let G be a pseudocompact abelian group and let D be a subgroup of G. Then: (a) D is Gδ-dense in G if and only if N + D = G for every N ∈ Λ(G); (b) if D is Gδ-dense in G and N ∈ Λ(G), then G/D ∼= N/(D ∩N); (c) if D is Gδ-dense in G and N ∈ Λ(G), then D∩N is Gδ-dense in N. Proof. (a) follows from the fact that D is Gδ-dense in G if and only if (x + N)∩D 6= ∅ for every x ∈ G and every N ∈ Λ(G). (b) By (a) G = N + D, hence G/D ∼= N/(D∩N). (c) follows from (a). � Lemma 2.8. Let G be a pseudocompact abelian group. Then: (a) if N ∈ Λ(G), then N ∈ Λ(G̃); (b) if G is dense in G1 and N ∈ Λ(G), then N G1 ∈ Λ(G1). Proof. (b) follows from (a) and G̃ = G̃1. � Corollary 2.9. Let G be a pseudocompact abelian group. If G is dense in G1 and M is a closed subgroup of G1 such that M∩G ∈ Λ(G), then M ∈ Λ(G1). In particular, M ∈ Λ(G̃) for every closed subgroup of G̃ such that M ∩G ∈ Λ(G). Lemma 2.10. Let G be a precompact abelian group, N ∈ Λ(G) and n ∈ N+. Then nN G̃ = nN G̃ and N ∩nN G̃ = nN G . Lemma 2.11. Let G be a topological group. Let τ and τ′ be pseudocompact group topologies on G such that τ′ ≥ τ. Then the following conditions are equivalent: (a) τ = τ′; (b) for every N ∈ Λ(G,τ) one has τ|N = τ ′|N ; (c) there exists N ∈ Λ(G,τ) such that τ|N = τ ′|N. Proof. (a)⇒(b) and (b)⇒(c) are obvious. (c)⇒(a) Let τq and τ ′ q be the quotient topologies induced on G/N by τ and τ′ respectively. Since τ ≤ τ′, Λ(G,τ) ⊆ Λ(G,τ′) and so in particular N ∈ Λ(G,τ′). This implies that τq and τ ′ q are compact group topologies on G/N. Moreover, τq ≤ τ ′ q and consequently τq = τ ′ q. Since τ|N = τ ′|N, Merzon’s lemma [29] implies that τ = τ′ on G. � Weakly metrizable pseudocompact groups 11 Remark 2.12. Let (G,τ) and H be topological groups and h : (G,τ) → H a homomorphism. Consider the map j : G → Γh such that j(x) = (x,h(x)) for every x ∈ G. Observe that j is a homomorphism such that j(G) = Γh, that is j is surjective; moreover j is injective and so j is an isomorphism. If p1 : G×H → H is the projection on the first component, p1 is continuous and also its restriction to Γh is continuous; since j is the inverse of p1|Γh, it follows that j is open. Endow Γh with the group topology induced by the product (G,τ) × H. We define the topology τh as the weakest group topology on G such that τh ≥ τ and for which j is continuous. Then j : (G,τh) → Γh is a homeomorphism. Note that the topology τh so defined is the weakest topology on G such that τh ≥ τ and for which h is continuous. Indeed, if p2 : G × H → H is the canonical projection on the second component, then its restriction to Γh is continuous. The homomorphism h is the composition of j and p2, in the sense that p2|Γh ◦j = h : (G,τh) → H and therefore, being j τh-continuous, h has to be τh-continuous too. Clearly, if h is τ-continuous then τh = τ. Lemma 2.13. Let (G,τ) be a topological group and let h : G → T be a homo- morphism such that (G,τh) is pseudocompact. Then the following conditions are equivalent: (a) h is continuous; (b) every N ∈ Λ(G,τ) is such that h|N : N → T is continuous; (c) there exists N ∈ Λ(G,τ) such that h|N : N → T is continuous. Proof. (a)⇒(b) and (b)⇒(c) are obvious. (c)⇒(a) Since h|N : N → T is continuous and τ ≤ τh, it follows that τh|N = τ|N . As τ and τh are pseudocompact by hypothesis, the assertion follows from Lemma 2.11. � The next property of pseudocompact groups was announced in [6, Remark 2.17] and proved in [21, Lemma 2.3]. Theorem 2.14. Let G be an infinite pseudocompact abelian group. Then either r0(G) ≥ c or G is bounded torsion with |G| ≥ c. Proof. If G is torsion, then G is bounded by [11, Lemma 7.4]. The inequality |G| ≥ c follows from van Douwen’s Theorem [31]. Assume G is not torsion. If G is compact, then the assertion follows from [20, Lemma 2.3]. Otherwise, pick a non-torsion element x ∈ G and consider the cyclic subgroup C = 〈x〉. Since C is countable, there exists a Gδ-set O around 0 that meets C in {0}. Find N ∈ Λ(G) contained in O, hence N ∩C = {0}. Then the quotient group G/N is compact and non-torsion. Hence the compact case applies. � The following results will be helpful in the sequel to produce Gδ-dense sub- groups of given pseudocompact groups. In particular, item (a) of the following lemma has been proved by Comfort and van Mill in [6, Lemma 2.13] and it has been announced with its corollary by Comfort, Gladdines and van Mill in [4, Lemma 4.1]. 12 D. Dikranjan, A. Giordano Bruno and C. Milan Lemma 2.15. Let G be a pseudocompact group. (a) [6, Lemma 2.13] If G = ⋃∞ n=0 An, where An is a subgroup of G for every n ∈ N, then there exist n ∈ N and N ∈ Λ(G) such that An ∩N is Gδ-dense in N. (b) [4, Lemma 4.1 (b)] If N ∈ Λ(G) and D is Gδ-dense in N, then there exists a subgroup E of G with |E| ≤ c such D+E is a Gδ-dense subgroup of G. (c) If G is infinite, then m(N) = m(G) for every N ∈ Λ(G). It directly follows from Lemma 2.15 that if G is a pseudocompact abelian group such that G = ⋃∞ n=0 An, where An are subgroups of G, then there exist n ∈ N, N ∈ Λ(G) and E ≤ G with |E| ≤ c such that (An ∩N) + E is Gδ-dense in G. In particular: Corollary 2.16. Let G be a pseudocompact abelian group such that G =⋃∞ n=0 An, where An ≤ G for every n ∈ N. Then there exists a subgroup E of G, with |E| ≤ c, such that An + E is Gδ-dense in G. 3. Extremal pseudocompact abelian groups 3.1. General properties of extremal groups. Lemma 3.1. Let G be a s (resp. d,c)-extremal pseudocompact abelian group and let N be a pseudocompact subgroup of G. (a) If N is closed, then G/N is s (resp. d,c)-extremal. (b) If N is an algebraic direct summand of G, then N is s (resp. d,c)- extremal. In particular, every divisible pseudocompact subgroup of G is s (resp. d,c)-extremal. Proof. (a) Let ψ : G → G/N be the canonical homomorphism and let H be a Gδ-dense subgroup of G/N. Then ψ −1(H) is a Gδ-dense subgroup of G (cf. [3, Theorem 5.3 (a)]). Hence G/ψ−1(H) = {0} (resp. G/ψ−1(H) is divisible, r0(G/ψ −1(H)) < c). Since (G/N)/H ∼= G/ψ−1(H) we conclude that (G/N)/H = {0} (resp. (G/N)/H is divisible, r0((G/N)/H) < c). (b) There exists a subgroup L of G such that G = N ⊕ L. Let D be a Gδ-dense subgroup of N. Then D1 = D ⊕L is a Gδ-dense subgroup of G. To see this, let M ∈ Λ(G). As D is Gδ-dense in N, by Lemma 2.7 (a) M +D ≥ N and consequently M +D1 ≥ N ⊕L = G. Since G is s (resp. d,c)-extremal, one has necessarily G/D1 = {0} (resp. G/D1 is divisible, r0(G/D1) < c). Since N/D ∼= G/D1, N is s (resp. d,c)-extremal. � Let us recall that stability of s- and r-extremality under taking quotients with respect to closed pseudocompact groups was proved by Comfort and Galindo ([23, Lemma 4.5], [3, Theorem 5.3]): Lemma 3.2 ([23, 3]). Let G be a pseudocompact abelian group and let N be a closed pseudocompact subgroup of G. (a) If G is r-extremal (resp., s-extremal), then G/N is r-extremal (resp., s-extremal). Weakly metrizable pseudocompact groups 13 (b) If G is doubly extremal, then G/N is doubly extremal. The Gδ-subgroups are particularly important since the following stability properties hold. Lemma 3.3. Let G be a pseudocompact abelian group and let N ∈ Λ(G). (a) If N is s (resp. d,c)-extremal, then G is s (resp. d,c)-extremal. (b) [3, Theorem 2.1 (b)] If N is r-extremal, then G is r-extremal. (c) If N is doubly extremal, then G is doubly extremal. Proof. (a) Let H be a Gδ-dense subgroup of G. Then H ∩ N is Gδ-dense in N and so N/H ∩N = {0} (resp. N/H ∩N is divisible, r0(N/H) < c). Since G = N + H, the conclusion follows from G/H = (N + H)/H ∼= N/H ∩N. (b) Let τ′ be a pseudocompact group topology on G finer than τ. Since N ∈ Λ(G,τ), then N ∈ Λ(G,τ′) and consequently (N,τ′|N) is pseudocompact. Then τ|N = τ ′|N, since N is r-extremal by hypothesis. By Lemma 2.11 it follows that τ = τ′, hence G is r-extremal. (c) It follows directly from (a) and (b). � Corollary 3.4. If G is a r (resp. s,d,c)-extremal pseudocompact abelian group and K is a compact metrizable group, then also G × K is a r (resp. s,d,c)- extremal pseudocompact group. Remark 3.5. Note that strong extremality and strong d-extremality are pre- served by taking closed, Gδ-subgroups by Corollary 2.6(d). In the next theorem we extend to c-extremal groups [4, Corollary 4.6] and [23, Theorem 7.3] (proved in [23] and [3, Theorem 5.10 (b)]), where the same result was announced respectively for s- and r-extremal groups. Our proof uses ideas from the proof of [4, Proposition 4.4] and is very similar to that of [3, Theorem 5.10 (b)]. We include it here for the sake of completeness. Theorem 3.6. Let G be a pseudocompact abelian group. If G is c-extremal, then r0(G) ≤ c. Proof. Let κ = r0(G). Let M be a maximal independent subset of G consisting of non-torsion elements. Then |M| = κ and there exists a partition M =⋃∞ n=1 Mn such that |Mn| = κ for each n. Let Un = 〈Mn〉, Vn = U1 ⊕···⊕Un and An = {x ∈ G : n!x ∈ Vn} for every n ∈ N+. Then G = ⋃∞ n=1 An. By Corollary 2.16 there exist n ∈ N+ and a subgroup E of G such that à = An +E is Gδ-dense in G and |E| ≤ c. Hence |E/(An ∩E)| ≤ c. So the isomorphism Ã/An = (An + E)/An ∼= E/(An ∩E) yields |Ã/An| ≤ c. Since G is c-extremal, r0(G/Ã) < c, so r0(G/An) ≤ c by the isomorphism (G/An)/(Ã/An) ∼= G/Ã. On the other hand, r0(G/An) = κ, as Un+1 embeds into G/An (note that An∩Un+1 = {0} since every x ∈ An∩Un+1 satisfies n!x ∈ Vn ∩ Un+1 = {0}, so x = 0 as Un+1 is a free group), hence κ ≤ c. � 14 D. Dikranjan, A. Giordano Bruno and C. Milan 3.2. The dense graph theorem and weakly extremal groups. The dense graph theorem (see Theorem 3.8 below) was announced in [11]. It gives a sufficient condition for a group to be neither s-extremal nor r-extremal. A lot of useful results in the study of extremality follow from this theorem. The following lemma has been announced in a similar form in [23] without a proof. Lemma 3.7. Let G be a topological abelian group and let H be a compact metrizable abelian group with |H| > 1. Let h : G → H be a surjective homo- morphism. Then Γh is Gδ-dense in G × H if and only if kerh is proper and Gδ-dense in G. By means of Lemma 3.7 one can prove that for a pseudocompact abelian group G the following statements are equivalent: (a) there exists a pseudocompact abelian group H with |H| > 1 and a homomorphism h : G → H with Γh Gδ-dense in G×H; (b) there exists a compact metrizable abelian group H with |H| > 1 and a surjective homomorphism h : G → H with Γh Gδ-dense in G×H; (c) there exists a compact metrizable abelian group H with |H| > 1 and a surjective homomorphism h : G → H with kerh Gδ-dense in G. Theorem 3.8 (of the dense graph [11, Theorem 4.1]). Let (G,τ) be a pseudo- compact abelian group. Suppose that there exist a pseudocompact abelian group H with |H| > 1 and h ∈ Hom(G,H) such that Γh is a Gδ-dense subgroup of G×H. Then (a) there exists a pseudocompact group topology τ′ on G such that τ′ > τ and w(G,τ′) = w(G,τ); (b) there exists a proper dense pseudocompact subgroup D of G such that w(D) = w(G,τ). Corollary 3.9. Let G be a pseudocompact abelian group. Assume that there exists a surjective homomorphism h : G → H, where H is a non trivial closed subgroup of T. If Γh is Gδ-dense in G×H, then G is neither s-extremal nor r-extremal. The next corollary directly follows from Lemma 3.7 and Corollary 3.9. Corollary 3.10. Let G be a pseudocompact abelian group. Suppose that there exists a surjective homomorphism h : G → H, where H is a closed non-trivial subgroup of T. If kerh is a proper Gδ-dense subgroup of G, then G is neither s-extremal nor r-extremal. Remark 3.11. Note that the hypotheses of Theorem 3.8 and Corollary 3.10 imply that the group G cannot be metrizable. Indeed, if G were metrizable, then kerh = G, that is h ≡ 0 and so the graph Γh of h would not be Gδ-dense in G×H. Thanks to the previous results it is possible to prove Theorem A of the introduction showing that both s- and r-extremality imply weak extremality (it can be deduced from [3, Theorem 4.4]): Weakly metrizable pseudocompact groups 15 Theorem 3.12. If a pseudocompact abelian group G is either s-extremal or r-extremal, then G is weakly extremal. Proof. Suppose for a contradiction that G is not weakly extremal. Then there exists a Gδ-dense subgroup H of G such that either G/H is not divisible or r0(G/H) ≥ c. In both cases, H is a proper subgroup of G. Moreover, H is dense and pseudocompact, hence G is not s-extremal. To find a contradiction it remains to prove that G is not even r-extremal. Let ψ : G → G/H be the canonical projection. Case 1. If G/H is not divisible, then there exists a prime p and a non-trivial homomorphism f : G/H → Z(p). Composing with the canonical projection ϕ : G → G/H we get a surjective homomorphism h = ϕ◦f : G → Z(p). Since kerh ⊇ H, it follows that kerh is a proper Gδ-dense subgroup of G, then Corollary 3.10 applies to conclude that G is not r-extremal. Case 2. If r0(G/H) ≥ c, then there exists a surjective homomorphism η : G/H → T (as r0(G/H) ≥ c = r0(T)). Define the surjective homomorphism h = η ◦ ψ : G → T and observe that kerh ⊇ H. Therefore kerh is a proper Gδ-dense subgroup of G. As before it follows from Corollary 3.10 that G is not r-extremal. � 4. Singular groups 4.1. d-extremality. The next lemma offers an alternative form for singular- ity of pseudocompact abelian groups (mG is compact metrizable for some m ∈ N+). It is useful when checking stability of this property under taking subgroups and quotients. Lemma 4.1. Let G be a topological abelian group and m ∈ N+. (a) If mG is metrizable, then G[m] ∈ Λ(G). (b) If G is pseudocompact, then G[m] ∈ Λ(G) implies that mG is metrizable (hence compact). Proof. Let ϕm : G → G be the continuous homomorphism defined by ϕm(x) = mx for every x ∈ G. Then kerϕm = G[m] and ϕm(G) = mG. Let i : G/G[m] → mG be the continuous isomorphism such that i ◦ π = ϕm, where π : G → G/G[m] is the canonical homomorphism. (a) If mG is metrizable, then ψ(mG) = ω and so ψ(G/G[m]) = ω. This implies that G[m] is a Gδ-set of G; then G[m] ∈ Λ(G). (b) Suppose that G[m] ∈ Λ(G). Then the quotient G/G[m] is metrizable, hence compact by Theorem 2.5. By the open mapping theorem the isomor- phism i is also open and consequently it is a topological isomorphism. Then the group mG is metrizable and so compact. � Remark 4.2. It immediately follows from Lemma 4.1 that singularity is stable under taking pseudocompact subgroups and quotients w.r.t. closed subgroups. The next lemma characterizes the singular groups in terms of the free rank of their closed Gδ-subgroups. 16 D. Dikranjan, A. Giordano Bruno and C. Milan Lemma 4.3. A pseudocompact abelian group G is not singular if and only if r0(N) ≥ c for every N ∈ Λ(G). In such a case, w(G) > ω and r0(N) = r0(G) ≥ c for every N ∈ Λ(G). Proof. Assume that G is not singular and let N ∈ Λ(G). Suppose for a con- tradiction that r0(N) < c. Since N is pseudocompact, by Theorem 2.14 N is bounded torsion, i.e., nN = {0} for some n ∈ N+. In particular, N ⊆ G[n] and therefore Corollary 2.6 implies G[n] ∈ Λ(G), i.e., G is singular, against the hypothesis. The converse implication is immediate. To prove the second part of the lemma, assume that G is not singular. Then obviously w(G) > ω. Let N ∈ Λ(G). Then the quotient G/N is compact and metrizable, hence |G/N| ≤ c. Since r0(N) ≥ c, it follows that r0(G) = r0(G/N) ·r0(N) ≤ c ·r0(N) = r0(N), i.e., r0(G) = r0(N). � We start by an example showing that singular pseudocompact abelian groups need not be d-extremal (see Theorem 4.6 for a general result). Example 4.4. Let p be a prime and let H be the subgroup of Z(p)c defined by H = ∑ {Z(p)I : I ⊆ c, |I| ≤ ω} (i.e., the Σ-product of c-many copies of the group Z(p)). If we denote by T the circle group, then the group G = T ×H is a singular non-metrizable pseudocompact abelian group with r0(G) = c. Thus G is not d-extremal by Theorem 4.6. Lemma 4.5. Let G be a pseudocompact abelian group. If for some n ∈ N+ w(G/nG) > ω, then G is not d-extremal. Proof. Let H = G/nG. The group H is pseudocompact, non-metrizable and bounded torsion, hence H is not s-extremal by Theorem 1.1. Let H1 be a proper Gδ-dense subgroup of H. Then H/H1 is bounded torsion, so H/H1 cannot be divisible. Thus H is not d-extremal. The subgroup nG of G is pseudocompact as a continuous image of G, hence also its closure nG is pseudocompact by Lemma 2.2. Now Lemma 3.1 (a) implies that also G is not d-extremal. � It was proved by Comfort and Robertson that questions (A) and (B) have positive answer in the case of torsion pseudocompact abelian groups [11, Corol- lary 7.5]. This can easily be generalized to d-extremal groups replacing “tor- sion” by a much weaker condition: Theorem 4.6. A singular pseudocompact abelian group is d-extremal if and only if it is metrizable. Proof. If G is metrizable then it is weakly extremal, hence d-extremal. Suppose that G is not metrizable, i.e., w(G) > ω. By Lemma 4.1 there exists m ∈ N+ such that mG is compact and metrizable. Let us consider the quotient G/mG, that is pseudocompact. Since w(mG) = ω and w(G) = w(G/mG) · w(mG), it follows that w(G/mG) = w(G) > ω. Then G is not d-extremal by Lemma 4.5. � Weakly metrizable pseudocompact groups 17 It follows from Theorem 4.6 that for a singular pseudocompact abelian group also weak extremality, s-extremality, r-extremality and double extremality are all equivalent to metrizability. 4.2. c-extremality. We see now that the impact of singularity on c-extremali- ty, compared to that on d-extremality, is quite different. Let us start by proving the immediate implication singular ⇒ c-extremal. Proposition 4.7. Every singular pseudocompact abelian group G is c-extremal. In particular, r0(G) ≤ c. Proof. Assume that G is singular. Then there exists a positive integer m such that mG is metrizable by Lemma 4.1. Let H be a Gδ-dense subgroup of G. We have to see that r0(G/H) < c. Since the homomorphism ϕm : G → mG is continuous, the subgroup mH of mG is Gδ-dense and so mH = mG as mG is metrizable. Therefore mG ≤ H, hence the quotient G/H is bounded torsion. In particular r0(G/H) = 0. The last assertion follows from Theorem 3.6. For a direct argument note that metrizability and compactness of mG yield r0(mG) ≤ c and r0(G/mG) = 0. Therefore, r0(G) = r0(mG) ≤ c. � To partially invert the implication (see Theorem 4.13), we need first some preparation. The construction used in the next lemma follows standard ideas of transfinite induction carried out in similar situations in [6] and [20]. Lemma 4.8. Let κ be an infinite cardinal, G be an abelian group and {Hα : α < κ} be a family of subgroups of G such that r0(Hα) ≥ κ for every α < κ. Then for every subset X = {xα : α < κ} of G there exists a subgroup L of G such that (a) L∩ (xα + Hα) 6= ∅ for every ordinal α < κ; (b) r0(G/L) ≥ κ. Proof. Our aim will be to build by transfinite recursion two increasing chains {Lα : α < κ} and {L ′ α : α < κ} of subgroups of G such that the following conditions are fulfilled for every α < κ: (aα) Lα ∩ (xα + Hα) 6= ∅; (bα) Lα ∩L ′ α = {0}; (cα) L ′ α/ ⋃ β<α L′ β is non-torsion when α > 0; (dα) r0(Lα) ≤ |α| and r0(L ′ α) ≤ |α|. If α = 0, take x0 = 0 and L0 = L ′ 0 = {0}. We need the following easy to prove claim for our construction. Claim. Let G be an abelian group and x ∈ G. Let H, L and N be subgroups of G such that L 6= {0}, r0(N) ≥ ω and r0(H) + r0(L) < r0(N). If H ∩L = {0}, then there exists a non-torsion element y ∈ x+N such that (H +〈y〉)∩L = {0} and 〈y〉∩H = {0}. 18 D. Dikranjan, A. Giordano Bruno and C. Milan Now suppose that α > 0 and Lβ and L ′ β are built for all β < α so that the conditions (aβ)-(dβ) are satisfied for all β < α. In order to define Lα,L ′ α let M = ⋃ β<α Lβ and M ′ = ⋃ β<α L′ β . Then r0(M) + r0(M ′) < κ, so by the claim applied with x = 0, N = Hα, H = M ′ and L = M, there exists a zα ∈ Hα such that M ∩ (M ′ + 〈zα〉) = {0} and 〈zα〉 ∩ M ′ = {0}. Put L′α = M ′ + 〈zα〉, hence r0(L ′ α) = r0(M ′) + 1 ≤ |α + 1| and L′α/M ′ is non- torsion. Now r0(M) + r0(L ′ α) < κ, so it is possible to apply again the claim with N = Hα, x = xα, H = M and L = L ′ α. Then there exists yα ∈ xα + Hα such that (M + 〈yα〉) ∩ L ′ α = {0}. Put Lα = M + 〈yα〉 and observe that (aα)-(dα) hold. Finally, let L = ⋃ α<κ Lα and L ′ = ⋃ α<κ L′α. Since (aα) is true for all α, this implies (a). From (bα) and (cα) it follows respectively that the subgroup L′ satisfies L ∩ L′ = {0} and r0(L ′) ≥ κ. Let ϕ : G → G/L be the canonical homomorphism. Then the restriction ϕ|L′ : L′ → G/L is injective, hence r0(G/L) ≥ r0(ϕ(L ′)) = r0(L ′) = κ. � A subgroup L of a group G is said to be a complement of H if G = H + L. The co-rank of L is r0(G/L). Lemma 4.8 will be used in two different ways. In the proof of Theorem 4.13 we use it to build a Gδ-dense subgroup of a pseudocompact group with large co-rank. In the next corollary we use it to build a complement of a given subgroup with large co-rank. Corollary 4.9. Let H be a subgroup of an abelian group G, such that r0(H) ≥ [G : H]. Then there exists a subgroup L of G such that L + H = G and r0(G/L) ≥ r0(H). Proof. Let κ = r0(H). Then we can write G = ⋃ α<κ (xα+H). Then by Lemma 4.8 applied to X = {xα : α < κ} and to the family {Hα : Hα = H, α < κ}, there exists a subgroup L of G with (a) and (b). By (a) L + H ⊇ xα + H for every α, that is L + H ⊇ G, while (b) ensures r0(G/L) ≥ r0(H). � Remark 4.10. (a) Obviously, the subgroup L in the above corollary has co- rank precisely r0(H) as G/L ∼= H/H ∩ L holds for every complement L of H. (b) One cannot remove the condition r0(H) ≥ [G : H]. Indeed, for every n ∈ N the only complement L of the subgroup H = Zn of G = Qn is L = G. The next theorem shows that c-extremality is stable under taking closed Gδ-subgroups. Theorem 4.11. Let G be a c-extremal pseudocompact abelian group. Then every pseudocompact subgroup of G of index ≤ c is c-extremal. In particular, every closed Gδ-subgroup of G is c-extremal. Proof. Aiming for a contradiction, assume that there exists a pseudocompact subgroup H of G with |G/H| ≤ c such that H is not c-extremal. Then there exists a Gδ-dense subgroup D of H with r0(H/D) ≥ c, so |G/H| ≤ r0(H/D). Set G1 = G/D and let ϕ : G → G1 be the canonical projection. Applying to ϕ(H) = H/D Corollary 4.9, we find a subgroup L of G1 such that L+ϕ(H) = Weakly metrizable pseudocompact groups 19 G1 and r0(G1/L) ≥ r0(ϕ(H)). Let H0 = ϕ −1(L). Then H + H0 = G. Since D is a subgroup of H0 which is Gδ-dense in H, the closure of H0 w.r.t. the Gδ-topology contains H + H0 = G and so H0 is Gδ-dense in G. Moreover, r0(G/H0) = r0(G1/L) ≥ r0(ϕ(H)) = r0(H/D) ≥ c (the first equality is due to G/ϕ−1(L) ∼= G1/L). We have produced a Gδ-dense subgroup H0 of G with r0(G/H0) ≥ c, a contradiction. Now assume that H is a closed Gδ-subgroup of G. Then G/H is a compact metrizable group, so |G/H| ≤ c. Hence the above argument applies to H. � Corollary 4.12. Let G be a c-extremal pseudocompact abelian group of size c. Then every pseudocompact subgroup of G is c-extremal. The next theorem establishes a sufficient condition for a c-extremal pseudo- compact abelian group to be singular. Theorem 4.13. Let G be a c-extremal pseudocompact abelian group. If w(G) ≤ c or G is compact, then G is singular. Proof. By Theorem 3.6 we have r0(G) ≤ c. If r0(G) < c, then G is bounded torsion by Theorem 2.14. Hence G is singular. Suppose r0(G) = c and w(G) ≤ c. Assume for a contradiction that G is not singular. By Lemma 4.3 r0(N) = r0(G) = c for every N ∈ Λ(G). Let {xα + Hα : α < c} be a list of all possible cosets of subgroups Hα ∈ Λ(G) (this is possible as |Λ(G)| ≤ w(G)ω = c). Since r0(Hα) = c for all α < c, by Lemma 4.8 there exists a subgroup L of G with (a) and (b). Clearly, (a) means that L is a Gδ-dense subgroup of G and (b) is r0(G/L) ≥ c. This proves that G is not c-extremal, against the hypothesis. Assume that G is compact. According to [18], for every non-singular com- pact abelian group G there exists a surjective homomorphism G → ∏ i∈I Ki, where I is uncountable and each Ki is a compact non-torsion abelian group. Obviously, such a product cannot be c-extremal, so G is not c-extremal ei- ther. � An alternative proof of the first part of the above theorem (w(G) ≤ c) can be derived from [3, Lemma 6.7] and Lemma 4.3. Now we can prove as a corollary of Theorem 4.13 a result that generalizes [9, Theorem 3.4 and Remark 4.3] where it is proved that questions (A) and (B) have positive answer in the case of compact groups. Corollary 4.14. A compact abelian group G is metrizable if and only if G is weakly extremal. Proof. It suffices to note that if G is weakly extremal, then G is singular by Theorem 4.13 and consequently metrizable by Theorem 4.6. � Using the properties of non singular groups it is possible to prove: Theorem 4.15. A pseudocompact abelian group G is metrizable if and only if G is weakly extremal and w(G) ≤ c. 20 D. Dikranjan, A. Giordano Bruno and C. Milan Proof. If G is metrizable then G is weakly extremal and w(G) ≤ ω. Assume that G is weakly extremal and w(G) ≤ c. Then G is singular by Theorem 4.13, hence Theorem 4.6 applies to conclude that G is metrizable. � 5. Almost connected groups In this section we study the properties of the connected component and of the subgroups N ∈ Λ(G) of an almost connected group G. Remark 5.1. Let G be an almost connected, pseudocompact abelian group. Then: (a) the quotient G/c(G) is metrizable and compact; (b) the connected component c(G) of G is a pseudocompact group (by Corollary 2.6 (a)); if G is not metrizable, then w(c(G)) = w(G) (by Corollary 2.6 (b)). Lemma 5.2. Let G be a pseudocompact abelian group. Then the following conditions are equivalent: (a) G is almost connected; (b) G̃ is almost connected; (c) if G is dense in a topological group G1, then G1 is almost connected. Proof. (a)⇒(c) By Lemma 2.2 the group G1 is pseudocompact and so G is Gδ- dense in G1. Since G is almost connected, c(G) ∈ Λ(G). As c(G) ⊆ c(G1) ∩G and c(G1)∩G is closed in G, it follows from Corollary 2.6 (c) that c(G1)∩G ∈ Λ(G). Now Corollary 2.9 applies to conclude that c(G1) ∈ Λ(G1), i.e., G1 is almost connected. (c)⇒(b) Is obvious. (b)⇒(a) Suppose that G̃ is almost connected. Then c(G̃) ∈ Λ(G̃). Since G is Gδ-dense in G̃, by Lemma 2.7 (c) c(G̃) ∩G is Gδ-dense in c(G̃), which is connected. Hence c(G̃)∩G is connected too and consequently c(G̃)∩G ⊆ c(G). As c(G̃) ∈ Λ(G̃), it follows that c(G̃)∩G ∈ Λ(G) and so c(G) ∈ Λ(G). � Corollary 5.3. If G is an almost connected pseudocompact abelian group and G is dense in a topological group G1, then c(G) = c(G1) ∩ G. In particular, c(G) = c(G̃)∩G. Proof. The inclusion c(G) ⊆ c(G1) ∩G holds in general. Since G is almost connected, Lemma 5.2 yields that also G1 is almost con- nected, i.e., c(G1) ∈ Λ(G1). Moreover, as G is Gδ-dense in G1, c(G1) ∩ G is Gδ-dense in c(G1), that is connected. Hence also c(G1) ∩ G is connected and so c(G1) ∩G ⊆ c(G). � Now we prove a remarkable property of the almost connected pseudocompact abelian groups. For a topological group G denote by q(G) the quasi component of G (i.e., the intersection of all clopen sets of G [16]). Usually, c(G) ≤ q(G), but strict inequality may hold even for pseudocompact abelian groups (see [16] for various levels of the failure of the equality c(G) = q(G)). Weakly metrizable pseudocompact groups 21 Corollary 5.4. Let G be an almost connected pseudocompact abelian group. Then q(G) = c(G). Proof. It is known that q(G) = c(G̃) ∩G [17]. Hence Corollary 5.3 yields that c(G) = c(G̃)∩G. Then q(G) = c(G). � Next we prove that the connected component of a pseudocompact abelian group can be computed in the same way as in the case of compact abelian groups. Corollary 5.5. If G is an almost connected pseudocompact abelian group, then c(G) = ⋂∞ n=1 nG G . Proof. By Corollary 5.3 c(G) = G ∩ c(G̃). Since for every compact abelian group K one has c(K) = ⋂∞ n=1 nK [19], Lemma 2.10 yields that c(G) = G∩ ∞⋂ n=1 nG̃ = ∞⋂ n=1 (G∩nG̃) = ∞⋂ n=1 nG G . � The next proposition shows that for singular (resp. totally disconnected) pseudocompact abelian groups almost connectedness and metrizability are e- quivalent. Proposition 5.6. Let G be a pseudocompact abelian group that is either to- tally disconnected or singular. Then G is almost connected if and only if G is metrizable. Proof. If G is metrizable, then G is almost connected. To prove the converse implication, suppose that G is almost connected. If G is totally disconnected, then {0} = c(G) ∈ Λ(G). Hence G is metrizable by Lemma 2.4. If G is singular, then there exists m ∈ N+ such that mG is compact and metrizable by Lemma 4.1(b). Since c(G) ⊆ mG by Corollary 5.5, it follows that w(c(G)) = ω. On the other hand, c(G) ∈ Λ(G) by hypothesis, hence w(G) = w(c(G)) = ω, i.e., G is metrizable. � Lemma 5.7. [6, Theorem 3.4] Let G be a connected pseudocompact abelian group. Then c(N) ∈ Λ(G) for every N ∈ Λ(G), i.e., N is almost connected for every N ∈ Λ(G). The above lemma can be generalized to almost connected pseudocompact abelian groups as follows. Theorem 5.8. Let G be a pseudocompact abelian group. Then the following conditions are equivalent: (a) G is almost connected; (b) there exists N ∈ Λ(G) such that N is almost connected; (c) every N ∈ Λ(G) is almost connected; (d) nG ∈ Λ(G) for every n ∈ N+; 22 D. Dikranjan, A. Giordano Bruno and C. Milan (e) there exists a Gδ-dense subgroup H of G such that H is almost con- nected; (f) every Gδ-dense subgroup H of G is almost connected. Proof. (a)⇒(c) Let N ∈ Λ(G) and let c(N) be the connected component of N. Then c(N) ⊆ c(G) so in particular c(N) ∈ Λ(c(G)) by Lemma 5.7 applied to c(G). Hence c(N) ∈ Λ(G) by Corollary 2.6 (d). (c)⇒(b) Is obvious. (b)⇒(a) If N ∈ Λ(G) is almost connected, then c(N) ∈ Λ(G). Since c(G) is closed in G and c(N) ⊆ c(G), Corollary 2.6 implies c(G) ∈ Λ(G). (a)⇒(d) If G is almost connected, then by Corollary 5.5 c(G) = ⋂∞ n=1 nG ∈ Λ(G). Hence c(G) ⊆ nG for every n ∈ N+ and consequently nG ∈ Λ(G) by Corollary 2.6 (c). (d)⇒(a) If nG ∈ Λ(G) for every n ∈ N+, then ⋂∞ n=1 nG ∈ Λ(G). The compactness of G̃ implies c(G̃) = ⋂∞ n=1 nG̃. Therefore c(G̃) ∩G = ∞⋂ n=1 nG̃∩G = ∞⋂ n=1 nG ∈ Λ(G) and so by Corollary 2.9 one has that c(G̃) ∈ Λ(G̃), i.e., G̃ is almost connected. Lemma 5.2 applies to conclude that G is almost connected. (a)⇒(f) Assume that G is almost connected and let H be a Gδ-dense sub- group of G. Since c(G) ∈ Λ(G), it follows that c(G) ∩H is Gδ-dense in c(G). Then c(G)∩H is connected and so c(G)∩H ⊆ c(H). Since c(G)∩H ∈ Λ(H), Corollary 2.6 applies to conclude that c(H) ∈ Λ(H). (f)⇒(e) Is obvious. (e)⇒(a) Suppose that H is an almost connected Gδ-dense subgroup of G. Then H is pseudocompact and c(H) ∈ Λ(H). Since c(H) ⊆ c(G) ∩ H and c(G) ∩H is closed in H, it follows from Corollary 2.6 that c(G) ∩H ∈ Λ(H). Then Corollary 2.9 implies that c(G) ∈ Λ(G), i.e., G is almost connected. � Theorem 5.8 can be applied to show that the same conclusion of Proposition 5.6 is true if we suppose that there exists N ∈ Λ(G) such that N is totally disconnected. Corollary 5.9. Let G be an almost connected pseudocompact abelian group. If there exists N ∈ Λ(G) such that N is totally disconnected, then G is metrizable. Proof. Since G is almost connected, Theorem 5.8 (d) implies that also N ∈ Λ(G) is almost connected. On the other hand, N is totally disconnected, hence metrizable by Proposition 5.6. Then also the group G is metrizable. � In the next corollary we give two opposite properties of the almost connected groups (as far as the similarity with connected groups is concerned). The first one shows that almost connectedness cannot be destroyed by taking direct products with a compact metrizable group, while the second one is a stability property usually possessed by connected groups. Weakly metrizable pseudocompact groups 23 Corollary 5.10. Let G be a pseudocompact abelian group. (a) For every compact metrizable group M, the product G × M is almost connected if and only if G is almost connected. (b) Let N be a closed subgroup of G: (b1) if G is almost connected, then also G/N is almost connected; (b2) if both N and G/N are almost connected, then also G is almost connected. The next lemma and its corollary are used in the proof of Theorem 5.20 to produce Gδ-dense subgroups of connected pseudocompact groups. Lemma 5.11. Let G be a precompact connected group. Then mD is dense in G for every m ∈ N+ and for every dense subgroup D of G. In particular, mG is dense in G for every m ∈ N+. Proof. Since G̃ is connected and compact, G̃ is divisible. Then mG̃ = G̃ for every m ∈ N+. Since D is dense in G, mD is dense in mG. Analogously, the density of G in G̃, implies that mG is dense in mG̃ = G̃. Then also mD is dense in G̃. As mD ⊆ G one has that mD is dense in G. The last assertion follows from the density of G in G̃. � Corollary 5.12. Let G be a connected pseudocompact abelian group. Then mA is Gδ-dense in G for every m ∈ N+ and for every Gδ-dense subgroup A of G. Theorem 5.13. Let G be an almost connected pseudocompact abelian group of uncountable weight α. Then: (a) r0(c(G)) = r0(G) ≥ c and α ≤ 2 r0(G); (b) |c(G)| = |G| ≤ 2α ≤ 22 r0(G) ; (c) m(c(G)) = m(G) ≤ r0(G). Proof. (a) Being almost connected and non-metrizable, the group G is not singular (cf. Proposition 5.6), hence r0(c(G)) = r0(G) ≥ c by Lemma 4.3. To prove α ≤ 2r0(G) fix a free subgroup F of size r0(G) of G. Then w(F) = w(F) ≤ 2r0(G). Since G/F is torsion, the pseudocompact group G/F is torsion, hence bounded by Theorem 2.14. So there exists n > 0 such that nG ⊆ F . So w(nG) ≤ w(F) ≤ 2r0(G). Since G is almost connected, w(nG) = w(G) = α by (d) of Theorem 5.8. This proves α ≤ 2r0(G). (b) If |G| = c, then by the first part of (a) c ≤ r0(c(G)) = r0(G) ≤ |G| ≤ c, hence |c(G)| = |G| = c. Suppose now that |G| > c. Since c(G) ∈ Λ(G), the quotient G/c(G) is compact and metrizable (cf. Remark 5.1) and so |G/c(G)| ≤ c. Now |G| = |G/c(G)| · |c(G)| implies |c(G)| = |G|. (c) The equality m(c(G)) = m(G) follows from (c) of Lemma 2.15 as G is almost connected. Moreover, w(c(G)) = w(G) > ω and |G/c(G)| ≤ c according by Remark 5.1 and r0(c(G)) ≥ c by (a). Let M be a maximal independent subset of c(G) and let A = 〈M〉. Then the quotient c(G)/A is torsion, so setting An = {x ∈ c(G) : nx ∈ A} for every 24 D. Dikranjan, A. Giordano Bruno and C. Milan n ∈ N+, one has c(G) = ⋃∞ n=1 An. By Corollary 2.16 there exist n ∈ N+ and E ≤ c(G), with |E| ≤ c, such that An + E is Gδ-dense in c(G). Then by Corollary 5.12 also n(An + E) is Gδ-dense in c(G). So for F := nE one has |F | ≤ c and n(An + E) ⊆ nAn + nE ⊆ A + F ⊆ c(G). Thus A+F is Gδ-dense in c(G). By Lemma 2.15 (b) there exists a subgroup D of G with |D| ≤ c such that L = A + F + D is Gδ-dense in G. Since |F + D| ≤ c ≤ |A|, we have |L| ≤ |A| = r0(c(G)) = r0(G). Hence m(G) ≤ r0(G). � The next theorem shows the relation between d-extremality and almost con- nectedness. Theorem 5.14. Every d-extremal pseudocompact abelian group is almost con- nected. In particular, weakly extremal pseudocompact abelian groups are almost connected. Proof. Aiming for a contradiction, assume that G is not almost connected. Then by Theorem 5.8 (d), there exists n ∈ N+ such that nG 6∈ Λ(G), i.e., nG is not a Gδ-set. Hence w(G/nG) > ω and G is not d-extremal by Lemma 4.5, a contradiction. � Let us see now that for compact groups the above implication can be in- verted. Theorem 5.15. For a compact abelian group G the following are equivalent: (a) G is almost connected; (b) G is d-extremal; (c) G is strongly d-extremal. Proof. The implication (b) ⇒ (a) follows from Theorem 5.14. Conversely, if G is almost connected, then c(G) ∈ Λ(G) is compact and connected, hence divisible. In particular, c(G) is d-extremal and consequently G is d-extremal too by Lemma 3.3 (a). By the equivalence of (a) and (b) and by Theorem 5.8, (b) ⇒ (c). � Remark 5.16. Let us note here that one cannot invert the implication d- extremal⇒almost connected even for connected pseudocompact abelian groups. By [21] there exists a dense pseudocompact subgroup G of Tc of size c. Note that |G| = w(G) = c. Since r0(T c) = 2c > c, there exists an infinite cyclic subgroup C of Tc such that C ∩G = {0}. Then the pseudocompact subgroup G1 = G + C of T c is connected (as Tc is connected) and G is Gδ-dense in G1. Since G1/G ∼= C is not divisible, G1 is not d-extremal. In contrast with Remark 5.16 the following corollary shows that for totally disconnected pseudocompact abelian groups d-extremality and almost connect- edness are equivalent. Let us recall that Questions (A) and (B) of the introduc- tion have positive answer in the case of zero-dimensional groups [11, Theorem 7.3] and in the case of totally disconnected groups [3]. The following corollary generalizing these results covers Theorem B from the Introduction. Weakly metrizable pseudocompact groups 25 Corollary 5.17. Let G be a pseudocompact abelian group that is either singular or totally disconnected. Then the following conditions are equivalent: (a) G is almost connected; (b) G is d-extremal; (c) G is metrizable. Proof. (c)⇒(b) is obvious, (b)⇒(a) follows from Theorem 5.14 and (a)⇒(c) follows from Proposition 5.6. � In particular one has: Corollary 5.18. A torsion pseudocompact abelian group G is weakly extremal iff G is metrizable. Corollary 5.19. Let G be a pseudocompact abelian group with w(G/c(G)) > ω. Then G is not d-extremal. Proof. If G were d-extremal, then by Theorem 5.14 G would be almost con- nected, i.e., c(G) ∈ Λ(G) and consequently w(G/c(G)) = ω. � The corollary applies also to pseudocompact abelian groups G with w(c(G))< w(G) (since then w(G/c(G)) > ω). Theorem 5.20. Let G be an almost connected pseudocompact abelian group. Then c(G) is divisible if and only if c(G) is d-extremal (so also G is d-extremal). Proof. Since G is almost connected the subgroup c(G) is pseudocompact by Remark 5.1 (b). If c(G) is divisible, then obviously c(G) is d-extremal. Assume now that c(G) is not divisible. Then there exists n0 ∈ N+ such that n0c(G) is a proper subgroup of c(G). By Corollary 5.12 n0c(G) is also Gδ- dense in c(G). Since the quotient c(G)/n0c(G) is bounded torsion, it cannot be divisible and so c(G) is not d-extremal. � There is a similar result for strong extremality: for a strongly extremal (strongly d-extremal) pseudocompact abelian group G, c(G) is divisible if and only if c(G) is strongly extremal (strongly d-extremal). Remark 5.21. For every s-extremal (resp. weakly extremal, d-extremal) pseu- docompact abelian group G the subgroup c(G) is divisible if and only if it is s-extremal (resp. weakly extremal, d-extremal). Indeed, by Theorem 5.20, if c(G) is s-extremal, then c(G) is divisible. Vice versa, assume that c(G) is di- visible. Note that, c(G) ∈ Λ(G) since G is almost connected by Theorem 5.14. Now Lemma 3.1 (b) applies. The case when c(G) is weakly extremal (resp., d-extremal) is analogous. Corollary 5.22. A d-extremal pseudocompact abelian group G is divisible if and only if G is connected. Proof. If G is divisible and pseudocompact, then G is connected too [33]. If G = c(G) is connected, Theorem 5.20 applies. � 26 D. Dikranjan, A. Giordano Bruno and C. Milan 6. Various characterizations of extremality 6.1. The closure of the graph of a homomorphism. If G and H are abelian topological groups and h : G → H is a homomorphism, the graph Γh of h is the subset {(x,h(x)) : x ∈ G} of G × H. Then Γh is a subgroup of G×H such that (1) if p1 : G×H → G is the canonical projection on the first component, then p1(Γh) = G; (2) Γh ∩ ({0}×H) = {(0,0)}. Consequently G×H = Γh ⊕ ({0}×H). Let Vh be the vertical component of Γh in G×H, that is Vh = Γh∩({0}×H). Then Γh splits also as Γh = Γh⊕Vh. Since Vh is a subgroup of {0}×H, it is possible to identify Vh with a closed subgroup of H. The fact that Vh is a subgroup follows also from the next lemma. Lemma 6.1. Let G and H be topological abelian groups and h : G → H be a homomorphism. Then Vh = {t ∈ H : ∃ a net {xα}α∈A ⊆ G such that xα → 0 and h(xα) → t}. Proof. If t ∈ H, then (0, t) ∈ Γh ∩ ({0}× H) if and only if there exists a net {(xα,h(xα))}α∈A in Γh such that 0 = lim xα and t = limh(xα). � Remark 6.2. (a) If G and H are abelian topological groups and h : G → H is a homomorphism, then (1) Γh is closed in G×H if and only if Vh = {0}. (2) Γh is dense in G×H if and only if Vh = H. Hence it follows that if H is compact, h is not continuous and Γh is not dense in G×H, then Vh is a proper closed subgroup of H. (b) If G is a pseudocompact abelian group, H is a compact subgroup of T and h : G → H is a non-continuous and surjective homomorphism such that Γh is not dense in G × H, then by (a) there exists n ∈ N, n > 1, such that Vh = Z(n). (c) One can easily prove that in the above lemma Vh = ⋂ {h(U) : U ∈ VG(0)}. For H = T the latter subgroup was introduced also in [3, Notation 3.3]. Lemma 6.3. Let G be an abelian topological group and let H be a compact abelian group. Let h : G → H be a homomorphism. Then for every k ∈ N: (a) Vkh = kVh; (b) kh is continuous if and only if kVh = 0. Proof. (a) First we prove that kVh ⊆ Vkh. Let t ∈ Vh. Then by Lemma 6.1 there exists a net {xα}α∈A in G such that 0 = lim xα and t = limh(xα). Hence kt = limk(h(xα)) = lim(kh)(xα) and consequently kt ∈ Vkh by Lemma 6.1. To prove the converse inclusion consider t ∈ Vkh. In particular, t ∈ H and by Lemma 6.1 there exists a net {xα}α∈A in G such that 0 = limxα and t = lim(kh)(xα). Thus {h(xα)}α∈A is a net in the compact group H, hence Weakly metrizable pseudocompact groups 27 there exists a subnet {h(xαβ )}β∈B of {h(xα)}α∈A which converges to s ∈ H. Since {xαβ}β∈B converges to 0, by Lemma 6.1 s ∈ Vh. Moreover it follows that {kh(xαβ )}β∈B converges to ks and so for the uniqueness of limits t = ks ∈ kVh. (b) It follows directly from (a) that Γkh = Γkh ⊕kVh for every k ∈ N. By the closed graph theorem kh is continuous if and only if Γkh is closed, i.e., kVh = 0. � Corollary 6.4. Let G be a topological abelian group and for n ∈ N+ let h : G → Z(n) be a homomorphism. Then the following conditions are equivalent: (a) Γh is dense in G×Z(n); (b) the homomorphism kh : G → Z(n) is not continuous for every integer k such that 0 < k < n; (c) kVh 6= {0} for every integer k such that 0 < k < n. Proof. (a)⇒(b) If Γh is dense in G× Z(n) then Vh = Z(n). Applying Lemma 6.3 we conclude that the homomorphism kh is not continuous for every k ∈ N+ such that k < n. (b)⇔(c) Follows directly from Lemma 6.3 (b). (c)⇒(a) By hypothesis Vh = Z(n), hence Remark 6.2 (2) applies. � Lemma 6.5. Let G be a topological abelian group and let H be a closed subgroup of T. Let h : G → H be a surjective homomorphism such that (G,τh) is pseudocompact and Vh = Z(n) for some n ∈ N+. Let N = kernh. Then: (a) N = h−1(Vh); (b) Γh|N is Gδ-dense in N ×Vh. (c) kerh is a proper Gδ-dense subgroup of N. Proof. (a) Let x ∈ N = kernh. Then (nh)(x) = 0, that is n(h(x)) = 0. It follows that h(x) ∈ Z(n) = Vh. This proves that h(N) ⊆ Vh, that is N ⊆ h−1(Vh). To prove the opposite inclusion, take y ∈ Vh. Observe that Vh ⊆ H = h(G). Then there exists x ∈ G such that h(x) = y. As y ∈ Vh and nVh = {0}, in particular ny = 0. Consequently 0 = ny = n(h(x)) = (nh)(x) and hence x ∈ N, that is y ∈ h(N). This proves that h−1(Vh) ⊆ N. (b) By (a) h|N : N → Vh = Z(n) is a surjective homomorphism. It fol- lows from Lemma 6.3 that the homomorphism kh is not continuous for every k ∈ N such that 0 < k < n. Corollary 6.4 implies that Γh|N is dense in N × Vh. Since(G,τh) is pseudocompact by hypothesis and nh is continuous, N ∈ Λ(G,τh) and (N,τh|N) = (N,(τ|N )h|N ) is pseudocompact. Moreover, (N,(τh|N)h|N ) is homeomorphic to Γh|N and then also Γh|N is pseudocompact. Hence Γh|N is Gδ-dense in N ×Vh. (c) It follows from (b) that Γh|N is Gδ-dense in N × Vh. Then by Lemma 3.7 kerh|N = kerh is proper and Gδ-dense in N. � Remark 6.6. Let (G,τ) be a topological abelian group and h : G → T be a homomorphism such that H = h(G) is a closed subgroup of T. (a) If Vh = Z(n) for some n ∈ N+, then Lemma 6.3 implies 〈h〉∩ Ĝ = 〈nh〉. 28 D. Dikranjan, A. Giordano Bruno and C. Milan (b) Combining appropriately (a) and the results of this section, it is possible to prove the following theorem, announced in [1, Theorem 5.10] and [23, Lemma 3.6] and proved in [3, Theorem 3.10]. If (G,τ) is pseudocompact and h is not τ-continuous, then: (i) if Ĝ∩〈h〉 = {0}, then (G,τh) is pseudocompact if and only if h(G) is closed in T and kerh is Gδ-dense in (G,τ); (ii) if Ĝ ∩〈h〉 = 〈nh〉 for some n ∈ N+, then (G,τh) is pseudocompact if and only if h(G) is closed in T and kerh is Gδ-dense in kernh. Since we are not going to use here this property, the interested reader is invited to consult [3] for a detailed proof. 6.2. Applications to r-extremality. Now we are able to prove several im- portant results on the relations among different kinds of extremality. Theorem 6.7. Let G be a pseudocompact abelian group. Then the following conditions are equivalent: (a) G is metrizable; (b) every N ∈ Λ(G) is s-extremal; (c) G is s-extremal and strongly d-extremal. Proof. (a)⇒(b) is obvious, (b) ⇒ (a) is proved in [3]. (b)⇒(c) is obvious. (c)⇒(b) Let N ∈ Λ(G). Since G is weakly extremal, G is almost con- nected by Theorem 5.14, hence c(N) ∈ Λ(G). Therefore, c(N) is a connected, d-extremal pseudocompact abelian group and consequently it is divisible by Corollary 5.22. Thus c(N) is s-extremal by Lemma 3.1 (b). Since c(N) ∈ Λ(N) by Theorem 5.8, N is s-extremal too according to Lemma 3.3 (a). � Here we see that the stronger version of weak extremality, imposed on all N ∈ Λ(G), gives strong extremality. Theorem 6.8. A pseudocompact abelian group G is strongly extremal if and only if every N ∈ Λ(G) is weakly extremal. Proof. If G is strongly extremal, then by Theorem 3.12 every N ∈ Λ(G) is weakly extremal. To prove the converse implication, it suffices to show that if every N ∈ Λ(G) is weakly extremal, then G is r-extremal. Indeed, if this is true, then every N ∈ Λ(G) is r-extremal (cf. Remark 3.5). Suppose for a contradiction that G is not r-extremal. Then there exists a pseudocompact group topology τ′ on G such that τ′ > τ. Since τ and τ′ are, in particular, precompact group topologies, there exists a homomorphism h ∈ (̂G,τ′) \ (̂G,τ), i.e., h : G → T is τ′-continuous but not τ-continuous. Let H = h(G). Then H is a compact subgroup of T, as the image of the pseudocompact group (G,τ′) under the τ′-continuous homomorphism h. Since h is not τ-continuous, Γh is not closed in (G,τ) ×H. There are two cases. Weakly metrizable pseudocompact groups 29 Case 1. Suppose that Γh is dense in (G,τ) ×H. Since h is τ ′-continuous, τh ≤ τ ′ and so (G,τh) is pseudocompact (as (G,τ ′) is pseudocompact by hy- pothesis). Since Γh is homeomorphic to (G,τh), also Γh is pseudocompact. Then Γh is Gδ-dense in (G,τ) ×H and hence kerh is a Gδ-dense subgroup of G by Lemma 3.7. Moreover, the quotient G/ kerh is algebraically isomorphic to H. Being a compact subgroup of T, the group H is all T or it is finite. If H = T then r0(H) = r0(T) = c and so r0(G/ kerh) = r0(H) = c. If H is finite, then also G/ kerh is finite so in particular it cannot be divisible. In both cases G is not weakly extremal, against the hypothesis. Case 2. If Γh is not dense in (G,τ) × H, then by Remark 6.2 (b) there exists n ∈ N+, n > 1, such that Vh = Z(n). Then nh is τ-continuous by Lemma 6.3 and so N = kernh ∈ Λ(G). As observed in Case 1, (G,τh) is pseudocompact, hence kerh is a proper Gδ-dense subgroup of N by Lemma 6.5. Since N/ kerh ∼= Z(n) it cannot be divisible, i.e. N is not weakly extremal, a contradiction. � Combining with Theorem 4.11 we obtain: Corollary 6.9. Let G be a c-extremal, strongly d-extremal pseudocompact abelian group. Then G is strongly extremal. Now we can prove Theorem C from the Introduction. We use essentially some ideas from the proof of Theorem 4.4 (b) of [3]. Theorem 6.10. Let G be a s-extremal pseudocompact abelian group. Then G is doubly extremal if and only if G[p] = G̃[p] for every p ∈ P. Proof. Assume that G[p] = G̃[p] for every p ∈ P and suppose for a contradiction that (G,τ) is not r-extremal. Then there exist a pseudocompact group topology τ′ on G such that τ′ > τ and a homomorphism h : G → T which is τ′-continuous but not τ-continuous (cf. the proof of Theorem 6.8). Take H = h(G); then H is a compact subgroup of T as h is τ′-continuous. Since h is not τ-continuous, Γh is not closed in (G,τ) ×H. There are two cases: Case 1. Suppose that Γh is dense in (G,τ) ×H. Since h is τ ′-continuous, τh ≤ τ ′ and so (G,τh) is pseudocompact. Since Γh is homeomorphic to (G,τh), also Γh is pseudocompact, hence Gδ-dense in (G,τ) × H. Now Theorem 3.8 applies to conclude that G is not s-extremal, a contradiction. Case 2. If Γh is not dense in (G,τ) × H, then by Remark 6.2 (b) there exists n ∈ N+, n > 1, such that Vh = Z(n). It follows from Lemma 6.3 that nh is continuous. Let p ∈ P such that p|n. Take m = n p ∈ N+; we have that m < n. Define h1 := mh. Since m < n the homomorphism h1 is not τ-continuous. On the other hand, h1 is τ ′-continuous (as h is τ′-continuous) and ph1 is τ-continuous as ph1 = pmh = nh. Moreover Vh1 = Vmh = mVh = mZ(n) = Z(p). 30 D. Dikranjan, A. Giordano Bruno and C. Milan Consider now f = ph1 : (G,τ) → pH ≤ H. Since f = ph1 = nh, one has f ∈ (̂G,τ). Note that f|G[p] = 0 and consider a continuous extension f̃ of f to G̃ (that is, the continuous homomorphism f̃ : G̃ → T such that f̃|G = f). Since f̃|G[p] = f|G[p] = 0 and f̃ is continuous, we have that f̃|G[p] = 0. By hypothesis G[p] = G̃[p], hence f̃| G̃[p] = 0, i.e., f̃ ∈ A(G̃[p]). Since A(G̃[p]) = p ̂̃ G, there exists a continuous homomorphism g̃ : G̃ → T such that f̃ = pg̃. If we take g := g̃|G, then f = pg. Since f = ph1 by definition, we have p(h1 − g) = 0 and so t := h1 − g is a torsion element of Hom(G,T) of order p. Clearly, as h1 is not τ-continuous, also t is not τ-continuous. Then t : G → Z(p) is surjective and Γt is not closed. Since by Remark 6.2 (1) Vt is a non trivial subgroup of Z(p), it follows that Vt = Z(p). Note that (G,τt) is a pseudocompact group, since t is τ ′- continuous and so τ′ ≥ τt. We can now apply Lemma 6.5 and conclude that kert is a proper Gδ-dense subgroup of kerpt = G. This proves that G is not s-extremal, a contradiction. Hence G is r-extremal, and consequently doubly extremal. The converse implication follows from Theorem 1.2 (b) since G is s-extremal. � For a compact abelian group G the dimension of G coincides with the free rank r0(Ĝ) of the group Ĝ of characters of G. In what follows it will be denoted by dimG. Definition 6.1. Let G be an almost connected pseudocompact abelian group. For N ∈ Λ(G) let codimN = dimG/N, and Λn(G) = {N ∈ Λ(G) : codimN = n}. Let us briefly list some properties of this new invariant: (a) Λ0(G) consists precisely of those N ∈ Λ(G) that contain c(G) (first note that if c(G) ∈ Λ(G), then for N ≥ c(G) the quotient G/N is also a quotient of the compact totally disconnected group G/c(G), so G/N is zero-dimensional); (b) If N1,N2 ∈ Λn and N1 ≤ N2, then N2/N1 is a compact totally discon- nected metrizable group; (c) Let Λ̃1(G) = {N ∈ Λ(G) : G/N ∼= T}. Clearly, Λ̃1(G) ⊆ Λ1(G) as dim T = 1. By (b), if N1,N2 ∈ Λ̃1(G) and N1 ≤ N2, then N2/N1 is finite cyclic (as the only compact totally disconnected subgroups of T are the finite ones). The next theorem will make use of the following condition: (d̃) every N ∈ Λ̃1(G) is d-extremal. Note that (d̃) is obviously weaker than strong d-extremality, but implies d- extremality by Lemma 3.3 (a). We prove now Theorem D which shows that Weakly metrizable pseudocompact groups 31 (d̃) in conjunction with weak extremality is equivalent to r-extremality for connected pseudocompact abelian groups. Theorem 6.11. Let G be a connected pseudocompact abelian group. Then G is r-extremal if and only if G is weakly extremal and (d̃) holds. Proof. Since r-extremality implies weak extremality, it suffices to assume that G is weakly extremal and prove that G is r-extremal if and only if (d̃) holds true. We shall prove that their negations are equivalent. Since G is weakly extremal and connected, we conclude by Corollary 5.22 that G is divisible. Denote by τ the topology of G and assume that (G,τ) is not r-extremal. Then there exists a discontinuous character h : (G,τ) → T such that (G,τh) is pseudocompact. Since h : (G,τh) → T is continuous and G is divisible, the image h(G) must be a non-trivial divisible compact subgroup of T (by the pseudocompactness of τh). Hence h is surjective. According to Remark 6.6 (b) one has the two alternatives (i) and (ii). In (i) the subgroup H = kerh of G is Gδ-dense and G/H ∼= T has free rank c. This contradicts weak extremality of G. Hence we are left with (ii); i.e., there exists n > 0 such that nh is τ-continuous and H is Gδ-dense in N = kernh. Since obviously N ∈ Λ̃1(G) and since N/H is bounded torsion, we conclude that N is not d-extremal, i.e. (d̃) fails. Now suppose that some N ∈ Λ̃1(G) is not d-extremal. Then there exists a Gδ-dense subgroup H of N such that N/H is not divisible. It is not restrictive to assume that N/H ∼= Z(p) for some prime p. To prove that G is not r- extremal we shall find a surjective, discontinuous character h : (G,τ) → T such that kerph = N and kerh = H. Then τh will be pseudocompact by Remark 6.6 and so (G,τ) will be not r-extremal. To build such an h we fix first a continuous surjective homomorphism l : G → T with ker l = N witnessing N ∈ Λ̃1(G). Let π : G → G/H be the canonical homomorphism. Let N′ = π(N). Then N′ ∼= Z(p). Our first aim is to prove now that G/H ∼= T. For simplicity write X = G/H and recall that X is divisible as a quotient of G. Hence the homomorphism ϕp : X → X is surjective and obviously N′ ≤ kerϕp. Hence ϕp factorizes as ϕp = γ ◦ λ, where λ : X → X/N ′ is the canonical homomorphism. As X/N′ ∼= G/N ∼= T and γ : X/N′ → X is surjective, we conclude that X is isomorphic to a quotient of T. Moreover, the homomorphism γ has kernel K = λ(X[p]), hence pK = 0. Since X/N′ ∼= T, this means that K is finite (cyclic), hence X ∼= (X/N′)/K ∼= T. Fix an isomorphism ξ : T → X and consider the composition s = λ ◦ ξ : T → T. Then kers = Z(p). Let us split T = Z(p∞) ⊕ T and note that s(Z(p∞)) = Z(p∞). Since End(Z(p∞)) = Zp is a discrete valuation domain, we can write s|Z(p∞) = pη1, where η1 is an automorphism of Z(p ∞). As T has no elements of order p, the restriction s|T : T → s(T) (1) is an isomorphism and therefore s(T) ∩ Z(p∞) = {0}. Then we can find an isomorphism η2 : T → s(T) such that s|T = pη2 (note that ϕp : s(T) → s(T) is an isomorphism as s(T) is divisible). Now let η : T → T be the isomorphism 32 D. Dikranjan, A. Giordano Bruno and C. Milan defined by η = η1 +η2. So, setting h : G → T to be the composition η◦ξ −1◦π, we can claim that ph = l is continuous and kerph = N. Since kerh = H, this ends up the proof. � 7. The cardinal invariants of extremal pseudocompact groups Comfort and van Mill [6, Theorem 6.1] proved that |t(G)| ≤ c for connected s-extremal pseudocompact abelian groups (here t(G) denotes the subgroup of torsion elements of G). Later Comfort, Gladdines and van Mill [4] proved that r0(G) ≤ c for an s-extremal pseudocompact abelian group G (given as Theorem 3.6 here). This allowed them to prove that s-extremal pseudocompact abelian groups have size ≤ c [4, Theorem 4.8]. The next theorem extends these results to all weakly extremal pseudocompact abelian groups (as s-extremal groups G obviously satisfy m(G) = |G|). Theorem 7.1. Let G be an infinite almost connected c-extremal pseudocompact abelian group. Then |G| ≤ c if and only if m(G) = |G|. Proof. Since m(G) ≥ c, obviously |G| ≤ c implies m(G) = |G|. Now assume m(G) = |G| holds. By Theorem 3.6 r0(G) ≤ c. If r0(G) < c, then G is bounded torsion according to Theorem 2.14. In particular, G is singular. By Proposition 5.6 G is metrizable, so |G| ≤ c. If r0(G) = c (so in particular G is not torsion), it is sufficient to apply Theorem 5.13 (c). � Corollary 7.2. Let G be a weakly extremal pseudocompact abelian group. Then |G| = c if and only if m(G) = |G|. In particular, r0(G) = |G| = c holds true for every non-torsion weakly extremal pseudocompact abelian group G with m(G) = |G|. An elegant application of the inequality |G| ≤ c for s-extremal groups was found by Comfort and Galindo [3, Theorem 6.1] (announced in [23, Lemma 8.1] and [1, Theorem 5.16 (b)]), namely (b) ⇒ (a) of Theorem 6.7 (which shows that for pseudocompact abelian groups the strong s-extremality is equivalent to metrizability). In the sequel we shall apply Theorem 7.1 to obtain further connections among the cardinal invariants of the extremal pseudocompact groups. Comfort and van Mill [6, Theorem 5.5] proved that a non-metrizable con- nected pseudocompact abelian group G with |G| ≥ w(G)ω is not s-extremal. As an immediate consequence of Theorem 7.1 we obtain the following stronger result. Corollary 7.3. An infinite weakly extremal pseudocompact abelian group G is metrizable if and only if m(G) = |G| ≥ w(G). Proof. By Theorems 7.1 and 2.14 we have |G| = c. Hence G is weakly extremal with w(G) ≤ c. Now Theorem 4.15 applies to conclude that G is metrizable. � Corollary 7.4. Let G be a weakly extremal pseudocompact abelian group. Then w(G) ≤ 2c. Weakly metrizable pseudocompact groups 33 Proof. If G is singular, then G is metrizable by Proposition 5.6, so w(G) = ω. If G is not singular, then w(G) > ω and r0(G) ≥ c by Lemma 4.3. On the other hand, r0(G) = c by Theorem 3.6, as G is c-extremal. Now Theorem 5.13 (a) applies as G is almost connected. � Let us recall that according to [17], a group G is hereditarily pseudocom- pact when every closed subgroup of G is pseudocompact. It was proved in [3] that hereditary pseudocompactness has a strong impact on the s-extremal pseudocompact abelian groups: (a) ([3, Theorem 6.9]) every finitely generated subgroup of G is metrizable. (b) ([3, Theorem 6.10], under the assumption of Lusin’s hypothesis) G itself is metrizable. Here we strengthen (a) by replacing “s-extremal” by “weakly extremal and m(G) = |G|” (Corollary 7.9). Moreover, we show that under this weaker hy- pothesis one can strengthen further (a) by replacing “finitely generated” by “countable”, if G is connected. This implies that hereditary pseudocompact- ness coincides with ω-boundedness for weakly extremal pseudocompact groups with m(G) = |G| (Corollary 7.11). We were not able to remove Lusin’s hy- pothesis in (b), but in Corollary 7.7 we give a stronger version of (b), with “G s-extremal” replaced by “G weakly extremal and m(G) = |G|”. Lemma 7.5. Let G be a c-extremal, hereditarily pseudocompact abelian group of size c. Then there exists m ∈ N+ such that mH is metrizable for every subgroup H of G with w(H) ≤ c. Proof. Let H be a subgroup of G with w(H) ≤ c. Since G is hereditarily pseudocompact, the closure L of H in G is a pseudocompact subgroup of G so L is c-extremal by Corollary 4.12. Since w(L) = w(H) ≤ c, L is singular by Theorem 4.13. Then there exists m > 0 such that mL (and so also mH) is metrizable. It remains to see that such an m can be chosen uniformly for all subgroups H ≤ G with w(H) ≤ c. Assume the contrary. Then for every m > 0 there exists a subgroup Hm of G such that w(Hm) ≤ c and mHm is not metrizable. Consider the subgroup H = ∑∞ m=1 Hm of G. To see that w(H) ≤ c it suffices to recall that H is precompact, so w(H) = |Ĥ|. If ρm : Ĥ → Ĥm is the restriction homomorphism, then the diagonal homomorphism ρ : Ĥ → ∏∞ m=1 Ĥm is injective. Since |Ĥm| = w(Hm) ≤ c, we conclude that |Ĥ| ≤ c. By the first part of the argument there exists m0 > 0 such that m0H is metrizable. Then m0Hm0 is metrizable as well, a contradiction. � Lemma 7.6. Let G be a connected pseudocompact abelian group of weight > c. Then there exists a subgroup H of G of size ≤ ω1 such that mH is not metrizable for any m > 0. Proof. Following the idea from the proof of [3, Theorem 6.10] one can construct by transfinite induction of length ω1 a subgroup H of G of size ≤ ω1 such that r0(Ĥ) ≥ ω1. (For every countable subgroup A of G the closure L of A has 34 D. Dikranjan, A. Giordano Bruno and C. Milan w(L) ≤ c, so G/L is a non-trivial connected pseudocompact abelian group. Hence there exists a non-torsion continuous character ξ : G/L → T, that produces a non-torsion continuous character χ : G → T such that χ|L = 0. The characters χα produced in this way, along with the elements xα ∈ G witnessing χα is non-torsion give rise to the subgroup H = 〈xα : α < ω1〉 of G that has character group Ĥ with r0(Ĥ) > ω1, witnessed by the independent family {χα|H}.) Let m > 0. Since m̂H = Ĥ/Ĥ[m] ∼= mĤ, it follows that w(mH) = |mĤ| > ω, i.e. mH is not metrizable. � Corollary 7.7. Under the assumption of the Lusin’s hypothesis every weakly extremal hereditarily pseudocompact abelian group G with m(G) = |G| is metriz- able. Proof. It is not restrictive to assume that G is infinite. According to Theorems 7.1 and 2.14 |G| = c, hence we can apply Lemma 7.5 to produce an m > 0 such that mH is metrizable for every subgroup H of G of weight ≤ c. Assume now that G is not metrizable. Then w(G) > c by Theorem 4.15. Since G is almost connected, w(c(G)) = w(G) > c according to Remark 5.1. Now by Lemma 7.6 we can find a subgroup H of c(G) with |H| ≤ ω1 and mH non-metrizable. This means that w(H) > c. Since w(H) ≤ 2ω1, we conclude that the Lusin’s hypothesis fails. � Corollary 7.8. Let G be a c-extremal, hereditarily pseudocompact abelian group of size c. Then there exists m ∈ N+ such that every countable subgroup of mG is metrizable. In particular, mG is ω-bounded. Proof. Let m ∈ N+ as in Lemma 7.5 and let D be a countable subgroup of mG. For every d ∈ D pick an element gd ∈ G such that d = mgd and let D1 be the subgroup of G generated by the subset X = {gd ∈ G : d ∈ D}. Then D1 is countable and w(D1) ≤ c so mD1 is metrizable by Lemma 7.5. Since D ⊆ mD1, D is metrizable as well. The last assertion follows from the fact that mG is hereditarily pseudocompact (as continuous image of G), hence the closure of D in mG is a metrizable pseudocompact (hence compact) subgroup of mG containing D. � The next corollary generalizes Theorem 6.9 from [3], where s-extremality of G is assumed instead of “c-extremal of size c”. Corollary 7.9. Let G be a c-extremal, hereditarily pseudocompact abelian group of size c. Then every finitely generated subgroup of G is metrizable. Proof. Let F be a finitely generated subgroup of G. By Corollary 7.8 there exists m > 0 such that the (finitely generated) subgroup mF of mG is metriz- able. Then F is metrizable since the quotient F/mF is finite (as every torsion finitely generated abelian group). � Corollary 7.10. Let G be a connected, weakly extremal, hereditarily pseu- docompact abelian group of size c. Then every countable subgroup of G is metrizable. In particular, G is ω-bounded. Weakly metrizable pseudocompact groups 35 Proof. Follows from Corollary 7.8 as G is divisible by Corollary 5.22. � This corollary implies that for weakly extremal connected pseudocompact group of size c hereditary pseudocompactness coincides with ω-boundedness. In the next corollary we isolate this fact for the smaller class of s-extremal groups. Corollary 7.11. A connected s-extremal pseudocompact abelian group is hered- itarily pseudocompact if and only if it is ω-bounded. 8. Further comments and open questions Even if the principal problems (A)-(C) (see Introduction) on metrizability of the extremal pseudocompact abelian groups have not been solved, the class of extremal pseudocompact abelian groups has been restricted to groups with specific properties. For the benefit of the reader we list below the most relevant ones: (1) |G| = r0(G) = c for every almost connected c-extremal pseudocompact abelian group G with m(G) = |G| (this includes all s-extremal groups); in case G is also hereditarily pseudocompact, then mG must be ω- bounded for some m > 0; (2) if G is weakly extremal and non-metrizable, then c < w(G/N) ≤ 2c for every closed pseudocompact subgroup N 6∈ Λ(G) of G; (3) if G is d-extremal, then G is almost connected (so non-totally discon- nected if G is not metrizable); moreover, if G is connected, then G is divisible; (4) if G is doubly extremal and non-metrizable, there exists a prime p such that G[p] is dense but not Gδ-dense in G̃[p]. The following conjecture, supported by positive evidence in two cases (see Theorem 4.13), plays a central role: it obviously implies (F) (and consequently (A)-(E)). Main Conjecture. Every c-extremal pseudocompact abelian group is singular. Since weakly extremal groups are c-extremal, the Main Conjecture implies that every weakly extremal pseudocompact abelian group G is singular. Note that this fact is equivalent, by Corollary 5.17, to Question (F), as weakly ex- tremal singular groups are metrizable according to Theorem 4.6. The Main Conjecture immediately gives “Yes” to the following question (suggested by Corollary 5.22) since connected groups are almost connected. Question 8.1. Let G be a connected c-extremal pseudocompact abelian group. Must G be divisible (or, equivalently, d-extremal)? Note that “Yes” to this question implies that the following holds true: For an s-extremal pseudocompact abelian group G, c(G) is s-extremal. The same holds true for r-extremality, for double extremality and for d-extremality. In- deed, the subgroup c(G) is c-extremal by Theorem 4.11, so would be divisible. 36 D. Dikranjan, A. Giordano Bruno and C. Milan Now Lemma 3.1 (b) implies that c(G) is s-extremal. The same argument holds for d-extremality. Along with the Main Conjecture one can consider also its restricted form: Restricted Main Conjecture. Every c-extremal pseudocompact abelian group of size c is singular. By Theorem 7.1 the Restricted Main Conjecture implies that every weakly extremal pseudocompact group G with m(G) = |G| is metrizable. So the Restricted Main Conjecture implies a positive answer to Questions (A), (C) and (D). Note that neither the Restricted Main Conjecture nor (F) follow from a positive answer to all questions (A)–(E), since a singular pseudocompact abelian group of size c need not be weakly extremal (nor metrizable). The following weaker form of Question (E) (as s-extremal groups satisfy m(G) = |G| and has size c) is open too: Question 8.2. Does every r-extremal pseudocompact abelian group G satisfy m(G) = |G|? Observe that a positive answer to Question 8.2 implies a positive answer to the next question by Theorem 7.1: Question 8.3. Is every infinite r-extremal pseudocompact abelian group of size c? On the other hand, a positive answer to Question 8.3, along with the Re- stricted Main Conjecture, implies a positive answer to (A)-(E). One can expect that connected weakly extremal pseudocompact abelian groups are r-extremal. A positive answer to this conjecture implies positive answer to (D) in the case of connected groups. In terms of Theorem 6.10, for the latter implication it suffices to check that G[p] = G̃[p] for every prime p. Question 8.4. Does weak extremality coincide with the disjunction of s-extre- mality and r-extremality? If “Yes”, then a positive answer to Question 8.3, along with the Restricted Main Conjecture, yields a positive answer to (F). Note Added December 2005. Recently Comfort and van Mill [8] obtained the following impressive result: Theorem 8.5. A pseudocompact abelian group is s-extremal or r-extremal if and only if it is metrizable. Clearly, this theorem solves the principal open problems related to extremal- ity: (A)–(E) and 8.2, 8.3, 8.4. Nevertheless, the Restricted Main Conjecture as well as questions 8.1 and (F) are left open by Theorem 8.5. Let us see now that an appropriate modification of the argument from [8] can prove our Main Conjecture, hence all remainig questions formulated in our paper. Weakly metrizable pseudocompact groups 37 Let G be a c-extremal pseudocompact abelian group. Theorem 3.6 yields r0(G) ≤ c. By Theorem 2.14 either G is bounded torsion or r0(G) = c. In the former case G is singular. Let r0(G) = c and assume that G is not singular. Let D = ⊕ S Q, with |S| = c, be the divisible hull of the torsion-free quotient G/t(G) and let π : G → D be the composition of the canonical projection G → G/t(G) and the inclusion G/t(G) →֒ D. For a subset A of S let G(A) = π−1( ⊕ A Q) and A = {A ⊆ S : G(A) contains a subgroup N ∈ Λ(G)}. Then A has the countable intersection property and |A| = c for all A ∈ A as r0(N) = c for every N ∈ Λ(G) by Lemma 4.3. By [8, Lemma 3.2] there exists a partition {Pn}n∈N of S such that |A ∩ Pn| = c for every A ∈ A and for every n ∈ N. Define Vn = G(P0 ∪ ·· ·∪Pn) for every n ∈ N and note that G = ⋃∞ n=0 Vn. By Lemma 2.15(a) there exist m ∈ N and N ∈ Λ(G) such that H = Vm ∩ N is Gδ-dense in N. By Theorem 4.11, to get a contradiction it suffices to show that r0(N/H) = c. Let F be a torsion-free subgroup of N such that F ∩H = {0} and maximal with this property. Suppose for a contradiction that |F | = r0(N/H) < c. So π(F) ⊆ ⊕ S1 Q for some S1 ⊆ S with |S1| < c and W = P0 ∪ ·· · ∪ Pm ∪ S1 has |W ∩ Pm+1| < c. Consequently W 6∈ A and so Λ(G) ∋ N 6⊆ G(W). Take x ∈ N \G(W). Since G/G(W) is torsion-free, 〈x〉∩ G(W) = {0}. But H + F ⊆ G(W) and so 〈x〉∩ (H + F) = {0}, that is (F + 〈x〉) ∩H = {0}, this contradicts the maximality of F . Acknowledgements. 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Received September 2004 Accepted January 2006 Dikran Dikranjan (dikranja@dimi.uniud.it) Dipartimento di Matematica e Informatica, Università di Udine, via delle Scienze, 206 - 33100 Udine, Italy Anna Giordano Bruno (giordano@dimi.uniud.it) Dipartimento di Matematica e Informatica, Università di Udine, via delle Scienze, 206 - 33100 Udine, Italy Chiara Milan (milan@dimi.uniud.it) Dipartimento di Matematica e Informatica, Università di Udine, via delle Scienze, 206 - 33100 Udine, Italy