Universit`adegli Studi di Udine Facolt`adi Scienze Matematiche, Fisiche e Naturali Corso di Dottorato di Ricerca in Matematica Ciclo XX Tesi di Dottorato di Ricerca Cardinal invariants of topological groups and applications to κ-pseudocompact subgroups Relatore: Dottoranda: Prof. Dikran Dikranjan Anna Giordano Bruno 14 febbraio 2008 ii Contents Introduction v 1 General results, notation and terminology 1 1.1 Limit cardinals . 1 1.2 Abelian groups . 2 1.2.1 Ranks . 3 1.2.2 Abelian p-groups . 5 1.3 Topological spaces . 7 1.4 Topological groups . 8 1.4.1 Pontryagin-van Kampen duality . 9 1.4.2 Pseudocompact and precompact groups . 11 1.5 General properties of compact groups . 12 1.5.1 Properties of compact Zp-modules . 14 1.5.2 Essential dense and totally dense subgroups . 15 2 κ-Pseudocompactness 21 2.1 Characterization . 22 2.2 The family Λκ(G)................................ 25 2.3 The Pκ-topology . 28 3 The divisible weight 33 3.1 The power of singularity . 33 3.2 The divisible weight . 35 3.3 w-Divisibility . 38 3.4 κ-Singularity . 40 3.4.1 Characterization and properties . 41 3.4.2 Measuring κ-singularity . 43 3.5 The stable weight . 44 3.6 Super-κ-singularity . 47 4 Projection onto products 51 4.1 The \local" case . 53 4.2 Projection onto w-divisible products . 55 4.3 Projection onto w-divisible powers . 59 iii iv CONTENTS 4.4 The non-abelian case . 63 5 The free rank of abelian pseudocompact groups 65 5.1 Measuring dense pseudocompact subgroups . 65 5.2 The case of w-divisible pseudocompact abelian groups . 68 5.3 The general case . 71 6 Extremality 73 6.1 First results . 74 6.2 Construction of Gκ-dense subgroups . 75 6.3 The dense graph theorem . 77 6.4 Torsion abelian groups and extremality . 80 6.5 Proof of the main theorem . 83 7 Dense compact-like subgroups 87 7.1 Totally dense κ-pseudocompact subgroups . 87 7.1.1 κ-Boundedness . 87 7.1.2 The property TDκ ........................... 89 7.1.3 Main theorems . 91 7.2 Essential dense κ-pseudocompact subgroups . 93 7.3 Either totally dense or essential dense subgroups . 96 7.4 Small essential pseudocompact subgroups . 97 7.4.1 CR-cardinals . 97 7.4.2 Measuring essential subgroups . 98 7.4.3 Independence results . 102 7.5 Open problems . 104 Bibliography 107 Index 112 List of Symbols 115 Introduction Pseudocompactness was introduced by Hewitt with the aim to weaken compactness in the spirit of Weierstraß theorem: a Tychonov topological space X is pseudocompact if ev- ery real valued continuous function of X is bounded [49]. Pseudocompactness coincides with compactness for metric spaces. Pseudocompact groups were characterized by Com- fort and Ross [18, Theorem 4.1] (see Theorem 2.7). Moreover pseudocompact groups are precompact, that is their completion is compact [18, Theorem 1.1]. All topological groups in this thesis are Hausdorff. A relevant problem involving pseudocompact groups is that of extremality, which was introduced and studied by Comfort and co-authors since 1982 [14, 20]. The following are the main two levels of extremality. Definition 1. [7, 16, 29] A pseudocompact group is: • s-extremal if it has no proper dense pseudocompact subgroup; • r-extremal if there exists no strictly finer pseudocompact group topology. It was immediately observed that every pseudocompact metrizable (so compact) group is s- and r-extremal. So arose the natural question of whether every pseudocom- pact group that is either s- or r-extremal is metrizable [14, 20]. This question, posed in 1982, turned out to be very difficult and many papers in the following twenty-five years proposed partial solutions [7, 9, 11, 12, 13, 14, 16, 20, 21, 22, 29, 43]. Recently Comfort and van Mill proved that the answer to this question is positive: Theorem A. [22, Theorem 1.1] For a pseudocompact abelian group G the following conditions are equivalent: (a) G is s-extremal; (b) G is r-extremal; (c) G is metrizable. Studying this topic, we introduced singular groups in [29, Definition 1.2]. Definition 2. A topological abelian group G is singular if there exists a positive integer m such that w(mG) ≤ !. v vi INTRODUCTION The condition given in [29, Definition 1.2] to define singular abelian groups G was (1) there exists a positive integer m such that G[m] is a Gδ-set of G, and it was given for pseudocompact abelian groups. Anyway for pseudocompact abelian groups these properties are equivalent and they are equivalent also to a third one, that is (2) G admits a closed torsion normal Gδ-subgroup. The condition in (2) can be given also for non-necessarily abelian topological groups. The equivalence of these three conditions for pseudocompact abelian groups is proved by Lemma 3.35 with κ = !. So singularity has various aspects and this is a reason for which it is useful in different topics, as we are going to describe. In [29] for example we proved Theorem A in the case when G is singular. Moreover we saw that singularity is a necessary condition for a pseudocompact abelian group to be either s- or r-extremal. Recently singular groups turned out to be useful in another case: in Section 3.1 we show that a counterexample for a recent conjecture which was in a preliminary version of [21] (see http://atlas-conferences.com/cgi-bin/abstract/cats-72) can be found by making use of singular groups. The form in (2) was already used in [32], where the problem of the characterization of compact groups admitting a proper dense subgroup with some compactness-like property was considered. We give a historical panoramic of the general problem and we focus our attention on the role of singular groups. We begin recalling some definitions. A subgroup H of a topological group G is strongly totally dense if H densely intersects every closed subgroup of G, and it is totally dense if H densely intersects every closed normal subgroup of G [63]. These two concepts coincide in the abelian case. The totally dense subgroups of a compact group K are precisely those dense subgroups of K that satisfy the open mapping theorem [30, 31, 48] (see Theorem 1.28), according to the \total minimality criterion" (see Theorem 1.50). The groups with this property were introduced in [30] under the name totally minimal: a topological group G is totally minimal if for every topological group H and for every continuous surjective homomorphism f : G ! H, f is open. A subgroup H of a topological group G is essential if H non-trivially intersects every non-trivial closed normal subgroup of G [59, 64]. A totally dense subgroup is necessarily dense and essential. A topological group G is minimal if there exists no strictly coarser group topology on G. A totally minimal group is minimal. A description of the dense minimal subgroups of compact groups was given in [59, 64] in terms of essential subgroups. According to the \minimality criterion" given in [31, 59, 64] (see Theorem 1.50) a dense subgroup H of a compact abelian group K is minimal if and only if H is essential in K. vii A particular case of the previous mentioned problem has been largely studied, that is the description of compact groups admitting proper totally dense subgroups with some other compactness-like property. To better explain this problem and the known results about it, we recall some definitions of compactness-like properties which did not appear until here: Definition 3. A Tychonov topological space X is: • !-bounded if every countable subset of X is contained in a compact subset of X; • countably compact if every countable open cover of X has a finite subcover; • strongly pseudocompact if X contains a dense countably compact subspace [2]. For topological groups we have the following chain of implications: compact ) !-bounded ) countably compact ) strongly pseudocompact ) pseudocompact ) precompact. It became clear that countable compactness and !-compactness have to be imme- diately ruled out, as no compact group can contain a proper strongly totally dense countably compact subgroup [32, Theorem 1.4] (see also [25] for stronger results). So one has to limit the compactness-like property within (strong) pseudocompactness. We study the problem of the existence of proper totally dense pseudocompact sub- groups of compact abelian groups. In view of our previous observation and of Comfort and Ross theorem about pseudocompact groups (see Theorem 2.7), this is equivalent to look for proper totally minimal Gδ-dense subgroups of compact abelian groups. This problem was studied for the first time by Comfort and Soundararajan [20], and they solved it in case K is a connected compact abelian group: the answer is if and only if K is non-metrizable. A topological group admitting some dense pseudocompact subgroup is necessarily pseudocompact (see Corollary 2.14 with κ = !). A necessary condition for a topological group G to have a strongly totally dense pseudocompact subgroup was given in [32, Theorem 1.7]: G does not admit any closed Gδ-subgroup, that is G is non-singular. As noted above, this is the first time in which this concept appeared. In the case of compact abelian groups this condition, namely non-singularity, was proved to be also sufficient under the Lusin's Hypothesis, which states that 2@1 = 2@0 (it obviously negates the Continuum Hypothesis) [32, Theorem 1.8]. In [32, Problem 1.11] it was asked if it is possible to remove this set-theoretical condition. In the same paper it was proved that the compact abelian groups K with non- metrizable connected component have the following stronger property TD! relaxing countable compactness: there exists a proper totally dense subgroup H of K that con- tains a dense !-bounded subgroup of K [32, Theorem 1.9].