Pseudocompact group topologies with no infinite compact subsets

PSEUDOCOMP A CT GR OUP TOPOLOGIES WITH NO INFINITE COMP A CT SUBSETS JORGE GALINDO A ND SERGIO MACARIO Abstra ct. W e sho w that every Ab elian group satisfying a mild cardi- nal inequality admits a pseudo compact group top ology from which all countable subgroups inherit the maximal totally b ounded top ology (we sa y that such a top ology satisfies property ♯ ). Every pseudo compact Ab elian group G with cardinalit y | G | ≤ 2 2 c satisfies this inequalit y and therefore admits a pseudo compact group top ology with prop erty ♯ . Under the Singular Cardinal H yp othesis (SCH) this criterion can be com bined with an analysis of the algebraic structure of pseudo compact groups to pro ve that every pseudo compact Ab elian g roup a dmits a pseudo compact g roup topology with property ♯ . W e also observe that pseudo compact A b elian groups with prop erty ♯ conta in no infinite compact sub sets and are examples of Pon tryag in re- flexive p recompact groups that are not compact. 1. Introduction A top ological space X is pseudo compact if eve ry real-v alued con tin uous function on X is b ounded. Pseud o compactness is greatly e nhanced by th e addition of algebraic structure. This fact w as disco v ered in 1966 by Com- fort and Ross [9] who p ro v ed that pseudo compact top ologica l groups are totally b ounded or, wh at is the same, th at they alw a ys app ear as sub gr oups of compact group s. They w en t ev en furth er and precisely ident ified ps eu - do compact groups among subgroup s of top ologica l groups: a s ubgroup of a compact group is pseudo compact if , and only if, it is G δ -dense in its closure (i.e., meets eve ry nonempt y G δ -subset of its clo sure). A p o we rful tool t o study tota lly b ounded topologies on Ab elian groups is P ontry agin dualit y . T his is b ecause a totally b oun ded group top ology is 2000 Mathematics Subje ct Classific ation. Pri mary 54H11,20K99 , Secondary 22A10,22B 05. Key wor ds and phr ases. G δ -dense, h- embedded, ♯ -p roperty , compact Ab elian group, SCH, torsion-free rank, dominant rank . Researc h supp orted by the Sp anish Ministry of Science (including FED ER fund s), gran t MTM2008 -04599/MTM and F un daci´ o Caixa Castell´ o-Bancaixa, gran t P1.1B2008 -26. 1 2 JORG E GALINDO AND SERGI O MACARIO alw a ys in duced by a group of c haracters [8] and P on try agin dualit y is based on relating a top ological group with its group of contin u ous c haracters. W e recall here that a c haracter of a group G is nothing but a homomorphism of G in to the multiplicati ve group T of co mplex n u m b ers of mo d u lus one. If G is an Ab elian top ological group, the top ology of un if orm con v ergence on compact subsets of G mak es the group of co nti nuous c haracters of G , denoted G ∧ , into a top ologic al group. Ev aluations then define a homomor- phism α G : G → G ∧∧ b et we en G and the group of all con tin uous c haracters on the dual group, the s o-called bidual group G ∧∧ . When α G is a top ologi- cal isomorph ism w e sa y that G is Pon tryag in reflexiv e. It will b e n ecessary for the dev elopment of this pap er to k eep in mind that c h aracter groups of discrete groups are compact groups. Ev en if it is not relev an t for our pur- p oses we cannot resist here to add that charact er groups of compact groups are agai n discrete, and that the P ontry agin v an-Kamp en theorem pr o v es that all lo cally compact Ab elian group s (discrete and compact ones are th us comprised) are reflexive. In the present pap er P ontry agin duality will app ear b oth a s a to ol for constructing pseudo compact group top ologies and as an ob j ective itself. T o b e precise, this pap er is motiv ated b y the follo wing t wo questions Question 1.1 ([3]) . Is eve ry Pontr yagin r eflexive total ly b ounde d Ab elian gr oup a c omp act gr oup? Question 1.2 ([12], Question 25 of [13]) . Do es every pseudo c omp act A b elian gr oup admit a pseudo c omp act gr oup top olo gy with no infinite c omp act sub- sets? In this pap er w e obtain a negativ e answ er to Question 1.1 and a p ositiv e answ er, v alid under the S ingular Card inal Hyp othesis (S CH), to Question 1.2. The fo cu s of the paper will b e on Qu estion 1.2 with the analysis of Question 1.1 and its rela tion with Question 1.2 deferred to S ection 6. It sh ould b e noted, in a direction o pp osite to Question 1.2, that ev ery pseudo compact group admits pseudo compact group top ology with n on trivial con v ergen t sequen ces, see [19]. Our approac h to Qu estion 1.2 consists in com b ining te c hniques that can b e traced back at least to [25] with the id eas of [18]. Our constru ction actu- ally prod uces ps eudo compact Ab elian groups with all coun table subgroups h -em b edded. This is stronger (see Section 2) that findin g pseud o compact group top ologies with no infinite compact subsets. With the aid of results PSEUDOCOMP ACT GR OUPS 3 from [23] t his construction will yield a wide range of n egativ e answ ers to Question 1.1. As p ointed to us b y M. G. Tk ac hen k o, Question 1.1 has b een answ ered indep endently in [1]. On notat ion a nd terminology. All groups considered in this pap er will b e Ab elian. S o, th e sp ecification Ab elian gr oup to b e found at some p oin ts will resp ond only to a matter of emphasis. T o further a void the cum b ersome use of the word ”Ab elian”, f ree Ab elian grou p s will simply b e termed as fr e e gr oups . The symb ol P will d enote the set of all p r ime num b ers. F aute de mieux , w e will use th e u n usual sym b ol P ↑ to denote the set of all prime p ow ers, i.e., an in teger k ∈ P ↑ if, and only if, k = p n for some p ∈ P and some p ositiv e in teger n . F or a set X and a cardinal n um b er α , [ X ] α stands for the collection of all subsets of X w ith cardinalit y α . F ollo w ing Tk ac h enk o [25], w e sa y that a subgroup H of a top ological group G is h -em b ed ded if ev ery h omomorphism of H to the un it circle T c an b e exte nded to a c ontinuous homomorphism of G to T . If G is totally b oun ded and H is h -em b edded in G , then the top ology of H m ust equal th e maximal totally b ounded top ology of H (o r, using v an Dou wen’s terminology , H = H ♯ ). The cardinal function m ( α ) will b e often u sed. The cardinal m ( α ) is defined for ev ery infinite cardinal α as the least cardinal n umber of a G δ - dense subset of a compact group K α of w eigh t α . It is p ro v ed in [7] that this definition do es not dep end on the choice K α and therefore mak es sense. The same reference co nta ins pro ofs of th e follo win g basic essential fea tures of m ( α ): log( α ) ≤ m ( α ) ≤ (log( α )) ω and cf ( m ( α )) > ω , for ev ery α ≥ ω . These inequalities ha v e a muc h simpler f orm if Singular Car dinal Hyp othesis (SCH) is assum ed. SCH is a condition consisten t with ZFC that follo ws f rom (but is muc h w eak er th an) the Gener alize d Continuum Hyp othesis (GCH). Under SCH ev ery infinite cardinal α satisfies m ( α ) = (log( α )) ω . It is w ell kno wn that ev ery compact g roup has cardinalit y 2 κ for some car- dinal κ . The question on whic h cardinals can app ear as the cardin al of a pseudo compact group is n ot so readily answ ered. W e will sa y that a cardinal 4 JORG E GALINDO AND SERGI O MACARIO κ is admissible p r o vided there is a pseudo compact group of cardinal κ . The first obstructions to admissibilit y we re found b y v an Dou w en [15], the main one b eing that the card inalit y | G | of a p seudo compact group cann ot b e a strong limit cardinal o f co untable cofinalit y; see [11, Chapter 3] for more information on admiss ib le cardinals. Most of our results concern constructing pseud o compact group top ologies on a giv en Ab elian group G . As indicated in the introd uction, ev ery p seu- do compact group top ology is totally b ounded and a totally b ounded group top ology T on an Abelian group G is alw ays in duced by a u nique group of c haracters H ⊂ H om ( G, T ), [8, 9]. T o stress this latter fact we will usually refer to T as T H . Recall that the top ology T H is Hausdorff if, and only if , the subgroup H separates points of G . W e ha v e also in tro duced abov e the symbol G ∧ to denote the group of all cont inuous c haracters of a top ologic al Ab elian group equipp ed with the compact-open to p ology . W e will use in this con text th e su bscript d to indicate that G carries the discrete topology . Th us ( G d ) ∧ equals the set H om ( G, T ) of all homomorphisms into T . B eing a closed subgroup of T G , ( G d ) ∧ is alw a ys a co mpact group . Sev eral p urely algebraic n otions from the theory of infi nite Ab elian group s will b e necessary , as for instance the n otion of basic su bgroup and the related one of p ure subgroup. W e r efer to [17] for th e meaning and signifi cance of these pr op erties. As usual, th e symbol t ( G ) stands for the torsion subgroup of the group G and r 0 ( G ) denotes the torsion-free rank of G . 2. The dual proper ty to pse udocomp actness The follo wing theorem is at the h eart of the relationship b et w een questions 1.2 and 1.1. Theorem 2.1 ([23]) . L et ( G, T H ) , H ⊂ Hom( G, T ) , b e a Hausdorff Ab elian total ly b ounde d gr oup. ( G, T H ) is pseudo c omp act i f, and only if, every c ount- able su b gr oup of ( H, T G ) is h -emb e dde d in ( G d ) ∧ . Definition 2.2. W e sa y that a top ological g roup G h as prop erty ♯ if e v ery coun table subgrou p of G is h -em b edded i n G . Th us prop ert y ♯ is, in t he terminology of [23], the d u al prop erty of p seu- do compactness. The r elation b etw een pr op ert y ♯ and Question 1.2 is clear from the fol- lo wing Lemma. Although a com b ination of P r op ositions 3.4 and 4.4 of [23] PSEUDOCOMP ACT GR OUPS 5 w ould pro vide an ind irect p ro of, we offer a d irect pro of f or the reader’s con v enience. Lemma 2.3. L et ( G, T H ) denote a total ly b ounde d g r oup with pr op erty ♯ . Then ( G, T H ) has no infinite c omp act subsets. Pr o of. W e first see that all countable subgroup s of G are T H -closed. Sup p ose otherwise t hat x ∈ cl ( G, T H ) N \ N w ith N a coun table subgroup of G . Th e subgroup e N = h N ∪ { x } i is also countable and , b y h yp othesis, inherits its maximal totally b ound ed group top ology fr om ( G, T H ). Since su bgroups are necessarily closed in that to p ology , it follo ws that N is closed in e N , which go es aga inst x ∈ e N \ N . No w supp ose K is an infi nite co mpact subset of G and let S ⊂ K b e a coun table s ubset of K . Define e G = h S i and denote by e G and ( e G, T H ) the completions of e G ♯ and ( e G, T H ) r esp ectiv ely . Since h S i is h -em b edded the iden tit y function j : e G ♯ → ( e G, T H ), extends to a top ological isomor- phism ¯  : b e G → ( e G, T H ). Then ¯ (cl b e G S ) = cl ( e G, T H ) j ( S ) ⊂ K , therefore cl ( e G, T H ) j ( S ) = cl ( e G, T H ) S and, it follo ws fr om the preceding paragraph that cl b e G S = ¯ (cl b e G S ) ⊂ h S i . But a w ell kno wn theorem of v an Douw en [16] (see a lso [20] and [2, The- orem 9.9.51] for different pr o ofs and [21] for extensions of th at result) states that | cl b ( e G ) S | = 2 c and therefore it is imp ossible that ¯ (cl b e G S ) S ⊂ h S i .  W e establish next some easily deduced p ermanence pr op erties. Prop osition 2.4. The class of gr oups having pr op erty ♯ is close d for finite pr o ducts. Pr o of. Let G 1 and G 2 b e tw o topological Abelian group s with p rop erty ♯ and let N b e a countable subgroup of G 1 × G 2 . Let h b e a homomorph ism from N to T . By considering an arbitrary extension of h to G 1 × G 2 w e ma y assume that h is actually defined on G 1 × G 2 . Since b oth π 1 ( N ) and π 2 ( N ) are coun table there w ill b e con tin uous homomorphisms h i : G i → T , i = 1 , 2, with h 1 ( x ) = h ( x, 0) and h 2 ( y ) = h (0 , y ) for all x ∈ π 1 ( N ) and y ∈ π 2 ( N ). The homomorphism ¯ h : G 1 × G 2 → T giv en b y ¯ h ( x, y ) = h 1 ( x ) · h 2 ( y ) is then a con tin uous extension of h .  Lemma 2.5. L et π : K → L b e a c ontinuous surje ction b etwe en two c omp act Ab elian gr oups K and L and supp ose tha t N is a sub gr oup of L th at, as subsp ac e of L , c arries the maximal total ly b ounde d top olo gy. If M is a 6 JORG E GALINDO AND SERGI O MACARIO sub gr oup of K such tha t π ↿ M is a gr oup isomor phism b etwe en M and N , then M also inh erits fr om K the maximal total ly b ounde d top olo gy. Pr o of. Denote b y T K and T L the top ologies that M inh erit from K and L resp ectiv ely (the latter obtained through π ↿ M ). Since π is con tin uous, the top ology T K is finer than T L , b ut T K is the maximal totally b oun ded top ology , therefore T K = T L .  3. Proper ty ♯ on tors ion-free and bounde d gr oups W e will mak e a hea vy use o f p o w ers of groups in the sequel. I f σ is a cardinal n umb er, K σ stands f or such p ow ers. W e use ca lligraphical letters, to d enote sets of coordin ates, that is, sub sets of σ . If D ⊂ σ , we will denote b y π K D the p ro jection fr om K σ to K D , if no confu sion is p ossible we will simply use π D . Lemma 3.1. L et G b e a metrizable gr oup and let σ ≥ c and α b e c ar dinal numb ers with m ( σ ) ≤ α , and α ω ≤ σ . Then ther e exists an indep endent G δ -dense subset D ⊆ G σ with c ar dinality m ( σ ) , D = { d η : η < m ( σ ) } , and two families of sets of c o or dinates {S θ : θ ∈ [ α ] ω } , {N η : η < α } ⊂ σ such that : (1) |S θ | = σ . (2) S θ ∩ S θ ′ = ∅ , if θ 6 = θ ′ . (3)       S θ \ [ η ∈ θ N η       = σ for every θ ∈ [ α ] ω . (4) Every su b set { g η : η < α } of G σ with π N η ( g η ) = π N η ( d η ) , for al l η < α is G δ -dense. Pr o of. Let A β = { a γ : γ < σ } b e a set with |A β | = σ an d consider the disjoin t union A = S β < c A β . W e identify G σ with G A and α with [ c ] ω × α . Since α ω ≤ σ , w e can a s well decomp ose ea c h A β as a disjoin t union A β = [ e θ ∈ [[ c ] ω × α ] ω A β , e θ of sets of card inalit y |A β , e θ | = σ . F or eac h N ∈ [ c ] ω , let next F N = { f ( N ,η ) : η < α } b e an indep endent G δ -dense subset of the prod uct G ∪ γ ∈ N A γ (note that m ( σ ) ≤ α and that G is met rizable). Ass u me that eac h f ( N ,η ) actually b elongs to G A b y putting π A γ ( f ( N ,η ) ) = 0 if γ / ∈ N . W e no w order α = [ c ] ω × α lexico graphically and define th e sets N e η , e η ∈ [ c ] ω × α and S e θ , e θ ∈ [[ c ] ω × α ] ω . F or e η = ( N , η ) ∈ [ c ] ω × α defi ne N ( N ,η ) = S γ ∈ N A γ , e η and giv en e θ = { ( N k , η k ) : k < ω , ( N k , η k ) ∈ [ c ] ω × α } , PSEUDOCOMP ACT GR OUPS 7 w e define S e θ = A β 0 , e θ where β 0 is suc h that β ∈ N k for some k , implies β < β 0 (recall that c has uncount able c ofinalit y). By construction of the sets A β , e θ , we hav e S e θ ∩ S e θ ′ = ∅ , when e θ 6 = e θ ′ . Condition (3) obvi ously holds, since S e θ and S e η ∈ e θ N e η are ev en d isjoin t. Define fin ally D = { f e η : e η ∈ [ c ] ω × α } = ∪ N ∈ [ c ] ω F N . Supp ose e D = { g e η : e η ∈ [ c ] ω × α } is suc h that π N e η ( g e η ) = π N e η ( f e η ), for all e η ∈ [ c ] ω × α . T o c h ec k that e D is indeed G δ -dense w e c ho ose a G δ -subset U of G A . There will b e then N = { α n : n < ω } ∈ [ c ] ω and a G δ -set V ⊂ G ∪A α n suc h that { ¯ x ∈ G A : π ∪ n A α n ( ¯ x ) ∈ V for eac h n < ω } ⊂ U . Sin ce F N is G δ - dense in G ∪ γ ∈ N A γ = G ∪ n A α n , th ere will b e an elemen t f ( N ,η ) ∈ F N with π ∪ n A α n ( f ( N ,η ) ) ∈ V for ev ery α n ∈ N . As g ( N ,η ) and f ( N ,η ) ha v e th e same ∪ γ ∈ N A γ -coord inates, w e conclud e that g ( N ,η ) ∈ U ∩ e D .  If χ is a h omomorphism b etw een t w o group s G 1 and G 2 and σ is a cardinal n umber, we denote by χ σ the pro du ct homomorphism χ σ : G σ 1 → G σ 2 defined b y χ σ (( g η ) η<σ ) = ( χ ( g η )) η<σ . It is easil y ve rified th at, for an y D ⊆ σ , the pro jections π G i D : G σ i → G D i , i = 1 , 2 satisfy π G 2 D ◦ χ σ = χ D ◦ π G 1 D Corollary 3.2. L et χ : G 1 → G 2 b e a surje ctive homomorphism b etwe en two metriza ble gr oups G 1 and G 2 . If σ and α ar e c ar dinal numb e rs with m ( σ ) ≤ α and α ω ≤ σ , then it is p ossible to find an indep endent G δ -dense subset D of G σ 1 satisfying the pr op erties of Pr op osition 3.1 such that in addition χ σ ( D ) is an indep endent subset of G σ 2 . Pr o of. It su ffices to rep eat the pro of of Lemma 3.1 taking care to c ho ose the sets F N in such a w ay that χ ∪ γ ∈ N A γ ( F N ) is also ind ep endent.  Prop osition 3.3. L et χ : G → T b e a surje ctive c har acter of a c omp act metrizable gr oup G . If σ and α ar e c ar dinal numb ers with m ( σ ) ≤ α , and α ω ≤ σ , then the top olo gic al gr oup G σ c ontains an indep endent G δ -dense subset F of c ar dinality α suc h that F and χ σ ( F ) gener ate isomorphic gr oups with p r op erty ♯ . Pr o of. W e b egin with a G δ -dense subset of G σ , D = { d η : η < α } , with the prop erties of Lemma 3.1 and Corollary 3.2. W e ha v e th us t w o families of sets {S θ , : θ ∈ [ α ] ω } , {N η , : η < α } ⊂ σ with the prop erties (1) through (4) of that L emma. 8 JORG E GALINDO AND SERGI O MACARIO Next, for ev ery θ ∈ [ α ] ω , w e c ho ose and fix a set of co ordinates D θ ⊆ σ of cardinalit y |D θ | = σ in suc h a w ay that D θ ⊆ S θ \ [ η ∈ θ N η (recall that by Lemma 3.1,    S θ \ S η ∈ θ N η    = σ ) Giv en eac h θ ∈ [ α ] ω , we consider the free subgroup h χ σ ( d η ) : η ∈ θ i and equip it with its maximal totally b ounded top ology . Denoting the resulting top ological group as h χ σ ( d η ) : η ∈ θ i ♯ , and taking into acco unt that it has w eigh t c , we can find an emb edding j θ : h χ σ ( d η ) : η ∈ θ i ♯ ֒ → T D θ . (3.1) F or eac h θ ∈ [ α ] ω and eac h η ∈ θ , let g η,θ denote an elemen t of G D θ with χ D θ ( g η,θ ) = j θ ( χ σ ( d η )). Observe that t he set { g η,θ : η ∈ θ } is in d ep endent. W e fi nally define the element s f η , η < α , b y the ru les: π G D θ ( f η ) = g η,θ , if θ ∈ [ α ] ω is such that η ∈ θ , and π G γ ( f η ) = π G γ ( d η ) if γ / ∈ D θ for an y θ ∈ [ α ] ω with η ∈ θ . Let us see that F = { f η : η < α } satisfies the desired prop erties: (1) F and χ σ ( F ) ar e indep endent. Supp ose that P m k =1 n k f η k = 0 with n k ∈ Z . Cho ose then θ ∈ [ α ] ω with η 1 , . . . , η m , ∈ θ . S in ce π G D θ ( f η k ) = g η k ,θ and the set { g η,θ : η ∈ θ } is indep endent, the in dep end ence of F follo w s. S ince π D θ ( χ σ ( f η )) = χ D θ ( g η,θ ), χ σ ( F ) is also indep endent. It is easy to see , no w , that h F i and h χ σ ( F ) i are isomorphic. (2) The sub gr oup h χ σ ( F ) i has pr op erty ♯ . Let N b e a coun table subgroup of h χ σ ( F ) i . Let θ ∈ [ α ] ω b e suc h that N ⊆ h χ σ ( f η ) : η ∈ θ i and define N θ := h f η : η ∈ θ i . Observe finally that π T D θ ( N ) = χ D θ ( π G D θ ( N θ )). This last su b- group is just j θ ( h χ σ ( d η ) : η ∈ θ i ) and the latter carries b y construc- tion its maximal totally b ounded top ology , since the restriction of π T D θ : T σ → T D θ to N is a group isomorp hism ont o π T D θ ( N ) = χ D θ ( π G D θ ( N θ )), Lemma 2.5 app lies. (3) h F i has pr op erty ♯ . T ak e π = χ σ , K = G σ and L = T σ . Bearing in mind that the restriction to h F i is an iso morph ism b ecause F and χ σ ( F ) are indep en d en t sets, Lemma 2.5 applies ag ain. (4) F i s a G δ -dense subset of G σ . Observ e that, for ev ery η < α , f η coincides w ith d η on the set of co ord in ates N η , for D θ ⊆ S θ \ [ η ∈ θ N η . PSEUDOCOMP ACT GR OUPS 9 Since D has the prop er ties of Lemma 3.1, w e conclude that F is G δ -dense.  Prop osition 3.4. L et σ and α b e c ar dinal numb ers with m ( σ ) ≤ α , a nd α ω ≤ σ . The top olo gic al gr oup Z ( p ) σ c ontains an indep endent G δ -dense subset H with pr op erty ♯ . Pr o of. Pro ceed exactly as in Prop osition 3.3 an d constru ct an em b edd ing in to Z ( p ) σ . T o obtain the ♯ -prop ert y w e iden tify coun table subgroup s with Bohr groups of the form ( ⊕ ω Z ( p )) ♯ .  4. The algebraic structure of pseudocomp a ct Abel ian groups W e obtain here some results on the algebraic structure of pseudo compact that will b e useful in the next section. The first of them is inspired (and shares a part o f its pr o of ) f r om th e fir st part of th e proof of Lemma 3.2 of [18 ]. W e ske tc h h er e the pro of for the reader’s con v enience. W e thank Dikran Dikranjan for p oin ting a misguiding sentence in a previous version of this pro of. Lemma 4.1. Every Ab elian gr oup adm its a de c omp osition G =    M p k ∈ P ↑ 0 M γ ( p k ) Z ( p k )    M H wher e P ↑ 0 is a finite subset of P ↑ and H is a sub gr oup of G with | nH | = | H | , for al l n ∈ N . Pr o of. Decompose t ( G ) = L p G p as a d irect sum of p -groups G p and let B p denote a basic subgrou p of G p for eac h p . This in particular m eans that B p is a direct s um of cyclic p -group s, B p = M n<ω B p,n with B p,n ∼ = M β p n Z ( p n ) and that G p /B p is divisible. Define D = {| B p,n | : p n ∈ P ↑ } . If D has no maximum or β 0 = max D is attained at an in finite num b er of | B p,n | ’s w e s top here. If, otherwise, β 0 = max D = | B p 1 ,n 1 | = . . . = | B p r ,n r | and | B p j ,n j | < β 0 for al l the r emaining p n j j ∈ P ↑ w e repeat the p ro cess w ith t he set D \ | B p 1 ,n 1 | . After a finite num b er of steps w e obtain in this m anner a finite collectio n of cardinals F ⊂ D such th at either: 10 JORG E GALINDO AND SERGI O MACARIO (1) Case 1: the sup rem um β := su p ( D \ F ) is not attained, or (2) Case 2: the s u premum β := sup ( D \ F ) is atta ined infinitely often, i.e., there is an in finite subset I ⊂ P ↑ with | B p,n | = β for all p n ∈ I . Define P ↑ 0 = { p n ∈ P ↑ : | B p,n | ∈ F } (observ e that P ↑ 0 is necessarily fi nite), and set γ ( p n k k ) = | B p k ,n k | if p n k k ∈ P ↑ 0 . Sin ce the subgroups B p k ,n k are b ound ed pure subgroup s, there will b e [17, Theorem 27.5] a subgroup H of G suc h that G =    M p n k k ∈ P ↑ 0 M γ ( p n k k ) B p k ,n k    M H , F or eac h p rime p , consider a p -basic su bgroup B p,H = ⊕ n B p,n,H of H p , th e p -part of t ( H ), it is immediate ly c hec k ed that either B p,H itself (if p 6∈ P ↑ 0 ) or B p,H L L p n k k ∈ P ↑ 0 p k = p L γ ( p n k k ) B p k ,n k ! (if p ∈ P ↑ 0 ) is also p -basic in G . Since differen t b asic sub groups are necessarily isomorphic [17, Th eorem 35], we ha v e t hat B p,H or B p,H L L p n k k ∈ P ↑ 0 p k = p L γ ( p n k k ) B p k ,n k ! is isomorph ic to B p . W e ha v e th erefore that, for eac h p , either sup | B p,n,H | is n ot attained (case 1 ab ov e) or attained at infinitely man y p n ’s (case 2). Let no w n b e any natural num b er. Then | nB p k ,n k ,H | = | B p k ,n k ,H | unless p n k k divides n . Since this will only happ en for finitely p n k k ’s, we conclude, in b oth cases 1 and 2 that | nB p,H | = | B p,H | . Using that B p,H is pu re in H p and that H p /B p,H is divisible we hav e that, | nH p | =     nH p nB p,H     + | nB p,H | =     n  H p B p,H      + | B p,H | =     H p B p,H     + | B p,H | = | H p | . Since | H | = P p H p + r 0 ( H ) | for ev ery infinite group H and r 0 ( nH ) = r 0 ( H ) w e ha v e finally th at | H | = | nH | , for ev ery n ∈ Z .  The terminology in tro d uced in the next definition is motiv ated, in the present con text, b y Theorem 4.4 b elo w. Definition 4.2. If G is an Ab elian group, the set P ↑ 0 of Lemma 4 .1 c an be partitioned as P ↑ 0 = P ↑ 1 ∪ P ↑ 2 with p n i i ∈ P ↑ 1 if, and only if, γ ( p n i i ) > r 0 ( G ). PSEUDOCOMP ACT GR OUPS 11 The cardin al num b ers γ ( p n i i ) with p n i i ∈ P ↑ 1 will b e called the dominant r anks o f G . Lemma 4.3. If G is a nonto rsion pseudo c omp act gr oup, then ther e is a p ositive inte ger such tha t: m ( w ( n G )) ≤ r 0 ( nG ) ≤ 2 w ( nG ) . (4.1) Pr o of. If nG is metrizable f or some n ∈ N , then n G is a compact metrizable group. Therefore r 0 ( nG ) = c and th e inequalities in (4.1 ) hold for this n . If nG is not metrizable for any n ∈ N , th en G is, in the terminology of [10], nonsingular . Com bining Lemma 3.3 and Theorem 1.15 of [10], there m ust b e n ∈ N suc h that r 0 ( nG ) is th e cardinal of a pseudo compact group of w eigh t w ( nG ). Th erefore m ( w ( n G )) ≤ r 0 ( nG ) ≤ 2 w ( nG ) .  Theorem 4.4. L et G b e an A b elian gr oup. If G admits a pseudo c omp act gr oup top olo gy, then G c an b e de c omp ose d as G =    M p k ∈ P ↑ 1 M γ ( p k ) Z ( p k )    ⊕ G 0 wher e γ ( p k i i ) , p k i i ∈ P ↑ 1 , ar e the dominant r anks of G and ther e is a c ar dinal ω d ( G ) such that m ( ω d ( G )) ≤ r 0 ( G ) ≤ | G 0 | ≤ 2 ω d ( G ) . (4.2) Pr o of. Since ev ery p seudo compact torsion group m ust b e of b ound ed order, the theorem is trivial (and v acuous) for su ch groups, w e ma y assume that G is nonto rsion. Decomp ose G a s in Lemm a 4.1:    M p k ∈ P ↑ 0 M γ ( p k ) Z ( p k )    M H with P ↑ 0 a finite sub set of P ↑ and | nH | = | H | f or a ll n ∈ N . 12 JORG E GALINDO AND SERGI O MACARIO Split P ↑ 0 = P ↑ 1 ∪ P ↑ 2 as in Definition 4. 2 and define G 0 = M p k i i ∈ P ↑ 2 M γ ( p k i i ) Z ( p k i i ) M H . W e will p ro v e that th e inequalities 4.2 hold for w d ( G ) = w ( nG 0 ). Lemma 4.3 pr o v es that there is s ome n ∈ N with m ( w ( n G 0 )) ≤ r 0 ( G 0 ) ≤ 2 w ( nG 0 ) . (4.3) If | G 0 | = γ ( p k i i ) f or so me p k i i ∈ P ↑ 2 , it follo ws from t he definition o f P ↑ 2 that | G 0 | = r 0 ( G ) and (4.2) is deduced from (4.3). If, otherwise, | G 0 | = | H | , then | nG 0 | ≥ | nH | = | H | = | G 0 | and w e deduce that | G 0 | = | nG 0 | and th us that | G 0 | ≤ 2 w ( nG 0 ) . Th is toge ther with (4.3) giv es ag ain (4 .2) with w d ( G ) = w ( nG 0 ).  R emark 4.5 . T he cardinal w d ( G ) used in Theorem 4.4 is p recisely the divisi- ble weight of G that was in tro duced and stud ied b y Dikranjan and Giordano- Bruno [10]. W e refer the r eader to that pap er to get an idea of the imp ortan t role pla y ed by th e divisible w eigh t in the stru cture of pseu do compact groups. One of its applications (Theorem 1.19 loc. cit.) is to pr o v e that r 0 ( G ) is an admissible cardinal for ev ery p seudo compact group G , a fact fi rst pro v ed by Dikranjan and Sh akhmato v in [14]. 5. Pseudo c omp act groups with proper ty ♯ The r esults of the previous sections w ill b e u sed here to obtain sufficient conditions for the existence of pseud o compact g roup top ologies with prop- ert y ♯ . Lemma 5.1. L et π : G 1 → G 2 b e a quotient homomorph ism b etwe en two Ab elian top olo gic al gr oups G 1 and G 2 and let L b e a c omp act A b e lian gr oup. Assume tha t the fol lowing c onditions hold: (1) G 1 c ontains a fr e e G δ -dense sub gr oup H 1 such that H 1 and π ( H 1 ) ar e isomorphic and have pr op erty ♯ . (2) G 1 c ontains another fr e e sub gr oup H 2 such that H 1 ∩ H 2 = { 0 } , H 1 + H 2 and π ( H 1 + H 2 ) ar e isomorphic and have p r op erty ♯ . (3) m ( w ( L )) ≤ | H 2 | . Under these c onditions the pr o duct G 1 × L c ontains a G δ -dense sub gr oup e H such that b oth e H and π  p 1 ( e H )  have pr op erty ♯ , wher e p 1 : G 1 × L → G 1 denotes th e first pr oje ction. PSEUDOCOMP ACT GR OUPS 13 Pr o of. W e first enumerate the elemen ts of H 1 and H 2 as H 1 = { f β : κ < β } and H 2 = { g η : η < α } . Since m ( w ( L )) ≤ α = | H 2 | , w e can also enumerate a G δ -dense subgroup D of L (allo wing rep etitions if necessary) as D = { d η : η < α } . W e no w define the subgroup e H of G 1 × L as e H = h ( f κ + g η , d η ) : η < α, κ < β i . It is easy to chec k th at e H is a G δ -dense subgroup of G 1 × L with e H ∩ { 0 }× L = { (0 , 0) } . Since the homomorphism p 1 is con tin uous and establishes a group isomor- phism b et w een e H and H 1 + H 2 , Lemma 2.5 shows t hat e H has prop ert y ♯ . The same argument applies to the group π  p 1 ( e H )  = π ( H 1 + H 2 ).  Definition 5.2. Let α ≥ ω b e a cardinal. W e sa y that α satisfies pr op- erty ( ∗ ) if: there is a c ardinal κ with κ ω ≤ α ≤ 2 κ (*) Ev ery cardinal α with α ω = α satisfies prop ert y (*). T his condition is equiv alen t to the condition ( m ( α )) ω ≤ α . T o app ly Lemma 5.1 we need the follo wing r esult: Theorem 5.3 (Theorem 4.5 of [4]) . L et G = ( G, T 1 ) b e a p seudo c omp act Ab elian gr oup with w ( G ) = α > ω , and set σ = min { r 0 ( N ) : N is a close d G δ -sub gr oup o f G } . If α ω ≤ σ and if λ ≥ ω satisfies m ( λ ) ≤ σ , then G admits a pseudo c omp act gr oup top olo gy T 2 such that w ( G, T 2 ) = α + λ and T 1 W T 2 is pseudo c omp act. Mor e over, every close d G δ -sub gr oup o f ( G, T 1 ) is G δ -dense ( G, T 2 ) . Corollary 5.4. L et σ , α and λ b e c ar dinals with α ω ≤ σ and m ( λ ) ≤ σ . If H is a fr e e, d ense sub gr oup of T σ with pr op erty ♯ and c ar dinality α , then T σ c ontains another su b gr oup H 2 with H ∩ H 2 = { 0 } , | H 2 | = λ + α and such t hat H + H 2 has p r op erty ♯ . Pr o of. Let F ( σ ) denote the free Abelian group of rank σ . W e apply Theo- rem 5.3 to the pseudo compact group ( F ( σ ) , T H ) defined by H . W e obtain th us a pseud o compact top ology T H 2 on F ( σ ) induced b y a subgroup H 2 of T σ of card inalit y | H 2 | = α + λ suc h th at T H W T H 2 = T H + H 2 is pseudo com- pact. By Theorem 2.1 the subgroup H + H 2 has pr op ert y ♯ and, since closed G δ -subgroups of T H are G δ -dense in T H 2 , we also ha v e that H ∩ H 2 = { 0 } .  14 JORG E GALINDO AND SERGI O MACARIO Theorem 5.5. L et G b e a pseudo c omp act Ab elian g r oup with dominant r anks γ ( p n 1 1 ) , . . . , γ ( p n k k ) and supp ose that γ ( p n i i ) , 1 ≤ i ≤ k , satisfy pr op- erty (*) . If r 0 ( G ) also satisfies pr op erty (*) for some κ w ith m ( | G 0 | ) ≤ 2 κ , then G admits a pseudo c omp act top olo gy with pr op erty ♯ . Pr o of. Decompose, foll o wing Theorem 4.4, G as a d irect sum G =    M γ ( p n 1 1 ) Z ( p n 1 1 ) M · · · M γ ( p n k k ) Z ( p n k k )    M G 0 Let F denote a free Ab elian grou p of cardin alit y r 0 ( G ) cont ained in G 0 and denote b y D ( F ) and D ( t ( G 0 )) divisible hulls of F and t ( G 0 ), resp ectiv ely . There is then a chai n of group em b edd ings (here w e use [17, Lemmas 16.2 and 24.3]) F j 1 → G 0 j 2 → D ( F ) ⊕ D ( t ( G 0 )) (5.1) Denote by χ the quotien t homomorphism ob tained as the du al map of the canonical em b eddin g Z → Q . Observe that iden tifyin g F with ⊕ r 0 ( G ) Z and D ( F ) with ⊕ r 0 ( G ) Q , the dual map of j 2 ◦ j 1 is exactly χ r 0 ( G ) . T aking σ = r 0 ( G ), G = Q ∧ d and α = κ ω , we can apply Prop osition 3.3 to get a G δ -dense su bgroup H 1 of ( D ( F ) d ) ∧ =  Q ∧ d  r 0 ( G ) with | H 1 | = κ ω and suc h that H 1 and χ r 0 ( G ) ( H 1 ) are isomorphic and ha ve prop ert y ♯ (notice that κ ω and r 0 ( G ) satisfy the h yp othesis of that Pr op osition). W e no w apply Corollary 5.4 to χ r 0 ( G ) ( H 1 ) to obtai n another free sub- group H ′ 2 of T r 0 ( G ) with χ r 0 ( G ) ( H 1 ) ∩ H ′ 2 = { 0 } , | H ′ 2 | = 2 κ and such that χ r 0 ( G ) ( H 1 ) + H ′ 2 has pr op ert y ♯ . By lifting (through χ r 0 ( G ) ) the f ree gener- ators of H ′ 2 to ( D ( F ) d ) ∧ , w e obtain a free subgroup H 2 of ( D ( F ) d ) ∧ suc h that H 1 ∩ H 2 = { 0 } and | H 2 | = 2 κ . Clearly H 1 + H 2 is isomorphic to χ r 0 ( G ) ( H 1 ) + H ′ 2 and therefore H 1 + H 2 has prop ert y ♯ by Lemma 2.5. W e finally apply Lemma 5.1. The role of G 1 × L is play ed b y ( D ( F ) d ) ∧ ×  D ( t ( G 0 )) d  ∧ ; G 2 is here identified with T r 0 ( G ) and π is χ r 0 ( G ) . Lemma 5.1 then pro vides a G δ -dense subgroup e H of  D ( F ) d  ∧ ×  D ( t ( G 0 )) d  ∧ suc h that b oth e H and χ r 0 ( G ) ( p 1 ( e H )) ha v e prop er ty ♯ . Th is subgroup generates a pseudo compact top ology T e H on D ( F ) ⊕ D ( t ( G 0 )) with pr op ert y ♯ that mak es F ps eudo compact (the ind u ced top ology on F is just th e top ology inh erited PSEUDOCOMP ACT GR OUPS 15 from χ r 0 ( G ) ( p 1 ( e H ))). Since G 0 sits b etw een F and D ( F ) ⊕ D ( t ( G 0 )), it follo w s that the r estriction of T e H to G 0 is pseu d o compact and has prop ert y ♯ . By Prop osition 3 .4 the boun ded group M α ( p n 1 1 ) Z ( p n 1 1 ) M · · · M α ( p n k k ) Z ( p n k k ) also admits a pseudo compact group top ology with prop erty ♯ and the theo- rem follo ws.  Dikranjan and Shakmato v [12] pro v e under a set-theoretic axiom call ed ∇ κ (that implies c = ω 1 and 2 c = κ with κ b eing an y c ardinal κ ≥ ω 2 ) that ev ery pseudo compact group of cardinalit y at most 2 c has a pseudo- compact group top ology with no infinite compact subsets. It follo ws from Theorem 5.5 that the result is true in ZFC, ev en for larger cardinalities. Corollary 5.6. L et G b e a pseudo c omp act Ab elian gr oup of c ar dinality | G | ≤ 2 2 c . Then G admits a pseudo c omp act top olo gy with pr op erty ♯ (and thus a pseudo c omp act top olo gy with no infinite c omp act subsets). Pr o of. Since a pseudo compact group with r 0 ( G ) < c is a b ound ed group it will suffice to c hec k that ev ery cardinal α with α ≤ 2 2 c satisfies prop ert y (*). Theorem 5.5 will then b e applied. W e consider the follo wing t wo cases: Case 1: c ≤ α ≤ 2 c . In this case we put κ = c . Case 2: α > 2 c . Cho ose κ = 2 c for this case. Observe that in b oth cases | m ( | G | ) | ≤ 2 κ and hence th at all hyp othesis of Theorem 5.5 are fu lfilled.  By v an Dou w en’s theorem [15], a strong limit admissible cardinal m ust ha v e uncoun table cofin ality . Un der mild set-theoreti c assum ptions this im- plies that admissible cardinals must ha v e prop ert y (*). It suffi ces, for in- stance, to assume the Singular Car dinal Hyp othesis SC H. Theorem 5.7 (Theorem 3.5 of [6] and Lemma 3.4 of [11]) . If SCH is as- sume d, then every admissible c ar dinal has pr op erty (*) . Com bining Theorem 4.4 and Theorem 5.5, it tur ns out that, und er SCH, ev ery pseudo compact group admits a pseudo compact group top ology with prop erty ♯ . Theorem 5.8 (SC H) . Every pseudo c omp act A b e lian gr oup G admits a pseu- do c omp act gr oup top olo gy with p r op erty ♯ . Pr o of. Let γ ( p n 1 1 ) ≥ · · · ≥ γ ( p n k k ) b e the dominant ranks of G . Then | G | = γ ( p n 1 1 ) and , γ ( p n 1 1 ) is admissible. Since w e can assu me that n i < n j when 16 JORG E GALINDO AND SERGI O MACARIO j > i and p i = p j , p 1 G w ill b e a ps eudo compact group of cardinality | p 1 G | = γ ( p n 2 2 ). Pro ceeding in the same w a y w e obtain that the dominant ranks are admissible cardinals. By Theorem 5.7 all these ca rdinals must satisfy prop erty (*). Th eorem 4.4 shows, on the other hand , that the cardin al r 0 ( G ) is also admissible a nd, actually: m ( w d ( G 0 )) ≤ r 0 ( G 0 ) = r 0 ( G ) ≤ | G 0 | ≤ 2 w d ( G 0 ) In order to apply T h eorem 5.5 and fi nish the pro of, w e must show that r 0 ( G ) also satisfies pr op ert y (* ) for some cardinal κ with m ( | G 0 | ) ≤ 2 κ . W e h a v e t w o p ossib ilities: Case 1: m ( w d ( G 0 )) ≤ r 0 ( G ) ≤ ( w d ( G 0 )) ω . In th is case, w e put κ = log( w d ( G 0 )). Then, b earing in mind that, under SCH, we h a v e m ( α ) = (log( α )) ω for ev ery infi nite cardinal α , w e g et: κ ω =  log  w d ( G 0 )   ω = m  w d ( G 0 )  ≤ r 0 ( G ) and r 0 ( G ) ≤  w d ( G 0 )  ω ≤  2 log  w d ( G 0 )   ω = (2 κ ) ω = 2 κ . So prop ert y (*) is c hec k ed. On the other hand, m ( | G 0 | ) ≤ m  2 w d ( G 0 )  =  log  2 w d ( G 0 )   ω ≤  w d ( G 0 )  ω ≤ 2 κ Case 2:  w d ( G 0 )  ω ≤ r 0 ( G ) ≤ 2 w d ( G 0 ) . In this case, p rop erty (*) a nd condition m ( | G 0 | ) ≤ 2 κ are obvi ously fulfilled with κ = w d ( G 0 ).  Theorem 5.8 relies qu ite strongly on S C H. It uses the constr u ction of Theorem 5.5 made applicable to all admissible cardinals b y T heorem 5.7. W e do not kno w whether SCH is essen tial for T heorem 5.8, i.e., whether the theorem is true for pseudo compact group s whose cardinal d o es not satisfy prop erty (*). Indeed, admissible cardinals not satisfying prop ert y (*) are hard to find in the literature. The follo wing (consistent ) example, suggested to us by W.W. Comfort and based on a construction due to Gitik and Shelah, pr o duces one suc h ca rdin al. W e refer to Remark 3.14 of the forthcoming pap er [5] for additional remarks concerning th e Git ik-Shelah m o dels. This same pap er con tains related results concerning the cardinals m ( α ) and, more generally , the densit y characte r of p o w ers of d iscrete groups in t he κ -b ox top ology . Example 5.9 . A pseudo compact group G wh ose cardinalit y d o es not satisfy prop erty (*). PSEUDOCOMP ACT GR OUPS 17 Pr o of. Gitik and Sh elah, [22], construct a mo del where m ( ℵ ω ) = ℵ ω +1 while 2 ℵ ω = ( ℵ ω ) ω = ℵ ω +2 . This means that the compact g roup { 1 , − 1 } ℵ ω has a G δ -dense subgroup G of cardinalit y | G | = ℵ ω +1 . Let us denote for simp licit y α = ℵ ω +1 . Supp ose that α satisfies pr op ert y (* ) . There is then a cardin al κ with κ ω ≤ α ≤ 2 κ . (5.2) Since α ω ≥ ( ℵ ω ) ω = ℵ ω +2 > α , w e see that κ ω 6 = α . It follo ws then from (5.2) that κ ω ≤ ℵ ω ≤ 2 κ . But then m ( ℵ ω ) ≤ m (2 κ ) ≤ κ ω ≤ ℵ ω , w hereas, b y construction, m ( ℵ ω ) = ℵ ω +1 . This con tradiction sho ws that α do es not satisfy prop erty (*).  6. Proper ty ♯ and the du a lity of tot all y bounde d Abelian groups P on try agin dualit y was designed to w ork in locally compact Ab elian groups and u sually wo rks b etter for complete groups. This b eha viour raised the question (act ually our fir st motiv ating Q u estion 1.1) as to whether all totally b ounded r eflexiv e group sh ou ld b e compact, [3]. W e see next that this is not the case. Theorem 6.1 . If a pseudo c omp act A b elian gr oup c ontains no infinite c om- p act subsets, then it is Pontrya gin r eflexive . Pr o of. Let G = ( G, T H ) b e a pseudo compact group with no infi nite compact subsets. Th e group of con tin uous c haracters of G is th en precisely H and since G has no in finite compact sub sets, the top ology of this du al group will equal the topology of p oint wise conv ergence on G , therefore G ∧ = ( H , T G ) (see in this connectio n [24]). By Theorem 2.1, ( H , T G ) must b e again a totally boun ded group with prop ert y ♯ and hence with no infinite compact subsets, the same argument as ab o v e then sho ws that G ∧∧ = ( H , T G ) ∧ = ( G, T H ) and ther efore that G is reflexiv e.  This last theorem com bined with Lemma 2.3 and the results of Section 5 pro vides a wide range of examples that answer negati v ely Question 1.1. This question has also b een ans w ered ind ep endently in [1] where anot her collect ion of examples has b een obtained. Corollary 6.2 (SCH) . Every infinite pseudo c omp act Ab elian gr oup G sup- p orts a nonc omp act, pseudo c omp act gr oup top olo gy T H such that ( G, T H ) is r eflexive. 18 JORG E GALINDO AND SERGI O MACARIO Corollary 6.3. Every infinite pseudo c omp act A b elian gr oup G with | G | ≤ 2 2 c supp orts a nonc omp act, pseudo c omp act gr oup top olo gy T H such that ( G, T H ) is r eflexive. Ac kno w ledgemen ts. W e h eartily thank M.G. Tk ac henko for sh aring w ith us a preprin t cop y of [1] and D. Dikranjan for h is remarks on a previous v ersion of this p ap er and for making us a ware of Lemma 4.3. W e are also indebted t o W. W. C omfort an d to D. Dikranjan for their help concernin g Example 5.9. Referen ces [1] S. Ardanza-T revijano, M.J. Chasco, X. Dom ´ ınguez, and M. G. Tk achenko . Precom- pact noncompact refl exive ab elian groups. F orum Mathematicum . T o app ear. [2] A. Arhangel ′ skii and M. G. Tk achenko. T op olo gic al gr oups and r elate d st ructur es , vol ume 1 of Atlantis Studies in M athematics . Atla ntis Press, P aris, 2008. [3] M. J. Chasco and E. Mart ´ ın-Peinador. An approach to duality on ab elian precompact groups. Journal of Gr oup The ory , 11:5:63 5–643, 2008. [4] W. W. Comfort and J. Galindo. Pseudocompact top ological group refin ements of maximal w eigh t. Pr o c. Amer. Math. So c. , 131(4):1311– 1320 (electronic), 2003. [5] W. W. Comfort and I. Gotchev. Cardinal in v arian ts for κ -b o x pro ducts: weigh t, density , c haracter and S ouslin number. Preprint., 2010. [6] W. W. Comfort and D. Remus. Imp osing pseudocompact group top ologies on ab elian groups. F und. Math. , 142(3):221– 240, 1993. [7] W. W. Comfort and L. C. Rob ertson. Cardinalit y constraints for pseudo compact and for totally dense subgroup s of compact topological groups. Paci fic J. Math. , 119(2):265 –285, 1985. [8] W. W. Comfort and K. A. Ross. T op ologies induced by group s of characters. F und. Math. , 55:283– 291, 1964. [9] W. W . Comfort and Kenneth A. Ross. Pseudocompactn ess and unifo rm contin uity in topological groups. Pacific J. Math. , 16:483–496, 1966. [10] D. Dikranjan and A. Giordano Brun o. w-divisible groups. T op olo gy Appl . , 155(4):252– 272, 2008. [11] D. Dikranjan and D. Sh akhmatov. Algebraic structure of pseudo compact groups. Mem. Amer. Math. So c. , 133(633 ):x+83, 1998. [12] D. Dikranjan and D. S hakhmatov. F orcing hereditarily separable compact-like group top ologies on ab elian groups. T op olo gy Appl. , 151(1-3):2–54, 2005. [13] D. Dikranjan and D. Shakh mato v. S elected topics from the structure theory of top o- logical groups. I n Elliott P earl, editor, Op en pr oblems in top olo gy , pages 389–406. Elsevier, 2007. [14] D. Dik ranjan and D. Shakhmatov. Algebraic structure of pseudocompact ab elian groups. Preprint, 2008. [15] E. K. v an Douw en. T he w eigh t o f a pseudo comp act (homogeneous) space whose cardinalit y has countable cofin alit y . Pr o c. Amer. Math. So c. , 80(4):678– 682, 1980. PSEUDOCOMP ACT GR OUPS 19 [16] E. K. v an Douw en. The maximal totally b ounded group topology on G and the b iggest minimal G -space, for ab elian groups G . T op olo gy Appl. , 34(1):69–9 1, 1990. [17] L. F uchs. I nfinite ab eli an gr oups. Vol. I . A cademic Press, New Y ork, 1970. [18] J. Galindo and S . Garc ´ ıa-F erreira. Compact groups con taining dense pseudo compact subgroups without n on-trivial conv ergent sequ ences. T op olo gy Appl. , 154(2):476–49 0, 2007. [19] J. Galindo, S. Garc ´ ıa-F erreira, and A. H . T omita. Pseudo compact group top ologies with p rescribed top ological subspaces. Sci. Math. Jpn. , 70(3):269–279, 2009. [20] J. Galindo and S. Hern´ andez. On a theorem of v an D ouw en. Extr acta Math. , 13(1):115– 123, 1998. [21] J. Galindo and S. Hern´ andez. The concept of b oun dedness and the Bohr compactifi- cation of a MAP abelian group. F und. M ath. , 159(3):195–218, 1999. [22] M. Gi tik and S. Shelah. On den sities of b ox pro d ucts. T op olo gy Appl . , 88 :219–237, 1998. [23] S. H ern´ andez and S. Macario. D ual prop erties in t otally boun ded abelian groups. Ar ch. Math. (Basel) , 80(3):271–283, 2003. [24] S. U. Raczko wski and F. Javier T rigos-Arrieta. Dualit y of totally bounded ab elian groups. Bol. So c. Mat. Mexic ana (3) , 7(1):1–12, 2001. [25] M. G. Tk aˇ cenko. Compactness type properties in topological groups. Cze choslovak Math. J. , 38(113)(2):324–3 41, 1988. Dep ar tmento de Ma tem ´ aticas, Unive rsit a t Ja ume I, Ca mpus R iu Sec, 12071, Castell ´ on Sp ain E-mail addr ess : jgalindo @mat.uji.es, macario@mat.uj i.es