Metrization criteria for compact groups in terms of their dense subgroups

METRIZA TION CRITERIA F OR COMP A CT GR OUPS IN TERMS OF THEIR DENSE SUBGR OUPS DIKRAN DIKRAN JAN AND DMITRI SHAKHMA TO V De dic ate d to Pr ofessor A. V. Arh angel’ski ˘ ı on the o c c asion of hi s 73r d anniversary Abstra ct. According to Comfort, Raczko wski and T rigos -Arrieta, a dense subgroup D of a com- pact abelian group G determines G if the restriction homomorphism b G → b D of the dual groups is a top ologica l isomorphism. W e introduce four conditions on D that are necessary for it to determine G and w e resolve the follo wing question: If one of these conditions holds for every dense (or G δ - dense) subgroup D of G , must G be metrizable? In particular, w e pro ve (in ZF C) t h at a compact abelian group determined by all its G δ -dense subgroups is metrizable, thereby resolving Question 5.12(iii) from [S. Hern´ andez, S. Macario and F. J. T rigos-Arrieta, Uncountable produ cts of deter- mined groups need not b e determined, J. Math. Anal. Appl. 348 (2008), 834–842]. (Un der the additional assumption of the Contin uum Hyp othesis CH, the same statemen t was pro ved recently b y Bruguera, Chasco, Dom ´ ı nguez, Tk ac henko and T rigos-Arrieta.) As a tool, we develop a mac hinery for building G δ -dense subgroups without uncountable compact subsets in compact groups of w eight ω 1 (in ZFC). The construction is delicate, as these subgroups must hav e non-trivial con vergen t sequences in some mo dels of ZFC. Al l sp ac es and top olo gic al gr oups ar e assume d to b e Hausdorff. Recall that a topological space X is calle d: • κ -b ounde d (for a giv en cardinal κ ) if the closure of ev ery sub s et of X of cardin alit y at most κ is compact , • c ountably c omp act if ev ery coun table op en co v er of X has a finite sub cov er, • pseudo c omp act if every real-v alued contin uous function defi n ed on X is b ounded. It is w ell kno wn that compact → κ -b ounded → ω -b ou n ded → co untably compact → pseudo compact for ev ery infinite cardinal κ . Sym b ols w ( X ), nw ( X ) and χ ( X ) denote th e w eigh t, the net work weig ht and the charac ter of a space X , resp ectiv ely . All u ndefined top ologic al terms can b e found in [21]. As usual, N denotes the set of n atural n umber s , P denote s the set of all prime n um b ers, Z denotes the group of intege rs, Z ( p ) = Z /p Z denotes the cycl ic group of order p ∈ P with the discrete top ology and T d en otes the circle group with its usual top ology . The symbol c denotes the cardinalit y of the cont inuum, ω 1 denotes the fir st uncounta ble cardinal and ω = | N | . Clearly , ω < ω 1 . By Canto r’s theorem, ω 1 ≤ c . The Con tin uum Hyp othesis CH sa ys that ω 1 = c . W e recall that this equalit y is b oth consistent with and indep endent of the usual Zermelo-F raenke l axioms ZF C of set theory [28 ]. Recall that a cardinal τ is str ong limit if 2 σ < τ for ev ery cardinal σ < τ . F or an ordinal (in particular, for a cardinal) α , we denote by cf ( α ) th e cofinalit y of α . F or a cardin al κ and a set Key wor ds and phr ases. dual group, determined group, qu asi-con vexely dense set, pseud ocompact, coun tably compact, ω - b ounded, Bernstein set. The fi rst named author was p artially sup p orted by SRA, grants P1-0292-0101 and J1-964 3-0101, and by gran t MTM2009 -14409-C02-01. The second named author w as partially supp orted b y the Gran t-in-Aid for Scientific Research (C) N o. 22540089 by th e Japan Society for the Promotion of Science (JSPS). 1 2 D. DIKRANJAN AND D. SHAKHMA TOV X , the sym b ol [ X ] ≤ κ denotes the family of all subsets of set X havi ng cardinalit y at most κ . All undefin ed set-theo retic terms can b e found in [28]. 1. Introduction Let G b e an ab elian top ological group . W e denote by b G th e d ual group of all contin uous c h aracters endo wed with the compact-op en topology . F ollo wing [9, 10], w e sa y that a dense sub group D of G determines G if the restriction homomorphism b G → b D of the dual group s is a topological isomorphism. According to [9, 10], G is said to b e determine d if ev ery dense subgroup of G determines G . T he cornerstone in this topic is the follo wing theorem due to Chasco and Außenhofer: Theorem 1.1. [2, 7] Every metrizable ab e lian gr oup is determine d. A remark able partial inv erse of this theorem w as pro ve d b y Hern´ andez, Macario and T rigos- Arrieta. (Under the assum ption of the Contin uum Hyp othesis, this wa s established earlier by Comfort, Raczk o wski and T rigos-Arrieta in [9 , 10]). Theorem 1.2. [2 4, Corollary 5.11] Every c omp act determine d ab elian gr oup is metriza ble. While Theorem 1.1 sa ys that every dense subgroup of a metrizable ab elian group determines it, Theorem 1.2 asserts that ev ery non-metrizable compact ab elian group n ecessarily con tains some dense su bgroup that does not determine it. A subgroup D of a top ological group G is called G δ -dense in G if D ∩ B 6 = ∅ f or ev ery n on-empt y G δ -subset B of G [12]. The follo wing classica l result is due to C omfort and Ross [12]: Theorem 1.3. A dense sub gr oup D of a c omp act g r oup G is pseudo c omp act if and only if D is G δ -dense in G . The follo wing question w as asked by Hern´ andez, Macario and T rigos-Arrieta in [24, Question 5.12(ii i)]: Question 1.4. Do es th er e exist (in ZF C) a non-metrizable compact ab elian group G su ch that ev ery G δ -dense subgroup D of G determines G ? This question w as also rep eated in [14, Question 4.1 2]. It is useful to state explicitly the n egatio n of the statemen t in Q uestion 1.4: Question 1.5. Let G b e a compact ab elian group such that ev ery G δ -dense sub group of G deter- mines G . Must G b e metrizable (in ZF C)? By Theorem 1.3, one can replace “ G δ -dense” by “den s e pseud o compact” in b oth questions to get th eir equiv alen t v ersions. Theorem 1.2 says that a compact ab elian group G is metrizable pro v id ed that every dense subgroup of G determines it. Since G δ -dense subgroups of G are d ense in G , a p ositiv e answ er to Question 1.5 (equiv alen tly , a n egativ e answer to Question 1.4) would pro vide a strengthening of Theorem 1.2, b ecause one would get the same conclusion und er a we ak er assum ption of requirin g only a muc h smaller family of G δ -dense su bgroups of G to determine it. O ne of the goals of this pap er is to accomplish precisely this, without recourse to an y add itional set-theoretic assum ptions b ey ond Zermelo-F rae nkel axioms ZF C of set theory . Remark 1.6. Chasco, Dom ´ ınguez and T rigos-Arrieta pro v ed r ecently that ev ery co mpact abelian group G with w ( G ) ≥ c has a G δ -dense subgroup wh ic h do es not determine G [8, Theorem 14]. Indep en den tly , Br u guera and Tk ac henko pro v ed that every compact ab elian grou p G with w ( G ) ≥ c con tains a p rop er G δ -dense reflexiv e subgroup D [6, Th eorem 4. 7]. As m entioned in th e the end of [8, Section 3], this D cann ot determine G . (In deed, b b D = D 6 = G = b b G implies b D 6 = b G .) It is clear that, under the assumption of the Continuum Hyp othesis, these r esults yield a c onsistent p ositive answer to Question 1.5 and ther efor e, a c onsistent ne gative answer to Question 1.4. METRIZA TION CRITERIA F OR COMP ACT GROUPS IN TER MS OF THEIR DENSE SUBGROUPS 3 An ov erview of the pap er follo ws. Inspired by Questions 1.4 and 1.5, in Section 2 we in tro duce four prop erties th at every dense subgroup determinin g a compact ab elian group must ha v e (see Diagram 1), thereby making a firs t attempt to clarify the “fine structure” of th e n otion of determination. Section 3 collects some basic facts ab out the introd uced prop erties that h elp the reader in b etter understand ing of these n ew notions. In Sect ion 4, we in v estigate what happ ens to a compact group wh en all its dense (or all it s G δ -dense) subgroups are assumed to hav e one of the four prop erties in tro d u ced in Section 2. Our results in S ection 4 su bstan tially clarify the “fine structure” of the notion of a determined group b y addressing the follo win g question systematically: “Ho w muc h determination” of a compact group is really necessary in o rder t o get its met rization in the spirit of T heorem 1.2 or Question 1.5? As it turns out, suc h metrization criteria can b e obta ined under muc h w eak er conditions than full determination; see Theorems 4.2 and 4.5. In turn, Theorems 4.1, 4.3 and 4.4 ser ve to demonstrate that the conditions equiv alen t to the metrization of a compact group in Theorems 4.2 and 4.5 are the b est p ossible, thereb y pinp oint ing the exa ct prop ert y among the four necessary conditions “resp onsible” for b oth the v alidit y of Th eorem 1.2 and the p ositiv e answer to Qu estion 1.5. The answ er to Question 1.5 itself comes as a particular coroll ary of the main result; see Corollary 4.6. An added b on u s of our approac h is that man y results in this section hold for n on -ab elian compact groups as w ell, whereas the notio n of a d etermined group is restricted to the a b elian ca se. (A non-comm utativ e version of a determined group w as introd uced recentl y in [22].) In Section 5 w e d evelo p a mac hinery for constructing G δ -dense sub groups D without u ncount able compact su bsets in compact groups G of weigh t ω 1 . F urthermore, when G b elongs to a fi xed v ariery V of group s, th e su bgroup D can b e c hosen to b e a free group in the v ariet y V . Our mac hinery works in ZFC alone. As in Section 4, results in this section do not require G to b e ab elian. T h e primary no v elt y here is our ability to hand le successfully small weigh ts o f G (lik e ω 1 ) at the exp ense of “killing” only u nc ountable compact subsets of D . All known constru ctions in the literature usually “kill” all infinite compact subsets, thereb y eliminating also all non-trivial conv ergen t sequences in D , b ut this stronger conclusion is accomplished at the exp ense of ha ving b een able to handle only groups G of w eight c . In f act, this difference is in heren t in the n ature of the problem and not purely coinciden tal. Indeed, Remark 5.6 sho w s that the group D w e construct must h a ve non-trivial conv ergen t sequences un der some additional set- theoretic assumptions. As a particular corollary of our results, we pro duce a pseudo compact group top ology on the fr ee group F c with c man y generators without uncountable compact subsets. A recen t resu lt by Thom [33] implies that suc h a top ology on F c m ust n ecessarily conta in a non-trivial conv ergen t sequence; see Remark 5.7. Sections 6, 7 and 8 are dev oted to th e pro ofs of the main results from Sections 4 and 5. Section 9 con tains some examples sho win g th e limits of our results, and Section 10 lists op en pr oblems related to the topic of this pap er. 2. F our nec e ssar y conditions for d e termina tion of a comp act abelian group In this section we introd uce four conditions and sho w that they are all necessary for d etermination of a co mpact ab elian group . Definition 2.1. Let X b e a space. (i) W e sh all say that X is w -c omp act if th ere exists a compact su bset C of X su c h that w ( C ) = w ( X ). (ii) W e shall sa y th at X has the A rhangel’ski ˘ ı pr op erty (or is an Arhangel’ski ˘ ı sp ac e ) pro vided that w ( X ) ≤ | X | . The letter w in front of “compact” in item (i) is in tended to abb r eviate the w ord “w eigh t”, but one can also view it as an abbreviation of the word “w eak”, as ev ery compact sp ace is ob viously w -compact. 4 D. DIKRANJAN AND D. SHAKHMA TOV The name f or the class of sp aces in item (ii) was c hosen to pay tribute to the first manuscript of Professor Arhangel’ski ˘ ı [1] where he int ro du ced the notion of n et work we igh t an d demonstrated its imp ortance in the stu dy of compact spaces. A celebrated result of Arhangel’ski ˘ ı fr om [1] sa ys that w ( X ) = nw ( X ) ≤ | X | for every compact space X . In our terminology , this means that ev ery compact sp ace h as the Arhangel’ski ˘ ı prop ert y . In fact, a bit more can b e said. Indeed, let X b e a w -compact space. Then X con tains a co mpact subset C suc h that w ( C ) = w ( X ). Com bining this with the ab o ve result of Arhangel’ski ˘ ı, w e obtain w ( X ) = w ( C ) ≤ | C | ≤ | X | . Therefore, X h as the Arhangel’ski ˘ ı prop erty . This argumen t sho ws that ( α ) a w -compact space has the Arhangel’ski ˘ ı prop ert y . Definition 2.2. Let G b e a top ological group. (i) W e shall sa y that G is pr oje ctively w -c omp act if every cont inuous homomorphic image of G is w -compact. (ii) W e shall sa y that G is pr oje ctively A rhangel’ski ˘ ı if ev ery con tin uous homomorphic image of G has the Arhangel’ski ˘ ı prop erty . Since compactness is p reserv ed by contin uous images and compact spaces are w -compact, all compact group s are p ro jectiv ely w -compact. F rom ( α ) and De finition 2.2(ii) we get ( β ) p ro jectiv ely w -compact group s are pro jectiv ely Arhangel’ski ˘ ı. The follo w ing n ecessary condition for determination w as foun d b y the authors in [16]. S ince it pla y s a crucial role in the p resen t pap er, we pro vide a shorter self-cont ained pro of of this result requiring n o recourse to the notion of qc-densit y that w as essential in [1 6]. Theorem 2.3. [16, Corollary 2. 4] If a su b gr oup D of an infinite c omp act ab elian gr oup G determines G , then D c ontains a c omp act subset X such that w ( X ) = w ( D ) . Pr o of. F or a subset X of G and an op en neighbour ho o d V o f 0 in T , let W ( X , V ) = { χ ∈ b G : χ ( X ) ⊆ V } . Since D d etermines G and b G is discrete, there exists a compact sub set X of D and an op en neig hbour ho o d V of 0 suc h that W ( X , V ) = { 0 } . Let π : b G → C ( X , T ) b e th e restriction homomorphism defined b y π ( χ ) = χ ↾ X for χ ∈ b G , wh ere C ( X , T ) denotes the group of all co nt inuous functions from X to T equip p ed w ith the co mpact-op en top ology . Since k er π ⊆ W ( X , V ) = { 0 } , π is a monomorph ism, and so w ( G ) = | b G | = | H | , where H = π ( b G ). F urtherm ore, U ∩ H = { 0 } , where U = { f ∈ C ( X, T ) : f ( X ) ⊆ V } is an op en subset of C ( X , T ), so H is a discrete subgroup of C ( X, T ). Therefore, | H | = w ( H ) ≤ w ( C ( X , T )) = w ( X ) + ω by [21, Prop osition 3.4.16]. This prov es that w ( G ) ≤ w ( X ) + ω . T o finish the p ro of of th e inequalit y w ( G ) ≤ w ( X ), it s u ffices to show that X is infinite. Indeed, assume that X is finite. T hen C ( X , T ) = T X is compact, and so the discrete subgroup H of C ( X , T ) m ust b e finite. This con tradicts the fact that | H | = | b G | ≥ ω , as G is infinite. F inally , the r everse inequalit y w ( X ) ≤ w ( G ) is clear.  The relev ance of the four notions int ro du ced in Definitions 2.1 and 2.2 to the topic of our pap er is evident from the follo wing corollary of this theorem. Corollary 2.4. If a sub gr oup D of a c omp act ab elian g r oup determines it, then D is pr oje ctively w - c omp act. Pr o of. Let D b e a dens e subgroup of a compact ab elian group G th at determines G , an d let f : D → N b e a contin uous h omomorphism on to some top ological group N . Then f can b e extended to a con tin uous group homomorphism f rom G to the completion H = b N of N , and w e denote this extension by the same letter f . S ince D determines G , the dense sub group f ( D ) of the compact group f ( G ) = H d etermines H [10, C orollary 3.15 ]. If H is finite, th en f ( D ) = H is compact, so trivially w -compact. If H is infinite, w e apply Theorem 2.3 to conclude that f ( D ) con tains a compact set X with w ( X ) = w ( H ) = w ( f ( D )). That is, f ( D ) is w -compact. Th is sh o ws that D is pro j ective ly w -compact.  METRIZA TION CRITERIA F OR COMP ACT GROUPS IN TER MS OF THEIR DENSE SUBGROUPS 5 The relations b et w een the p rop erties introdu ced ab o v e in the class of precompact ab elian group s can b e summarized in the follo wing diagram: compact / / determining the completion ( 2.4 )   metrizable ( 1.1 ) o o pro jectiv ely w -compact   ( β ) / / pro jectiv ely Arhangel’ski ˘ ı   w -compact ( α ) / / Arhangel’ski ˘ ı Diagram 1. This diagram s ho w s that four pr op erties from Defin itions 2.1 and 2.2 are necessary for deter- mination of the completion of a precompact abelian group. With an exception of the arro w (2.4), none of the other arr o ws in Diagram 1 are inv ertible. A d ense subgroup D of a compact group G that d etermines G need not b e either compact or metrizable. T o see this, it suffices to r ecall that the dir ect su m L α<ω 1 T of ω 1 copies of T determines T ω 1 ; see [10, Corollary 3.12]. In Example 9.1, we exh ib it a pseudo compact p ro jectiv ely Arhangel’ski ˘ ı group D that is not w - compact. (F urthermore, under the assum p tion of the CH, D can b e c hosen to b e ev en coun tably compact.) In particular, n either the arrow ( α ) nor the arro w ( β ) is rev ersible. F or ev ery infinite cardinal κ , there exists a κ -b ounded w -compact (thus, Arhangel’ski ˘ ı) ab elian group th at is not p ro jectiv ely Arhangel’ski ˘ ı (and so is not p ro jectiv ely w -compact); see E x amp le 9.2. W e do not know if the arro w (2.4) in Diagram 1 is inv ertible. In fact, it is tempting to conjecture that Corolla ry 2.4 giv es not on ly a n ecessary b ut also a sufficien t condition for determination of a compact ab elian group by its d ense sub group. Question 2.5. Do es e v ery dense pro jectiv ely w -compact sub grou p of a compact ab elian group determine it? W e refer the reader to Remark 10.2(ii) for a partial p ositiv e answer to this question. 3. Pr oper ties of Arhan ge l ’sk i ˘ ı sp aces and projectivel y Arhangel ’ski ˘ ı gr oups Our fir st remark shows that the Arhangel’ski ˘ ı prop ert y is “local”. Remark 3.1. F or every sp ac e X , the ine qualities w ( X ) ≤ | X | and χ ( X ) ≤ | X | ar e e q uivalent. Prop osition 3.2. (i) L o c al ly c omp act sp ac es have the A rhangel’ski ˘ ı pr op erty. (ii) First c ountable (in p articular, metric) sp ac es have the Arhangel’ski ˘ ı pr op erty. (iii) The class o f A rhangel’ski ˘ ı sp ac es is close d under taking p erfe ct pr eimages; that is, if f : X → Y is a p erfe ct map fr om a sp ac e X onto an Ar hangel’ski ˘ ı sp ac e Y , th en X has the Arh angel’ski ˘ ı pr op erty. (iv) If w ( X ) is a str ong limit c ar dinal, then X has the Arhangel’ski ˘ ı pr op erty. Pr o of. (i) Let X b e a lo cally compact s pace. If X is finite, then X h as the Arhangel’ski ˘ ı pr op ert y . Supp ose that X is infi nite. Since the one-p oin t compactification Y of X is compact, it h as th e Arhangel’ski ˘ ı pr op ert y , so w ( Y ) ≤ | Y | . Since X is infinite and Y \ X is a s ingleton, | Y | = | X | . Since X is a subspace of Y , we get w ( X ) ≤ w ( Y ). Th is pro ves that w ( X ) ≤ | X | . (ii) F or finite spaces X , this follo ws from (i). If X is infi nite, then the conclusion follo ws from Remark 3.1. 6 D. DIKRANJAN AND D. SHAKHMA TOV (iii) Sin ce finite sp aces hav e the Arhangel’ski ˘ ı pr op ert y by (i), w e shall assume that X is infinite. There exists a one-to-one con tin uous map g : X → Z onto a space Z suc h that w ( Z ) ≤ n w ( X ) ≤ | X | [1]. Let h : X → Y × Z b e the diagonal pro d uct of f and g defined b y h ( x ) = ( f ( x ) , g ( x )) for all x ∈ X . Sin ce f is a p erfect map, so is h [21, Th eorem 3.7.9]. Since g is one-to-one, h is an injection. It follo ws that X and h ( X ) are homeomorphic, so (1) w ( X ) ≤ w ( h ( X )) ≤ w ( Y × Z ) = max { w ( Y ) , w ( Z ) } ≤ m ax { w ( Y ) , | X |} . Since Y has the Arhangel’ski ˘ ı pr op ert y , w ( Y ) ≤ | Y | = | f ( X ) | ≤ | X | . Combining this with (1), we conclude th at w ( X ) ≤ | X | . Th us, X has the Arhangel’ski ˘ ı prop ert y . (iv) Since d ( X ) ≤ w ( X ), w ( X ) ≤ 2 d ( X ) and w ( X ) is a strong limit cardinal, w ( X ) = d ( X ) ≤ | X | .  Prop osition 3.3. If a top olo g i c al gr oup G c ontains a dense sub gr oup H with th e Arh angel’ski ˘ ı pr op erty, then G itself has the Arhan gel’ski ˘ ı pr op erty. Pr o of. Since H is dense in G , χ ( H ) = χ ( G ). Since H h as the Arhangel’ski ˘ ı pr op ert y , χ ( H ) ≤ w ( H ) ≤ | H | . Since H is a subgroup of G , | H | ≤ | G | . Th is sho ws that χ ( G ) ≤ | G | . Therefore, G has the Arhangel’ski ˘ ı prop ert y by Remark 3.1.  This p rop osition do es n ot hold for spaces s in ce one ma y ha v e w ( Y ) < w ( X ) when Y is a dense subspace of X . Prop osition 3.4. Eve ry pseudo c omp act gr oup G such tha t w ( G ) ≤ c is pr oje ctively Arh angel’ski ˘ ı. Pr o of. Indeed, let f : G → H b e a con tin uous su rjectiv e h omomorp hism of G on to a top ological group H . Th en H is pseudo compact, as a contin uous image of the pseudo compact space G . If H is finite, then H has the Arhangel’ski ˘ ı p rop ert y by Pr op osition 3.2(i). Assume no w that H is infinite. Then | H | ≥ c [20, Pr op osition 1.3(a)]. T o sho w that H has the Arhan gel’ski ˘ ı pr op ert y , it suffices to note that w ( H ) ≤ c . Indeed, let b f : b G → b H b e the extension of f ov er the completion b G of G . Since b G is compact and b f is surjectiv e, w ( H ) = w ( b H ) ≤ w ( b G ) = w ( G ) ≤ c .  Item (i) of our next prop osition sho ws that th e restriction on w eight in Prop osition 3.4 is the b est p ossible, w hile item (ii) of Prop osition 3.5 shows that ev en groups “arbitrarily close” to compact need not ha ve the Ar hangel’ski ˘ ı prop erty . (Compare this with Prop osition 3.2(i).) Prop osition 3.5. (i) Every c omp act gr oup G with w ( G ) = c + has a dense c ountably c omp act sub gr oup without the Arhangel’ski ˘ ı pr op erty. (ii) F or every infinite c ar dinal κ , e ach c omp act gr oup G of weight τ = 2 2 2 κ has a dense κ - b ounde d sub gr oup without the Arhangel’ski ˘ ı pr op erty. Pr o of. (i) Since c + ≤ 2 c , app lying [26, Theorem 2.7] w e can c ho ose a dense subgroup H of G suc h that | H | = c . By the stand ard closing-o ff argumen t, w e can find a coun tably compact subgroup D of G s uc h that H ⊆ D and | D | ≤ c . Since H is d ense in G , so is D . Since | D | = c < c + = w ( G ) = w ( D ), D do es not ha ve the Arhangel’ski ˘ ı prop ert y . (ii) By [26, Theorem 2.7], G con tains a dense su bgroup H of size 2 2 κ . L et D b e the κ -closure of H in G ; th at is, D = S  A : A ∈ [ H ] ≤ κ  , where A denotes the closure of A in G . Clearly , D is a subgroup of G conta ining H , so D is d ense in G . Since   [ H ] ≤ κ   ≤ 2 2 κ and | A | ≤ 2 2 κ for ev ery A ∈ [ H ] ≤ κ , w e conclude that | D | ≤ 2 2 κ < 2 2 2 κ = w ( D ). Therefore, D do es not ha v e the Arhangel’ski ˘ ı prop erty .  4. Metrizability of comp act gr o ups via conditions on their dense s ubgroups Our first th eorem demonstrates that th e w eak est condition in Di agram 1 is not sufficient for getting the metrizabilit y of a compact grou p G even when th is condition is imp osed on all dens e subgroups of G . METRIZA TION CRITERIA F OR COMP ACT GROUPS IN TER MS OF THEIR DENSE SUBGROUPS 7 Theorem 4.1. Every dense sub gr oup of a c omp act gr oup G has the Arh angel’ski ˘ ı pr op erty if and only if w ( G ) is a str ong limit c ar dinal. Our second theorem sho ws that the pro jectiv e v ersion of the wea k est condition in Diag ram 1 imp osed on al l dense subgroups of a compact group G suffices to obtain its metrizabilit y . Theorem 4.2. Ev ery dense sub gr oup of a c omp act gr oup G is pr oje ctively A rhangel’ski ˘ ı if and only if G is metrizable. Since a dense determining subgroup of a compact ab elian group is pro jectiv ely Arhangel’ski ˘ ı (see Diagram 1), in the ab elian case the “only if ” part of this result strengthens Theorem 1.2 b y offering the same co nclusion under a m uch w eak er assump tion. F or a cardinal σ , the minimum cardinalit y of a pseudo compact group of w eigh t σ is denoted by m ( σ ) [1 1]. The next theorem is a coun terpart of Theorem 4.1 for G δ -dense sub groups. Theorem 4.3. Every G δ -dense sub g r oup of a c omp act gr oup G has the A rhangel’ski ˘ ı pr op erty if and only if m ( w ( G )) ≥ w ( G ) . Our next result is the coun terpart of Theorem 4.2 with “dense” replaced b y “ G δ -dense”. Theorem 4.4. F or a c omp act gr oup G , the fol lowing c onditions ar e e quivalent: (i) every G δ -dense (e q uivalently, e ach dense pseudo c omp act) sub gr oup of G is pr oje ctively Arh angel’ski ˘ ı; (ii) al l dense c ountably c omp act sub gr oups of G ar e pr oje ctively A rhangel’ski ˘ ı; (iii) w ( G ) ≤ c . This theorem sho ws th at ha ving all G δ -dense su bgroups of a compact group G p ro jectiv ely Arhangel’ski ˘ ı is n ot s u fficien t for ob taining metrizabilit y of G . Our next theorem sh o w s that strengthening “pro jectiv ely Arhangel’ski ˘ ı” to “pro jectiv ely w -compact” yields metrizabilit y of G in case when G is either connected or ab elian. Theorem 4.5. L et G b e a c omp act gr oup that is either ab elian or c onne cte d. If al l G δ -dense (e q uiv- alently, al l dense pseudo c omp act) sub gr oups of G ar e pr oje ctively w -c omp act, then G is metrizable. Com bining this result with Corollary 2. 4, we obtain the follo win g corollary s olving Question 1.4 in the negativ e and Question 1. 5 in the p ositiv e. Corollary 4.6. If al l G δ -dense sub g r oups of a c omp act ab elian gr oup G determine it, then G is metrizable. Under the assu m ption of the Con tin uum Hyp othesis, the follo wing stronger v ersion of Theorem 4.5 can b e obtained in the ab elian case . Theorem 4.7. Assume CH. If al l dense c ountably c omp act sub gr oups of a c omp act ab elian gr oup G ar e pr oje ctively w -c omp act, then G is metrizable. Since coun table compactness is stronger then p seudo compactness and a dense pseudo compact subgroup of a compact ab elian group is G δ -dense in it (Th eorem 1.3), our Theorem 4.7 strengthens also the consisten t result typeset in italics in Remark 1.6. The pro ofs of Theorems 4.1, 4.2, 4.3, 4.4 are p ostp oned u n til Section 6 , wh ile the pro ofs of Theorems 4.5 and 4.7 are p ostp oned until Section 8. Let G b e any compact ab elian group of weigh t ω 1 . It follo w s fr om Theorem 4.4 that all G δ -dense subgroups of G are pro jectiv ely Arhangel’ski ˘ ı, ev en though G is not metrizable. This sh o ws that “pro jectiv ely w -compact” ca nnot b e w eak en ed to “pro j ectiv ely Arhangel’ski ˘ ı” in the assumption of Theorems 4.5 and 4.7. F u rthermore, since ω 1 is not a strong limit cardinal, Theorem 4.1 implies that G has a dense subgroup without the Arhangel’ski ˘ ı pr op ert y . C om b ining Theorem 4.1 with 8 D. DIKRANJAN AND D. SHAKHMA TOV Example 9.3(ii) b elo w, we obtain compact ab elian groups G of arbitrarily large weigh t such that ev ery G δ -dense subgroup of G has the Arhangel’ski ˘ ı pr op ert y , but there exists a d ense subgroup of G without the Arhangel’ski ˘ ı pr op ert y . W e fi n ish this section with the follo wing corolla ry of its main results. Corollary 4.8. F or a c omp act ab elian gr oup G , the fol lowing c onditions ar e e quivalent: (i) G is metriza ble; (ii) every dense sub gr oup of G determines G ; (iii) every G δ -dense (e quivalently, e ach dense pseudo c omp act) sub gr oup of G determines G ; (iv) every dense sub gr oup of G is pr oje c tively A rhangel’ski ˘ ı; (v) every G δ -dense (e quivalently, e ach dense pseudo c omp act) sub gr oup of G is pr oje ctively w - c omp act. F urthermor e, under CH, the fol lowing two items c an b e adde d to the list of e qu ivalent c onditions (i)–(v): (vi) every dense c ountably c omp act sub gr oup of G determines G ; (vii) every dense c ountably c omp act sub gr oup of G is pr oje ctively w -c omp act. Pr o of. (i) → (ii) is Theorem 1.1, (ii) → (iv) follo ws from Diagram 1, (iv) → (i) f ollo ws from Theorem 4.2. (i) → (iii) follo w s from T h eorem 1.1, (iii) → (v) follo ws from Corollary 2.4, (v) → (i) is Th eorem 4.5. (i) → (vi) f ollo ws fr om Theorem 1.1, (vi) → (vii) follo ws fr om Corollary 2.4. Finally , (vii) → (i) is Theorem 4.7. (W e note that only the last implication n eeds CH.)  5. Pseudocomp act gr oups o f sm a ll weight without unc ount able comp act subsets F or a subset X of a group G we denote by h X i the su b group of G generated by X . By a variety of gr oups w e mean, as u sual, a class of groups closed und er taking Cartesian pro du cts, subgroups and quotien ts (i.e., a close d class in the sense of Birkh off [5]). Another, equiv alen t, wa y of d efining a v ariet y is by giving a fixed family of ident ities satisfied b y all group s of the v ariet y ([5]; see also [30, Th eorem 15.51]). Definition 5.1. Let V b e a v ariet y of groups. (a) Recall that a subset X of a group G is called V - indep endent pro vided that the follo w ing t wo conditions are satisfied: (i) h X i ∈ V ; (ii) for ev ery map f : X → G with G ∈ V , there exists a homomorphism ˜ f : h X i → G extending f . (b) F or ev ery group G ∈ V the ca rdin al r V ( G ) = sup {| X | : X is V -indep end ent subset of G } is called the V -r ank of G . (c) A group G is V -fr e e if G is generated by its V -indep end ent su bset X . W e call this X th e gener ating set (or the set of gener ator s of G and we write G = F V ( X ). Theorem 5.2. L et V b e a variety of gr oups and L b e a c omp act metric gr oup that b elongs to V such that r V ( L ω ) ≥ ω . L et I b e a set such that ω 1 ≤ | I | ≤ c . Then the gr oup L I c ontains a G δ -dense (so dense pseudo c omp act) V -fr e e sub gr oup D of c ar dinality c such that al l c omp act subsets of D ar e c ountable; in p articular D is not w-c omp act. The pro of of this theorem is p ostp oned unt il Secti on 7. Corollary 5.3. L et L b e a c omp act simple Lie gr oup. Then for every unc ountable set I o f size at most c , the gr oup L I c ontains a G δ -dense fr e e sub gr oup D of c ar dinality c such that al l c omp act subsets of D ar e c ountable; in p articular D is not w-c omp act. METRIZA TION CRITERIA F OR COMP ACT GROUPS IN TER MS OF THEIR DENSE SUBGROUPS 9 Pr o of. By [3, Theorem 2], r G ( L ω ) ≥ r G ( L ) ≥ ω , where G is the v ariet y of all groups. Now w e can apply Th eorem 5.2 with V = G .  Corollary 5.4. F or every non-trivial c omp act metric ab elian gr oup L and every u nc ountable set I of size at most c , the gr oup L I c ontains a G δ -dense sub gr oup D of c ar dinality c such th at al l c omp act subsets of D ar e c ountable; in p articular D is not w-c omp act. F urthermor e , if L is unb ounde d, then D c an b e chosen to b e fr e e. Pr o of. W e consider t w o cases. Case 1 . L is b ounde d . Let n be th e ord er of L , and let A n b e the v ariet y of ab elian groups of order n . Then L ∈ A n and r A n ( L ω ) ≥ ω , so the conclusion follo ws f rom Th eorem 5.2 app lied to V = A n . Case 2 . L is unb ounde d . Let A b e the v ariet y of all abelian groups . Then r A ( L ω ) ≥ ω , so the conclusion follo ws from Theorem 5. 2 applied to V = A .  F ollo wing [15, Definition 5.2] , we say that a v ariet y V is pr e c omp act if V is g enerated b y its finite group s . One can find a host of conditions equiv alen t to precompactness of a v ariet y in [15, Lemma 5.1] . In p articular, it is worth noting in connection with Theorem 5.2 that the existence of a compact group L ∈ V with r V ( L ) ≥ ω is equiv alen t to precompactness of th e v ariet y V [15, Lemma 5.1]. Most of the w ell-kno wn v arieties are precompact; see [15, Lemma 5.3] and the commen t follo wing this lemma. The Burnsid e v ariet y B n for odd n > 665 is not pr ecompact [13]. Corollary 5.5. F or a variety V , the fol lowing c onditions ar e e quivalent: (i) V is pr e c omp act; (ii) for ev e ry c ar dinal σ with ω 1 ≤ σ ≤ c , the V -fr e e gr oup with c many gener ators admits a pseu- do c omp act gr oup top olo gy of weight σ without unc ountable c omp act subsets; in p articular, this top olo gy is not w - c omp act. Pr o of. (i) → (ii) Supp ose that V is precompact. By [15, Lemma 5.1], there exists a co mpact metric group L ∈ V with r V ( L ) ≥ ω . Since r V ( L ω ) ≥ r V ( L ), applyin g Theorem 5.2 w e get (ii) . (ii) → (i) Th is follo w s f r om [15, Theorem 5.5].  Our next remark s ho w s th at Theorem 5.2 and its Corollaries 5.3, 5.4 and 5.5 are the b est p ossible results that one can obtain in ZF C. Remark 5.6. Assu me MA+ ¬ CH, w h ere MA stands for Martin’s Axiom. In Th eorem 5.2 a nd Corollaries 5.3, 5.4, tak e I to b e a set of size ω 1 , and let D b e the group as in the conclusion of these results. In Corollary 5.5, let σ = ω 1 and let D denote the V -free group with c many generators. Then D is a top ological group of w eigh t ω 1 < c . Since MA holds, ev ery coun table subgroup of D is F r ´ ec het-Urysohn [29]; in p articular, D cont ains man y non-trivial con v ergen t sequences. Therefore, “all compact subsets of D are coun table” cannot b e strengthened to “all compact subsets of D are finite” in co nclusions of Theorem 5.2 and Corollaries 5.3, 5.4, and “without un coun table compact subsests” cann ot b e strengthened to “without infinite compact subsests” in the the conclusion of Corollary 5.5. Recall that the strongest tota lly b ou n ded group top ology on a group is called its Bohr top olo gy . Remark 5.7. Thom recen tly prov ed that the free grou p with t w o generators equipp ed w ith its Bohr top ology con tains a n on-trivial con verge nt sequence [33]. T h is easily implies that every p recompact group top ology on the free group with t wo generato rs conta ins a non-trivial con ve rgen t sequence. Since pseud o compact groups are p recompact, it follo ws that every pseudo c omp act fr e e g r oup of size c c ontains a non-trivial c onver gent se quenc e . Combining this with Theorem 1.3, we conclude that the group D as in the co nclusion of C orollary 5.3 con tains a non-trivial con v ergen t sequence. This 10 D. DIKRANJAN AND D. SHAKHMA TOV sho ws that “all compact sub sets of D are counta ble” cann ot b e strengthened to “all compact sub sets of D are fin ite” in the conclusion of Corollary 5.3 and “without uncounta ble compact sub sests” cannot b e str en gthened to “without in fi nite compact sub sests” in the the conclusion of Corollary 5.5 when V is the v ariet y of all groups. 6. Pr oofs of Theore ms 4.1, 4.2, 4.3 , 4.4. Pro of of Theorem 4.1: Su pp ose that w ( G ) is not a strong limit cardinal. Then there exists a dense su bgroup D of G such that | D | = d ( G ) < w ( G ) = w ( D ); see [26, Theorem 2.7]. Hence, D do es not ha ve the Arhangel’ski ˘ ı p rop erty . Supp ose n o w that w ( G ) is a strong limit cardinal. Let D b e a dense subgroup of G . Since w ( D ) = w ( G ), the cardinal w ( D ) is strong li mit. Hence, D has the Arhangel’ski ˘ ı prop ert y by Prop osition 3.2 (iv).  Pro of of T he orem 4.3: Assume that every G δ -dense sub group of G has the Ar hangel’ski ˘ ı prop- ert y . Ac cording to [11], G h as a G δ -dense subgroup D of size m ( σ ). Since D has the Arhangel’ski ˘ ı prop erty , th is yields m ( σ ) = | D | ≥ w ( D ) = w ( G ) = σ . C onv ersely , if m ( σ ) ≥ σ holds, then for ev ery G δ -dense subgroup D of G , one has | D | ≥ m ( σ ) ≥ σ = w ( D ), so D h as the Arhangel’ski ˘ ı prop erty .  F act 6.1. [27, Lemma 1.5] L et G b e an infinite c omp act gr oup. F or every infinite c ar dinal τ ≤ w ( G ) ther e exists a c ontinuous homomorp hism f : G → H of G onto a c omp act g r oup H with w ( H ) = τ . F act 6.2 . Su pp ose that f : G → H is a c ontinuous surje ctive homomorp hism of c omp act ab elian gr oups, D is a sub gr oup of H and D 1 = f − 1 ( D ) . (i) If D is dense i n H , then D 1 is dense in G . (ii) If D is pseudo c omp act (c ountably c omp act, κ - b ounde d for some infinite c ar dinal κ ), then D 1 has the same pr op erty. (iii) If D is not (pr oje ctively) w -c omp act, then D 1 is not pr oje ctively w -c omp act e i ther. (iv) If D is not (pr oje ctively) Arha ngel’ski ˘ ı, then D 1 is not pr oje ctively Arhangel’ski ˘ ı either. Pr o of. (i) Let L b e the closur e of the subgroup D 1 in G . Since L ⊇ D 1 ⊇ ker f , f ( L ) is a closed subgroup of H . Since it con tains the d ense sub group D , we deduce that f ( L ) = H . Using again L ⊇ k er f , we deduce that L = G , i.e., D 1 is d ense in G . (ii) Since the map f is p erfect, the conclusion follo ws from the well-kno wn fact that the p rop erties listed in item (ii) are preserv ed by taking full preimages un der p erfect m aps. (iii) and (iv) are straightfo rward.  Pro of of Theorem 4.2: The “if ” part follo ws from Theorem 1.1 and Diagram 1. Let us pr o ve the “only if ” p art. Let G b e a non-metrizable compact group. By F act 6.1, there exists a contin uous group homomorphism f : G → H onto a compact group H suc h that w ( H ) = ω 1 . S ince ω 1 is not a strong limit cardinal, we can use Theorem 4.1 to find a dense sub group D of H without the Arhangel’ski ˘ ı prop erty . By F act 6.2, D 1 = f − 1 ( D ) is a d ense subgroup of G that is not pro jectiv ely Arhangel’ski ˘ ı.  Pro of of Theorem 4.4: (i) → (ii) T his implication is trivial, as all coun tably compact groups are pseudo compact. (ii) → (iii) Let G b e a compact ab elian grou p such that w ( G ) ≥ c + . By F act 6.1, ther e exists a con tinuous sur jectiv e h omomorphism f : G → H on to a compact group H suc h that w ( H ) = c + . By P r op osition 3.5(i), H has a dense coun tably compact subgroup D without the Arh angel’ski ˘ ı prop erty . By F act 6.2, D 1 = f − 1 ( D ) is a dense co untably compact subgroup of G that is not pro jectiv ely Arhangel’ski ˘ ı. This con tradicts (ii). METRIZA TION CRITERIA F OR COMP ACT GROUPS IN TER MS OF THEIR DENSE SUBGROUPS 11 (iii) → (i) In deed, let D b e a G δ -dense su bgroup of G . Then D is pseudo compact. Since w ( D ) = w ( G ) ≤ c , from Prop osition 3.4 we conclude that D is pr o j ectiv ely Arhangel’ski ˘ ı.  7. Pr oof of Theorem 5.2 Lemma 7.1. L et X b e a set. F or every g ∈ F V ( X ) \ { e } ther e exists the unique non-empty finite set F ⊆ X such that g ∈ h F i and g 6∈ h F ′ i for every pr op er subset F ′ of F . Pr o of. The existence of suc h an F is clear. S upp ose that F 0 and F 1 are fi nite s u bsets of X suc h that g ∈ h F i i and g 6∈ h F ′ i i for ev ery prop er subs et F ′ i of F i ( i = 0 , 1). Let F ′ = F 0 ∩ F 1 , so th at F ′ ⊆ F i for i = 0 , 1. Fix i = 0 , 1. Let f : X → F V ( X ) b e the map that coincides with the identit y on F i and sends ev ery elemen t x ∈ X \ F i to e ∈ F V ( X ). Since X is V -ind ep endent, F V ( X ) = h X i ∈ V by item (i) of Defin ition 5.1(a), so we can us e item (ii) of the same definition to fi nd a homomorph ism ˜ f : F V ( X ) → F V ( X ) extending f . Since g ∈ h F i i and f is the identit y on F i , we conclude that ˜ f ( g ) = g . Since g ∈ h F 1 − i i , we ha v e (2) g = ˜ f ( g ) ∈ h f ( F 1 − i ) i = h f ( F 1 − i ∩ F i ) ∪ f ( F 1 − i \ F i ) i = h f ( F ′ ) ∪ { e }i = h f ( F ′ ) i = h F ′ i . Since F ′ ⊆ F i , from f ∈ h F i i , (2) and our assump tion on F i w e conclude that F i = F ′ = F 0 ∩ F 1 = F i ∩ F 1 − i . This pro v es that F i ⊆ F 1 − i . Since the last inclusion h olds for b oth i = 0 , 1, it follo ws that F 0 = F 1 , as required.  F or ev ery g ∈ F V ( X ) \ { e } w e d enote b y supp X ( g ) the un ique set F ⊆ X as in the conclusion of Lemma 7.1. W e shall call a space X semi-Bernstein p r o vid ed that eve ry compact su bset of X is countable. A m otiv atio n for this d efinition comes fr om the classical notion of a Bernstein su bset in the real line. One can easily see that a s ubset X of the real line R is a Berns tein set if and only if b oth X and its complemen t R \ X are semi-B ernstein spaces, in our terminology . Lemma 7.2. Assume that V is a variety of gr oups and X is a V - i ndep endent subset of a sep ar able metric gr oup K such that | X | = c . Th en ther e exists X ′ ⊆ X such th at | X ′ | = c a nd h X ′ i is semi-Bernstein. Pr o of. Since X is V -indep endent, h X i is isomorphic to F V ( X ), so we can use the notatio n su pp X ( g ) for all g ∈ h X i . Since K is separable metric, the family C = { C ⊆ h X i : C is compact and | C | = c } has size at most c , so w e can fix an en umeration C = { C α : α < c } of C . By tr ansfinite r ecursion on α < c we shall choose x α , y α ∈ X s atisfying conditions (i α )–(iii α ) b elo w. (i α ) x α 6∈ { x β : β < α } , (ii α ) { x β : β ≤ α } ∩ { y β : β ≤ α } = ∅ , (iii α ) y α ∈ supp X ( g α ) for some g α ∈ C α . Basis of r ecursion . Let g 0 ∈ C 0 \ { e } . Ch o ose arbitrary y 0 ∈ supp X ( g 0 ) and x 0 ∈ X \ { y 0 } . No w conditions (i 0 )–(iii 0 ) are satisfied. Recursiv e step . S upp ose th at α < c and x β , y β ∈ X we re already c hosen for all β < α so that conditions (i β )–(iii β ) are satisfied. W e shall c ho ose x α , y α ∈ X s atisfying conditions (i α )–(iii α ). Let (3) H α = h{ x β : β < α } ∪ { y β : β < α }i . Then | H α | ≤ | α | · ω < c , Since | C α | = c , we can c ho ose (4) g α ∈ C α \ H α . F rom (3) and (4) it follo ws that supp X ( g α ) 6⊆ { x β : β < α } , so w e can c ho ose (5) y α ∈ supp X ( g α ) \ { x β : β < α } . 12 D. DIKRANJAN AND D. SHAKHMA TOV F rom (4) and (5) w e co nclude that (iii α ) holds. Since | X | = c and | H α | < c , we can c ho ose (6) x α ∈ X \ ( H α ∪ { y α } ) . No w (i α ) is satisfied by (3) an d (6). It remains only to c hec k condition (ii α ). Since (ii β ) holds for ev ery β < α , we ha ve { x β : β < α } ∩ { y β : β < α } = ∅ . C ombining this with (5) and (6), w e get (ii α ). The recursiv e construction b eing co mplete, w e claim that X ′ = { x α : α < c } ⊆ X is as required. Since (i α ) holds for ev ery α < c , w e ha ve | X ′ | = c . Since (ii α ) holds for ev ery α < c , for Y = { y α : α < c } we ha v e X ′ ∩ Y = ∅ . It remains only to sho w that h X ′ i con tains no uncountable compact sub sets. Indeed, sup p ose that C is an uncoun table compact subset of h X ′ i . By [21, Exercise 1.7.11 ], every separable metric space is a union of a p erfect set and a co untable set. S ince a p erfect set has size c , it follo ws th at | C | = c . Since C ⊆ h X ′ i ⊆ h X i , we obtain C ∈ C , and so C = C α for some α < c . F rom (iii α ), there exists g α ∈ C α suc h that y α ∈ supp X ( g α ). Since y α ∈ Y and Y ∩ X ′ = ∅ , we conclude that y α 6∈ X ′ . Therefore, y α ∈ supp X ( g α ) \ X ′ . Since X ′ ⊆ X , this means that g α 6∈ h X ′ i . W e obtained a cont radiction with g α ∈ C α = C ⊆ h X ′ i .  Lemma 7.3. L et V b e a variety of gr oups and let I b e a set with ω 1 ≤ | I | ≤ c . Assume that K is a c omp act metric gr oup, X ⊆ K I and ϕ : X → K is an inje ction such that: (i) ϕ ( X ) is V -indep endent, (ii) h ϕ ( X ) i i s semi-Bernstein, (iii) h X i ∈ V , (iv) for every x ∈ X ther e exists J x ∈ [ I ] ≤ ω such that π i ( x ) = ϕ ( x ) for e ach i ∈ I \ J x , wher e π i : K I → K is the pr oje ction on i th c o or dinate. Then X is V -i ndep endent and h X i is semi-Bernstein. Pr o of. F rom (iv) one imm ed iately gets the follo wing claim. Claim 1. F or every Y ∈ [ X ] ≤ ω , the fol lowing holds: (a) the set I Y = I \ S x ∈ Y J x is unc ountable; (b) π i ↾ Y = ϕ ↾ Y for al l i ∈ I Y . Let Y b e a finite subset of X . Since h Y i ⊆ h X i ∈ V by (iii), it follo ws that h Y i ∈ V . By Claim 1 (a), we can choose i ∈ I Y . By Claim 1 (b), π i ↾ Y = ϕ ↾ Y . Since ϕ is an inj ection, π i ↾ Y is an injection as w ell. Since π i ( Y ) = ϕ ( Y ) ⊆ ϕ ( X ) and ϕ ( X ) is V -ind ep endent b y (i), w e conclude that Y is V -indep enden t [15, Lemma 2. 4]. Since this holds for ev ery finite subset Y of X , it follo ws that X is V -ind ep endent [15, Lemma 2.3]. Since X and ϕ ( X ) are b oth V -ind ep endent, there exists a u nique isomorphism Φ : h X i → h ϕ ( X ) i extending ϕ . The next cl aim is immediate fr om Claim 1 (b) and our definition of Φ. Claim 2. F or every Y ∈ [ X ] ≤ ω one has π i ↾ h Y i = Φ ↾ h Y i for al l i ∈ I Y . F or every subs et J of I let p J : K I → K J denote the pro jection. Assume that C is an un coun table compact su bset of h X i . Th en Φ( C ) is an u ncoun table subset of h ϕ ( X ) i , so the closure F of Φ( C ) is an uncounta ble compact subset of K . By (ii ), F \ h ϕ ( X ) i 6 = ∅ , so w e can c h o ose b ∈ F \ h ϕ ( X ) i ⊆ F \ Φ( C ). Since K is a metric space, b ∈ F \ Φ( C ) and Φ( C ) is dense in F , w e can c h o ose a faithfully indexed sequ en ce { c n : n ∈ N } ⊆ C such that the sequence { Φ( c n ) : n ∈ N } con v erges to b in K . Fix Y ∈ [ X ] ≤ ω suc h that { c n : n ∈ N } ⊆ h Y i . F r om Claim 2 w e conclud e that (7) { π i ( c n ) : n ∈ N } = { Φ( c n ) : n ∈ N } for all i ∈ I Y . Use Claim 1 (a) to fix j ∈ I Y . Since the sequence { c n : n ∈ N } is faithfu lly indexed and Φ is an injection, it follo ws from (7) that the s equence { π j ( c n ) : n ∈ N } is faithfully indexed. Therefore, METRIZA TION CRITERIA F OR COMP ACT GROUPS IN TER MS OF THEIR DENSE SUBGROUPS 13 the sequ ence { p S ( c n ) : n ∈ N } is faithfully in dexed as well, wh ere S = { j } ∪ S x ∈ Y J x . Since K S is compact, the sequence { p S ( c n ) : n ∈ N } has an acc umulatio n p oin t y ∈ K S . D efine g ∈ K I b y (8) g ( i ) = ( y ( i ) if i ∈ S b if i ∈ I \ S for all i ∈ I . Claim 3. g b elongs to the closur e of the set { c n : n ∈ N } in K I . Pr o of. Let W b e an op en neigh b ourho o d of g in K I . Th en there exist an op en set U ⊆ K S and an op en set V ⊆ K I \ S suc h that g ∈ U × V ⊆ W . Since I \ S ⊆ I Y and the sequence { Φ( c n ) : n ∈ N } con verges to b in K , applying (7 ) and (8) w e ca n find n 0 ∈ N suc h that p I \ S ( c n ) ∈ V for all n ∈ N with n ≥ n 0 . Since y is an ac cumulat ion p oin t of { p S ( c n ) : n ∈ N } , there exists an in teger m ≥ n 0 suc h that p S ( c m ) ∈ U . No w c m ∈ U × V ⊆ W .  Since C is compact, it is closed in K I . F r om { c n : n ∈ N } ⊆ C and Claim 3 we get g ∈ C . Since C ⊆ h X i , it follo ws that g ∈ h X i . Let E b e a finite sub set of X with g ∈ h E i . Since I E is uncounta ble b y Claim 1 (a) and S is co untable, we can c ho ose i ∈ I E \ S . Then b = π i ( g ) = Φ( g ) b y (8) and Claim 2. Thus, b = Φ ( g ) ∈ Φ( h X i ) = h ϕ ( X ) i , in contradict ion with our choi ce of b . This p ro ves that all compact su bsets of h X i are coun table.  Lemma 7.4. L et V b e a variety of gr oups and let I b e a set with ω 1 ≤ | I | ≤ c . Assume that K ∈ V is a c omp act metric gr oup and Z is a V -indep endent subset o f K such tha t | Z | = c and h Z i is semi-Bernstein. Then ther e exists a subset X of H = K I with the fol lowing pr op erties: (a) X is a V -indep endent subset of H of size c ; (b) h X i is semi-Bernstein; (c) X is G δ -dense in H . Pr o of. F or ev ery J ∈ [ I ] ≤ ω let K J = { y α,J : α < c } b e an en umeration of K J . F rom | I | ≤ c it follo ws that   [ I ] ≤ ω   ≤ c , so w e can fix a f aithfu l enumeration Z = { z α,J : α < c , J ∈ [ I ] ≤ ω } of Z . F or α < c and J ∈ [ I ] ≤ ω define x α,J ∈ H by (9) x α,J ( i ) = ( y α,J ( i ) if i ∈ J z α,J if i ∈ I \ J for all i ∈ I . W e claim that X = { x α,J : α < c , J ∈ [ I ] ≤ ω } has the desired prop erties. Define the bijection ϕ : X → Z b y ϕ ( x α,J ) = z α,J for ( α, J ) ∈ c × [ I ] ≤ ω . Th en ite ms (i), (ii) and (iv) of Lemm a 7.3 are satisfied. Since h X i is a subgroup of H = K I and K ∈ V , it follo ws that h X i ∈ V , so item (iii) of Lemma 7.3 is satisfied as w ell. App lying this lemma, w e conclude that X is V -indep endent and (b) holds. S ince ϕ : X → Z is a bijection, | X | = | Z | = c . Th us, (a) also holds. It remains only to c hec k (c). T o ac hieve this, it suffices to sho w that π J ( h X i ) = K J for eve ry J ∈ [ I ] ≤ ω , where π J : K I → K J is the p ro jection. Fix such a J . Let y ∈ K J . Th ere exists α < c suc h that y = y α,J . No w π J ( x α,J ) = y α,J = y by (9). Since x α,J ∈ X , w e are done.  Pro of of Theorem 5.2: Let K = L ω . T hen K ∈ V and K conta ins a V -indep en den t set of size c [15, Lemm a 4.1]. T herefore, K satisfies the assumptions of Lemma 7.2. The conclusion of th is lemma s ays that K satisfies the assumptions of Lemma 7.4. Let X b e the set as in the conclusion of this lemma. Then D = h X i is a G δ -dense subgroup of K I suc h that ev ery compact subset C of D is countable; in particular, w ( C ) ≤ | C | ≤ ω . Since D is dense in K I , w e ha ve w ( D ) = w ( K I ) = | I | ≥ ω 1 . This sho ws that D is not w -compact. Since D = h X i and X is a V -indep end en t set of cardinalit y c , D is a V -free group with c many generators. Note that K I ∼ = L I , as I is un countable.  14 D. DIKRANJAN AND D. SHAKHMA TOV 8. Pr oofs of Theore ms 4.5 and 4.7 The pro of of th e follo wing w ell-kno wn fact can b e found, for example, in [10, Theorem 4.15 and Discussion 4.14]. F act 8.1. L et G b e a c omp act ab elian gr oup. (i) If G is c onne cte d, then ther e exists a c ontinuous surje ctive homomorp hism of G onto T w ( G ) . (ii) If τ is a c ar dinal such that ω < cf ( τ ) ≤ τ ≤ w ( G ) , then ther e exists a c ontinuous su rje ctive homomo rphism f : G → H = K τ , wher e K = T or K = Z ( p ) for some prime numb er p . The pro of of the follo wing fact can b e found in [25]. F act 8.2. If N is a total ly disc onne cte d close d normal sub gr oup of a c omp act c onne cte d gr oup K , then w ( K / N ) = w ( K ) . W e denote by G ′ the comm u tator sub group of a group G . Recall th at a group G is p erfe ct if G = G ′ . A semisimple group is a p erfect compact connected group [25, Defin ition 9.5]. F or a top ologica l group G , w e use c ( G ) to d enote the connected comp onen t of G and w e use Z ( G ) for denoting th e cen ter of G . W e need the follo wing w ell-known fact. F act 8.3. L et G b e a non-trivial c omp act c onne cte d gr oup and let A = c ( Z ( G )) . (i) G = A · G ′ and ∆ = A ∩ G ′ is total ly disc onne cte d; (ii) G ∼ = ( A × G ′ ) / ∆ and G/ ∆ ∼ = A/ ∆ × G ′ / ∆ ; (iii) w ( G ) = max { w ( A ) , w ( G ′ ) } ; (iv) w ( A ) = w ( A/ ∆) = w ( G/G ′ ) ; (v) if G = G ′ is semisimple, then A = ∆ = { e } , G/ Z ( G ) is a pr o duct of c omp act simple Lie gr oups and w ( G/ Z ( G )) = w ( G ) ; (vi) the gr oup G/ ∆ admits a c ontinuous surje ctive homomorphism onto T w ( A ) × Q i ∈ I L i , wh er e e ach L i is a c omp act simple Lie gr oup and w ( G ′ ) = ω · | I | ; (vii) if cf ( w ( G )) > ω , then G admits a c ontinuous surje ctiv e homomor phism onto T w ( G ) or onto L w ( G ) , for some c omp act simple Lie gr oup L . Pr o of. (i) Th is can b e found in [25, Th eorem 9.2 4]. (ii) Since A is a cen tral subgroup of G , the map f : A × G ′ → G defined by f ( a, g ) = a − 1 g for ( a, g ) ∈ A × G ′ , is a con tinuous group homomorphism. C learly , f is surjectiv e. Since k er f = ∆ ∗ = { ( x, x ) : x ∈ ∆ } ⊆ A × G ′ and ∆ ∗ ∼ = ∆, we conclude that G ∼ = ( A × G ′ ) / k er f = ( A × G ′ ) / ∆ ∗ ∼ = ( A × G ′ ) / ∆ . Moreo ve r, since (∆ × ∆) / ∆ ∗ = (∆ × ∆) / k er f = f (∆ × ∆) = ∆, we obtain A/ ∆ × G ′ / ∆ ∼ = ( A × G ′ ) / (∆ × ∆) ∼ = (( A × G ′ ) / ∆ ∗ ) / ((∆ × ∆) / ∆ ∗ ) ∼ = G/ ∆ . (iii) F rom (i) it follo ws that G is a contin uous image of A × G ′ , so w ( G ) ≤ w ( A × G ′ ) = max { w ( A ) , w ( G ′ ) } . Since b oth A and G ′ are sub grou p s of G , max { w ( A ) , w ( G ′ ) } ≤ w ( G ). This establishes (iii). (iv) Since A is connected, the first equalit y follo ws from (i) and F act 8.2. F r om (i) one easily gets the isomorphism G/G ′ ∼ = A/ ∆, wh ic h giv es the second equalit y . (v) This is a particular case of a theorem of V arop oulos [35]. The equalit y w ( G/ Z ( G )) = w ( G ) follo ws from F act 8.2 since Z ( G ) is totall y disconnected [25, Theorem 9.19]. (vi) By (iv) and F act 8 .1(i), the connected compact ab elian group A/ ∆ admits a cont inuous surjectiv e h omomorphism on to T w ( A ) . Since ∆ ⊆ Z ( G ) ⊆ Z ( G ′ ), the group G ′ / ∆ has G ′ / Z ( G ′ ) as its quotien t. Since G ′ is s emisimp le [25, Corollary 9.6], from this and item (v) it follo w s that G ′ / ∆ a dmits a conti nuous surjectiv e homomorphism on to a p ro duct Q i ∈ I L i , where eac h L i is a compact simple Lie group and w ( G ′ ) = w ( G ′ / Z ( G ′ )) = ω · | I | . METRIZA TION CRITERIA F OR COMP ACT GROUPS IN TER MS OF THEIR DENSE SUBGROUPS 15 Since G/ ∆ ∼ = A/ ∆ × G ′ / ∆ by (ii), we get the conclusion of ite m (vi). (vii) F ollo ws fr om (iii), (vi) and the fact that there are only coun tably many pairwise non- isomorphic (as topological groups) compact simple Lie group s.  Pro of of Theorem 4.5: Supp ose that G is not metrizable. If G is ab elian, w e can use F act 8.1(ii) to find a con tinuous surjectiv e homomorph ism f : G → H = L ω 1 , where L is either T or Z ( p ) for some prime n u m b er p . If G is connected, we first u se F act 6.1 to find a con tin uous homomorph ism of G on to (compact connected) group of wei gh t ω 1 , and then w e apply F act 8.3 (vii) to find a con tinuous surjectiv e homomorph ism f : G → H = L ω 1 , where L is either T or a compact simple Lie group. When L is ab elian, we apply C orollary 5.4 with I = ω 1 to get a subgroup D of H as in the conclusion of this corollary . Wh en L is a compact s imple Lie group, we apply Corollary 5.3 with I = ω 1 to get a subgroup D of H as in the conclusion of this corolla ry . In b oth cases, w e use F act 6.2 to conclude that D 1 = f − 1 ( D ) is a G δ -dense subgroup of G that is not pro jectiv ely w -compact. This con tradicts the assu mption of our theorem. Therefore, G m ust b e metrizable.  Lemma 8.4. Assume CH . If K = T or K = Z ( p ) for some prime numb er p , then H = K ω 1 has a dense c ountably c omp act sub gr oup D without infinite c omp act subsets. Pr o of. W e consider t w o cases. Case 1 . K = T . T k ac hen ko [34] constructed a dense countably compact subgroup D of K ω 1 suc h that | D | = c = ω 1 and D h as no non-trivial con v ergen t sequences. Case 2 . K = Z ( p ) for some prime numb er p . In this case w e can argue as follo ws. Since CH implies Martin’s Axiom MA, and the group L = Z ( p ) ω is compact (in the Tyc honoff pro d u ct top ology), by the implicatio n (a) → (c) of [19, Th eorem 3.9], the g roup L admits a counta bly compact group top ology without non-trivial conv ergen t s equences. An analysis of this pro of shows that this top ology comes from a monomorphism j : L → Z ( p ) c suc h that D = j ( L ) is a den se subgroup of Z ( p ) c . Under CH, w e conclude that H = K ω 1 has a dense count ably compact sub grou p D without non-trivial con vergen t sequences. 1 The rest of th e pro of is common for b oth cases. S upp ose th at X is an infinite compact subset of D . Since D has no non-trivial con v ergen t sequ ences, X do es not ha v e any p oin t of coun table c h aracter. Then | X | ≥ 2 ω 1 > ω 1 = c by the ˇ Cec h -P osp i ˇ sil theorem. This con tradicts th e inequalit y | X | ≤ | D | = c . T his pro v es that ev er y compact subset X of D is fi nite.  Pro of of Theorem 4.7: Su p p ose that G is not metrizable. Use F act 8.1(ii) to find a con tinuous surjectiv e homomorphism f : G → H = K ω 1 , where K is either T or Z ( p ) for some p rime n u mb er p . Let D b e a dense coun tably compact subgroup of H without infinite compact subsets constru cted in Lemma 8.4. Since D is dense in H , w ( D ) = w ( H ) = ω 1 . Th is sho ws that D is not w -compact. By F act 6.2, D 1 = f − 1 ( D ) is a dense countably compact subgroup of G that is n ot pro jectiv ely w -compact. This con tradicts the assu mption of our theorem. Therefore, G m ust b e metrizable.  9. Examples Example 9.1. F or every c ar dinal τ such that ω 1 ≤ τ ≤ c , ther e exists a pseudo c omp act pr oje ctively Arh angel’ski ˘ ı gr oup D of weight τ that i s not w -c omp act. F urthermor e, under CH, D c an b e chosen to b e e v en c ountably c omp act. Indeed, let K = T or Z ( p ) for some prime num b er p . Ap p ly Corollary 5.4 to L = K and I = τ to find a G δ -dense s ubgroup D of K τ suc h that all compact subsets of D are coun table; in particular, D is not w -compact. By Theorem 1.3, D is pseud o compact. Und er CH, 1 In case p = 2, one can also make a recourse to an old result of Ha jnal and Juh´ asz [23] claiming the existence of a subgroup D of K ω 1 that is an HFD set. Such D is a dense countably compact subgroup of K ω 1 without infinite compact subsets. 16 D. DIKRANJAN AND D. SHAKHMA TOV w e can use L emma 8.4 to c ho ose D to b e ev en coun tably compact. S ince w ( D ) = w ( K τ ) = τ ≤ c , from Pr op osition 3.4 w e conclude that D is pro jectiv ely Arhangel’ski ˘ ı. Recall that a subgroup D of a topological abelian group G is called e ssential in G if D ∩ N = { 0 } implies N = { 0 } for every closed sub grou p N of G [4, 31, 32]. A top ologic al group G is called minimal if there exists no Hausdorff group top ology on G strictly coarser than the top ology of G . A dense subgroup D of a compact ab elian group G is minimal if and only if D is essentia l in G [4, 31, 32]. Example 9.2. L et p b e a prime numb e r and κ b e an infinite c ar dinal. Define τ = 2 2 2 κ . Then ther e exists a dense essential (=minimal) κ -b ounde d w -c omp act sub gr oup of Z ( p 2 ) τ that is not pr oje ctively Arh angel’ski ˘ ı . Ind eed, let G = Z ( p 2 ) τ and let f : G → G b e the (contin uous) map defin ed f ( g ) = pg for g ∈ G . Let H = f ( G ). Then H ∼ = Z ( p ) τ . F rom Prop osition 3.5 (ii), we get a dense κ -bou n ded subgroup D of H ∼ = Z ( p ) τ without the Arh an gel’ski ˘ ı pr op ert y . Applying F act 6.2, we conclude that D 1 = f − 1 ( D ) is a dense κ -b oun ded subgroup of G that is not p ro jectiv ely Arh angel’ski ˘ ı. Sin ce pG = ker f is easily seen to b e an essen tial sub group of G , from k er f ⊆ D 1 it follo w s that D 1 is an essen tial subgroup of G . Finally , note that k er f ∼ = Z ( p ) τ is a compact subset of D 1 suc h that w (ker f ) = w ( Z ( p ) τ ) = τ = w ( G ) = w ( D 1 ), wh ic h sho ws that D 1 is w -compact. F or an infinite cardinal σ , d efine log σ = min { τ ≥ ω : σ ≤ 2 τ } . Let i 0 = ω , and let i α +1 = 2 i α for ev ery ordinal α and i β = sup { i α : α < β } for ev ery limit ordinal β > 0. Example 9.3. L et G b e a c omp act gr oup of weight σ > ω . (i) If cf (log σ ) = ω and σ = (log σ ) + , th en every G δ -dense sub gr oup of G has the A rhangel’ski ˘ ı pr op erty. In deed, by Theorem 4.3, it su ffices to sh ow that m ( σ ) ≥ σ . It is kn o wn th at log σ ≤ m ( σ ) and cf ( m ( σ )) > ω [11, Theorem 2.7]. Therefore, m ( σ ) > log σ and m ( σ ) ≥ (log σ ) + = σ b y our h yp othesis. (ii) If α is an or dinal of c ountable c ofinality and σ = i + α , then al l G δ -dense sub gr oups of G have the A rhangel’ski ˘ ı pr op erty. Indeed, it suffices to chec k that σ = i + α satisfies the h y p othesis of item (i). Ob vious ly , log σ = i α , so cf (log σ ) = cf ( i α ) = cf ( α ) = ω and σ = i + α = (log σ ) + . Here is an alternativ e pr o of of item (ii) of this example that mak es n o recourse to its item (i) and the cardinal fu n ction m ( − ). Assume that D is a G δ -dense subgroup of G without the Arhangel’ski ˘ ı prop ert y . Then | D | < w ( D ) = w ( G ) = i + α , so | D | ≤ i α . Sin ce i α is strong limit and i + α = w ( D ) ≤ 2 | D | , we deduce that | D | = i α . Th erefore, D is a pseudo compact group such that | D | a strong limit cardinal of countable cofinalit y . This contradict s a well-kno wn theorem of v an Dou wen [20]. 10. Final re m arks and open q u estions Remark 10.1. While “pro jectiv ely w -compact” and “p r o jectiv ely Arhangel’ski ˘ ı” are differen t prop erties wh en restricted to a single group, the equiv alence of items (ii) and (iv) of Corolla ry 4.8 sho ws that these tw o pr op erties and the prop erty “dete rmining the co mpletion” co incide wh en imp osed u niformly on al l dense subgrou p s of a giv en compact ab elian group. Similarly , while it is unclear whether “determining th e completion” and “pro jectiv ely w -compact” are differen t p rop er- ties for an y giv en group, the equiv alence of items (iii) and (v) of Corollary 4. 8 sh o w s that these t w o prop erties coincide w h en imp osed u niformly on al l G δ -dense subgroups of a giv en compact ab elian group. Recall that a top ological group G is calle d total ly minimal if all (Ha usdorff ) quotien t groups of G are minimal. Remark 10.2. (i) In a forthcoming pap er [17] w e p ro v e that every dense total ly minimal sub gr oup of a c omp act ab elian gr oup G determines G . Th is sho ws that, in con trast with METRIZA TION CRITERIA F OR COMP ACT GROUPS IN TER MS OF THEIR DENSE SUBGROUPS 17 the results in Section 4, a weak er form of “determinatio n” asking all dense totally minimal subgroups of G to determine G imp oses no restrictions whatso ev er on a compact ab elian group G . (ii) In a forthcoming pap er [18] w e prov e that total ly minimal ab elian gr oups ar e pr oje ctively w - c omp act . Therefore, the italiciz ed statemen t in item (i) sho w s that the answ er to Qu estion 2.5 is p ositiv e for this (pr op er) sub class of the class of p ro jectiv ely w -compact groups. Question 10.3. What can one sa y ab out a compact (ab elian) group G suc h that all dense sub- groups of G are w -compact? F rom Theorem 4.1 and Diagram 1 it follo w s that w ( G ) m ust b e a strong limit cardinal, but w e do not kno w if G must b e metrizable. Question 10.4. What is the minimal w eigh t σ of an ω -b ound ed ab elian group that is not pro jec- tiv ely Arhangel’ski ˘ ı? Is σ = c + ? W e only know that c + ≤ σ ≤ 2 2 c . The first inequalit y follo ws from P r op osition 3.4 and the second inequalit y follo ws from Example 9.2 (with κ = ω ). Question 10.5. Does Theorem 4.5 h old for all compact groups? Question 10.6. Do es Theorem 4.7 hold in ZF C? Does the implication (vi) → (i) of Corollary 4.8 hold in ZF C? As an in term ed iate step to solving this question, one may also w ond er if CH ca n b e w eak ened to Martin’s Axiom MA in Theorem 4.7 and the implication (vi) → (i) of Corollary 4.8. W e conjecture that the follo wing question has a negativ e answe r (although w e h a ve no counter- example at hand): Question 10.7. If ev ery ω -b ounded d ense sub grou p of a compact ab elian group G determines it, m ust G b e metrizable? Here come the coun terp art of Question 10.3 for G δ -dense sub groups: Question 10.8. Describ e the compact (ab elian) group s G suc h that ev ery G δ -dense subgroup of G is w -compact. Question 10.9. Let K = T or K = Z ( p ) for some p rime num b er p . In ZFC, do es there exist a dense countably compact subgroup D of K ω 1 without un coun table compact s ubsets? 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Shakhmatov) Divi sion of Ma them a tics, Physi cs and Ear th S ciences, Gradua te School of Science and Engineering, Ehime Uni versity, Ma tsuy ama 790-8577, Jap an E-mail addr ess : dmitri.shakhmatov @ehime-u.ac.jp