Radon-Nikodym compact spaces of low weight and Banach spaces

RADON-NIKOD ´ YM COMP A CT SP A CES OF LOW WEIGHT AND BANA CH SP A CES ANTONIO A VIL ´ ES Abstract. W e prov e that a con tin uous image of a Radon-Nikod´ ym compact of we ight less th an b is Radon-Nikod´ ym compact. As a Banac h space coun ter- part, subspaces of Asplund generated Banac h spaces of density characte r l ess than b are Asplund generated. In this case, in addition, there exists a sub- space of an Asplund generat ed space which is not Asplund generated which has densit y character exactly b . The concept of Radon-Niko d´ ym compact, due to Reynov [1 2], has its o rigin in Banach space theory , and it is defined as a top olo gical spa c e which is homeo mo r- phic to a weak ∗ compact subset of the dual of a n Asplund space, that is, a dual Banach spac e with the Radon- Niko d´ ym prop erty (topolo gical spaces will be here assumed to be Hausdor ff ). In [9], the following characterization of this class is given: Theorem 1. A c omp act sp ac e K i s R adon-Niko d ´ ym c omp act if and only if t her e is a lower semic ontinuous metric d on K which fr agments K . Recall that a ma p f : X × X − → R on a top olo gical space X is said to fr agment X if for e a ch (closed) subset L of X and each ε > 0 there is a nonempty relative op en subset U of L o f f -diameter less than ε , i.e. sup { f ( x, y ) : x, y ∈ U } < ε . Also, a map g : Y − → R fro m a top olo gical space to the r eal line is lower semic ontinuous if { y : g ( y ) ≤ r } is closed in Y for every real n umber r . It is an op en problem whether a contin uous image of a Radon- Nikod´ ym compac t is Radon-Nikod´ ym. Arv anitakis [2] has made the following appr oach to this pro b- lem: if K is a Ra don-Nikod ´ ym compact and π : K − → L is a contin uous sur jection, then we hav e a lower semicon tinuous frag menting metric d on K , and if we want to prov e that L is Ra don-Nikod ´ ym compact, we s hould find such a metric on L . A natural candida te is: d 1 ( x, y ) = d ( π − 1 ( x ) , π − 1 ( y )) = inf { d ( t, s ) : π ( t ) = x, π ( s ) = y } . The map d 1 is low er semico nt inuous and fragments L and it is a quasi metric , that is, it is symmetric and v anishes o nly if x = y . But it is not a metric b ecause, in g eneral, it lacks triangle inequality . Cons equently , Arv anitakis [2] introduce d the 2000 Mathematics Subje ct Classific ation. Primary: 46B26. Secondary: 46B22, 46B50, 54G99. Key wor ds and phr as es. Radon-Nikod´ ym compact, quasi R adon-Ni k o d´ ym compact, coun tably low er fragment able compact, As plund generated space, weakly K -analytic space, w eakly compactly generated space, cardinal b , Martin’s axiom. Author supported by FPU grant of SEEU-ME C D of Spain. 1 2 ANTONIO A VIL ´ ES following co ncept: Definition 2 . A compact spac e L is sa id to b e quasi R adon-Niko d´ ym if there ex ists a lower semico nt inuous q uasi metr ic whic h fra gments L . The class of qua si Radon-Nikod´ ym compac ta is c losed under co ntin uous im- ages but it is unknown whether it is the same class as that of Radon-Nikod ´ ym compacta o r even the class of their contin uous images. A t least other t wo su- per classes of contin uous images o f Radon-Nikod ´ ym compa cta app ear in the liter- ature. Reznichenk o [1, p. 104 ] defined a c ompact space L to b e str ongly fr ag- mentable if there is a metric d which fragments L such that each pair o f different po ints of L p os sess disjoint neig hbourho o ds at a p o s itive d -distanc e . It has b een noted by Namiok a [10] that the cla sses o f quas i Rado n- Nikod´ ym and strongly frag- men table compacta are equal. The other sup er class of contin uous images of Radon- Nikod´ ym compacta, called c ountably lower fr a gmentable compacta, was introduced by F abian, Heisler and Ma tou ˇ sk ov´ a [5]. In sectio n 3, we recall its definition and we prov e that this class is eq ua l to the other tw o. The main result in section 1 is the following: Theorem 3. If K is a quasi Rad on-Niko d´ ym c omp ac t sp ac e of weight less than b , then K is R adon-Niko d´ ym c omp act. The w eight of a to po logical space is the least cardinality of a ba se for its top ol- ogy . W e also recall the definition o f car dinal b . In the set N N we c o nsider the order relation given by σ ≤ τ if σ n ≤ τ n for a ll n ∈ N . Cardinal b is the least cardinality of a subset of N N which is not σ -b ounde d for this order (a set is σ -b ounde d if it is a count able union of bounded subsets ). It is consistent that b > ω 1 . In fact, Ma r- tin’s axio m and the nega tion of the contin uum hypothesis imply that c = b > ω 1 , cf. [6 , 11D and 1 4B]. It is also p ossible that c > b > ω 1 , cf. [17, section 5 ]. On the other hand, cardina l d is the least car dinality of a cofinal subset of ( N N , ≤ ), that is, a set A such that for each σ ∈ N N there is some τ ∈ A such that σ ≤ τ . In a sense, the following prop osition puts a rough b ound on the size of the class o f quasi Radon-Nikod ´ ym co mpa cta with resp ect to Ra don-Nikod ´ ym compa cta. Prop ositi on 4. Every quasi R adon-Niko d ´ ym c o mp act sp ac e emb e ds into a pr o d uct of R adon-Niko d´ ym c omp act sp ac es with at most d factors. In section 2 w e discuss the Banach space counterpart to Theorem 3. A Banach space V is Asplund gener ated, or GS G , if there is some Asplund space V ′ and a bo unded linear op erator T : V ′ − → V such that T ( V ′ ) is dense in V . O ur main result for this class is the following: 3 Theorem 5. L et V b e a Banach sp ac e of density char a cter less than b and such that the dual unit b al l ( B V ∗ , w ∗ ) is quasi R adon-Niko d´ ym c omp act, t hen V is As- plund gener ate d. The density character of a Ba nach spa ce is the le a st car dina l of a norm-dense subset, and it equals the weigh t of its dual unit ball in the weak ∗ top ology . Examples co nstructed by Ro senthal [1 3] and Argy ros [4, section 1 .6] show that there ex is t Banach spa c e s which are subspaces of Asplund generated spaces but which are not Asplund genera ted. Howev er , since the dual unit ball of a subspace of a n Asplund generated space is a contin uous ima g e of a Radon-Nikod ´ ym compact [4, Theo rem 1 .5.6], we hav e the following corolla ry to Theorem 5: Corollary 6. If a Banach sp ac e V is a subsp ac e of an Asplund gener ate d sp ac e and the density char acter of V is less t han b , t hen V is Asplund gener ate d. Also, a Banach space is weakly co mpactly generated (W CG) if it is the closed linear s pa n o f a weakly compact subset. The sa me e x amples mentioned ab ov e show that neither is this pro p erty inherited b y subspaces. A Banach space V is weakly compactly g e ne r ated if and only if it is Asplund generated a nd its dual unit ball ( B V ∗ , w ∗ ) is Co rson compa ct [11], [1 4]. Having Cors on dual unit ball is a heredi- tary prop erty since a contin uous ima g e of a Cor son compact is Corso n co mpact [7], hence: Corollary 7. If a Banach sp ac e V is a subsp ac e of a we akly c omp actly gener ate d sp ac e and the density char acter of V is less than b , then V is we akly c omp actly gener ate d. Corollar y 7 ca n a lso be obtained from the following theor em, essentially due to Mercoura kis [8 ]: Theorem 8. If a Banach sp ac e V is we akly K -analytic and the density char acter of V is less than b , then V is we akly c omp actly gener ate d. The class of weakly K - a nalytic spaces is larger than the class of subspaces of weakly compactly g enerated s paces. W e reca ll its definition in section 2. The result of Mer courakis [8 , Theorem 3.13 ] is that, under Martin’s axiom, weakly K -a nalytic Banach spaces o f density character less tha n c are weakly compactly genera ted, but his arguments prov e in fact the more genera l Theo r em 8. W e give a mor e elemen- tary pro of of this theo rem, o btaining it as a co ns equence of a purely top olog ical result: An y K -ana lytic top olo gical spa c e of density character less than b contains a dense σ - compact subset. W e also remark that it is not po ssible to generalize Theorem 8 for the cla ss of weakly co untably de ter mined Banach spaces . Cardinal b is b est p oss ible for Theo rem 5, Theorem 8 and their co r ollaries , as it is shown by slig ht mo difications of the men tioned example of Ar gyros [4, section 1.6] 4 ANTONIO A VIL ´ ES and of the exa mple of T ala grand [1 5] o f a weakly K -analytic Bana ch spac e which is no t weakly compactly genera ted, so that we get examples of density character exactly b . F or infor ma tion a b out cardinals b and d we refer to [17]. Co nc e rning Banach spaces, our main reference is [4]. I w ant to expr ess my gratitude to J os´ e Orihuela for v alua ble discussions and suggestions and to Witold Marciszews ki, from w ho m I lear nt abo ut cardinals b and d . I also thank Isaa c Namiok a a nd the refer ee for sugge stions which hav e improv ed the fina l version of this article. 1. Quasi Radon-Nikod ´ ym comp act a of low weight In this sec tion, we characterize quasi Rado n- Nikod´ ym co mpacta in terms of em- bedding s into cub es [0 , 1] Γ and from this, we will derive proo fs of Theorem 3 and Prop os itio n 4. T echniques of Arv anitakis [2] will pla y an important role in this section, a s well as the follo wing theorem of Namiok a [9]: Theorem 9. L et K b e a c omp act sp ac e. The fol lowing ar e e quivalent. (1) K is R adon-Niko d´ ym c omp act. (2) Ther e is an emb e dding K ⊂ [0 , 1] Γ such that K is fr agmente d by t he u ni- form metric d ( x, y ) = sup γ ∈ Γ | x γ − y γ | . Let P ⊂ N N be the set o f all strictly increasing sequences of p ositive integers. Note that this is a cofinal subs e t of N N . F or ea ch σ ∈ P w e cons ider the low er semicontin uo us non decreasing function h σ : [0 , + ∞ ] − → R given by: • h σ (0) = 0 , • h σ ( t ) = 1 σ n whenever 1 n +1 < t ≤ 1 n . • h σ ( t ) = 1 σ 1 whenever t > 1. Also, if f : X × X − → R is a map and A, B ⊂ X , w e will use the notation f ( A, B ) = inf { f ( x, y ) : x ∈ A, y ∈ B } . Theorem 10. L et K b e a c omp act subset of t he cu b e [0 , 1] Γ . The following ar e e quivalent: (1) K is quasi Ra don-Niko d´ ym c omp act. (2) Ther e is a m ap σ : Γ − → P su ch that K is fr agmente d by f ( x, y ) = sup γ ∈ Γ h σ ( γ ) ( | x γ − y γ | ) which is a lower semic ontinuous quasi m etric. 5 PROOF: Observe that f in (2) is e x pressed a s a supr emum o f lower s emicon- tin uous functions, and therefor e, it is lower semicontin uous. Also, f ( x, y ) = 0 if and only if h σ ( γ ) ( | x γ − y γ | ) = 0 for all γ ∈ Γ if and only if | x γ − y γ | = 0 for all γ ∈ Γ. Hence, f is indeed a low er s emicontin uous quas i metric a nd it is clear that (2) implies (1). Assume now that K is qua si Radon-Nikod ´ ym compact and let g : K × K − → [0 , 1] b e a lower semico nt inuous qua s i metric which fragments K . F or γ ∈ Γ, w e call p γ : K − → [0 , 1] the pro jection on the co o rdinate γ , p γ ( x ) = x γ , and we define a qua si metr ic g γ on [0 , 1] b y the rule: g γ ( t, s ) = ( g ( p − 1 γ ( t ) , p − 1 γ ( s )) if p − 1 γ ( t ) and p − 1 γ ( s ) a re nonempt y , 1 otherwise . Note that g γ is lower semicontin uous beca use for r < 1 { ( t, s ) : g γ ( t, s ) ≤ r } = \ r ′ >r ( p γ × p γ ) { ( x, y ) ∈ K 2 : g ( x, y ) ≤ r ′ } Observe also that if x, y ∈ K , then g γ ( x γ , y γ ) = g γ ( p γ ( x ) , p γ ( y )) ≤ g ( x, y ). Hence, K is fragmented by g ′ ( x, y ) = sup γ ∈ Γ g γ ( x γ , y γ ) ≤ g ( x, y ) The pro of finishes by ma k ing use of the following lemma, where we put g 0 := g γ : Lemma 11. L et g 0 : [0 , 1] × [0 , 1] − → [0 , 1] b e a lower semic ontinuous quasi metric on [0 , 1] . Then, ther e exists τ ∈ P such that h τ ( | t − s | ) ≤ g 0 ( t, s ) for al l t, s ∈ [0 , 1] . PROOF: W e define τ recursively . Suppo se that we hav e defined τ 1 , . . . , τ n in such a wa y that if | t − s | > 1 n +1 , then h τ ( | t − s | ) ≤ g 0 ( t, s ). Let K m =  ( t, s ) ∈ [0 , 1] × [0 , 1] : | t − s | ≥ 1 n + 2 and g 0 ( t, s ) ≤ 1 m  Then, { K m } ∞ m =1 is a decreasing se q uence of compact subsets of [0 , 1] 2 with empty int ersec tio n. Hence, there is m 1 such that K m is empty for m ≥ m 1 . W e define τ n +1 = max { m 1 , τ n + 1 } .  Now, we state a lemma which is just a piece of the pr o of of [2, Pro p osition 3.2]. W e include its pro of for the sake of completenes s. Lemma 12. L et K , L b e c omp act sp ac es, let f : K × K − → R b e a symmetric m ap which fr agments K and p : K − → L a c ontinuous surje ction. Then L is fr agmente d by g ( x, y ) = f ( p − 1 ( x ) , p − 1 ( y )) and in p articular, L is fr agment e d by any g ′ with g ′ ≤ g . PROOF: Let M b e a closed subs e t o f L and ε > 0. By Zorn’s lemma a set N ⊂ K ca n be found s uch that p : N − → M is onto and irreducible (that is, for every N ′ ⊂ N closed, p : N ′ − → M is not onto). W e find U ⊂ N a relative open subset of N of f - diameter less than ε . By ir reducibility , p ( U ) has nonempty relative int erior in M . This interior is a nonempty relative o p en subset o f M o f g -dia meter less tha n ε .  6 ANTONIO A VIL ´ ES In the sequel, we use the following notation: If A ⊂ Γ a re sets, d A states for the pseudometric in [0 , 1] Γ given by d A ( x, y ) = sup γ ∈ A | x γ − y γ | . Lemma 13. L et K b e a c omp act subset of the cub e [0 , 1] Γ and let σ : Γ − → P b e a map such that the quasi metric f ( x, y ) = sup γ ∈ Γ h σ ( γ ) ( | x γ − y γ | ) fr agments K and su ch that σ (Γ) is a σ - b ounde d subset of N N . Then, K is Rad on- Niko d´ ym c omp act. In addition, ther e exist set s Γ n ⊂ Γ su ch that Γ = S n ∈ N Γ n and e ach d Γ n fr agments K . PROOF: There is a decompositio n Γ = S n ∈ N Γ n such that each σ (Γ n ) ha s a bo und τ n in ( N N , ≤ ). W e cho ose τ n ∈ P. First, we prov e that each d Γ n fragments K . F or ev ery n ∈ N , K is frag mented by the map f n ( x, y ) = s up γ ∈ Γ n h σ ( γ ) ( | x γ − y γ | ) ≤ f ( x, y ) and f n ( x, y ) = sup γ ∈ Γ n h σ ( γ ) ( | x γ − y γ | ) ≥ s up γ ∈ Γ n h τ n ( | x γ − y γ | ) = h τ n  sup γ ∈ Γ n | x γ − y γ |  = h τ n ( d Γ n ( x, y )) . Hence, a set of f n -diameter less than 1 τ n in K is a set of d Γ n -diameter less than 1 n and therefore, since f n fragments K , also d Γ n fragments K . Consider now p n : [0 , 1] Γ − → [0 , 1] Γ n the natural pro jection and K n = p n ( K ). By Lemma 12, since K is fragmented by f n , K n is fragmented by g n ( x, y ) = sup γ ∈ Γ n h σ ( γ ) ( | x γ − y γ | ) . and hence, K n is Ra don-Nikod ´ ym co mpact. Mo r eov er, since Γ = S n ∈ N Γ n , K embeds into the pr o duct Q n ∈ N K n and the class of Ra don-Nikod ´ ym co mpacta is closed under ta king countable pro ducts a nd under taking clo sed subspaces [9], so K is Radon-Nikod ´ ym compact.  PROOF OF THEO REM 3: If the weigh t of K is less than b , then K can b e embedded into a c ub e [0 , 1] Γ with | Γ | < b . Any subset of N N of c a rdinality less than b is σ -b ounded, so Theo rem 3 follows directly from Theorem 10 and Lemma 13.  PROOF OF PROPOSITION 4: Let K b e quasi Radon-Niko d´ y m compa ct, sup- po se K is em b edded into some cub e [0 , 1] Γ and let σ : Γ − → P b e as in Theor e m 10. Let A ⊂ P be a cofinal s ubs et of P of cardinality d . F or α ∈ A , let Γ α = { γ ∈ Γ : σ ( γ ) ≤ α } , let p α : [0 , 1] Γ − → [0 , 1] Γ α be the natural pr o jection, and let K α = p α ( K ). Again, since Γ = S α ∈ A Γ α , K em b eds in to the pro duct Q α ∈ A K α . B y Lemma 12, K α is 7 fragmented by g α ( x, y ) = sup γ ∈ Γ α h σ ( γ ) ( | x γ − y γ | ) The s e t { σ ( γ ) : γ ∈ Γ α } is a bounded, and hence σ -bounded, set. Hence, by Lemma 1 3, K α is Radon-Nikod´ ym compact.  W e note tha t from Lemma 13, we o btain so mething stronger than Theorem 3: Theorem 14. F or every quasi R adon-Niko d´ ym c omp act subset of a cub e [0 , 1] Γ with | Γ | < b ther e is a c oun t able de c omp osition Γ = S n ∈ N Γ n such that d Γ n fr agments K for al l n ∈ N . A s imila r result holds also fo r g e neralized Cantor cubes (cf. [5, Theor em 3], [2, Theorem 3.6 ]): If K is a qua si Radon-Niko d´ y m compact subset of { 0 , 1 } Γ , then there is a decomp osition Γ = S n ∈ N Γ n such that d Γ n fragments K for all n ∈ N . W e give now an exa mple which sho ws that this pheno menon do es not ha pp en for general cub es, even if the compact K has w eight less than b or it is zero -dimensional: Prop ositi on 1 5 . Ther e exist a set Γ of c ar dinality b and a c omp act su bset K of [0 , 1] Γ home omorphic to t he metr izable Cantor cub e { 0 , 1 } N such that for any de- c omp osition Γ = S n ∈ N Γ n ther e ex ists n ∈ N such that d Γ n do es not fr agment K . PROOF: First, we take Γ a subset of N N of car dina lity b which is not σ -b ounded. W e call A = { γ n : γ ∈ Γ , n ∈ N } the set of all terms of elements of Γ. W e define K ′ = { x ∈ { 0 , 1 } Γ × N : x γ , n = x γ ′ ,n ′ whenever γ n = γ ′ n ′ } . Observe that K ′ is homeo morphic to { 0 , 1 } N : namely , for each a ∈ A c ho ose some γ a , n a ∈ Γ × N such that γ a n a = a ; in this case we have a ho meomorphism K ′ − → { 0 , 1 } A given by x 7→ ( x γ a ,n a ) a ∈ A . Now, we consider the em b edding φ : { 0 , 1 } Γ × N − → [0 , 1] Γ given by φ ( x ) = X n ∈ N  2 3  n x γ , n ! γ ∈ Γ W e cla im that the space K = φ ( K ′ ) ⊂ [0 , 1] Γ verifies the statement. Let Γ = S n ∈ N Γ n any countable deco mp o sition of Γ. Since Γ is not σ -b ounded, there is some n ∈ N s uch that Γ n is not b ounded. F or this fixed n , since Γ n is no t b ounded, there is some m ∈ N suc h that the s e t S = { γ m : γ ∈ Γ n } ⊂ A is infinite. W e consider K 0 = { x ∈ K ′ : x γ , k = 0 whenever γ k 6∈ S } ⊂ K . By the same a rguments as for K ′ , K 0 is homeomor phic to the C a ntor cub e { 0 , 1 } N by a map K 0 − → { 0 , 1 } S given by x 7→ ( x γ a ,n a ) a ∈ S . Now, we ta ke tw o different elements x, y ∈ K 0 . Then, ther e m ust exist so me γ ∈ Γ n such that x γ , m 6 = y γ , m , and this implies that | φ ( x ) γ − φ ( y ) γ | ≥ 3 − m and therefore d Γ n ( φ ( x ) , φ ( y )) ≥ 3 − m . This means that any nonempty subset of φ ( K 0 ) of d Γ n -diameter less than 3 − m m ust be a singleton. If d Γ n fragmented K , this would imply that φ ( K 0 ) has an isolated po int, which co nt radicts the fact that it is homeomorphic to { 0 , 1 } N .  8 ANTONIO A VIL ´ ES 2. Banach sp aces of low density character In this sec tio n we find that ca rdinal b is the lea st p oss ible density character of Banach spaces which are co un terexa mples to several questions. First, we introduce some notation: If A is a subset o f a Ba nach space V , w e call d A to the pseudo- metric d A ( x ∗ , y ∗ ) = sup x ∈ A | x ∗ ( x ) − y ∗ ( x ) | on B V ∗ . Also, we rec all the following definition [4, Definition 1.4.1]: Definition 16. A nonempt y b ounded subset M of a Bana ch spa ce V is called an Asplund set if for each countable se t A ⊂ M the pseudometric space ( B V ∗ , d A ) is separable. By [3, Theor em 2.1], M is an Asplund subset of V if and only if d M fragments ( B V ∗ , w ∗ ). Also, b y [4, Theorem 1.4.4], a Banach s pace V is Asplund ge ne r ated if and only if it is the closed linear span of an Asplund subset. PROOF OF THEOREM 5 : Let Γ be a dense subset of the unit ball B V of V of cardinality les s than b . Then, we hav e a natural embedding ( B V ∗ , w ∗ ) ⊂ [ − 1 , 1] Γ . Since ( B V ∗ , w ∗ ) is quasi Radon-Niko d´ y m compact, we apply Theore m 14 and we hav e Γ = S Γ n and ea ch d Γ n fragments ( B V ∗ , w ∗ ). This means that for each n , Γ n is a n Asplund set, and by [4, Lemma 1.4.3 ], M = S n ∈ N 1 n Γ n is a n Asplund set to o. Finally , since the closed linea r span o f M is V , by [4, Theor e m 1.4.4], V is Asplund generated.  W e re c all now the concepts that we need for the pro of of T he o rem 8. W e follow the ter minology and notation of [4, sections 3.1, 4.1 ]. Le t X and Y b e top olo gical spaces. A map φ : X → 2 Y from X to the subsets of Y is said to be an usco if the following co nditions hold: (1) φ ( x ) is a compact subset of Y for all x ∈ X . (2) { x : φ ( x ) ⊂ U } is op en in X , for every op en set U of Y . In this situation, for A ⊂ X we denote φ ( A ) = S x ∈ A φ ( x ). A completely r egular top olog ical space X is sa id to b e K -analytic if there exists an usco φ : N N → 2 X such that φ ( N N ) = X . A Banach space is weakly K - analytic if it is a K -a nalytic space in its w eak topo lo gy . W e note that if a Banach space V contains a weakly σ -co mpact s ubs et M w hich is dense in the w eak top ology , then V is WCG. This is b eca use if M = S ∞ n =1 K n being K n a weakly compact s e t b ounded by c n > 0, then { 0 } ∪ S 1 nc n K n is a weakly compact s ubset of V whose linear span is (weakly) dens e in V . Hence, Theore m 8 is deduced from the following: Prop ositi on 17. If X is a K -analytic t op olo gic al sp ac e which c ontains a dense subset of c ar dinality less than b , then X c ontains a dense σ -c omp act subset. PROOF: W e hav e a n usco φ : N N − → 2 X with φ ( N N ) = X a nd a lso a set Σ ⊂ N N such that | Σ | < b and φ (Σ) is dense in X . Any subse t of N N of cardinal les s than 9 b is con tained in a σ -compact subset o f N N [17, Theorem 9.1]. Uscos send compact sets o nto compa c t sets, s o if Σ ′ ⊃ Σ is σ -co mpact, then φ (Σ ′ ) is a dense σ -co mpact subset of X .  W e recall that a completely regular top o logical space X is K -countably deter- mined if there exists a subset Σ of N N and an usco φ : Σ − → 2 X such that φ (Σ) = X and that a Banach space is weakly countably determined if it is K -co untably deter- mined in its weak top olo gy . T alagr and [16] has constructed a Ba na ch s pa ce whic h is weakly countably deter mined but which is not weakly K - analytic. A slight mo d- ification o f this exa mple gives a similar one with density character ω 1 . This shows that no analog ue of Theorem 8 is p ossible for weakly countably deter mined Banach spaces. The change in the exa mple consists in substituting the set T consider ed in [16, p. 78] by any subset T ′ ⊂ T o f cardina l ω 1 such that { o ( X ) : X ∈ T ′ } is uncountable and A by A ′ = { A ⊂ T ′ : A ∈ A 1 } (the notations ar e explained in [16]). Now, w e turn to the fact tha t cardinal b is b es t p ossible in Theo rem 5, Theo- rem 8 and their cor o llaries. W e fix a subset S of N N of car dinality b which is not σ -b ounded. F ollowing the exp osition of the example of Arg yros in [4, sectio n 1.6] we just sub- stitute the space Y = span { π σ : σ ∈ N N } in [4, Theorem 1.6.3] b y Y ′ = sp an { π σ : σ ∈ S } and we obtain a Banach space of densit y character b which is a subspace of a W CG space C ( K ) but which is not Asplund g enerated. The same arguments in [4 , section 1 .6] hold just changing N N by S where neces s ary . O nly the pro of of [4, Lemma 1 .6 .1] is not g o o d fo r this case. It must b e substituted b y the following: Lemma 18. L et Γ n , n ∈ N , b e any subsets of S such t hat S n ∈ N Γ n = S . Then ther e exist n, m ∈ N and an infinite set A ∈ A m such that A ⊂ Γ n . Here, a s in [4 , sec tio n 1.6], A m is the family of a ll subsets A ⊂ N N such that if σ, τ ∈ A and σ 6 = τ , then σ i = τ i if i ≤ m and σ m +1 6 = τ m +1 . Also, A = S ∞ m =1 A m . PROOF OF LE MMA 18: W e consider Γ i,j = { σ ∈ Γ i : σ 1 = j } , i, j ∈ N . Note that S = S i,j Γ i,j . Since S is not σ -b ounded, there exist n, l with Γ n,l un b ounded. This implies that for so me m , the set { σ m : σ ∈ Γ n,l } is infinite. W e take m the least integer with this prop er ty ( m > 1). Let B ⊂ Γ n,l be an infinite set such that σ m 6 = σ ′ m for σ , σ ′ ∈ B , σ 6 = σ ′ . Since a ll σ k with σ ∈ B , k < m , lie in a finite set, an infinite set A ⊂ B ca n b e chose such that A ∈ A m − 1 .  On the other hand, if we fo llow the pr o of in [4, section 4.3] that the Banach space C ( K ) of T alagrand is weakly K - a nalytic but not WCG, a nd we change K in [4, p. 76] b y K ′ = { χ A : A ∈ A , A ⊂ S } ⊂ { 0 , 1 } S then C ( K ′ ) still verifies this conditions a nd has density character b . Obs erve that C ( K ′ ) is weakly K -analy tic bec ause K ′ is a retract of the original K . The fact that C ( K ′ ) is not W CG (not even a subspace o f a WCG space) follows from [4, Theo r em 4.3.2] a nd Lemma 1 8 ab ov e by the s ame a rguments as in [4, p. 7 8 ]. 10 ANTONIO A VIL ´ ES 3. Count abl y lower fragment able comp act a In this section we prov e that the concept of quasi Radon Nikod´ ym compact [2] is equiv alent to that of countably low er fragmentable compact [5]. The main result for this class in [5] is that if K is co un tably low er fragmentable, then so is ( B C ( K ) ∗ , w ∗ ). W e no te that, with these tw o facts at hand, tog ether with the fact tha t if C ( K ) is Asplund gene r ated, then K is Radon-Nikod´ ym [4, Theor em 1.5.4], Theorem 3 is deduced from Theor e m 5. W e ne e d so me no tation: if K is a compa ct space and A ⊂ C ( K ) is a bo unded se t of contin uo us functions o ver K , we define the ps eudometric d A on K a s d A ( x, y ) = sup f ∈ A | f ( x ) − f ( y ) | . If X is a top o logical spa ce, d : X × X − → R is a map, and ∆ is a p ositive real num b er, it is said that d ∆- fr agments X if for each subset L of X there is a relative o p e n subset U of L of d -diameter less than or equal to ∆. Definition 1 9. A compa ct space K is said to b e countably lo wer fragmentable if there are bounded subsets { A n,p : n, p ∈ N } of C ( K ) suc h that C ( K ) = S n ∈ N A n,p for every p ∈ N , and the pseudometric d A n,p 1 p -fragments K . This is the definition as it app ear s in [5]. Howev er , v aria ble p is sup erfluous in it. If the sets A n, 1 exist, it is sufficient to define A n,p = { 1 p f : f ∈ A n, 1 } . On the other hand, we recall a concept introduced b y Namiok a [9]: F or a to p o - logical space K , a set L ⊂ K × K is said to b e an almost neighb orho o d of the diagonal if it contains the dia gonal ∆ K = { ( x, x ) : x ∈ K } and satis fie s that fo r every nonempty subset X of K there is a nonempty rela tive ope n subset U of X such that U × U ⊂ L . The use of this was sugge s ted to us b y I. Namiok a and simplifies o ur origina l pro of. Theorem 20. F or a c omp act subset K of [0 , 1] Γ the fol lowing ar e e qu ivalent: (1) K is quasi Ra don-Niko d´ ym c omp act (2) K is c oun tably lower fr agmentable. (3) Ther e ar e subsets Γ n,p , n, p ∈ N , of Γ su ch that d Γ n,p 1 p -fr agments K for every n, p ∈ N . PROOF: Suppose K is quasi Radon-Niko d ´ ym co mpa ct and let φ b e a low er semicontin uo us quasi metric whic h fra g ments K . Then, we just define A n,p =  f ∈ C ( K ) : | f ( x ) − f ( y ) | < 1 p whenever φ ( x, y ) ≤ 1 n  ∩ { f : k f k ∞ ≤ n } Clearly , d A n,p 1 p -fragments K b eca use any s ubs et of K of φ -diameter less than 1 n has d A n,p -diameter les s than 1 p , and we know that φ frag ments K . On the other hand, for a fix e d p ∈ N , in order to prove that C ( K ) = S n ∈ N A n,p , observe that, if f ∈ C ( K ), then C n =  ( x, y ) ∈ K × K : | f ( x ) − f ( y ) | ≥ 1 p and φ ( x, y ) ≤ 1 n  11 is a decreasing seq uence of compac t s ubsets of K × K with empty intersection so there is some n > k f k ∞ such that C n is empty , a nd then, f ∈ A n,p . That (2) implies (3) is e v ident, just to tak e Γ n,p = A n,p ∩ Γ whenever A n,p , n, p ∈ N ar e the sets in the definition of co untably low er frag ment ability . Now, supp ose (3). F or every n, p ∈ N , since d A n,p 1 p -fragments K , this means that the set C n,p = { ( x, y ) ∈ K × K : d Γ n,p ( x, y ) ≤ 1 p } is an almost neighbor ho o d of the diagonal which, in addition, is closed. On the other hand, observe that, fo r each n, p ∈ N , ( x, y ) ∈ C n,p if a nd only if | x γ − y γ | ≤ 1 p for a ll γ ∈ Γ n,p so tha t \ n,p ∈ N C n,p = \ p ∈ N ( ( x, y ) : | x γ − y γ | ≤ 1 p ∀ γ ∈ [ n ∈ N Γ n,p = Γ ) = ∆ K Now, K is quasi Radon-Nikod ´ ym by virtue o f [1 0, Theor em 1 ], which states that K is quasi Radon-Nikod ´ ym co mpact if and only if there is a co un table fa mily of closed a lmost neighbo r ho o ds of the diagonal whos e intersection is the diagonal ∆ K .  References [1] A. V. Arkhangel’ski ˘ ı, Gener al t op olo gy. II , Encyclopaedia of Mathematical Sciences, vol. 50, Springer-V erlag, Berli n, 1996. [2] A. D. Arv anitakis, Some r emarks on Rado n-Niko d´ ym co mp act sp ac es , F und. Math. 1 72 (2002), no. 1, 41–60. [3] B. Cascales, I. Namiok a, and J. Orihuela, The Lindel¨ of pr op erty in Banach sp ac e s , Studia Math. 1 54 (2003), no. 2, 165–192. [4] M . F abian, Gˆ ate aux differ entiability of c onvex funct ions and top olo gy , Canadian Mathemat- ical So ciet y Series of Monographs and Adv anced T exts, John Wiley & Sons Inc., New Y ork, 1997, W eak Asplund spaces, A Wiley-Interscience Publi cation. [5] M . F abian, M. Heisler, and E. Matou ˇ sko v´ a, R emarks on c ontinuous images of Radon- Niko d´ ym c omp acta , Comment . Math. U niv. Carolin. 3 9 (1998), no. 1, 59–69. [6] D. H. F remli n, Conse quenc es of Martin ’s axiom , Cambridge T r acts i n Mathematics, vol. 84, Camb ridge Universit y Press, Cambridge, 1984. [7] S.P . Gul’ko, On pr op ert ies of subsets of Σ -pr o ducts , So v. Math. Dokl. 18 (1977), no. 1, 1438–144 2. [8] S. M ercourakis and E. Stamati, A new class of we akly K -analytic Banach sp ac es , Mathe- matik a (to app ear). [9] I. Nami ok a, R adon-Niko d´ ym c omp act sp ac es and fr agmentability , M athematik a 34 (1987), no. 2, 258–281. [10] , On gener alizations of Radon-Niko d´ ym c omp act sp ac es , T op ology Pr o ceedings 26 (2002), 741–750. [11] J. Orihuela, W. Schac hermay er, and M. V aldivia, Every Radon-Niko d´ ym Corson c omp act sp ac e is Eb erlein c omp act , Studia Math. 98 (1991), no. 2, 157–174. [12] O. I. Reyno v, On a class of Hausdorff c omp acts and GSG Banach sp ac es , Studia M ath. 71 (1981/82 ), 294–300. [13] H. P . Rosenthal, The her e dity pr oblem for wea kly c omp actly ge ner ate d Banach sp ac es , Com- posi tio Math. 28 (1974), 83–111. [14] C. Stegall, Sp ac es of L ipschitz functions on Banach sp ac e s , F unctional analysis (Essen, 1991), Lecture Notes in Pur e and Appl. Math., v ol. 150, D ekker, New Y ork, 1994, pp. 265–278. [15] M. T al agrand, Esp ac es de Banach f aible ment K -analytiques , Ann. of Math. (2) 110 (1979), no. 3, 407–438. [16] , A new c ountably det e rmine d Banach sp ac e , Israel J. Math. 47 (1984), no. 1, 75–80. 12 ANTONIO A VIL ´ ES [17] E. K. v an Douw en, The inte gers and top olo gy , Handb o ok of set-theoretic topology , North- Holland, Amsterdam, 1984, pp. 111–167. Dep ar t am ento de M atem ´ aticas, Un iversidad de Murcia, 30100 Espinardo (Mu rcia), Sp ain E-mail addr e ss : avileslo@um.e s