Countable products of spaces of finite sets

COUNT ABLE PR ODUCTS OF SP A CES OF FINITE SETS ANTONIO A VIL ´ ES Abstract. W e consider the compact spaces σ n (Γ) of subsets of Γ of cardinality at most n and their countable products. W e g ive a complete classification of their Banach spaces of con tinuous functions and a partial top ologica l classification. F or an infinite set Γ and a natural n umber n , w e consider t he space σ n (Γ) = { x ∈ { 0 , 1 } Γ : | supp ( x ) | ≤ n } . Here supp ( x ) = { γ ∈ Γ : x γ 6 = 0 } . This is a closed, hence compact subset of { 0 , 1 } Γ , whic h is iden tified with the family of all subsets o f Γ of cardinal- it y at m ost n . In this w o rk w e will study the spaces whic h ar e countable pro ducts of spaces σ n (Γ), mainly their top o logical classification as well as the classification of their Banac h spaces o f contin uous functions. Let T b e the set of all sequenc es ( τ n ) ∞ n =1 with 0 ≤ τ n ≤ ω . When τ runs o ver T , σ τ (Γ) = Q ∞ 1 σ n (Γ) τ n runs ov er all finite and countable pro ducts of spaces σ k (Γ). F or τ ∈ T w e will call j ( τ ) to the suprem um o f all n with τ n > 0 and i ( τ ) to the suprem um of all n with τ n = ω . If τ n < ω for all n ≥ 1, then i ( τ ) = 0. Alw ay s 0 ≤ i ( τ ) ≤ j ( τ ) ≤ ω . Theorem 1 b elo w summarizes o ur know ledge ab out the top ological classification and it s pro of consists of a n um b er of lemmas along Section 2. Theorem 1. L et τ , τ ′ ∈ T and Γ an unc ountable set. (1) Supp ose j ( τ ) < ω . In this c ase, σ τ ′ (Γ) is h o me omorphic to σ τ (Γ) if and only if i ( τ ) = i ( τ ′ ) and τ n = τ ′ n for al l n > i ( τ ) . (2) Supp ose i ( τ ) = ω . In this c ase, if i ( τ ′ ) = ω , then σ τ (Γ) is home o- morphic to σ τ ′ (Γ) . This is not a complete classification and leav es the follo wing question op en: 2000 Mathe matics Subje ct Classi fic ation. 4 6B50, 46B2 6, 54B10, 54D30. Key wor ds and phr ases. Unifor m Eb erlein compact, regular a veraging op erator , count- able pro duct. This research w a s pa rtially s uppo rted b y the g rant BFM2002 -0171 9 of MCy T (Spain) and a FPU gr ant of MEC (Spa in). 1 2 ANTONIO A VIL ´ ES Problem 1. Let Γ b e an uncoun table set and τ , τ ′ ∈ T suc h that j ( τ ′ ) = j ( τ ) = ω , i ( τ ) < ω and there exists some n ≥ i ( τ ) with τ n 6 = τ ′ n . Is σ τ (Γ) homeomorphic to σ τ ′ (Γ)? F or ex ample, one particular instance of the problem is whethe r Q ∞ i =1 σ i (Γ) is homeomorphic to Q ∞ i =2 σ i (Γ). Ab out the spaces of con tinuous functions, it has b een recen tly prov ed by Marciszews ki [6] that a Banac h space C ( K ) is isomorphic to c 0 (Γ) if and only if K ⊂ σ n (Γ) for some n < ω . This is the case of an y compact of the form K = Q n i =1 σ k i (Γ) whic h can b e em b edded in to σ P k i ( S n 1 Γ × { i } ) by x 7→ S n 1 x i × { i } . Hence, it is a consequenc e of M arciszews ki’s result that the Banac h spaces of contin uous functions ov er finite pro ducts of spaces σ k (Γ) o ver a fixed Γ are all isomorphic. In Section 1 w e prov e a similar result for coun table pro ducts: Theorem 2. L et Γ b e an infinite set and ( k n ) b e any se quenc e of p osi- tive inte gers. Then the Banach sp ac es C ( Q n<ω σ k n (Γ)) and C ( σ 1 (Γ) ω ) ar e isomorphic. The tec hniques that w e will use are ba sed o n the use of regula r a v era ging op erators and the so called P e lczy´ nski’s decomp osition m etho d, dev elop ed in [8] and [9] in o rder to ac hiev e M iljutin’s res ult that the spaces of contin uous functions ov er uncoun table metrizable compacta are all isomorphic. Definition 3. Let φ : L − → K b e a con tin uous surjection b et w een compact spaces. A regular a ve raging op erator for φ is a b ounded p o sitiv e linear op erator T : C ( L ) − → C ( K ) with T (1 L ) = 1 K and T ( x ◦ φ ) = x for all x ∈ C ( K ). The coun table pro ducts of spaces of the form σ n (Γ) are uniform Eb erlein compact spaces, cf. [3]. This class consists of the w eakly compact subsets of the Hilb ert spaces , or equiv alen tly of the compact subse ts of the space B (Γ) = { x ∈ [ − 1 , 1] Γ : X γ ∈ Γ | x γ | ≤ 1 } ∼ ( B ℓ 2 (Γ) , w ) for some set Γ. Indeed, σ n (Γ) is homeomorphic to B (Γ) ∩ { 0 , 1 n } Γ . W e establish the follow ing result: Theorem 4. L et K b e a uniform Eb erlein c om p ac t of weig ht κ . Then ther e is a close d subsp ac e L of σ 1 ( κ ) N and an onto c ontinuous map f : L − → K which admits a r e gular aver aging op er ator. This improv es a result of Argyros and Arv anitakis [1] that for ev ery uni- form Eb erlein compact space K there is a totally disconnected uniform Eb erlein compact space L of the same w eight and a con tin uous surjection 3 f : L − → K whic h admits a regular a veraging op erato r, a nd also a result of Ben y amini, Rudin and W ag e [2] that ev ery uniform Eb erlein compact of w eight κ is a contin uous image o f a closed subset of σ 1 ( κ ) N . W e note that there are man y totally disconnec ted uniform Eb erlein compact space s whic h cannot b e em b edded into σ 1 ( κ ) N , cf. Lemma 12 b elow. Not a tions All to p ological space s will b e assu med t o b e completely regular. By iden- tifying elemen ts of { 0 , 1 } Γ with subsets of Γ, the space σ n (Γ) ⊂ { 0 , 1 } Γ can b e view ed a s the fa mily of all subsets of Γ of cardinality less t han or equal to n , endo w ed with the to p ology whic h has a base the sets of the form Φ G F = { y ∈ σ n (Γ) : F ⊂ y ⊂ Γ \ G } for F and G finite subsets of Γ. W e will denote b y p : σ 1 (Γ) k − → σ k (Γ) the con tin uous surjection given by p ( x 1 , . . . , x k ) = x 1 ∪ · · · ∪ x k . Note that from the existence of such a function fo llo ws the fact tha t an y coun ta ble pro duct Q i<ω σ k i (Γ) is a con tin uous image o f σ 1 (Γ) ω . W e will also denote B + (Γ) = B (Γ ) ∩ [0 , 1] Γ . 1. Banach sp a ce classifica tion The follow ing Theorem 5 is the k ey result of this section. A somewhat similar fa ct can b e fo und in [10], namely t hat the natural surjection K 2 − → exp 2 ( K ) = {{ x, y } : x, y ∈ K } giv en by ( x, y ) 7→ { x, y } has a regular a veraging op erato r. Theorem 5. The map p : σ 1 (Γ) k − → σ k (Γ) admits a r e gular ave r agin g op er ator. Pro of: F or ev ery y ∈ σ k (Γ) let us denote by L ( y ) the subset of p − 1 ( y ) consisting of all ( x 1 , . . . , x k ) ∈ p − 1 ( y ) suc h that x i ∩ x j = ∅ for i 6 = j (that is, L ( y ) consists of those tuples o f p − 1 ( y ) in whic h no singleton app ears t wice). The regula r av eraging op erator T : C ( σ 1 (Γ) k ) − → C ( σ k (Γ)) is defined as follo ws: T ( f )( y ) = 1 | L ( y ) | X x ∈ L ( y ) f ( x ) The o nly difficult p oint is in proving that T ( f ) is a con tin uous function whenev er f is con tin uous. So fix f ∈ C ( σ 1 (Γ) k ) a nd a p oin t y ∈ σ k (Γ) and ε > 0. F or each x = ( x 1 , . . . , x k ) ∈ L ( y ), since f is contin uous a t x , there is a neigh b orho o d U x of x in σ 1 (Γ) k in whic h sup x ′ ∈ U x | f ( x ) − f ( x ′ ) | < ε . The set U x m ust con tain a basic neigh b orho o d of x of the form 4 ANTONIO A VIL ´ ES Φ G x 1 x 1 × · · · × Φ G x k x k ⊂ U x where G x i is a finite set of Γ disjoint with x i . W e define a neigh b orho o d of y as V = Φ S x ∈ L ( y ) S k i =1 G x i \ y y and w e shall see that | T ( f )( y ) − T ( f )( y ′ ) | < ε for eve ry y ′ ∈ V . So w e fix y ′ ∈ V (in particular y ⊂ y ′ ). First, w e define an on to map r : L ( y ′ ) − → L ( y ) in the follow ing w ay , if ( x 1 , . . . , x k ) ∈ L ( y ′ ) then r ( x ) = ( r ( x ) 1 , . . . , r ( x ) k ) where r ( x ) i = x i ∩ y . It is straigh tforw ard to c heck that all the fibers of r ha v e the same cardinality , call n = | r − 1 ( x ) | , so that | L ( y ′ ) | = n | L ( y ) | . The k ey fact ( used in the final inequalit y in t he ex pression b elo w) is that if x ∈ L ( y ) and x ′ ∈ r − 1 ( x ), then x ′ ∈ U x . T o see this, tak e x = ( x 1 , . . . , x k ) ∈ L ( y ) and x ′ = ( x ′ 1 , . . . , x ′ k ) ∈ r − 1 ( x ). W e che c k that x ′ i ∈ Φ G x i x i . If x ′ i ⊂ y then x ′ i = x i . If x ′ i = { γ } ⊂ y ′ \ y then x i = ∅ and since y ′ ∈ V , γ 6∈ G x i and a gain x ′ i ∈ Φ G x i x i . Finally , | T ( f )( y ′ ) − T ( f )( y ) | =       1 | L ( y ′ ) | X x ′ ∈ L ( y ′ ) f ( x ′ ) − 1 | L ( y ) | X x ∈ L ( y ) f ( x )       =       1 | L ( y ′ ) | X x ∈ L ( y ) X x ′ ∈ r − 1 ( x ) f ( x ′ ) − 1 | L ( y ) | X x ∈ L ( y ) f ( x )       =       1 n | L ( y ) | X x ∈ L ( y ) X x ′ ∈ r − 1 ( x ) f ( x ′ ) − 1 | L ( y ) | X x ∈ L ( y ) f ( x )       =       1 | L ( y ) | X x ∈ L ( y )     1 n X x ′ ∈ r − 1 ( x ) f ( x ′ )   − f ( x )         =       1 | L ( y ) | X x ∈ L ( y )   1 n X x ′ ∈ r − 1 ( x ) ( f ( x ′ ) − f ( x ))         ≤ 1 | L ( y ) | X x ∈ L ( y )   1 n X x ′ ∈ r − 1 ( x ) | f ( x ′ ) − f ( x ) |   < 1 | L ( y ) | X x ∈ L ( y )   1 n X x ′ ∈ r − 1 ( x ) ε   = ε 5  Lemma 6. (a) L et g : L − → K b e a c ontinuous surje ction b etwe en c omp act sp ac es which admits a r e gular aver aging op er ator and le t M b e a close d subse t of K . Then the r estriction g : g − 1 ( M ) − → M also admits a r e gular aver aging op er ator [1, Prop osition 1 8] . (b) L et { g i : L i − → K i } b e a fa mily of c ontinuous s urje ctions b etwe en c omp act sp ac es which admit r e gular aver aging op er ators. Then the pr o duct map Q g i : Q L i − → Q K i admits a r e gular aver aging op er- ator to o [9, Prop osition 4.7 ] . Pro of of Theorem 4: W e mak e the observ ation that the space B (Γ) can b e em b edded into B + (Γ × { a, b } ) ∼ B + (Γ) b y the map u ( x ) γ ,a = max(0 , x γ ) and u ( x ) γ ,b = max(0 , − x γ ). This observ ation allows to consider our K as a subset of B + (Γ) with | Γ | = κ . Let φ : { 0 , 1 } ω − → [0 , 1] giv en b y φ ( x ) = P r i x i where r i = 1 3  2 3  i . It is prov en in [1 ] that φ admits a regular a veraging op erator and hence by Lemma 6 also φ Γ : { 0 , 1 } ω × Γ − → [0 , 1 ] Γ and its restriction φ Γ : L ′ = ( φ Γ ) − 1 ( K ) − → K a dmit a regular a v eraging op erator. The space L ′ is a subspace of L 0 = ( φ Γ ) − 1 ( B + (Γ)) for whic h w e can giv e the follo wing description: x ∈ L 0 ⇐ ⇒ φ Γ ( x ) ∈ B + (Γ) ⇐ ⇒ X γ ∈ Γ φ Γ ( x ) γ ≤ 1 ⇐ ⇒ X γ ∈ Γ ∞ X n =0 r n x ( γ ,n ) ≤ 1 ⇐ ⇒ ∞ X n =0 r n N n ( x ) ≤ 1 , where N n ( x ) is the cardinalit y of supp ( x | Γ ×{ n } ). F rom this description, if M n denotes the in teger part of r − 1 n , then L ′ ⊂ L 0 ⊂ Q ∞ n =1 σ M n (Γ). F rom Theorem 5 a nd part (b) of Lemma 6 follows the existence of a contin u- ous surjection g : σ 1 (Γ) ω − → Q ∞ n =1 σ M n (Γ) whic h admits a regular av er- aging op erator. Making use of part (a) of Lemma 6 w e get a surjection g : L = g − 1 ( L ′ ) − → L ′ with regular a v erag ing op erator and the composition L − → L ′ − → K is the desired map.  W e shall need no w the so called P e lczy ´ nski’s decomposition metho d, wh ic h is used to establish the existence of isomorphisms b etw een Banac h spaces. F or Banach spaces X a nd Y w e shall write X | Y if t here exists a Ba nac h space Z suc h that X ⊕ Z is isomorphic to Y , shortly X ⊕ Z ∼ Y . Also, 6 ANTONIO A VIL ´ ES Y = ( X 1 ⊕ X 2 ⊕ · · · ) c 0 denotes the c 0 -sum of t he Banac h space s X 1 , X 2 , . . . , Y = { y = ( x n ) ∈ Y X n : lim k x n k = 0 } , k y k = sup n k x n k . Theorem 7 (cf. [9 ], § 8) . L et X and Y b e Ban a ch sp ac es such that X | Y , Y | X and ( X ⊕ X ⊕ · · · ) c 0 ∼ X , then X ∼ Y . If there exists a surjection φ : L − → K with regular av eraging op erator, then C ( K ) | C ( L ), cf. [9]. In part icular if L ⊂ K is a retract of K , since in this case the restriction op erator is a regular av eraging o p erator for the retraction. On the other hand, in order to guaran tee the last h yp o thesis in Theorem 7 w e shall use the criterion of Lemma 8 . F or to p ological spaces K n , K 1 ⊕ K 2 ⊕ · · · denotes the discrete to p ological sum, while α ( S ) is the one p oint compactification of a lo cally compact space S . Lemma 8. L et K b e a c omp act sp ac e which is home om orphic to α ( K ⊕ K ⊕ · · · ) . Then ( C ( K ) ⊕ C ( K ) ⊕ · · · ) c 0 ∼ C ( K ) . Pro of: W e apply Theorem 7 to X = ( C ( K ) ⊕ C ( K ) ⊕ · · · ) c 0 and Y = C ( K ). The only p oint is in ch ec king that X | Y . Let ∞ denote the infinity p oint of α ( K ⊕ K ⊕ · · · ) ∼ K . Then X ∼ Y ′ = { f ∈ C ( K ) : f ( ∞ ) = 0 } and Y ∼ Y ′ ⊕ R .  Pro of of Theorem 2: Set K = σ 1 (Γ) ω and L = Q σ k n (Γ). W e apply Theorem 7 to X = C ( K ) and Y = C ( L ). First, we already observ ed that from Theorem 5 and Lemma 6(b) fo llo ws the existence o f a surjection with regular av eraging op erat or f : K − → L and hence C ( L ) | C ( K ). On the other hand, K is a retra ct of L b ecause for an y k , σ 1 (Γ) is homeomorphic to a clop en subset of σ k (Γ): the family of all subsets whic h contain fixed elemen ts γ 1 , . . . , γ k − 1 . Therefore C ( K ) | C ( L ). By Lem ma 8, it only remains to sho w that α ( K ⊕ K ⊕ · · · ) ∼ K . F or this, fix γ ∈ Γ and set for n = 1 , 2 , . . . K n = { x ∈ K = σ 1 (Γ) ω : γ ∈ x 1 ∩ · · · ∩ x n − 1 \ x n } . The sets K n are disjoin t clop en sets homeomorphic to K and K is the one p oint compactification of their union with p oin t of infinity ( { γ } , { γ } , . . . ).  2. Topological classifica tion This section is devoted to the pro of of Theorem 1 . Before en tering this, w e p oin t out wh y w e assume Γ to b e uncoun table. The reasonings b elo w do not apply in the coun table case a nd the situation is indeed completely differen t. All p erfect totally disconnected metrizable compact spaces ar e homeomorphic [5 , Theorem 7.4] and this implies that all coun ta ble pro d- ucts of spaces σ k ( ω ) are homeomorphic. The finite pro ducts are countable 7 compacta, whose top olo gical classification is also w ell kno wn after the clas- sical pap er [7]: tw o of them are homeomorphic if and only if they ha v e same Can tor-Bendixson deriv ation index and the same cardinalit y of the last nonempt y Can tor-Bendixson deriv a tiv e. Straigh tfo rw ard computations giv e that these t wo in v arian ts for a finite pro duct Q n i =1 σ k i ( ω ) tak e the v alues 1 + P n 1 k i and 1 resp ectiv ely . F rom no w on, Γ will b e alw a ys an uncoun table set. Lemma 9. If m < n then σ m (Γ) × σ n (Γ) ω is home omorp hic to σ n (Γ) ω . Pro of: W e denote again by ( X 1 ⊕ X 2 ⊕ · · · ) the discrete top ological sum of the spaces X 1 , X 2 , . . . and b y α X the one-p oint compactification o f the lo cally compact space X . Fix γ 0 , . . . , γ n − 1 ∈ Γ. W e consider t he set L = ω × { 0 , . . . , n − 1 } endow ed with the lexicographical order: ( k , i ) < ( k ′ , i ′ ) whenev er either k < k ′ or k = k ′ and i < i ′ . F or ev ery ( k , i ) ∈ L w e define a clop en set of σ n (Γ) ω as A ( k, i ) = { x ∈ σ n (Γ) ω : γ i 6∈ x k , γ i ′ ∈ x k ′ ∀ ( k ′ , i ′ ) < ( k , i ) } = { x ∈ σ n (Γ) ω : γ i 6∈ x k ⊃ { γ 0 , . . . , γ i − 1 } , x j = { γ 0 , . . . , γ n − 1 }∀ j < k } . Notice that A ( k, i ) is homeomorphic to σ n − i (Γ) × σ n (Γ) ω and that { A l : l ∈ L } constitutes a disjoin t sequence of clop en subsets of σ n (Γ) ω with only limit p oint the sequence ξ ∈ σ n (Γ) ω constan tly equal t o { γ 0 , . . . , γ n − 1 } . Hence, σ n (Γ) ω ≈ α M l ∈ L A l ! ≈ α n − 1 M i =0 M j <ω ( σ n − i (Γ) × σ n (Γ) ω ) ! . On the other ha nd, w e can p erform a similar decomp osition in σ m (Γ) × σ n (Γ) ω defining, for j < m and ( k , i ) ∈ L : B ′ j = { ( y , x ) ∈ σ m (Γ) × σ n (Γ) ω : γ j 6∈ y , { γ 0 , . . . , γ j − 1 } ⊂ y } B ( k, i ) = { ( y , x ) ∈ σ m (Γ) × σ n (Γ) ω : γ i 6∈ x k , γ i ′ ∈ x k ′ ∀ ( k ′ , i ′ ) < ( k , i ) , { γ 0 , . . . , γ m − 1 } ⊂ y } Again B ′ j is homeomorphic to σ m − j (Γ) × σ n (Γ) ω , B ( k, i ) is homeomorphic to σ n − i (Γ) × σ n (Γ) ω and altog ether they constitute a disjoin t seque nce of clop en sets with a single limit p oint ( { γ 0 , . . . , γ m − 1 } , ξ ) o ut o f them, so σ m (Γ) × σ n (Γ) ω ≈ α M l ∈ L B l ⊕ m − 1 M j =0 B ′ j ! ≈ α n − 1 M i =0 M j <ω ( σ n − i (Γ) × σ n (Γ) ω ) ! . Lemma 10. If m < n < ω then σ m (Γ) ω × σ n (Γ) ω is home omorphic to σ n (Γ) ω . Pro of: σ m (Γ) ω × σ n (Γ) ω ≈ ( σ m (Γ) × σ n (Γ) ω ) ω ≈ ( σ n (Γ) ω ) ω ≈ σ n (Γ) ω .  8 ANTONIO A VIL ´ ES Lemma 11. L et m 1 , . . . , m r < n < ω and e 1 , . . . , e r ≤ ω . Then the sp ac e Q r i =1 σ m i (Γ) e i × σ n (Γ) ω is home omorp hic to σ n (Γ) ω . Pro of: F ollo ws from r ep eated application o f Lem mas 9 and 10 a b ov e.  F rom Lemma 11 it follows that an y space σ τ (Γ) with i ( τ ) = ω is home- omorphic to σ ( ω, ω ,... ) (Γ) (b ecause we can substitute eac h factor σ n (Γ) ω of σ τ (Γ) b y t he homeomorphic Q i ≤ n σ i (Γ) ω ) and this pro ves part (2) of The o- rem 1. L emma 11 also sho ws that it is irrelev an t in determining the home- omorphism class of σ τ (Γ) which are the v alues τ n for n < i ( τ ). Hence, in order to pro v e part(1) of Theorem 1 it remains to sho w that if j ( τ ) < ω and σ τ (Γ) is homeomorphic to σ τ ′ (Γ) then τ n = τ ′ n for all n > i ( τ ). W e recall that a family { S η } η ∈ H of sets is a ∆-system if there is a set S (called the ro ot of the ∆-system) suc h that S η ∩ S η ′ = S for all η 6 = η ′ . W e will make use of the fact that any uncoun table family o f finite sets has an uncoun table subfamily whic h is a ∆-system, cf.[4, Theorem 1.4 ] fo r κ = ω and α = ω 1 . The f ollo wing lemma includes as a particular case that σ n +1 (Λ) do es not em b ed in to σ n (Γ) ω . This fact, whose pro of corr esp onds to Steps 1 - 3 below w as show n to us by Witold Marciszewski, and it seems that it w a s kno wn to sev eral p eople b efore. Lemma 12. I f | Λ | > ω , n ≥ 0 , k ≥ 0 , then the sp ac e σ n +1 (Λ) k +1 do es n o t emb e d into σ n (Γ) ω × σ n +1 (Γ) k . Pro of: Supp ose that there exists suc h an em b edding. Step 1 . P assing to a suitable uncountable subset of Λ , w e can supp ose that there is an em b edding φ : σ n +1 (Λ) k +1 − → σ n (Γ) m × σ n +1 (Γ) k for some m < ω . T o see this, tak e ϕ : σ n +1 (Λ) k +1 − → σ n (Γ) ω × σ n +1 (Γ) k our original em b edding. In this s tep, we shall denote an elemen t x ∈ σ n +1 (Λ) k +1 as x = ( x 0 , . . . , x k ). F or each λ ∈ Λ and ev ery i ∈ { 0 , . . . , k } we find a clop en set A i λ of σ n (Γ) ω × σ n +1 (Γ) k whic h separates t he disjoint compact sets ϕ ( { x : λ ∈ x i } ) and ϕ ( { x : λ 6∈ x i } ). Asso ciated to A i λ w e ha v e a finite subset F i λ ⊂ ω suc h that A i λ = σ n (Γ) ω \ F i λ × B i λ with B i λ a clop en subset of σ n (Γ) F i λ × σ n +1 (Γ) k . W e ch o ose Λ ′ to b e an uncoun table subset of Λ suc h that S k i =0 F i λ = S k i =0 F i λ ′ = F for all λ , λ ′ ∈ Λ ′ and in this case the comp osition σ n +1 (Λ ′ ) k +1 ֒ → σ n +1 (Λ) k +1 − → σ n (Γ) ω × σ n +1 (Γ) k − → σ n (Γ) F × σ n +1 (Γ) k 9 is one-to- one. The r eason is that if x, y ∈ σ n +1 (Λ ′ ) k +1 are different then there exists i ∈ { 0 , . . . , k } and λ ∈ Λ ′ suc h that λ ∈ x i but λ 6∈ y i (or vicev - ersa). Then φ ( x ) ∈ A i λ and φ ( y ) 6∈ A i λ so either the co ordinate of σ n +1 (Γ) k or some co ordinate of F λ i ⊂ F m ust b e differen t for φ ( x ) and φ ( y ). Step 2 . F or i = 0 , . . . , k and λ ∈ Λ w e define e λ i ∈ σ n +1 (Λ) k +1 to b e the elemen t whic h has { λ } in co ordinate i and ∅ in all other co ordinates. Eac h φ ( e λ i ) will b e of the form φ ( e λ i ) = ( x λ i [1] , . . . , x λ i [ m ] , x λ i [ m + 1] , . . . , x λ i [ m + k ]) with x λ i [ j ] ∈ σ n (Γ) if j ≤ m and x λ i [ j ] ∈ σ n +1 (Γ) if m < j ≤ m + k . Pass ing to a suitable uncountable subset o f Λ, w e can assume that for every fixed i ∈ { 0 , . . . , k } and j ∈ { 1 , . . . , m + k } the fa mily { x λ i [ j ] : λ ∈ Λ } is a ∆- system of ro ot R i [ j ] formed by sets of the same cardinalit y c i [ j ]. Step 3 . W e claim that for i = 0 , . . . , n and j = 1 , . . . , m , the ∆-system { x λ i [ j ] : λ ∈ Λ } is constan t. Supp ose the contrary for some fixed i ≤ n a nd j ≤ m . Then x λ i [ j ] = R ∪ S λ ∈ σ n (Γ) where R ∩ S λ = ∅ , S λ 6 = ∅ , a nd S λ ∩ S λ ′ = ∅ for λ 6 = λ ′ . W e consider the sets A λ = { y = ( y [1] , . . . , y [ m + k ]) ∈ σ n (Γ) m × σ n +1 (Γ) k : y [ j ] ⊃ S λ } . The A λ ’s are neighborho o ds of the φ ( e λ i )’s with the prop ert y that for eve ry F ⊂ Λ with | F | > n , T λ ∈ F A λ = ∅ ( b ecause fo r y in that in tersection, | y [ j ] | > n and y [ j ] ∈ σ n (Γ)). Let ψ : σ n +1 (Λ) − → σ n +1 (Λ) k +1 b e the map defined b y ψ ( x ) i = x and ψ ( x ) i ′ ( x ) = ∅ if i ′ 6 = i . Then the ( φψ ) − 1 ( A λ )’s are neighborho o ds o f the { λ } ’s in σ n +1 (Λ) with the prop ert y that for ev- ery F ⊂ Λ with | F | > n , T λ ∈ F ( φψ ) − 1 ( A λ ) = ∅ . This is a con tradiction since suc h a family of neigh b orho o ds cannot b e found. Namely , tak e basic neigh b orho o ds with { λ } ∈ Φ G λ { λ } ⊂ ( φψ ) − 1 ( A λ ) and tak e Λ ′ ⊂ Λ uncoun t- able with { G λ : λ ∈ Λ ′ } a ∆-system of ro o t R ′ . Then construct inductiv ely a finite seque nce F = { λ 1 , . . . , λ n +1 } ⊂ Λ ′ \ R ′ suc h that λ p 6∈ S q

0 w e need some extra w ork. F rom step 3, w e deduce that fo r eac h i ∈ { 0 , . . . , k } there m ust exist j ∈ { m + 1 , . . . , m + k } suc h that the family { x λ i [ j ] : λ ∈ Λ } is a nonconstan t ∆-system. Since i runs in a set of k + 1 eleme n ts a nd j in a set of k elemen ts, there must exist t w o different i, i ′ ∈ { 0 , . . . , k } such that for 10 ANTONIO A VIL ´ ES the same j , { x λ i [ j ] : λ ∈ Λ } and { x λ i ′ [ j ] : λ ∈ Λ } are nonconstan t ∆-systems. W e assume that c i [ j ] ≥ c i ′ [ j ] (these n umbers are defined in step 2). Again, for λ ∈ Λ w e consider the sets A λ = { ( y [1] , . . . , y [ m + k ]) ∈ σ n (Γ) m × σ n +1 (Γ) k : y [ j ] ⊃ x λ i [ j ] } , A ′ λ = { ( y [1] , . . . , y [ m + k ]) ∈ σ n (Γ) m × σ n +1 (Γ) k : y [ j ] ⊃ x λ i ′ [ j ] } . The A λ ’s and the A ′ λ ’s are neigh b orho o ds o f t he φ ( e λ i )’s and the φ ( e λ i ′ )’s resp ectiv ely with the prop erty that ( ∗ ) ∀ λ ∈ Λ ∀ F ⊂ Λ | F | > n ∧ x λ i [ j ] 6⊆ [ µ ∈ F x µ i ′ [ j ] ! ⇒ A λ ∩ \ µ ∈ F A ′ µ = ∅ . That interse ction is empty b ecause if y b elongs to it, then x λ i [ j ] ∪ [ µ ∈ F x µ i ′ [ j ] ⊂ y [ j ] ∈ σ n +1 (Γ) and the set in the left, if x λ i [ j ] 6⊆ S µ ∈ F x µ i ′ [ j ], has cardinalit y g reater than n +1, a contradiction. Since the ∆-systems are not constan t and c i [ j ] ≥ c i ′ [ j ], if x λ i [ j ] ⊆ S µ ∈ F x µ i ′ [ j ] holds, t here m ust b e some µ ∈ F and some γ ∈ x λ i [ j ] suc h that γ ∈ x µ i ′ [ j ] \ R i ′ [ j ]. F o r a fixed λ there are only finitely many µ ’s with ( x µ i ′ [ j ] \ R i ′ [ j ]) ∩ x λ i [ j ] 6 = ∅ . Hence fo r ev ery λ , we can find a cofinite subset Λ λ of Λ suc h that the hypothesis x λ i [ j ] 6⊆ S µ ∈ F x µ i ′ [ j ] of statemen t ( ∗ ) holds whenev er F ⊂ Λ λ . F or short, w e kno w that for ev ery λ ∈ Λ there exists a cofinite subset Λ λ of Λ suc h that ∀ F ⊂ Λ λ | F | > n ⇒ A λ ∩ \ µ ∈ F A ′ µ = ∅ . This con tradicts the following lemma for B λ = φ − 1 ( A λ ) and B ′ λ = φ − 1 ( A ′ λ ): Lemma 13. F or every λ ∈ Λ , let B λ and B ′ λ b e neighb orho o ds of e λ i and e λ i ′ r esp e ctively in σ n +1 (Λ) k +1 . Then ther e exists λ 0 ∈ Λ and an infinite set S ⊂ Λ such that for every F ⊂ S with | F | = n + 1 , B λ 0 ∩ \ µ ∈ F B ′ µ 6 = ∅ Pro of: F o r a simpler notat ion, w e will assume that i = 0 and i ′ = 1. Notice that a basic clop en set Φ G F of σ n +1 (Λ) is nonempt y if and o nly if F ∩ G = ∅ and | F | ≤ n + 1 . Eac h B λ and eac h B ′ µ con tain basic clop en sets of the form Φ G λ 0 { λ } × Φ G λ 1 ∅ × Φ G λ 2 ∅ × · · · × Φ G λ k ∅ ⊆ B λ Φ H µ 0 ∅ × Φ H µ 1 { µ } × Φ H µ 2 ∅ × · · · × Φ H µ k ∅ ⊆ B ′ µ 11 with all G λ l and H µ l finite su bsets of Λ and λ 6∈ G λ 0 and µ 6∈ H µ 1 . First, w e find M ⊂ Λ a countably infinite set suc h that µ ′ 6∈ H µ 1 for ev ery µ, µ ′ ∈ M . This can b e do ne as fo llo ws. W e b egin with an infinite M 1 ⊂ Λ suc h that the f amily { H µ 1 : µ ∈ M 1 } is a ∆-system o f ro ot R , and w e set M 2 = M 1 \ R . Then we can find recursiv ely a sequence ( µ p ) p<ω ⊂ M 2 suc h that µ p 6∈ S q

j . In order to finish the pro of of this part (1 ), w e m ust c hec k that i ( τ ) = i ( τ ′ ) = i and that τ k = τ ′ k for i < k < j . In order t o g et this, we shall lo ok at embeddabilit y of spaces σ n (Γ) k in to the clop en sets of σ τ (Γ). F or this purp ose, w e observ e that it is enough to lo ok a t some basic family of clop en sets, if the others are union of them: Lemma 14. L et X b e a c o m p act sp ac e and C 1 , . . . , C t op en subsets o f X . If σ n (Λ) k emb e ds in to S t 1 C i , then ther e exists i ≤ t such that σ n (Λ) k emb e ds into C i . Pro of: It reduces to pro v e that whe nev er w e expres s σ n (Λ) k as a union of op en sets as σ n (Λ) k = C 1 ∪ · · · ∪ C t then some C i m ust contain a cop y of σ n (Λ) k . Pic k i ∈ { 1 , . . . , t } suc h that x 0 = ( ∅ , . . . , ∅ ) ∈ C i . There a re finite sets G 1 , . . . , G k of Λ suc h that x 0 ∈ Φ G 1 ∅ × · · · × Φ G k ∅ ⊂ C i 12 ANTONIO A VIL ´ ES This finishes the pro of b ecause Φ G 1 ∅ × · · · × Φ G k ∅ is homeomorphic to σ n (Λ) k .  Let us denote no w b y K = Q s ∈ S σ n s (Γ) an y finite or countable pro duct of spaces of t yp e σ n (Γ). Any clopen set of K is a finite union of basic clop en sets of t he form C = Y s ∈ A Φ G s F s × Y s 6∈ A σ n s (Γ) where A is a finite s ubset of S and Φ G s F s a basic clop en se t of σ n s (Γ). Such a basic clopen set is homeomorphic to ( ⋆ ) C ∼ Y s ∈ A σ n s −| F s | (Γ) × Y s 6∈ A σ n s (Γ) No w, after Lemma 12, Lemma 14 and the top ological description ( ⋆ ) of the basic clop en sets given ab ov e, we are in a p osition to state that, in the situation o f part (1) of Theorem 1, the following hold: (A) i ( τ ) = i ( τ ′ ) = i is the greatest in teger n suc h tha t σ n (Γ) em b eds in to an y clop en set of σ τ (Γ). (B) F or n = j, j − 1 , j − 2 , . . . , i + 1, τ n = τ ′ n is the greatest integer k suc h that there is a clop en set C of σ τ (Γ) in whic h σ n +1 (Γ) cannot b e em b edded, but in w hic h nev ertheless σ n (Γ) k + P r >n τ r do es em b ed. This finishes the pro of of Theorem 1. F or statement (A), since σ i ( τ ) (Γ) ω is one of the factor s of σ τ (Γ), it is clear tha t still σ i ( τ ) (Γ) ω is a factor of an y clop en set lik e in ( ⋆ ). On the other hand, there a re only finitely man y factors of t yp e σ m (Γ), m > i ( τ ) in σ τ (Γ), hence a clop en set lik e in ( ⋆ ) can b e obtained so that all f actors in Q s ∈ A σ n s −| F s | (Γ) × Q s 6∈ A σ n s (Γ) are o f the form σ m (Γ) w ith m ≤ i ( τ ). By Lemma 12, σ k (Γ) does not em b ed in s uc h C if k > i ( τ ). Statemen t (B) is prov ed b y “down w ards induction” starting in j and finishing with i + 1. W e know, by Lemma 1 1, that σ τ (Γ) ∼ σ i (Γ) ω × j Y m = i +1 σ m (Γ) τ m No w statemen t ( B ) for n = j is a direct consequence of Lemma 12 since no clop en can contain σ j +1 (Γ) and the maximal expo nen t of σ j (Γ) inside σ τ (Γ) is τ j . W e pass to the case when i < n < j . The “biggest” p ossible basic clop en set C of σ τ (Γ) not con taining σ n +1 (Γ) is obtained by reducing 13 as necessary the factors σ m (Γ) with m > n : C ∼ σ i (Γ) ω × n Y m = i +1 σ m (Γ) τ m × j Y m = n +1 σ n (Γ) τ m The maximal exponent of σ n (Γ) in suc h a C is P j m = n σ τ m .  The presen t w ork w a s written during a visit to the Unive rsit y of W a r- sa w. The author wishes to t hank their hospitality , specially to Wito ld Mar- ciszews ki and Roman P ol, and to Rafa l G ´ orak, from the Polish Academy of Sciences. This w ork o w es v ery mu c h to the discussion with them a nd their suggestions. Reference s [1] S. A. Argyros and A. D. 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