Characterization of $ell_p$-like and $c_0$-like equivalence relations

CHARA CTERIZA TION OF ℓ p -LIKE AND c 0 -LIKE EQUIV ALENCE RELA TIONS LONGYUN DING Abstra ct. Let X b e a P olish space, d a pseudo-metric on X . If { ( u, v ) : d ( u, v ) < δ } is Π 1 1 for each δ > 0, we show th at either ( X, d ) is separable or there are δ > 0 and a p erfect set C ⊆ X such that d ( u, v ) ≥ δ for distinct u, v ∈ C . Gran ting th is dic hotomy , we c haracterize the p ositions of ℓ p -like and c 0 -like equiv alence relations in the Borel reducibility hierarch y . 1. Introduction Let ( X, d ) b e a pseudo-metric space . If X is not separable, by Zorn’s lemma, w e can easily p ro v e that, there are δ > 0 and a noncoun table set C ⊆ X suc h that d ( u, v ) ≥ δ for distinct u, v ∈ C . Ho w ev er, if w e do not assume C H, can w e fin d such a C w hose cardinal is of 2 N ? J. H. S ilv er [10] answered a similar pr ob lem for equiv ale nce relatio ns und er an extra assumption of coanalytici t y . A top ological space is called a Polish sp ac e if it is separable and com- pletely metrizable. As usual, W e denote the Borel, analytic and coana- lytic sets by ∆ 1 1 , Σ 1 1 and Π 1 1 resp ectiv ely . F or their effectiv e analogues, the Kleene p oin tclasses and the relativized Kleene pointcla sses are d enoted by ∆ 1 1 , Σ 1 1 , Π 1 1 , ∆ 1 1 ( α ) , Σ 1 1 ( α ) , Π 1 1 ( α ), etc. F or m ore d etails in d escrip tiv e s et the- ory , one can see [7] and [9]. Theorem 1.1 (Silv er) . L et E b e a Π 1 1 e quivalenc e r elation on a Polish sp ac e. Then E has either at most c ountably many or p erfe ctly many e quivalenc e classes. In section 2, we use the Gandy-Harrington top ology to e stablish the fol- lo wing dic hotom y . Theorem 1.2. L et X b e a Polish sp ac e, d a pseudo-metric on X . If { ( u, v ) : d ( u, v ) < δ } is Π 1 1 for e ach δ > 0 , then either ( X, d ) is sep ar able or ther e ar e δ > 0 a nd a p erfe ct set C ⊆ X such th at d ( u, v ) ≥ δ for d istinct u, v ∈ C . Let X , Y b e Polish spaces and E , F equiv alence relations on X, Y resp ec- tiv ely . A Bor el r e duction of E to F is a Borel function θ : X → Y suc h Date : Nov ember 20, 2018. 2000 Mathematics Subje ct Classific ation. Primary 03E15, 54E35 , 46A45. Researc h partially supported b y the National Natural Science F oundation o f China (Grant No. 107 01044). 1 2 LONGYUN DING that ( x, y ) ∈ E iff ( θ ( x ) , θ ( y )) ∈ F , for all x, y ∈ X . W e sa y that E is Bor el r e ducible to F , denoted E ≤ B F , if ther e is a Borel reduction of E to F . If E ≤ B F and F ≤ B E , we say that E and F are Bor el bir e ducible and denote E ∼ B F . W e refer to [4] for b ac kground on Borel redu cibilit y . In section 3, we will in tro duce notions of ℓ p -lik e and c 0 -lik e equiv alence relations. Granting the dic hotomy on ps eu do-metric spaces, we answ er that when is E 1 Borel reducible to an ℓ p -lik e or a c 0 -lik e equiv alence relation. In th e end , w e compare ℓ p -lik e and c 0 -lik e equiv alence relations with some remark able equiv alence relations E 0 , E 1 , E ω 0 . (a) F or x, y ∈ 2 N , ( x, y ) ∈ E 0 ⇔ ∃ m ∀ n ≥ m ( x ( n ) = y ( n )) . (b) F or x, y ∈ 2 N × N , ( x, y ) ∈ E 1 ⇔ ∃ m ∀ n ≥ m ∀ k ( x ( n, k ) = y ( n, k )) . (c) F or x, y ∈ 2 N × N , ( x, y ) ∈ E ω 0 ⇔ ∀ k ∃ m ∀ n ≥ m ( x ( n, k ) = y ( n, k )) . The follo wing dic hotomies sho w us wh y these equiv alence r elations are so remark able. Theorem 1.3. L et E b e a Bor el e quiv alenc e r elation. Then (a) (Harrington-Kec hr is-Louv eau [6]) either E ≤ B id( R ) or E 0 ≤ B E ; (b) (Kec h ris-Louv eau [8]) if E ≤ B E 1 , then E ≤ B E 0 or E ∼ B E 1 ; (c) (Hjorth-Kec h ris [5]) if E ≤ B E ω 0 , then E ≤ B E 0 or E ∼ B E ω 0 . 2. Sep arab le or not F or a Π 1 1 equiv alence relation E on X , le t u s co nsider th e follo w ing pseudo- metric on X : d E ( u, v ) =  0 , ( u, v ) ∈ E , 1 , ( u, v ) / ∈ E . F rom S ilv er’s theorem, w e can see that either d E is separable or there is a p erfect set C ⊆ X such that d E ( u, v ) = 1 for distinct u, v ∈ C . By the same sp irit of the Silve r dic h otom y theorem, w e define: Definition 2.1. Let X be a P olish space, d a pseudo-metric on X . If { ( u, v ) : d ( u, v ) < δ } is Π 1 1 for eac h δ > 0, w e sa y d is lower Π 1 1 . F or a pseudo-metric sp ace ( X , d ) and δ > 0, we sa y ( X, d ) is δ -sep ar able if there is a coun table set S ⊆ X suc h th at ∀ u ∈ X ∃ s ∈ S ( d ( u, s ) < δ ) . Hence ( X , d ) is separab le iff it is δ -separable for arbitrary δ > 0. Theorem 2.2. L et X b e a Polish sp ac e, d a lower Π 1 1 pseudo-metric. Then for δ > 0 , either ( X, d ) is δ - sep ar able or ther e is a p erfe ct set C ⊆ X su ch that d ( u, v ) ≥ δ / 2 for distinct u, v ∈ C . Pr o of. W e denote Q = { ( u, v ) : d ( u, v ) < δ } and R = { ( u, v ) : d ( u, v ) < δ / 2 } . W e s ee that b oth Q , R are Π 1 1 . Then the theorem follo ws from the next lemma.  Lemma 2.3. L et X b e a Polish sp ac e, Q, R ⊆ X 2 . A ssume that CHARACTERIZA TION OF ℓ p -LIKE AND c 0 -LIKE EQUIV ALENCE RELA TIONS 3 (i) Q is Π 1 1 and R is σ ( Σ 1 1 ) (the σ -algebr a gener ate d by the Σ 1 1 sets); (ii) ∆( X ) = { ( u, u ) : u ∈ X } c ontains in Q ; (iii) if ther e exists v ∈ X such that ( v , u ) ∈ R , ( v , w ) ∈ R , then ( u, w ) ∈ Q . Then one of the fol lowing ho lds: (a) ther e i s a c ountable set S ⊆ X suc h that ∀ u ∈ X ∃ s ∈ S (( u, s ) ∈ Q ) ; (b) ther e is a p erfe ct set C ⊆ X suc h that ( u, v ) / ∈ R for distinct u, v ∈ C . Pr o of. W e follo w the metho d as in Harr ington’s p r o of for Silver’s theorem. Without loss of generalit y we m a y assume X = N N and Q ∈ Π 1 1 . The pro of for Q ∈ Π 1 1 ( α ) with α ∈ N N is similar. Let τ b e the Gandy-Harrington top ology (the top ology generated by all Σ 1 1 sets) on N N . F or u ∈ X we denote Q ( u ) = { v ∈ X : ( u, v ) ∈ Q } . First w e define V = { u ∈ X : there is no ∆ 1 1 set U such that u ∈ U ⊆ Q ( u ) } . If V = ∅ , since th er e are only countably man y ∆ 1 1 set, w e can find a coun table subset S ⊆ X whic h meets ev ery nonempty ∆ 1 1 set at least one p oint . F or eac h u ∈ X there is a nonemp ty ∆ 1 1 set U ⊆ Q ( u ). Let s ∈ S ∩ U . Then s ∈ Q ( u ), i.e. ( u, s ) ∈ Q . F or the rest of the pro of we assume V 6 = ∅ . Note that u ∈ V ⇐ ⇒ ∀ U ∈ ∆ 1 1 ( u ∈ U → ∃ v ∈ U ( v / ∈ Q ( u ))) . With th e co ding of ∆ 1 1 sets (see [4] Theorem 1.7.4), there are Π 1 1 subsets P + , P − ⊆ N × N N and D ⊆ N such that (1) ∀ n ∈ D ∀ u (( n, u ) ∈ P + ⇔ ( n, u ) / ∈ P − ); (2) for an y ∆ 1 1 set A there is n ∈ D such that ∀ u ( u ∈ A ⇔ ( n, u ) ∈ P + ). Th us we ha v e u ∈ V ⇐ ⇒ ∀ n (( n ∈ D , ( n, u ) ∈ P + ) → ∃ v (( n, v ) / ∈ P − , ( u, v ) / ∈ Q )) . So V is Σ 1 1 . By a theorem of Nik o dym (see [7] Corollary 29.14), the class of sets with the Baire prop ert y in any top ologic al space is closed under the Suslin op er- ation. It is we ll kno wn that all Σ 1 1 sets are r esults o f the Suslin op eratio n applied on closed sets in the usual top ology (see [7] T heorem 25.7 ). Note that all closed sets in u sual top ology are also closed in τ , we see that ev ery σ ( Σ 1 1 ) su bset of N N (or N N × N N ) has Baire pr op ert y in τ (or τ × τ ). T o ward a contradict ion assume that f or some v ∈ V , R ( v ) is not τ -meager in V . Sin ce R ( v ) has Baire prop er ty in τ , there is a n onempt y Σ 1 1 set U ⊆ V suc h that R ( v ) is τ -comea ger in U . By Louv eau’s lemma (see [9] Lemma 9.3.2), R ( v ) × R ( v ) m eets an y nonemp ty Σ 1 1 set in U × U . W e denote ¬ Q = ( N N × N N ) \ Q . If ¬ Q ∩ ( U × U ) 6 = ∅ , since it is Σ 1 1 , we ha ve ( R ( v ) × R ( v )) ∩ ¬ Q ∩ ( U × U ) 6 = ∅ , 4 LONGYUN DING whic h contradicts to clause (iii). Th us w e ha ve U × U ⊆ Q . W e define W b y w ∈ W ⇐ ⇒ ∀ u ( u ∈ U → ( u, w ) ∈ Q ) . Fix a u 0 ∈ U . W e can s ee that W is Π 1 1 and U ⊆ W ⊆ Q ( u 0 ). By th e separation prop erty for Σ 1 1 sets there is U 0 ∈ ∆ 1 1 suc h that U ⊆ U 0 ⊆ W . Then w e h av e u 0 ∈ U 0 ⊆ Q ( u 0 ), which con tradicts u 0 ∈ U ⊆ V . Therefore, R ( v ) is τ -meage r in V for eac h v ∈ V . Since R h as Baire prop erty in τ × τ , by the Kurato wski-Ulam theorem (see [7] Theorem 8.41), R is τ × τ -meage r in V × V . By the definition of V and clause (ii), w e see that V con tains no ∆ 1 1 real, i.e . V has no isolate p oint in τ . Since the space N N with τ is strong Cho quet (see [4] Theorem 4.1.5), V is a p erfect Cho quet space. F rom [7] Exercise 19.5, w e can find a p erfect set C ⊆ V suc h that ( u, v ) / ∈ R for distinct u, v ∈ C .  3. Characteriza tion The n otion of ℓ p -lik e equiv alence relation was int ro duced in [2]. Definition 3.1. Let ( X n , d n ) , n ∈ N b e a sequence of pseudo-metric sp aces, p ≥ 1. W e d efine an equiv alence relation E (( X n , d n ) n ∈ N ; p ) on Q n ∈ N X n b y ( x, y ) ∈ E (( X n , d n ) n ∈ N ; p ) ⇐ ⇒ X n ∈ N d n ( x ( n ) , y ( n )) p < + ∞ for x, y ∈ Q n ∈ N X n . W e call it an ℓ p -like e quivalenc e r elation . I f ( X n , d n ) = ( X, d ) f or ev ery n ∈ N , w e write E (( X, d ); p ) = E (( X n , d n ) n ∈ N ; p ) for the sak e of brevit y . If X is a separable Banac h space, we ha ve E ( X ; p ) = X N /ℓ p ( X ) wh ere ℓ p ( X ) is the Banac h space whose un derlying space is { x ∈ X N : P n ∈ N k x ( n ) k p < + ∞} with the norm k x k =  P n ∈ N k x ( n ) k p  1 p . Th en E ( X ; p ) is an orbit equiv alence r elation ind uced by a P olish g roup action, th u s E 1 6≤ B E ( X ; p ) (see [4] Theorem 10.6.1). Let ( X , d ) b e a p s eudo-metric space, w e denote δ ( X ) = inf { δ : X is δ -separable } . Theorem 3.2. L et X n , n ∈ N b e a se quenc e of Polish sp ac es, d n a Bor el pseudo-metric on X n for e ach n and p ≥ 1 . Denote E = E (( X n , d n ) n ∈ N ; p ) . We have (i) P n ∈ N δ ( X n ) p < + ∞ ⇐ ⇒ E ≤ B E ( c 0 ; p ) ; (ii) P n ∈ N δ ( X n ) p = + ∞ ⇐ ⇒ E 1 ≤ B E . Pr o of. Because E 1 6≤ B E ( c 0 ; p ), w e only need to pro v e ( ⇒ ) for (i) and (ii). (i) By the defin ition of δ ( X n ), we see that X n is ( δ ( X n ) + 2 − n )-separable, i.e. ther e is a count able set S n ⊆ X n suc h that ∀ u ∈ X n ∃ s ∈ S n ( d n ( u, s ) < δ ( X n ) + 2 − n ) . CHARACTERIZA TION OF ℓ p -LIKE AND c 0 -LIKE EQUIV ALENCE RELA TIONS 5 Let S n = { s n m : m ∈ N } . Without loss of generalit y , w e assume that d n ( s n k , s n l ) > 0 for k 6 = l , i.e. d n is a metric on S n . F or u ∈ X we de- note m ( u ) the least m su c h that d ( u, s n m ) < δ ( X n ) + 2 − n . Then we d efine h n : X n → S n b y h n ( u ) = s n m ( u ) for u ∈ X . I t is easy to see that h n is Borel. Define θ : Q n ∈ N X n → Q n ∈ N S n b y θ ( x )( n ) = h n ( x ( n )) for x ∈ Q n ∈ N X n . Note that for eac h x we ha ve X n ∈ N d n ( x ( n ) , θ ( x )( n )) p < X n ∈ N ( δ ( X n )+2 − n ) p ≤ 2 p − 1 X n ∈ N ( δ ( X n ) p +2 − np ) < + ∞ , i.e. ( x, θ ( x )) ∈ E . It follo ws that ( x, y ) ∈ E ⇔ ( θ ( x ) , θ ( y )) ∈ E . Hence E ≤ B E (( S n , d n ) n ∈ N ; p ). Note that eac h ( S n , d n ) is a separable metric space. F rom Aharoni’s theorem [1], there are K > 0 and T n : S n → c 0 satisfying d n ( u, v ) ≤ k T n ( u ) − T n ( v ) k c 0 ≤ K d n ( u, v ) for eve ry u, v ∈ S n . Define θ 1 : Q n ∈ N S n → c N 0 b y θ 1 ( x )( n ) = T n ( x ( n )) for x ∈ Q n ∈ N S n . It is easy to chec k that θ 1 is a Borel reduction of E (( S n , d n ) n ∈ N ; p ) to E ( c 0 ; p ). (ii) Without loss of generalit y , we may assum e that δ ( X n ) > 0 f or eac h n . Select a sequence 0 < δ n < δ ( X n ) , n ∈ N suc h that P n ∈ N δ p n = + ∞ . Thus w e can find a strictly increasing sequen ce of natural n umbers ( n j ) j ∈ N suc h that n 0 = 0 and n j +1 − 1 X n = n j δ p n ≥ 2 p , j = 0 , 1 , 2 , · · · . Since ( X n , d n ) is n ot δ n -separable, f rom Theorem 2.2, there is a Borel injection g n : 2 N → X n suc h that d n ( g n ( α ) , g n ( β )) ≥ δ n / 2 for distinct α, β ∈ 2 N . Define ϑ : (2 N ) N → Q n ∈ N X n b y ϑ ( x )( n ) = g n ( x ( j )) , n j ≤ n < n j +1 . F or eac h x, y ∈ (2 N ) N , if x ( j ) 6 = y ( j ) for some j ∈ N , w e ha v e n j +1 − 1 X n = n j d n ( ϑ ( x )( n ) , ϑ ( y )( n )) p = n j +1 − 1 X n = n j d n ( g n ( x ( j )) , g n ( y ( j ))) p ≥ n j +1 − 1 X n = n j ( δ n / 2) p ≥ 1 . Therefore ( ϑ ( x ) , ϑ ( y )) ∈ E ⇐ ⇒ P j ∈ N P n j +1 − 1 n = n j d n ( ϑ ( x ) , ϑ ( y )) p < + ∞ ⇐ ⇒ ∃ k ∀ j > k ( x ( j ) = y ( j )) ⇐ ⇒ ( x, y ) ∈ E 1 . Th us ϑ witnesses that E 1 ≤ B E .  c 0 -lik e equiv alence relations were first studied by I. F arah [3]. 6 LONGYUN DING Definition 3.3. Let ( X n , d n ) , n ∈ N b e a sequence of pseudo-metric sp aces. W e defin e an equiv alence relation E (( X n , d n ) n ∈ N ; 0) on Q n ∈ N X n b y ( x, y ) ∈ E (( X n , d n ) n ∈ N ; 0) ⇐ ⇒ lim n →∞ d n ( x ( n ) , y ( n )) = 0 for x, y ∈ Q n ∈ N X n . W e c all it a c 0 -like e quivalenc e r elation . If ( X n , d n ) = ( X, d ) f or ev ery n ∈ N , we write E (( X, d ); 0) = E (( X n , d n ) n ∈ N ; 0) for th e sak e of brevit y . F arah mainly inv estigate d the case named c 0 -equalities that all ( X n , d n )’s are finite metric sp aces and d enoted it by D( h X n , d n i ). Theorem 3.4. L et X n , n ∈ N b e a se quenc e of Polish sp ac es, d n a Bor el pseudo-metric on X n for e ach n . De note E = E (( X n , d n ) n ∈ N ; 0) . We have (i) lim n →∞ δ ( X n ) = 0 ⇐ ⇒ E ≤ B R N /c 0 ; (ii) ( δ ( X n )) n ∈ N do es not c onver ge to 0 ⇐ ⇒ E 1 ≤ B E . Pr o of. W e clo sely follo ws the proof of Theorem 3.2 . S ome conclusions w ill b e made without pro ofs for brevit y , since they follo w by similar argumen ts. (i) Note that f or eac h x w e h a ve lim n →∞ d n ( x ( n ) , θ ( x )( n )) = lim n →∞ ( δ ( X n ) + 2 − n ) = 0 , i.e. ( x, θ ( x )) ∈ E . It follo ws that ( x, y ) ∈ E ⇔ ( θ ( x ) , θ ( y )) ∈ E . Hence E ≤ B E (( S n , d n ) n ∈ N ; 0). Fix a bijection h· , ·i : N 2 → N . W e define θ 2 : Q n ∈ N S n → R N b y θ 2 ( x )( h n, m i ) = T n ( x ( n ))( m ) for x ∈ Q n ∈ N S n and n , m ∈ N . It is easy to see that θ 2 is Borel. No w we c heck that θ 2 is a reduction. F or eve ry x, y ∈ Q n ∈ N S n , if ( x, y ) ∈ E (( S n , d n ) n ∈ N ; 0), then lim n →∞ d n ( x ( n ) , y ( n )) → 0 . So ∀ ε > 0 ∃ N ∀ n > N ( d n ( x ( n ) , y ( n )) < ε ). Since k T n ( x ( n )) − T n ( y ( n )) k c 0 ≤ K d n ( x ( n ) , y ( n )) < K ε , w e h av e ∀ n > N ∀ m ( | T n ( x ( n ))( m ) − T n ( y ( n ))( m ) | < K ε ) . F or n ≤ N , since T n ( x ( n )) , T n ( y ( n )) ∈ c 0 , we ha ve lim m →∞ | T n ( x ( n ))( m ) − T n ( y ( n ))( m ) | = 0 . Therefore, for all bu t finitely man y ( n, m )’s, we ha ve | θ 2 ( x )( h n, m i ) − θ 2 ( y )( h n, m i ) | = | T n ( x ( n ))( m ) − T n ( y ( n ))( m ) | < K ε. Th us lim h n,m i→∞ | θ 2 ( x )( h n, m i ) − θ 2 ( y )( h n, m i ) | = 0 . It follo w s that θ 2 ( x ) − θ 2 ( y ) ∈ c 0 . On th e other hand, for ev ery x, y ∈ Q n ∈ N S n , if θ 2 ( x ) − θ 2 ( y ) ∈ c 0 , th en ∀ ε > 0 ∃ N ∀ n > N ∀ m ( | θ 2 ( x )( h n, m i ) − θ 2 ( y )( h n, m i ) | < ε ) . CHARACTERIZA TION OF ℓ p -LIKE AND c 0 -LIKE EQUIV ALENCE RELA TIONS 7 Therefore, for n > N we ha ve d n ( x ( n ) , y ( n )) ≤ k T n ( x ( n )) − T n ( y ( n )) k c 0 = sup m ∈ N | T n ( x ( n ))( m ) − T n ( y ( n ))( m ) | = sup m ∈ N | θ 2 ( x )( h n, m i ) − θ 2 ( y )( h n, m i ) | ≤ ε. It follo w s that ( x, y ) ∈ E (( S n , d n ) n ∈ N ; 0). T o su m up, w e ha v e E ≤ B E (( S n , d n ) n ∈ N ; 0) ≤ B R N /c 0 . (ii) Assume that ( δ ( X n )) n ∈ N do es not con ve rge to 0. Th en there are c > 0 and a strictly increasing sequ ence of natural num b ers ( n j ) j ∈ N suc h that δ ( X n j ) > c for eac h j . F rom Theorem 2.2, there is a Borel injection g ′ j : 2 N → X n j suc h that d n j ( g ′ j ( α ) , g ′ j ( β )) ≥ c/ 2 f or distinct α, β ∈ 2 N . Fix an elemen t a n ∈ X n for ev ery n ∈ N . Define ϑ ′ : (2 N ) N → Q n ∈ N X n b y ϑ ′ ( x )( n ) =  g ′ j ( x ( j )) , n = n j , a n , otherwise . Then ϑ ′ witnesses that E 1 ≤ B E .  4. Fur ther remarks The follo win g cond ition w as intro d uced in [2] to in vestig ate the p osition of ℓ p -lik e equiv alence relations. ( ℓ 1) ∀ c > 0 ∃ x, y ∈ Q n ∈ N X n such that ∀ n ( d n ( x ( n ) , y ( n )) p < c ) and X n ∈ N d n ( x ( n ) , y ( n )) p = + ∞ . Let X n , n ∈ N be a sequence of P olish space s, d n a Borel pseudo-metric on X n for eac h n and p ≥ 1. Denote E = E (( X n , d n ) n ∈ N ; p ). It was p ro ved in [2] that (i) if ( ℓ 1) holds, then R N /ℓ 1 ≤ B E ; (ii) if ( ℓ 1) fails, then either E 1 ≤ B E , E ∼ B E 0 or E is trivial, i.e. all elemen ts in Q n ∈ N X n are equiv alen t. Th us w e ha ve a corollary of Theorem 3.2. Corollary 4.1. Denote E = E (( X n , d n ) n ∈ N ; p ) . We have (a) P n ∈ N δ ( X n ) p < + ∞ a nd ( ℓ 1 ) fails ⇐ ⇒ E ∼ B E 0 or E is trivial; (b) P n ∈ N δ ( X n ) p < + ∞ and ( ℓ 1) holds ⇐ ⇒ R N /ℓ 1 ≤ B E ≤ B E ( c 0 ; p ) ; (c) P n ∈ N δ ( X n ) p = + ∞ ⇐ ⇒ E 1 ≤ B E . Another condition was int ro duced b y I. F arah [3] for inv estigating c 0 - equalities. ( ∗ ) ∀ c > 0 ∃ ε < c ( ε > 0 and ∃ ∞ i ∃ u i , v i ∈ X i ( ε < d i ( u i , v i ) < c )) . It is easy to chec k that ( ∗ ) h olds iff for arbitrary c > 0, there exist x, y ∈ Q n ∈ N X n suc h th at ∀ n ( d n ( x ( n ) , y ( n )) < c ) and ( d n ( x ( n ) , y ( n ))) n ∈ N do es not con verge to 0. With similar arguments, w e get a corollary of T h eorem 3.4. 8 LONGYUN DING Corollary 4.2. Denote E = E (( X n , d n ) n ∈ N ; 0) . We have (a) lim n →∞ δ ( X n ) = 0 a nd ( ∗ ) fails ⇐ ⇒ E ∼ B E 0 or E is trivial; (b) lim n →∞ δ ( X n ) = 0 a nd ( ∗ ) ho lds ⇐ ⇒ E ω 0 ≤ B E ≤ B R N /c 0 ; (c) ( δ ( X n )) n ∈ N do es not c onver ge to 0 ⇐ ⇒ E 1 ≤ B E . Referen ces [1] I. Aharoni, Ever y sep ar able metric sp ac e is Lipschitz e quivalent to a subse t of c + 0 , Isreal J. Math. 19 (1974), 284–291. [2] L. Ding, A trichotomy f or a class of e quivalenc e r elations , preprin t, av ailable at http://arx iv.org/abs/ 1001.0834 . [3] I. F arah, Basis pr oblem for turbulent ac tions II: c 0 -e qualities , Proc. London Math. Soc. 82 (2001), no. 3, 1–30. [4] S. Gao, Inv arian t Descriptive Set Theory , Monographs and T extb ook s in Pure and Applied Mathematics, vol. 293, CRC Press, 2008. [5] G. Hjorth and A. S. Kechris, New dichotomies for Bor el e quivalenc e r elations , Bull. Symbolic Logic 3 (1997), no. 3, 329–3 46. [6] L. A. Harrington, A. S. Kechris, and A. Louveau, A Glimm-Effr os dichotomy for Bor el e quivalenc e r elations , J. Amer. Math. So c. 3 (1990), no. 4, 903–928. [7] A. S. Kec h ris, Clas sical Descriptive Set Theory , Graduate T exts in Mathematics, vol. 156, Springer-V erlag, 1995. [8] A. S. Kechris and A. Louveau, The classific ation of hyp ersmo oth Bor el e quivalenc e r elations , J. Amer. Math. So c. 10 (1997), no. 1, 215–242. [9] D. A . Martin and A. S. Kechris, Infinite games and effe ctive descriptive set the ory , in Analytic S ets, 403–470. Academic Press, 1980. [10] J. H. Silver, Counting the numb er of e quivalenc e classes of Bor el and c o analytic e quiv- alenc e r elations , Ann . Math. Logic 18 (1980), no. 1, 1–28. School of Ma them a tical Sciences and LPMC, Nankai Universi ty, Tianjin, 300071, P.R.China E-mail addr ess : dinglongyun @gmail.com