A characterisation of compact, fragmentable linear orders

A CHARACTERISA TION OF COMP ACT, FRAGM ENT ABLE LINEAR ORDERS R. J. SMITH Abstract. W e give a c haracterisation o f fragmentable, co mpact linearly or der spaces. In particular, we show that if K is a compact, frag men table, linearly ordered space then K is a Radon-Nikod´ ym compact. In addition, we obtain some corolla r ies in top ology a nd reno rming theory . 1. Intr oduction Throughout this note, a ll top ological spaces are assumed to b e Hausdorff. Definition 1.1. Le t K b e a compact space. (1) W e s a y that K is fr agmentable if there exists a me tric d : K × K − → [0 , ∞ ) with the prop ert y that giv en an y non-empt y set M ⊆ K and ε > 0, there exists an o p en set U ⊆ K satisfying M ∩ U 6 = ∅ and d -diam ( M ∩ U ) < ε . (2) W e sa y that K is an Radon-Nikod ´ ym compact, or RN compact, if there exists a metric that is lo w er semicon tinuous on K × K and satisfies the conditions in (1). F ragmentable and RN compact spaces hav e b een the sub ject of enduring study . The pap er o f Namiok a [2 ] contains many of the fundamen tal results on RN compact spaces. F or e xample, Namiok a sho w ed that the definition of R N compacta giv en ab ov e is equiv alent to the original definition of RN compact spaces, namely that K is RN compact if it is home omorphic to a w ∗ -compact subset of an As plund Banach space. The most w ell kno wn unsolve d problem in the theory of RN compacta is the question of whether the con tinu ous image of an RN compact space is again RN compact. In [1 ], it is prov ed that if K is a linearly ordered compact space and the con tinuous image of a RN compact, then K is RN compact. This result uses a necessary condition for K to b e fragmentable. W e sa y tha t a compact space M is almost total ly disc o n ne cte d if it is homeomorphic t o some A ⊆ [0 , 1] Γ in the p oint wise top ology , with the prop ert y tha t if f ∈ A then f ( γ ) ∈ (0 , 1) for at most countably man y γ ∈ Γ. The class o f almost tot a lly disconnected spaces con tains all Corson compact spaces and all totally disconnected spaces. Date : Nov em b er 2 008. 2000 Mathematics Subje ct Classific ation. 0 6 A05; 54 D3 0; 4 6 B50; 46B2 6. Key wor ds and phr ases. F ragmentabilit y , lin ear order, Radon-Nik o d´ ym compa ct, strictly co n vex norm. 1 Theorem 1.2. [1, Theorem 3] L et K b e a line arly or der e d fr agmentable c omp act. Then K is almo st total ly d i s c onne cte d. In [1], Avil ´ es asks whether a linearly ordered compact space K is RN compact whenev er K is fragmentable. In this note, w e c haracterise compact, frag mentable, linearly ordered spaces. Theorem 1.3. L et K b e a c omp act, line arly or der e d sp a c e. Then the fol lowing ar e e quivalent. (1) K is fr agmentable; (2) ther e is a fam ily L n , n ∈ N , of c omp act, sc atter e d subse ts of K , with unio n L , such that whe never u, v ∈ K and u < v , ther e exist x, y ∈ L satisfying u ≤ x < y ≤ v . (3) K is RN c omp act. In doing so, w e obtain Avil ´ es’s r esult concerning contin uous images. Corollary 1.4. [1, Corollary 4] L et K b e a c omp act, lin e arly or der e d sp ac e that is also a c ontinuous imag e of a RN c omp act. Then K is a RN c omp act. In fact, Corolla ry 1.4 is originally stated in [1] in terms of quasi-R adon-Niko d´ ym compact spaces , which we w on’t define here. All w e need to know is that if K is a contin uous image o f a RN compact then it is a quasi-RN compact, whic h in turn implies t hat K is fra gmen table. With this in mind, w e will see that the original statemen t also follo ws from Theorem 1.3. The pro of of Theorem 1 .3, (1) ⇒ (2), is the sub ject of Sections 2 and 3. As a b ypro duct of this inv estigation, w e o btain more results. W e denote the first un- coun ta ble cardinal by ω 1 . Prop osition 1.5. L et K a c omp a ct, fr agmentable, line arly or der e d sp ac e, and as- sume that K c ontains no or d e r-isomorphic c opy of κ , wher e κ is a r e g ular, unc ount- able c ar dinal. Then the top olo gic a l weight of K is strictly less than κ . I n p articular, if K c ontains no c op y of ω 1 then K is m e trisable. The next corollary extends a theorem in [6 ], which states that C ( ω 1 + 1) admits no equiv alen t nor m with a strictly con v ex dual no r m. Corollary 1.6. L et K b e a c omp act, line arly or der e d set, and supp ose that C ( K ) admits an e quival e nt no rm with strictly c onvex d ual norm. Then K is m etrisable. This leads directly to the final result. Corollary 1.7. L et K b e a c omp a ct, line arly or der e d, Gruenhage s p ac e. Then K is metrisable. It is worth noting that w e cannot simply demand that the union L in Theorem 1.3, part (2), is topo logically dense in K . If w e let K = [0 , 1] × { 0 , 1 } b e the lexicographically ordered ‘split interv al’, then K is separable. Ho we v er, it is w ell kno wn not to b e fr a gmen table. W e prov ide a pro of of this at the b eginning o f section 2. Alternativ ely , we can use Prop osition 1 .5, since it is easy to sho w tha t K is not 2 second coun table and do es no t contain any uncoun table we ll o rdered or con v ersely w ell ordered subsets. Inciden tally , since K is 0- dimensional, this example sho ws that the necessary condition of Theorem 1.2 is not sufficien t. Moreov er, it is no t p ossible to deduce Prop o sition 1.5 fro m Theorem 1.2. W e conclude this section by pro ving Theorem 1.3, (2) ⇒ (3). The argument is a straigh tforw a rd elab oratio n o f Namiok a’s pro of that the ‘extended long line’ is RN compact; see [2, Example 3.9, (b)] for details. Of course, Theorem 1.3, (3) ⇒ (1 ) , follo ws imme diately from Definition 1.1. W e shall use ( x, y ) to denote b oth ordered pairs and op en interv als. The interpretation of the notation should be clear from the con text. W e mak e use o f the following c haracterisation of RN compacta. Theorem 1.8 ([2, Corollary 3.8]) . A c omp act Hausdorff sp ac e K is RN c omp a ct if and only if ther e exists a norm b ounde d set Γ ⊆ C ( K ) such that (1) Γ sep ar ates p oints of K , and (2) for every c ountable set A ⊆ Γ , K is sep ar able, r el a tive to the ps e udo-metric d A , given by d A ( x, y ) = sup {| f ( x ) − f ( y ) | | f ∈ A } for x, y ∈ K . Pr o of of The or e m 1.3, (2) ⇒ (3 ) . Let L n b e as in (2). W e shall assume that min K , max K ∈ L 1 and L n ⊆ L n +1 for all n . Let ∆ n = { ( x, y ) ∈ L 2 n | x < y and ( x, y ) ∩ L n is empt y } . F or eac h ( x, y ) ∈ ∆ n , tak e an increasing function f x,y ∈ C ( K ) suc h that f ( w ) = 0 for w ≤ x and f ( z ) = n − 1 for z ≥ y . W e claim that the family f x,y , ( x, y ) ∈ ∆ n , n ∈ N separates p oin t s of K and, moreo v er, if A ⊆ S ∞ n =1 ∆ n is coun table, then K is d A -separable, where d A is the pseudo-metric defined by d A ( u, v ) = sup {| f x,y ( u ) − f x,y ( v ) | | ( x, y ) ∈ A } . First, let u, v ∈ K , with u < v . By the h ypo thesis and the fact t hat L n ⊆ L n +1 , there exists n and x, y ∈ L n suc h that u ≤ x < y ≤ v . If we tak e an isolated p oint w of [ x, y ] ∩ L n then at least o ne of the p oin ts inf { z ∈ [ x, y ] ∩ L n | w < z } , sup { z ∈ [ x, y ] ∩ L n | z < w } is in [ x, y ] ∩ L n and necessarily not equal to w . Therefore, w e can find ( x ′ , y ′ ) ∈ ∆ n with x ≤ x ′ < y ′ ≤ y . It follows that f x ′ ,y ′ separates u a nd v . No w let A ⊆ S ∞ n =1 ∆ n b e countable. W e set A n = A ∩ ∆ n and M n = [ ( x,y ) ∈ A n { x, y } . Since M n is compact, separable, scattered and linearly or dered, it is coun table. If ( x, y ) ∈ ∆ n then o bserv e t hat, for ev ery i ≤ n , there is a unique ( x i , y i ) ∈ ∆ i satisfying x i ≤ x < y ≤ y i . D efine Γ x,y ,n = { (( q 1 , r 1 ) , . . . , ( q n , r n )) ∈ ( Q 2 ) n | q k < r k and n \ i =1 f − 1 x i ,y i ( q i , r i ) 6 = ∅ } 3 and, for eve ry ( q , r ) = (( q 1 , r 1 ) , . . . , ( q n , r n )) ∈ Γ x,y ,n , take z x,y ,n,q ,r ∈ L ∩ n \ i =1 f − 1 x i ,y i ( q i , r i ) . W e claim t hat the coun table set D = { min K , max K } ∪ ∞ [ n =1 M n ∪ { z x,y ,n,q ,r | ( x, y ) ∈ A n , ( q , r ) ∈ Γ x,y ,n and n ∈ N } is d A -dense in K . If w ∈ K and n ∈ N , w e find z ∈ D suc h that d A ( w , z ) < n − 1 . W e assume w / ∈ D and let M = { min K , max K } ∪ n [ k =1 M k . If w ∈ ( x, y ) and ( x, y ) ∈ A k for some k ≤ n then let k b e maximal. O therwise, let k = 0. Since M is closed a nd min K, max K ∈ M , w e can find u, v ∈ M , suc h that ( u, v ) ∩ M is empty a nd w ∈ ( u , v ). Assume that k < j ≤ n and ( x, y ) ∈ A j . W e m ust hav e f x,y ( u ) = f x,y ( w ) = f x,y ( v ). Ind eed, if ( x, y ) ∈ A j then x, y ∈ M . Since ( u, v ) ∩ M is empty , either x ≤ u and v ≤ y , or y ≤ u , or v ≤ x . Ho w ev er, the first p ossibilit y cannot hold b ecause w / ∈ ( x, y ). No w let w ∈ ( x, y ), where ( x, y ) ∈ A k . F or i ≤ k , tak e rationals q i and r i suc h that q i < f x i ,y i ( u ) < r i and r i − q i < n − 1 . Let ( x ′ , y ′ ) ∈ A i , where i ≤ k . If ( x ′ , y ′ ) 6 = ( x i , y i ) then since w , z x,y ,n,q ,r ∈ ( x, y ) ⊆ ( x i , y i ), w e hav e f x ′ ,y ′ ( w ) = f x ′ ,y ′ ( z x,y ,n,q ,r ) . On the other hand, if ( x ′ , y ′ ) = ( x i , y i ) then we ha v e ensured tha t | f x ′ ,y ′ ( w ) − f x ′ ,y ′ ( z x,y ,n,q ,r ) | < n − 1 . If z x,y ,n,q ,r ≤ w then set z = max { u , z x,y ,n,q ,r } , and if w < z x,y ,n,q ,r then set z = min { v , z x,y ,n,q ,r } . By the construction, and the fact that the f x,y are increasing, w e ha v e made sure that | f x,y ( w ) − f x,y ( z ) | < n − 1 whenev er ( x, y ) ∈ S n i =1 A n . If ( x, y ) ∈ A m and n < m , then | f x,y ( w ) − f x,y ( z ) | ≤ m − 1 < n − 1 . Therefore d A ( w , z ) < n − 1 . That K is RN compact no w follows from Theorem 1.8.  2. Simple s ubsets of trees It is a standard result in elemen tary analysis that ev ery uncoun table set H ⊆ R con ta ins an uncountable subset E , with the prop ert y that each x ∈ E is a t w o-sided condensation p oint of E ; that is, giv en x ∈ E and ε > 0, bot h ( x − ε, x ) ∩ E and ( x, x + ε ) ∩ E are uncoun t a ble. As explained in [1], the abundance of condensation p oin ts in eac h uncoun table subset o f R can b e used to show that the split interv a l K = [0 , 1] × { 0 , 1 } is not frag men table. It is worth repeating the argumen t here. If d is a metric on K , then there ex ists an uncoun table subset H ⊆ [0 , 1] with the prop ert y that d (( x, 0) , ( x, 1)) ≥ n − 1 for all x ∈ H . If E ⊆ H is a s ab o ve, then for ev ery o p en subset U ⊆ K suc h that U ∩ ( E × { 0 , 1 } ) is non-empt y , d - diam ( U ∩ ( E × { 0 , 1 } )) ≥ n − 1 . 4 In this note, w e shall c onsider subs ets of more general linear orders w hic h b eha v e similarly to uncountable subsets of R in this w a y . In order to do so, we first define and inv estigate a family of subsets of trees. Linear o rders share a long and close rela- tionship with trees, and trees will f eat ure strongly in what follo ws. F or conv enience, w e la y do wn some of the basic definitions. A par t ia lly ordered set ( T , < ) is a tr e e if the set of pr edecessors of any giv en elemen t of T is w ell ordered. If x, z ∈ T , we define the interv al ( x, z ] = { y ∈ T | x < y ≤ z } . W e introduce elemen ts 0 a nd ∞ , not in T , with the prop ert y that 0 < x < ∞ fo r all x ∈ T , and define the interv als (0 , z ), [ x, ∞ ) and [ x, z ] in the ob vious manner. In general, w e say that s ⊆ T is an interval if y ∈ s whe nev er x ≤ y ≤ z and x, z ∈ s . W e let h t( x ) b e the order t yp e of (0 , x ) and, if α is an ordinal, w e let T α b e the leve l of T of order α ; that is, the set of all x ∈ T satisfying h t( x ) = α . The interval top olo gy of T tak es as a basis all sets of the form ( x, z ], where x ∈ T ∪ { 0 } and z ∈ T . This top olog y is lo cally compact and s cattered. W e shall sa y that a subset of T is open if it is so with respect to this top ology . If x ∈ T , w e let x + b e the set of immediate successors o f x . W e sa y that y ∈ T is a suc c esso r if y ∈ x + for some x or , equiv alen t ly , y ∈ T ξ +1 for some ξ . In this note, w e shall only consider trees with the Hausdorff prop ert y , i.e., if A ⊆ T is non-empt y and tota lly ordered, then A has at most one minimal upper b ound. If T is Hausdorff then t he inte rv al top olo g y on T is Hausdorff in the usual, t o p ological sense. If H is a subset of a tree T , the map π : H − → T is r e gr essive if π ( x ) < x for all x ∈ H . If α is a limit ordinal with cofinality κ = cf ( α ) , w e say that ( α ξ ) ξ <κ is a c ofinal se quenc e of ordinals if it is strictly increasing and conv erges to α . Definition 2.1. L et α b e a limit ordinal. W e sa y that H ⊆ T α is simple if, f or some cofinal sequence ( α ξ ) ξ <κ , there is a injectiv e, regressiv e map π : H − → S ξ <κ T α ξ . W e w ill tie trees to compact linear orders in the nex t se ction. Before doing so, w e explore the pro p erties of simple sets. Firstly , it is w orth noting that the c ho ice of cofinal sequence in the definition of simple sets is not imp ortant. Lemma 2.2. If ( α ξ ) ξ <κ and ( α ′ ξ ) ξ <κ ar e c ofinal s e quenc es, and π : H − → S ξ <κ T α ξ is a n inje ctive, r e gr essive map, then ther e exists a n inje c tive , r e gr essive map π ′ : H − → S ξ <κ T α ′ ξ . Pr o of. By taking a subs equence of ( α ′ ξ ) ξ <κ if necessary , w e can ass ume that α ξ ≤ α ′ ξ for all ξ < κ . If x ∈ H and π ( x ) ∈ T α ξ , let π ′ ( x ) b e the unique elemen t of [ π ( x ) , x ] ∩ T α ′ ξ . No w supp ose that π ′ ( x ) = π ′ ( y ). It follows tha t π ( x ) and π ( y ) are comparable, and since π ′ ( x ) and π ′ ( y ) share the same lev el, so do π ( x ) and π ( y ). Therefore, π ( x ) = π ( y ), giving x = y .  The next result rev eals an imp ortant permanence prop erty of simple sets. Prop osition 2.3. L et α b e a limit or dinal. Supp ose that H ⊆ T α , ( α ξ ) ξ <κ is a c ofinal se quenc e, and π : H − → S ξ <κ T α ξ is a r e gr essive mappin g, with the p r o p erty that every fibr e of π is sim ple. Then H is simp l e . Pr o of. F or e ac h w ∈ S ξ <κ T α ξ , let π w : π − 1 ( w ) − → S ξ <κ T α ξ b e a regressiv e, injectiv e map. By Lemma 2.2, w e can assume that π ( x ) ≤ π π ( x ) ( x ) for all x ∈ H . F or 5 ξ ≤ ξ ′ < κ , define Γ ξ ,ξ ′ = { x ∈ H | π ( x ) ∈ T α ξ and π π ( x ) ( x ) ∈ T α ξ ′ } . The sets Γ ξ ,ξ ′ , ξ ≤ ξ ′ < κ , are pairwise disjoint, and for ev ery x ∈ H , there exist suc h ordinals with the prop ert y that x ∈ Γ ξ ,ξ ′ . T ak e η ≤ η ′ < κ . W e define σ ( x ) for x ∈ Γ η,η ′ in the fo llo wing w ay . First, observ e that the set [ { (0 , x ] ∩ [ π π ( y ) ( y ) , y ] | y ∈ Γ ξ ,ξ ′ , ξ ≤ ξ ′ ≤ η ′ and ( ξ , ξ ′ ) 6 = ( η , η ′ ) } has an upp er b ound w < x . Indeed, supp ose first that ξ ≤ ξ ′ ≤ η ′ , ( ξ , ξ ′ ) 6 = ( η , η ′ ) and y , z ∈ Γ ξ ,ξ ′ are suc h that (0 , x ] ∩ [ π π ( y ) ( y ) , y ] and (0 , x ] ∩ [ π π ( z ) ( z ) , z ] are b oth non-empty . Then π ( y ) , π ( z ) < x , so they are comparable. As they o ccup y the same leve l T α ξ , w e hav e π ( y ) = π ( z ). Moreov er, π π ( y ) ( y ) , π π ( y ) ( z ) < x and o ccupy the same lev el T α ξ ′ , th us y = z , b ecause π π ( y ) is injectiv e. So (0 , x ] ∩ [ π π ( y ) ( y ) , y ] is non-empt y for at most one y ∈ Γ ξ ,ξ ′ . Because ( ξ , ξ ′ ) 6 = ( η , η ′ ) and T is Hausdorff, this in tersection has an upp er b ound w ξ ,ξ ′ < x . Sinc e ξ ≤ ξ ′ ≤ η ′ < κ = cf ( α ), the required upp er bo und w exists . Th us w e can tak e σ ( x ) ∈ S ξ <κ T α ξ , satisfying w , π π ( x ) ( x ) < σ ( x ). No w let x, y ∈ H . W e claim that if [ σ ( x ) , x ] ∩ [ σ ( y ) , y ] is non-empt y , then x = y . Indeed, take suc h x and y , and let ξ ≤ ξ ′ , η ≤ η ′ satisfy x ∈ Γ η,η ′ and y ∈ Γ ξ ,ξ ′ . Without loss of generalit y , w e can assume that ξ ′ ≤ η ′ . If ( ξ , ξ ′ ) = ( η , η ′ ) then, as ab ov e, π ( x ) and π ( y ) are comparable, as are π π ( x ) ( x ) and π π ( y ) ( y ). Since b oth pairs o ccup y the same levels resp ectiv ely , w e g et x = y . Ins tead, if ( ξ , ξ ′ ) 6 = ( η , η ′ ) t hen from the construction ab ov e, it follow s that [ σ ( x ) , x ] ∩ [ σ ( y ) , y ] ⊆ [ σ ( x ) , x ] ∩ [ π π ( y ) ( y ) , y ] is empt y . Th us w e mus t hav e ( ξ , ξ ′ ) = ( η , η ′ ) and x = y .  The next corolla ry follows immediately from the pro of of Prop osition 2.3 . Corollary 2.4. L et H ⊆ T α b e simple. T h en for every c ofinal se quenc e ( α ξ ) ξ <κ , ther e exists a r e gr essive map π : H − → S ξ <κ T α ξ with the pr op erty that x = y whenever [ π ( x ) , x ] ∩ [ π ( y ) , y ] is non-empty. Prop osition 2 .3 yields a nother straigh tforw a rd corollary . Corollary 2.5. If α is a limit or dinal, κ = cf ( α ) and H ξ ⊆ T α is simple for al l ξ < κ , then so is the union H = S ξ <κ H ξ . I n p articular, c ountable unio ns of sim ple subsets of T α ar e simple. Pr o of. Assume the H ξ are pairwise disjoin t. Let ( α ξ ) ξ <κ b e a cofinal sequence and define π : H − → S ξ <κ T α ξ b y letting π ( x ) b e the unique elemen t o f (0 , x ] ∩ T α ξ , whenev er x ∈ H ξ . Then π − 1 ( w ) ⊆ H ξ whenev er w ∈ T α ξ .  6 W e finish this section with a final result concerning simple subsets. W e say that W ⊆ T is an initial p art of T if x ∈ W whenev er x ∈ T , x ≤ y and y ∈ W . Prop osition 2.6. L et T b e a tr e e and supp ose that the level T α is simple for e v ery limit or dinal α . Then ther e is a p artition of T c on sisting entir ely of op en interval s . Pr o of. The pro of comes in tw o par t s. In the first part, w e pro v e the f ollo wing claim. Let W b e a family of initial parts of T , with union T , a nd suc h that, giv en V , W ∈ W , either V is an initial part of W , or vice-ve rsa. Supp ose further that eac h W has a partition P W consisting wholly of o p en interv als of W , with the prop ert y tha t if V is an initial segmen t of W then, f o r eve ry s ∈ P V , there exists t ∈ P W with s ⊆ t . Then T has a partition P consis ting of op en in terv als only . Define ∼ on T by declaring that x ∼ y if and only if x, y ∈ s for some s ∈ P W , W ∈ W . This is an equiv alence relation. Symmetry is immediate. If x ∈ T then x ∈ s for some s ∈ P W and W ∈ W , so ∼ is reflexiv e. If x ∼ y a nd y ∼ z then tak e s ∈ P V , t ∈ P W with x, y ∈ s and y , z ∈ t . Without loss of generality , w e may assume tha t V is an initial segmen t of W , and so there is u ∈ P W with s ⊆ u . Since y ∈ t ∩ u , w e ha v e x, z ∈ t , so transitivit y holds. If P is the correspo nding partition of T then it is clear that each s ∈ P is an op en in terv al. This completes the first part of the pr o of. F or the second part, fo r each initial segmen t W α = S ξ <α T ξ , we construct a partition P W α in suc h a wa y that the resulting family satisfies t he prop erty ab ov e. Assume ( P W ξ , ξ < α , hav e b een constructed with the prop ert y in question. If α is a limit ordinal then w e simply define P W α as in part one of t he pro of. If α is a success or ordinal η + 1 then there are tw o cases. If η is itself a successor ordinal then all w e need to do is set P W α = P W η ∪ {{ x } | x ∈ T η } . Instead, if η is a limit ordinal, w e use Corollary 2.4 to obtain a regressiv e map σ : T η − → S ξ <η T ξ +1 with the prop ert y that [ σ ( x ) , x ] ∩ [ σ ( y ) , y ] = ∅ whenev er x 6 = y . Notice that eac h σ ( x ) is a succes sor, so [ σ ( x ) , x ] is an op en in terv al. F or each x ∈ T η , let s x b e the unique elemen t of P W η con ta ining σ ( x ). Now define P W α = { s x ∪ [ σ ( x ) , x ] | x ∈ T η } ∪ { s ∈ P α | s ∩ [ x ∈ T η [ σ ( x ) , x ] = ∅ } . It is straigh tforw ard to c hec k that P W α has the required prop erty . This completes the induction and the pro of.  3. Comp a ct linear orders and p ar tition trees As men tio ned Section 2, linear orders a nd trees enjoy a close relationship. W e will employ t he established not ion of a partit io n tree of a linear order. Let K b e a compact, linear order. W e let an in t erv al a ⊆ K b e called trivial if it con t a ins a t most one p oin t. 7 Definition 3.1. W e shall sa y that a tree T with lev el T α of order α is an a d missible p artition tr e e of K if it satisfies the follow ing prop erties (1) eve ry a ∈ T is a non-trivial, closed in terv al of K ; (2) a ≤ b if and only if b ⊆ a ; (3) T 0 = { K } ; (4) if a ∈ T contains t w o elemen ts then a + is empt y; (5) if a ∈ T contains at least three elemen ts then a has exactly t w o immediate success ors b and c , satisfying min a = min b < max b = min c < max c = max a ; (6) if α is a limit ordinal then a ∈ T α if and o nly if there exist α ξ ∈ T ξ , ξ < α , with a = T ξ <α a ξ . The next lemma states some ob vious consequences of the definition ab ov e. Lemma 3.2. L et T b e an admissibl e p artition tr e e of K . (1) if a, b ∈ T α ar e distinct then a ∩ b is trivial; (2) if a ∩ b is non-trivial then a, b ∈ T ar e c o mp ar able; (3) if L = S a ∈ T { min a, max a } then give n u, v ∈ K , u < v , ther e exist x, y ∈ M such that u ≤ x < y ≤ v . I n p articular, { ( x, y ) | x, y ∈ L, x < y } is a b asis for the top olo gy of K ; (4) if s ⊆ T is total ly or der e d then { min a | a ∈ s } and { max a | a ∈ s } ar e w e l l or der e d and c on v ersely wel l or der e d r esp e ctively. Mor e over, if the c ar dinality κ of s is infi nite, then either the c ar dinality o f { min a | a ∈ s } , or that of { max a | a ∈ s } , is e qual to κ . Pr o of. W e prov e (1) b y induction on α . Supp ose that the result holds for all ξ < α . First, assume α = ξ + 1 and tak e distinct a, b ∈ T α . Let a ∈ a + 0 and b ∈ b + 0 for some a 0 , b 0 ∈ T ξ . If a 0 = b 0 then by Definition 3.1, part (5), a ∩ b is trivial. Otherwise, b y the inductiv e h yp othesis, a ∩ b ⊆ a 0 ∩ b 0 is trivial. Now assume that α is a limit ordinal, with a, b ∈ T α distinct. T ak e sequenc es ( a ξ ) , ( b ξ ) ξ <α as in D efinition 3.1, part ( 6 ). If a ξ = b ξ for all ξ then a = b , thus a ξ 6 = b ξ for some ξ . It follows that a ∩ b ⊆ a ξ ∩ b ξ . This completes the pro of of (1). T o pro ve (2), t a k e a, b ∈ T with a ∩ b non-trivial. Now let a 0 ≤ a and b 0 ≤ b with a 0 and b 0 in the same lev el, and either a 0 = a or b 0 = b . Then a 0 ∩ b 0 is non-trivial, so they mus t b e equal. (2) follows . F or (3), let u, v ∈ K with u < v . Let a b e the greatest elemen t o f T containing u and v . The existence of a follows b y compactness. If a = { u, v } is a tw o-elemen t set then w e are done. Otherwise, b y Definition 3.1 , part ( 5), a has distinct immediate success ors b and c , with x = max b = min c . The maximality of a ensures tha t u < x < v . Now rep eat with x and v . (4) If s ⊆ T is totally ordered then it is w ell ordered. Put E = { ξ | s ∩ T ξ 6 = ∅ } 8 and give n ξ ∈ E , let a ξ b e the unique elemen t of s ∩ T ξ . D efine also F = { η ∈ E | min a ξ < min a η whenev er ξ < η } and G = { η ∈ E | max a ξ > max a η whenev er ξ < η } . It is clear that { min a | a ∈ s } and F share the same or der t yp e, and lik ewise { max a | a ∈ s } has con v erse order t yp e equal to that of G . Moreo v er, E = F ∪ G . Indeed, if there exist ξ , ξ ′ < η suc h that min a ξ = min a η and max a ξ ′ = max a η then, assuming as w e can tha t ξ ≥ ξ ′ , w e hav e a ξ = a η , which is not allow ed in an admissible partition tree. The car dina lity assertion follow s immediately .  By recurs ion, it is clear that an admissible partition tree exists f or ev ery compact linear order K with at least t w o elemen ts. Moreo v er, for an y suc h T and an y branc h s ⊆ T , we ha v e that T s con tains at most tw o elemen ts. F rom now o n, we shall a ssume that all trees T are admiss ible partition trees of compact linear orders. Recall the discussion of the split in terv al at the b eginning of Section 2. Giv en a compact linear order K , an admissible partition tree T of K , a limit ordinal α and a non-simple subset H ⊆ T α , w e sho w that [ a ∈ H { min a, max a } b eha v es similarly to [0 , 1] × { 0 , 1 } . This a llows us to find a necessary condition for K to b e fragmen table. In the next three lemmas, w e shall assume that α is a limit ordinal and ( α ξ ) ξ <κ is a cofinal sequenc e. Lemma 3.3. L et H ⊆ T α . Th e n ther e exists a r e g r essive map π : H − → S ξ <κ T α ξ with the pr op erty that, for al l w ∈ S ξ <κ T α ξ , we c an find b ∈ H such that the set { min a | a ∈ π − 1 ( w ) } is b ounde d ab o v e by min b . Pr o of. If { min a | a ∈ H } has a maxim um elemen t then there is nothing to pro v e. No w supp ose otherwis e. Then, giv en a ∈ H , there exists b ∈ H with max a < min b . Since a is a limit elem en t, w e can tak e π ( a ) < a , π ( a ) ∈ S ξ <κ T α ξ , suc h t ha t max π ( a ) < min b . Now let w ∈ S ξ <κ T ξ , with π ( a ) = w . By definition, there exists b ∈ H such that max π ( a ) < min b . If a ′ ∈ π − 1 ( w ) then min a ′ < max a ′ ≤ max π ( a ′ ) = max π ( a ) < min b.  Lemma 3.4. L et H ⊆ T α . Define L = { b ∈ H | ther e is x b < min b such that { a ∈ H | a ⊆ ( x b , min b ] } is simpl e } . Then L is simple. Sim ilarly, if R = { b ∈ H | ther e is x b > max b such that { a ∈ H | a ⊆ [max b, x b ) } is simple } then R is simple. 9 Pr o of. Supp o se L is not simple. F or eac h b ∈ L , we can find π ( b ) ∈ S ξ <κ T α ξ suc h that π ( b ) < b and x b < min π ( b ). Since π is regressiv e and L is not simple, by Prop osition 2.3, there exists w ∈ S ξ <κ T α ξ suc h that E = π − 1 ( w ) is not simple. Observ e that whenev er a, b ∈ E satisfy min a < min b , w e hav e x b < min π ( b ) = min π ( a ) ≤ min a < max a ≤ min b whence a ⊆ ( x b , min b ]. It f ollo ws that whenev er b ∈ E , the set { a ∈ E | min a ≤ min b } is simple. By Lemma 3.3, there exists a regressiv e map σ : E − → S ξ <κ T α ξ with the prop ert y that whenev er w ∈ S ξ <κ T α ξ , the set { min a | a ∈ σ − 1 ( w ) } b ounded ab ov e by min b , for some b ∈ E . Hence σ − 1 ( w ) is simple for a ll w . Therefore E is simple b y Proposition 2.3, whic h is a con tradiction. Consequen tly , L is simple . It is clear that the ‘righ t hand’ v ersion of Lemma 3.3 holds, thus R is also simple.  Lemma 3.5. Supp ose that H ⊆ T α is non-simple . Then ther e exists a subset C ⊆ H with the pr o p erty that whene v er c ∈ C , x, y ∈ K and x < min c < ma x c < y , the sets { a ∈ C | a ⊆ ( x, min c ] } and { a ∈ C | a ⊆ [max c, y ) } ar e b oth non-sim ple. Pr o of. If M ⊆ T α is non- simple, let C M = M \ ( L ∪ R ) , where L and R are defined as in Lemma 3.4. W e kno w t hat C M is non-simple b y Lemma 3.4 and Corollary 2.5. Put C = C H . Let c ∈ C and x < min c . If we set M = { a ∈ H | a ⊆ ( x, min c ] } then M is non-simple. W e can see t hat C M , whic h is non-simple, is a subset of C ∩ M = { a ∈ C | a ⊆ ( x, min c ] } . Likew ise, if max c < y then { a ∈ C | a ⊆ [ma x c, y ) } is non-simple.  This allows us to give a necessit y (and sufficien t) condition for the fragmen tabilit y of K in terms of admissible partition trees. Prop osition 3.6. If K i s a c omp ac t, fr agmentable, line arly or der e d set and T is any admissible p artition tr e e o f K , then ther e is a p artition of T c o nsisting en tir ely of op en interval s . Pr o of. The first t hing to show is tha t if K is fragmentable then T α is simple for ev ery admissible partition tree T o f K and limit o rdinal α . Assume that H ⊆ T α is non-simple and let d b e a metric o n K . Set H n = { a ∈ H | d (min a, max a ) ≥ n − 1 } . By Corollary 2.5, G = H n is non-simple for some n . Let E = [ c ∈ C { min c, max c } 10 where C = C G \ { min K , max K } a nd C G is as in Lemma 3 .5 . Supp ose that U ∩ E is non-empt y . If min c ∈ U for some c ∈ C , then there exists x < min c with ( x, min c ] ⊆ U . F rom Lemma 3 .5, w e kno w that b ⊆ U fo r some b ∈ C , whence diam ( U ∩ E ) ≥ n − 1 . If max c ∈ U for some c ∈ C t hen w e r each the same conclu- sion. Therefore d do es not fra gmen t K . Since the metric w as arbitrary , we deduce that K is not f r a gmen table. T o finish the pro of, use Prop osition 2.6.  This result allow s us to complete the pro of of Theorem 1.3. Pr o of of The or e m 1.3, (1) ⇒ (2 ) . Let K b e a compact, f r a gmen table, linearly or- dered space. W e sho w that if T is an y admissible par t it io n tree and L = [ a ∈ T { min a, max a } is as in Lemma 3 .2, pa r t (3), then L is a countable union of compact, sc attered subsets. Let T b e an admissible partition tree. By Prop osition 3.6, let P be a partition of T consisting en tirely of op en in terv als. If a ∈ T then the set { s ∈ P | (0 , a ] ∩ s is non-empt y } is non-empt y and finite by compactness, and the fact that P is a partit ion. W e define rank a to b e the cardinality of this set. D efine L n = [ rank a ≤ n { min a, max a } . F or con v enience, we set L 0 = { min K , max K } . W e prov e by induction on n ≥ 0 that L n is closed a nd scattered. F or eac h n , define ∆ n = { ( x, y ) ∈ L 2 n | x < y and ( x, y ) ∩ L n is empt y } as in the pro of o f Theorem 1.3, (2) ⇒ (3 ). Assuming that L n is closed and scattered, and prov e that L n +1 shares these prop erties b y sho wing that L n +1 \ L n = [ ( x,y ) ∈ ∆ n ( x, y ) ∩ L n +1 and that each suc h set ( x, y ) ∩ L n +1 is scattered and closed in ( x, y ). Let w ∈ L n +1 \ L n . There exists b ∈ T with rank b = n + 1, suc h that w is an endp oint of b . If b ∈ s ∈ P then (0 , b ] \ s is a closed, bounded in terv al, so has a great est elemen t a , of rank n . Let a hav e immediate successors c and d , with min a = min c < max c = min d < max d = max a. Without lo ss of generalit y , we can assume tha t c ≤ b . Neces sarily , rank c = n + 1. There a re tw o cases: rank d = n or rank d = n + 1. If rank d = n then let x = min a = min c and y = max c = min d . Since b ⊆ c and w / ∈ L n , w e ha v e x < w < y . If v ∈ ( x, y ) and v is an endp o in t of some e ∈ T , then b y Lemma 3.2, part (2 ) , c and e are compar a ble, and moreo ver c ≤ e . Since rank c = n + 1, w e ha v e v / ∈ L n . This means that ( x, y ) ∈ ∆ n . Moreo v er, if v ∈ L n +1 then since c ≤ e and 11 rank e = n + 1 = rank c , w e ha v e e ∈ s . Con v ersely , if v ∈ ( x, y ) is an endp oin t of some e ∈ s , then v ∈ L n +1 . Therefore ( x, y ) ∩ L n +1 = ( x, y ) ∩ [ e ∈ s { min e, max e } . Since s is a closed interv a l, w e see fro m Lemma 3.2, part (4) a nd Definition 3.1, part (6), t hat ( x, y ) ∩ L n +1 is scattered and closed in ( x, y ). If ra nk d = n + 1 then let x = min a and y = max a . W e us e a similar argumen t to sho w that ( x, y ) ∈ ∆ n and ( x, y ) ∩ L n is scattered and closed in ( x, y ).  W e finish this section with pro ofs o f Prop osition 1.5 and Corollaries 1.4, 1.6 and 1.7. Pr o of of Pr op osition 1 . 5 . Let T be an admissible partition tree of K . Supp ose that T α is non-empt y for all α < κ . If α < κ is a limit ordinal, no te that, as T α is simple b y Prop osition 3.6, w e hav e card T α ≤ card S ξ <α T ξ . By a simple transfinite induction, it follow s that card T α < κ for all α < κ . No w, for ev ery limit α < κ , c ho o se a α ∈ T α . Since T splits into a partit ion P of op en sets, there exists σ ( a α ) < a α with σ ( a α ) , a α ∈ s α ∈ P . W e obta in a regressiv e map τ : L − → κ b y setting τ ( α ) = h t( σ ( a α )), where L is the set of limit ordinals in κ . By the pressing do wn lemma, there is an ordinal ξ and a stationa ry set E ⊆ L suc h that τ ( α ) = ξ for a ll α ∈ E . It is we ll know n in set theory that the union of strictly less than κ non- stationary subsets of κ is again non-stationar y . Th us, as card T ξ < κ , w e conclude that there is w ∈ T ξ and a stat io nary subs et F of E suc h that σ ( a α ) = w for all α ∈ F . Being stationary , F is un b ounded in κ , and since w ∈ s α for all α ∈ F , w e obtain a n inte rv al of length κ in T . Therefore K con t a ins a cop y of κ b y Lemma 3.2, part (4) . It fo llo ws that if K con t a ins no copy of κ , ev ery admissible partition tree T o f K has heigh t strictly less than κ . This, together with the fact tha t card T α < κ for all α < κ , means that card T < κ , so the w eight of K is strictly less than κ b y Lemma 3.2, part (3).  Pr o of of Cor ol la ry 1.4. If K is the contin uous image o f a RN compact then it is the con tinuous image of a fragmen table compact and thus f r a gmen table by [3, Proposi- tion 2.8]. Hence, by The orem 1.3, K is RN compact.  Pr o of of Cor ol la ry 1.6. By [4, Theorem 1.1], K is fragmen table. By [6], K con ta ins no cop y of ω 1 . Thus K is metrisable by Prop osition 1.5.  Pr o of of Cor ol la ry 1.7. By [5, Theorem 7], if K is Gruenhage then C ( K ) admits an equiv alen t norm with a strictly conv ex dual norm. No w apply Coro llary 1.6.  Reference s 1. A. Avil´ es Line arly or der e d R adon-Niko d´ ym c omp act sp ac es. T oplogy Appl. 154 (20 07), 404 – 409. 2. I. Namiok a R adon-Niko d´ ym c omp act s p ac es and fr agmentability. Ma thematik a 34 (1 987), 258 – 281. 3. N. K. Ribars k a Internal char acterization of fr agmentable sp ac es. Mathematik a 34 (19 87), 243 – 257. 12 4. N. K. Ribarsk a The dual of a Gˆ ate aux smo oth sp ac e is we ak s t ar fr agmentable. Pro c. Amer. Math. So c. 114 (1992 ), 1003– 1008. 5. R. J. Smith, Gruenhage c omp acta and strictly c onvex dual norms. F orthcoming in J. Math. Anal. App. doi:10.10 16/j.jmaa.20 08.07 .0 17 6. M. T alagra nd R enormages de quelques C ( K ) I s rael J. Ma th. 54 (19 86), 32 7 –334. Institute of Ma thema tics of the AS CR, ˇ Zitn ´ a 25, CZ - 115 67 Praha 1, Czech Republic E-mail ad dr ess : smit h@math .cas. cz 13