math.FA 2014-02-26 2

Strictly convex norms and topology

We introduce a new topological property called (*) and the corresponding class of topological spaces, which includes spaces with $G_\delta$-diagonals and Gruenhage spaces. Using (*), we characterise those Banach spaces which admit equivalent strictly…

Authors: Jose Orihuela, Richard J. Smith, Stanimir Troyanski

STRICTL Y CONVEX NORMS AND TOP OLOGY J. ORIHUELA, R. J. SMITH AND S. TRO Y ANSKI Abstract. W e int ro duce a new top ological prop ert y called ( ∗ ) and the cor re- sp onding class of top ologica l spaces, which includes spaces with G δ -diagona ls a nd Gruenhage spaces. Using ( ∗ ), we c haracter ise tho s e Banac h s paces whic h admit equiv alen t strictly co n v ex norms, and give an int ernal top ological characteris a tion of those sca ttered compact spaces K , for which the dual Banach spa ce C ( K ) ∗ admits an eq uiv alent strictly conv ex dual norm. W e establish some r elationships betw een ( ∗ ) and other topo lo gical concepts, and the p osition of sev eral well-known examples in this con text. F or instance, w e sho w that C ( K ) ∗ admits a n equiv a - lent strictly conv ex dual nor m, w her e K is K unen’s compa ct space. Also , under the contin uum h yp othesis CH, we giv e an example of a c o mpact scattered non- Gruenhage spa c e having ( ∗ ). 1. Intr oduction Hereafter, all Banac h spaces will b e assumed real and, unless explicitly stated otherwise, all top ological spaces will b e Hausdorff. Throughout this pap er we will b e defi ning new norms on existing Banac h spaces. These new norms will alw ays b e equiv alen t to the given canonical norms. Banac h space nota tion and terminology is standard throughout. A norm k · k on a Banac h space X is said to b e strictly c onvex ( o r rotund) if, giv en x, y ∈ X satisfying k x k = k y k = k 1 2 ( x + y ) k , w e ha v e x = y [5, p. 404]. Geometrically , this means that the unit sphere S X of X in this norm has no non- trivial line segmen ts, or, equiv a len tly , ev ery elemen t of S X is an extreme p oin t of the unit ball B X . Clearly , there are man y Banac h spaces whos e natural norms are not strictly con- v ex. Ho w ev er, b y app ealing to the linear and top ological prop erties of a g iven space, it is often p ossible to define a new norm that is strictly con vex . Changing the norm in this w a y is often called r enorming . In certain cases, w e would lik e the new norm Date : Apr il 7 , 201 8 . 2010 Mathematics Subje ct Cla ssific ation. 46B0 3, 4 6 B26, 54G12. Key wor ds and phr ases. Cont inuum h yp othesis, descriptive, frag men table, Gruenhage spa ce, G δ -diagona l, hereditarily separable, Kunen compact, Ostas z ewski space, ro tund, scattered, strictly conv ex. Some of this res e arc h was co nducted dur ing several visits of R. Smith to the Universit y of Murcia, Spain, from 20 07 to 2 010, and during a visit of S. T royanski to Universit y Colleg e Dublin, in 2 010. J. Orihuela and S. T r o yanski w ere suppor ted by MTM2008-053 96/MTM F ondos F eder and F un- daci´ on S ´ eneca 008848 /PI/08 C ARM. S. T royanski was also suppor ted by the Institute of Math- ematics and Informatics, Bulgarian Academ y of Sciences, Bulgaria n National F und for Scientific Research cont ract DO 02 -360/200 8. 1 to po ssess in addition some form of lo w er semicon tin uit y . F or instance, we may wish for a norm on a dua l space X ∗ to b e w ∗ -lo w er semicon tinuous, so tha t it is the dual of some norm on X . Alternatively , w e may like a norm on a C ( K )-space to b e lo w er semicon tin uous with resp ect to the top ology of p oint wise conv ergence. Suc h additional requiremen ts can mak e norms muc h more difficult to construct, but they do b esto w certain b enefits. F or ex ample, if X ∗ can b e endow ed with a strictly con v ex dual norm then the predual norm o n X is automat ically Gˆ ateaux smo oth, b y virtue of ˇ Sm uly an’s Lemma, cf. [8 , T heorem I.1.4]. Despite the natural and in tuitiv e nature of strict con v exit y , the ques tion of whether a Banach space m ay b e giv en suc h a norm turns out to b e rather diffic ult to answ er in general. A n um b er of mathematicians hav e sough t to establish more easily ve ri- fiable sufficien t conditio ns and necessary conditions fo r a space to admit a strictly con v ex norm. Before outlining this pap er, w e men tion some of the contributions to this collectiv e endea v our. Specialists will realise that it is p ossible to endo w man y (but no t all) o f the spaces b elo w with norms sp orting strong er prop erties than strict con v exit y , but w e prefer not to dwe ll on suc h prop erties here. F or a f uller discussion, w e refer the reader to [8, 36]. In [5, Theorem 9], it is sho wn that ev ery separable Banach space admits a strictly con v ex norm. By following C larkson’s pro of, Da y sho wed that if a Banac h space X is separable then X ∗ admits a strictly conv ex dual norm [6, Theorem 4]. If Γ is a set then c 0 (Γ) admits a strictly con v ex norm [6, Theorem 10] (see also [8, Definition I I.7.2]). On the other hand, if Γ is uncoun table then the space ℓ c ∞ (Γ) o f coun tably supp orted b ounded functions x : Γ − → R with the suprem um norm is simply to o big to admit a strictly conv ex norm [6, Theorem 8] ([8, Theorem I I.7.12]). Amir and Lindenstrauss sho w ed that if X is w eakly compac tly generated (WC G) then b oth X and X ∗ admit a strictly con v ex norm and a strictly con v ex dual norm, resp ec tiv ely [2, Theorem 3]. Thes e results rely on the fact that if a b ounded linear map T : X − → Y is injec tiv e and Y admits a s trictly con vex norm, then so do es X . If X is W CG then w e can find such maps on b oth X and X ∗ , where Y = c 0 (Γ) for some Γ. The n [6, Theorem 10] can b e applied. A t the time, suc h a ‘linear transfer’ in to some c 0 (Γ) w as the only w ay of showin g that spaces admitted strictly con v ex norms. Moreo v er, ℓ c ∞ (Γ), Γ uncoun table, was the ‘smallest’ space know n no t to admit a strictly con v ex norm. In [9], the author s construct an increasing transfinite seq uence ( X α ) 1 ≤ α<ω 1 of spaces of Baire-1 func- tions on [0 , 1], all admitting strictly con v ex norms, and no ne a dmitting a b ounded linear injectiv e map into a n y c 0 (Γ), pro vided α ≥ 2. Moreo v er, b y refining Day’s argumen t [6, Theorem 8], they show ed t hat the union Y = S α<ω 1 X α do es not ad- mit a strictly con v ex norm, and that there is no bounded linear injectiv e map from ℓ c ∞ ([0 , 1]) in to Y . The fa ct that the dual o f ev ery WC G space a dmits strictly conv ex dual norm, with a necessarily Gˆ ateaux s mo oth predual norm, prompted Lindenstrauss to conjecture that if X admits a Gˆ ateaux sm o oth norm then it m ust em b ed as a subspace of some W CG space [21]. Mercourakis pro vided a negativ e a nsw er to this conjecture b y showing that if X is a we akly c ountably determine d (W CD) space, then b oth X 2 and X ∗ admit strictly con v ex norms [22, Theorems 4.6 and 4.8], by virtue of linear transfers (although not into c 0 (Γ) in general). P ap ers suc h as [9, 22] suggest that there is no simple w a y of c haracterising strict con v exit y in terms of linear structure. Since then, the problem of classifying Bana ch spaces admitting strictly con v ex norms has b een approa c hed from a more top ological p erspectiv e, and particular a tten tion has been paid to strictly con v ex dual norms and C ( K )-spaces. An y Banac h s pace X em b eds isometrically in to C ( B X ∗ , w ∗ ), and this fact enables certain results a bout C ( K )-spaces t o b e generalised to all Ba nac h spaces, b y phrasing them in t erms of the topolo gical struc ture of ( B X ∗ , w ∗ ). F or example, if X ∗ admits a s trictly con vex dual norm then ( B X ∗ , w ∗ ) is f r ag- men table [3 0 , Theorem 1.1]. W e can sa y that a top ological space is fr agm entable if it admits, for eac h n ∈ N , a n increasing we ll ordered family of op en subsets ( U ξ ) ξ <λ n , with the prop ert y that giv en distinct p oin ts x and y , w e can find some n 0 and ξ < λ n 0 suc h that { x, y } ∩ U ξ is a singleton [29, Theorem 1.9]. The idea o f p oin t separation features throughout this pap er. Indeed, t he notion of strict con v exit y can b e view ed as a form of p oin t separation. The nece ssit y condition abov e is fa r from sufficie n t how ev er. The class of frag- men table space s is v ery large and includes , for instance, a ll scattered spaces. Recall that a to polo g ical space is sc atter e d if ev ery non- empt y subspace admits a relativ ely isolated p oin t. In t he y ear b efore [22] app eared, T ala grand sho w ed that the space C ( ω 1 + 1) ∗ do es not admit a strictly conv ex dual norm [38, Th ´ eor` eme 3], where ω 1 is the firs t uncountable ordinal conside red in its (scattered) order top ology . On the other hand, the dual unit ball ( B C ( ω 1 +1) ∗ , w ∗ ) is fragmentable [29, The orem 3.1]. The next significan t sufficiency condition w e men tion requires a definition. Definition 1.1. A compact space K is d e scriptive if it admits a σ -iso late d network , that is to sa y , a family N = S ∞ n =1 N n of subsets of K , satisfying (1) N ∩ S N n \ { N } is empt y whenev er N ∈ N n and n ∈ N , and (2) if x ∈ U ⊆ K , where U is op en, then there exists n ∈ N and N ∈ N n suc h that x ∈ N ⊆ U . This top ological co v ering prop ert y arose out of t he theory of ‘generalised metric spaces’ [12]. The class of descriptiv e compact spaces is large. F or example , if X is W CD then ( B X ∗ , w ∗ ) is descriptiv e ([3 7 , Th´ eor ` eme 3.6] and [28, Corolla ry 2.4]). In [28, The orem 3 .3], Ra ja sho w ed that if K is descrip tive then C ( K ) ∗ admits a strictly con v ex dual norm. T his result can b e adapted to giv e a suffic ien t condition whic h applies to a wide class of dual Banac h space s [26, Theorem 1 .3], including duals of W CD spaces . W e remark that a compact scattered space K is descriptiv e if and only if it is σ -d iscr ete , that is, K = S n =1 D n , whe re eac h D n is dis crete in its relativ e top ology . This fact follo ws from [28, Lemma 2.2]. Despite these adv ances, there is a v ery large gap b et w een the class of descriptiv e spaces a nd ω 1 + 1 and the more general class of fragmen table spaces. Some years prior to the publication of [2 8], Hay don constructed some strictly con v ex dual norms on spaces o f the form C ( K ) ∗ , where t he K are 1-p oint c ompactifications of certain trees in their in terv al top ologies [16, Theorem 7.1]. It turns out that some of these 3 spaces are not descriptiv e, so Ha ydon’s suffi cien t condition is not cov ered b y Ra ja ’s um brella. In [33, Theorem 6], t he second-named author generalised Hay don’s result b y c har- acterising those trees for whic h the asso ciated spaces C ( K ) ∗ admit strictly conv ex dual norms. Later, in [34], this order-theoretic c haracterisation w as repro v ed in in ternal, top ological terms. T o state this result w e need another definition. Definition 1.2. A compact space K is called Gruenhage if there exis ts a sequence ( U n ) ∞ n =1 of families of op en subsets of K , and sets R n , n ≥ 1, with the prop ert y tha t (1) if x, y ∈ K are distinct, then there exists n ∈ N and U ∈ U n , suc h that { x, y } ∩ U is a singleton, and (2) U ∩ V = R n whenev er U, V ∈ U n are distinct. This definition is equiv alen t to the origina l [13, p. 372 ] (see [34, Prop osition 2]). Ev ery descriptiv e compact space is Gruenhage [34, Corollary 4]. Theorem 1.3 ([34 , Theorems 7 and 16]) . L et K b e c omp act. Then the fol lowing statements hold. (1) If K is Gruenhage then C ( K ) ∗ admits a strictly c on v e x dual lattic e norm. (2) If K is the 1-p oint c omp actific ation of a tr e e and C ( K ) ∗ admits a strictly c onve x dual norm, then K is Gruenhage. Theorem 1.3 (1) can b e adapted to giv e a sufficien t condition ([34, Corollary 10]) whic h applies to class of dual Banach sp aces ev en wider t ha n that co v ered b y [26, Theorem 1 .3]. There are other instances o f nece ssit y b esides Theorem 1.3 (2). F or instance, if the Banac h space X has an (uncountable) unconditional basis then X ∗ admits a strictly con v ex dual norm if and only if ( B X ∗ , w ∗ ) is Gruenhage (equiv alen tly , if ( B X ∗ , w ∗ ) is descriptiv e) [35, The orem 6]. Des pite some courageous attempts, it w as not p ossible to prov e the conv erse implication of Theorem 1.3 (1). Man y of t he results of this pap er are the pro duct of e fforts to res olv e this difficult y . This pap er is o rganised as fo llo ws. In Section 2, w e in tro duce a generalisation of Gruenhage’s prop ert y , lab eled ( ∗ ) (Definition 2.6), a nd use it to giv e a char- acterisation of Banac h spaces whic h admit a strictly conv ex norm satisfying some additional low er semicon tin uit y prop ert y (Theorem 2.8). This characterisation at- tempts to top ologise as m uc h as p ossible the geometric condition of strict con v exit y . In Section 3, we use ( ∗ ) to find an analogue of Theorem 1.3 whic h applies to all scattered compact spaces (Theorem 3.1 ). This class is significan t in Banac h space theory because C ( K ) is an Asplund space if and only if K is scattered. In doing so, w e show that ( ∗ ) comes close to pro viding a complete top ological c haracterisation of t ho se K , for whic h C ( K ) ∗ admits a strictly conv ex dual norm. In Section 4, w e establish some of the top ological properties of ( ∗ ) and its p osition in the w ider con- text of co v ering prop erties, and pro vide some examples o f scattered compact spaces, some of whic h ha ving ( ∗ ) and others not . In particular, we giv e an example of a scattered non- Gruenhage compact space ha ving ( ∗ ) (Example 4.10). Th us, Theorem 3.1 do es not follow from previous results suc h as Theorem 1.3. Along the w ay , w e answ er an op en question concerning Kunen’s compact space: sp ecifically , w e sho w that it is G ruenhage (Prop osition 4.7 ) . In sev eral cases, inc luding E xample 4.10, w e 4 shall assume extra principles indep enden t of the us ual axioms of set theory . Finally , in Section 5, w e presen t some op en problems ste mming from this study . 2. A chara cterisa tion of strict convexity in Banach sp aces In this section, we provide a general characteris ation of strictly con v ex renormings in Banac h spaces. Throughout this section, X will b e a Banac h space (a nd o cca- sionally a general top ological space) and F ⊆ X ∗ a nor ming subspace. Recall that σ ( X , F ) denotes the coarsest top ology on X with respect to which ev ery elemen t of F is con tin uous. W e b egin b y presen ting a use ful folklore result, together with a brief sk etc h pro of. Prop osition 2.1. L et F ⊆ X ∗ b e a norming subsp ac e. Supp ose that ther e ex ists a se quenc e of σ ( X , F ) - l o wer semic ontinuous c onvex functions ϕ n : X − → [0 , ∞ ) such that given distinct x, y ∈ X , we c an fi nd n ∈ N satisfying (1) ϕ n ( 1 2 ( x + y )) < max { ϕ ( x ) , ϕ ( y ) } . Then X admits a σ ( X, F ) -lower semi c ontinuous strictly c onvex norm | | | · | | | . Inste ad, if X is a Banach lattic e, (1) holds whenever x, y ∈ X + ar e distinct, and ϕ n ( x ) ≤ ϕ n ( y ) whenever | x | ≤ | y | and n ∈ N , then | | | · | | | is a σ ( X, F ) -lower semic ontinuous strictly c onve x lattic e norm. Pr o of. Let k · k denote the origina l norm on X . W e define a new norm b y | | | x | | | 2 = X n,q c n,q k x k 2 n,q where k · k n,q is the Mink o wski f unctional of C n,q =  x ∈ X : ϕ n ( x ) 2 + ϕ n ( − x ) 2 ≤ q  whenev er q is a rational num b er satisfying q > 2 ϕ n (0) 2 , and where the constants c n,q > 0 are c hosen to ensure the uniform con v ergence of the sum on b ounded sets . By a standard con v exit y ar g umen t (cf. [8, F act I I.2.3]), it can b e sho wn that if | | | x | | | = | | | y | | | = 1 2 | | | x + y | | | then ϕ n ( x ) = ϕ n ( y ) = ϕ n ( 1 2 ( x + y )) fo r all n , whence x = y b y h yp othesis. If w e adopt the lattice hypotheses instead then clearly | | | · | | | is also a la ttice norm, and strictly con v ex on X + . T o see tha t the strict conv exit y extends to a ll of X , let x, y ∈ X and suppose that | | | x | | | = | | | y | | | = 1 2 | | | x + y | | | . Then 1 2 | | | | x | + | y | | | | = | | | x | | | as well, so strict con v exit y o n X + yields | x | = | y | . If we set w = 1 2 ( x + y ) then rep eating the a bov e gives us | x | = | w | . A simple lat t ice argumen t (e.g. [34, p. 749]) leads us to conclude t ha t x = y .  Our characterisation adopts sev eral ideas from [25, 24]. Recall that if A is a subset of a lo cally con v ex space then an op en slic e U of A is the in tersection of A with an op en half-space of X . The follow ing prop osition will b e our main to ol. 5 Prop osition 2.2. L et A b e a b ounde d subset of X an d U a family of non-empty σ ( X , F ) -op e n sli c es of A . Then ther e exists a σ ( X , F ) -lower semic ontinuous 1- Lipschitz c onvex function ϕ with the pr op erty that w henever x, y ∈ A , { x, y } ∩ S U is no n -empty and ϕ ( x ) = ϕ ( y ) = ϕ ( 1 2 ( x + y )) , we h a ve x, y ∈ U for some U ∈ U . Prop osition 2.2 is an immediate c orollary o f the ne xt result, dubb ed the ‘Slice Lo calisation Theorem’. Theorem 2.3 ([25, Theorem 3]) . L et A b e a b o unde d subset o f X a n d U a fa mily of non -empty σ ( X , F ) -op en slic e s of A . T hen ther e is an e quivalent σ ( X , F ) -lower semic ontinuous norm k · k s uch that for every se quenc e ( x n ) ∞ n =1 ⊆ X and x ∈ A ∩ S U , if 2 k x k 2 + 2 k x n k 2 − k x + x n k 2 → 0 , then ther e is a se quenc e of slic es ( U n ) ∞ n =1 ⊆ U an d n 0 ∈ N such that (1) x, x n ∈ U n whenever n ≥ n 0 and x n ∈ A ; (2) for every δ > 0 ther e is some n δ ∈ N such that x, x n ∈ (con v( A ∩ U n ) + δ B X ) σ ( X ,F ) for al l n ≥ n δ . The Slice Lo calisation Theorem can b e used to simplify the pro ofs of netw ork c haracterisations of Banach spaces whic h admit lo cally uniformly rotund norms. T o pro v e Prop osition 2.2, all w e need to do a pply Theorem 2.3 with x n = y for all n . Ho w ev er, there is a more transparen t proof of this prop osition which w e pro vide for completeness . Of k ey imp ortance to the pr o of is the concept o f F -distance, in tro duced in [24]. Let D ⊆ X b e a non-empt y , con v ex b ounded subset. Giv en ξ ∈ X ∗∗ , define (1) k ξ k F = sup { ξ ( f ) : f ∈ B X ∗ ∩ F } . It is clear that k · k F is σ ( X ∗∗ , F )-lo w er semicon tinuous ( σ ( X ∗∗ , F ) b eing the only generally non-Hausdorff top ology men tioned in t his pa per). No w set ϕ ( x ) = inf n k x − d k F : d ∈ D σ ( X ∗∗ ,X ∗ ) o . Definition 2.4. Giv en a non-empt y , con v ex b ounded subset D ⊆ X , we call ϕ ( x ) the F -distanc e from x ∈ X to D . W e pass to the bidual o f X in o rder t o control the lo w er semicon tinuit y prop erties of ϕ . The notion o f F -distance has a n um b er of useful prop erties whic h we list in the next lemma. Lemma 2.5. L et ϕ ( x ) b e the F -distance fr om x ∈ X to D . (1) ϕ is c onvex and 1 - Lipschitz; (2) ϕ is σ ( X, F ) -lower semic o ntinuous; (3) D σ ( X ,F ) = ϕ − 1 (0) . 6 Prop erties (1 ) and (2) ar e pro v ed in [24, Prop osition 2 .1 ] and the third is a straigh t- forw ard exercise in v olving the Hahn-Banac h separation theorem. Now w e can giv e our alternativ e pro of of Prop osition 2.2. Pr o of of Pr op osition 2.2. F or eac h U ∈ U and x ∈ X , define ϕ U ( x ) to b e the F - distance fro m x to (con v A ) \ U . Since A is b ounded, w e can define another con ve x, σ ( X , F )-low er semicon tin uous, 1-Lipsc hitz function b y ϕ ( x ) = sup { ϕ U ( x ) : U ∈ U } . Let x, y ∈ A with { x, y } ∩ S U non- empt y and supp ose that ϕ ( x ) = ϕ ( y ) = ϕ ( 1 2 ( x + y )) . Without loss of generalit y , w e c an assume tha t x ∈ U for some U ∈ U . Since U ∩ (con v A ) \ U σ ( X ,F ) is empt y , w e hav e ϕ ( x ) ≥ ϕ U ( x ) > 0 by Lemma 2.5, part (3 ) . Pic k ε > 0 suc h tha t ϕ ( x ) > 5 ε 2 and c ho ose V ∈ U with the prop ert y that ϕ ( 1 2 ( x + y )) 2 < ϕ V ( 1 2 ( x + y )) 2 + ε 2 . W e ha v e 0 = 1 2 ( ϕ ( x ) 2 + ϕ ( y ) 2 ) − ϕ ( 1 2 ( x + y )) 2 > 1 2 ( ϕ V ( x ) 2 + ϕ V ( y ) 2 ) − ϕ V ( 1 2 ( x + y )) 2 − ε 2 ≥ 1 2 ( ϕ V ( x ) 2 + ϕ V ( y ) 2 ) − 1 4 ( ϕ V ( x ) + ϕ V ( y )) 2 − ε 2 = 1 4 ( ϕ V ( x ) − ϕ V ( y )) 2 − ε 2 th us (2) | ϕ V ( x ) − ϕ V ( y ) | < 2 ε. Since ϕ V is conv ex, w e hav e max { ϕ V ( x ) , ϕ V ( y ) } ≥ ϕ V ( 1 2 ( x + y )). T ogether with (2), this implies min { ϕ V ( x ) , ϕ V ( y ) } ≥ max { ϕ V ( x ) , ϕ V ( y ) } − 2 ε ≥ ϕ V ( 1 2 ( x + y )) − 2 ε ≥  ϕ ( 1 2 ( x + y )) − ε 2  1 2 − 2 ε > 0 . Therefore ϕ V ( x ) , ϕ V ( y ) > 0. Since x, y ∈ A , w e get x, y ∈ V .  Prop osition 2.2 motiv ates the in tro duction of the cen tral top ological concept fea- turing in this pap er. Definition 2.6. W e sa y that a top ological space X ha s ( ∗ ) if there exists a sequence ( U n ) ∞ n =1 of families of op en subsets of X , with the prop ert y that give n any x, y ∈ X , there exists n ∈ N suc h that (1) { x, y } ∩ S U n is non-empt y , and (2) { x, y } ∩ U is at most a singleton for all U ∈ U n . 7 An y sequenc e ( U n ) ∞ n =1 satisfying the conditio ns of Definition 2.6 will b e called a ( ∗ )-se quenc e for X . In addition, if X is lo cally conv ex and A ⊆ X then w e say A has ( ∗ ) with slic es if A admits a ( ∗ )- se quence ( U n ) ∞ n =1 , with the prop ert y that ev ery elemen t of S ∞ n =1 U n is an op en slice of A . Remark 2.7. It will b e con v enien t to note that if A ⊆ X , then to say t ha t ( A, σ ( X , F )) has ( ∗ ) with slices is equiv alen t to t here b eing a family o f subsets G n ⊆ ( S X ∗ ∩ F ) × R , n ∈ N , suc h that given distinct x, y ∈ A , we hav e n ∈ N satisfying (a) max { f ( x ) , f ( y ) } > λ for some ( f , λ ) ∈ G n , and (b) min { g ( x ) , g ( y ) } ≤ µ for ev ery ( g , µ ) ∈ G n . Our c haracterisation follo ws. Theorem 2.8. L et F ⊆ X ∗ b e a 1-norming subsp ac e. Then the fol lowing ar e e quivalen t. (1) X a d mits a σ ( X, F ) -lowe r semic ontinuous strictly c onvex norm; (2) ( X , σ ( X , F )) has ( ∗ ) with slic es; (3) ( S X , σ ( X , F )) h a s ( ∗ ) with slic es; (4) ther e is a se quenc e of subsets ( X n ) ∞ n =1 of X , such that  ( x, y ) ∈ X 2 : x 6 = y  ⊆ ∞ [ n =1 X 2 n and wher e e ach ( X n , σ ( X , F )) h a s ( ∗ ) with slic es. Pr o of. (1) ⇒ (2): let k · k b e a σ ( X , F )-low er semicon tinuous strictly con v ex norm on X . Then F is also 1-nor ming for k · k . Let G q = ( S ( X, k·k ) ∗ ∩ F ) × { q } for each rationa l n um b er q > 0. W e v erify that ( X, σ ( X , F )) has ( ∗ ) b y sho wing that the G q satisfy (a) and (b) of R emark 2.7. Giv en distinct x, y ∈ X , assume that k x k ≤ k y k . The strict con v exit y o f k · k tells us that k 1 2 ( x + y ) k < k y k . Let rational q s atisfy k 1 2 ( x + y ) k < q < k y k . Since F is 1-norming f or k · k , w e kno w that f ( y ) > q for a pair ( f , q ) ∈ G q , giving (a ) . Now supp o se g ( y ) > q for some ( g , q ) ∈ G q . The n certainly g ( x ) ≤ q , else w e w ould ha v e q < 1 2 g ( x + y ) ≤ 1 2 k x + y k , whic h do esn’t mak e an y sense. This sho ws that (b) is also satisfied. (2) ⇒ (3) is trivial b ecause ( ∗ ) with slices is inherited by subspaces. (3) ⇒ ( 2 ): if ( S X , σ ( X , F )) ha s ( ∗ ) with slices then w e take sets G n , n ∈ N that satisfy (a) and (b) of Remark 2.7. W e can assume that G n ⊆ ( S X ∗ ∩ F ) × ( − 1 , 1 ) for ev ery n . Given rational q , r > 0 , set H q = ( S X ∗ ∩ F ) × { q } and L n,q ,r = { ( f , q ( λ + r )) : ( f , λ ) ∈ G n } . W e claim tha t the H q and L n,q ,r v erify that ( X , σ ( X , F )) has ( ∗ ) , using Remark 2.7. Let x, y ∈ X be distinct, with k x k ≤ k y k . If k x k < k y k the n w e choo se rational q to satisfy k x k < q < k y k . Since F is 1-norming, it is easy to c hec k that (a) and (b) 8 are fulfilled by H q . No w supp ose k x k = k y k . W e kno w that, with respect to x/ k x k and y / k y k , (a) and (b) a re satisfied b y some G n . Without loss of generalit y , assume f ( x ) > k x k λ , where ( f , λ ) ∈ G n . Our argumen t dep ends o n the sign of λ . If λ ≥ 0 then c ho ose rational q , r > 0 satisfying f ( x ) > k x k ( λ + r ) and k x k 1 + r < q < k x k . The constan ts ha v e b een arra nged to ensure (3) µ ( k x k − q ) < k x k − q < q r whenev er | µ | < 1. W e ha v e f ( x ) > k x k ( λ + r ) > q ( λ + r ). No w supp ose that g ( x ) > q ( µ + r ), where ( g , µ ) ∈ G n . The n g ( x ) > q ( µ + r ) > k x k µ b y equation (3 ) ab o v e. This means g ( x/ k x k ) > µ , w hence g ( y / k y k ) ≤ µ b y (b), giving g ( y ) < q ( µ + r ). In sum mary , we hav e sho wn that (a) and (b) o f Remark 2.7 are fulfilled by L n,q ,r . If instead λ < 0 , w e c ho ose r < − λ as ab ov e a nd ensure that q satisfies k x k < q < k x k 1 − r . By arguing similarly , we get what w e w an t. (2) ⇒ ( 4 ) follo ws easily by setting X n = X . W e finish by pro ving (4) ⇒ (1). By taking in tersections with mB X , m ∈ N , a nd reindex ing if necessary , w e can assume that each X n is b ounded. Let eac h X n ha v e a ( ∗ )- seq uence ( U n,m ) ∞ m =1 , where eac h elemen t of S ∞ m =1 U n,m is a (non-empty ) σ ( X , F )-op en slice of X n . Let ϕ n,m denote the con v ex function constructed b y applying Prop osition 2.2 to X n and the family U n,m . W e ha v e ensured that if x, y ∈ X are distinct then we can find n and m suc h that ϕ n,m ( 1 2 ( x + y )) < max { ϕ n,m ( x ) , ϕ n,m ( y ) } . The rest follo ws from Prop osition 2.1.  Note that Theorem 2.8 (1), (2) and (4) a re also equiv alent when F is simply a norming subspace, rather than a 1-norming subspace. W e end this section b y giving an example to sho w that the reliance on slices in the statemen t of Theorem 2.8 is necessary in general. Example 2.9. L et K b e the pro duct { 0 , 1 } ω 1 , endo w ed with the lexicographic order top ology . According to [17, Example 1], C ( K ) admits a Kadec norm k · k but no strictly con v ex norm. By the definition of Kadec norms, the w eak top ology agrees with the norm top ology o n S ( C ( K ) , k·k ) . I n particular ( S ( C ( K ) , k·k ) , w ) is metris able, meaning that it has a σ -discrete base and th us has ( ∗ ) as w ell. Ho wev er, since k · k cannot be strictly con v ex, Theorem 2.8 implies tha t ( S ( C ( K ) , k·k ) , w ) do es not hav e ( ∗ ) with slic es . W e conclude this section b y giving a sufficien t condition for constructing strictly con v ex norms. Theorem 2.11 b elo w can b e applied to many spaces of significance to the theory , suc h a s the Mercourakis spaces c 1 (Σ ′ × Γ) (see [8, Section VI.6]), Dashiell- Lindenstrauss spaces and spaces o f the f o rm C ( K ) ∗ , where K is Gruenhage. The idea, whic h go es back to the classical norm of D a y for c 0 (Γ) [6, Theorem 10], is to 9 ‘glue together’ strictly conv ex norms on finite-dimensional spaces (which are readily a v ailable) to obta in strictly con vex no r ms on larg er spaces. Elemen ts of Theorem 2.11 can be fo und in [11, Theorem 5]. Before giving the theorem, w e stat e a simple fact. F act 2.10. L et ξ : [0 , 1] − → R b e a function satisfying ξ (0) ξ (1) < 0 , and supp ose that ξ + and ξ − ar e c onvex. Then for every λ ∈ (0 , 1) , we have (4) ξ ± ( λ ) < λξ ± (1) + (1 − λ ) ξ ± (0) . Pr o of. Assume ξ (0 ) > 0. Since ξ ± are con v ex and ξ is necessarily contin uous, it is easy to see that there is a unique interv al [ a, b ], where 0 < a ≤ b < 1, suc h that ξ ( u ) > ξ ( v ) = 0 > ξ ( w ) whenev er u ∈ [0 , a ), v ∈ [ a, b ] and w ∈ ( b, 1]. If λ ∈ [ a, b ] then clearly equation (4) holds for ξ ± . Let λ < a . The n ξ − ( λ ) = 0 and, as ξ − (1) > 0 , (4) holds for ξ − . Since ξ + is con v ex, setting µ = λ/a giv es ξ + ( λ ) ≤ (1 − µ ) ξ + (0) + µξ + ( a ) = (1 − µ ) ξ + (0) < (1 − λ ) ξ + (0) so (4) holds for ξ + . The pro of fo r the case λ > b is similar.  Clearly , if ξ is linear then ξ ± are con vex . The same is true if ξ is p ositiv e and con v ex. Theorem 2.11. L et Θ n : X − → ℓ ∞ (Γ n ) b e a se quenc e of maps such that b oth functions x 7→ Θ n, ± ( x )( γ ) ar e σ ( X, F ) -lower semic ontinuous and c onvex f o r ev e ry γ ∈ Γ n and n ∈ N . L et us a s sume in a ddition that for al l distinct x, y ∈ X , ther e ar e λ ∈ (0 , 1) , n ∈ N and a fini te set A ⊆ Γ n , such that (1) Θ n ( x ) ↾ A 6 = Θ n ( y ) ↾ A , and (2) | Θ n ( z )( α ) | > | Θ n ( z )( γ ) | whenever α ∈ A and γ ∈ Γ \ A, wher e z = λx + (1 − λ ) y . Then X admi ts a σ ( X , F ) -lower semic ontinuous strictly c onve x norm | | | · | | | . Inste a d , if X is a Banac h latt ic e, Θ n, ± ( x ) ≤ Θ n, ± ( y ) whe never | x | ≤ | y | and e quations (1) and (2) app l y to distinct x, y ∈ X + , then | | | · | | | is a σ ( X , F ) -lower semic ontinuous strictly c onvex lattic e norm. Pr o of. Since Θ n, ± ( · )( γ ) are b oth con v ex and σ ( X, F )-low er semicon tin uous, the same is true of | Θ n ( · )( γ ) | . Define Θ n, 0 ( x )( γ ) = Θ 2 n ( x )( γ ) a nd Θ n, ± 1 ( x )( γ ) = Θ n, ± ( x )( γ ) . If Γ = S ∞ n =1 Γ n , u ∈ ℓ ∞ (Γ) and A ⊆ Γ is finite, set ϕ A ( u ) = X γ ∈ A u ( γ ) and put ϕ A,n,i = ϕ A ◦ Θ n,i for ev ery n ∈ N and i ∈ {− 1 , 0 , 1 } . Certainly , eac h ϕ A,n,i is σ ( X , F )-low er semicon tin uous, non-negativ e and conv ex. Finally , let ψ m,n,i ( x ) = sup { ϕ A,n,i ( x ) : A ⊆ Γ n has cardinalit y m } . 10 T o finish the pro of, w e shall sho w that for ev ery distinct pair x, y ∈ X , there is m, n ∈ N and i ∈ {− 1 , 0 , 1 } suc h that (5) ψ m,n,i ( 1 2 ( x + y )) < max { ψ m,n,i ( x ) , ψ m,n,i ( y ) } . holds. T hen we can appeal to Prop o sition 2.1. T ak e λ ∈ (0 , 1) , n ∈ N and A ⊆ Γ n satisfying (1) and (2). W e consider tw o cases. First supp ose that Θ n ( x )( β )Θ n ( y )( β ) < 0 for some β ∈ A . F rom (2) w e kno w that Θ n ( z )( β ) 6 = 0. Assu me for no w that Θ n ( z )( β ) > 0 a nd define the non-empt y set B = { α ∈ A : Θ n, + ( z )( α ) > 0 } . so that Θ n, + ( z )( α ) > Θ n, + ( z )( γ ) for ev ery α ∈ B and γ ∈ Γ \ B . Therefore ψ n,m, 1 ( z ) = P α ∈ B Θ n ( z )( α ), where m is the cardinalit y of B . Applying F act 2.10 to ξ ( t ) = Θ n ( tx + (1 − t ) y )( β ), t ∈ [0 , 1 ], w e get Θ n, + ( z )( β ) < λ Θ n, + ( x )( β ) + (1 − λ )Θ n, + ( y )( β ) whence ψ n,m, 1 ( z ) < λψ n,m, 1 ( x ) + ( 1 − λ ) ψ n,m, 1 ( y ) from whic h (5) quic kly follows for i = 1, b y conv exity . If Θ n ( z )( β ) < 0 then w e argue similarly with i = − 1. Let’s now cons ider the case (6) Θ n ( x )( α )Θ n ( y )( α ) ≥ 0 for all α ∈ A . Let m ∈ N b e the cardinalit y of A . Since t 7→ t 2 is strictly conv ex, from condition (1) w e hav e X α ∈ A ( λ Θ n ( x )( α ) + (1 − λ )Θ n ( y )( α )) 2 < X α ∈ A λ (Θ n ( x )( α )) 2 + (1 − λ )(Θ n ( y )( α )) 2 = λϕ A,n, 0 ( x ) + ( 1 − λ ) ϕ A,n, 0 ( y ) ≤ λψ m,n, 0 ( x ) + (1 − λ ) ψ m,n, 0 ( y ) ≤ max { ψ m,n, 0 ( x ) , ψ m,n, 0 ( y ) } . Giv en the con v exit y of | Θ n ( · )( α ) | and equation (6 ), w e obtain | Θ n ( z )( α ) | = | Θ n ( λx + (1 − λ ) y )( α ) | ≤ | λ Θ n ( x )( α ) + (1 − λ )Θ n ( y )( α ) | . This and condition (2) imply ψ m,n, 0 ( z ) = ϕ A,n, 0 ( z ) ≤ X α ∈ A ( λ Θ n ( x )( α ) + (1 − λ )Θ n ( y )( α )) 2 . Com bining these inequalities we see that ψ m,n, 0 ( z ) < max { ψ m,n, 0 ( x ) , ψ m,n, 0 ( y ) } from whic h (5) follows fo r i = 0, ag ain by con vex ity . If w e a dopt the lattice as- sumptions instead, then eac h ψ n,m,i satisfies the lattice assumptions in Prop osition 2.1.  11 In the first corollary b elo w is a sufficien t condition o f ‘Mercourakis t yp e’, which is formally more general tha n similar conditions giv en in the literature. Corollary 2.12. L e t X b e a subsp ac e or sublattic e of ℓ ∞ (Γ) and s upp o se that ther e ar e subse ts Γ n ⊆ Γ , n ∈ N , with the pr op erty that giv e n x ∈ X and α ∈ supp x , we c an find n and α ∈ Γ n , so that { γ ∈ Γ n : | x ( γ ) | ≥ | x ( α ) |} is finite. Then X adm i ts a p ointwise lower semic ontinuous strictly c onv ex norm or lattic e norm, r esp e ctively. Pr o of. Let P n ( x )( γ ) = | x ( γ ) | whenev er γ ∈ Γ n and n ∈ N . T he co ordinate maps are p ositiv e and con v ex. W e sho w that P n satisfies conditions (1 ) and (2) of Theorem 2.11. Giv en distinct x, y ∈ X , t ak e n ∈ N and β ∈ Γ n suc h that x ( β ) 6 = y ( β ). Then there is λ ∈ (0 , 1) suc h that λx ( β ) + (1 − λ ) y ( β ) is non-zero. Set z = λx + (1 − λ ) y and tak e n ∈ N suc h tha t A = { α ∈ Γ n : | z ( α ) | ≥ | z ( β ) |} is finite. E viden tly β ∈ A , so P n ( x ) ↾ A 6 = P n ( y ) ↾ A , and | P n ( z )( α ) | ≥ | z ( β ) | > | P n ( z )( γ ) | whenev er α ∈ A and γ ∈ Γ n \ A .  Corollary 2.13 ([34, Theorem 7]) . If K is Gruenha g e then C ( K ) ∗ admits a strictly c onve x dual lattic e norm. Pr o of. If K is Gr uenhage then (cf. [34, Lemma 6]), w e can find sequen ces ( U n ) ∞ n =1 and ( R n ) ∞ n =1 as in Definition 1.2, with the further prop ert y t ha t if µ ∈ C ( K ) ∗ and µ ( U ) = 0 for all U ∈ U n , n ∈ N , then µ = 0. Let Γ n = U n and define Θ n ( µ )( U ) = | µ | ( U ) , U ∈ U n . Since | λµ + (1 − λ ) ν | ≤ λ | µ | + (1 − λ ) | ν | whenev er λ ∈ [0 , 1], the co ordinate maps Θ n ( · )( U ) are p ositiv e and conv ex. If µ , ν ∈ C ( K ) ∗ are p ositiv e and distinct, then there exis ts n ∈ N and U ∈ U n suc h that µ ( U ) 6 = ν ( U ). If w e set τ = 1 2 ( µ + ν ) then w e hav e τ ( U ) > τ ( R n ). By considering Definition 1.2 part ( 2 ), w e see that for an y r > τ ( R n ), there are only finitely man y V ∈ U n satisfying τ ( V ) ≥ r . Therefore, conditions (1) and (2) of Theorem 2.11 apply to p ositiv e eleme n ts of C ( K ) ∗ . Now w e are able to apply Theorem 2.11 .  Dashiell-Lindenstrauss spaces can be sho wn to hav e strictly conv ex lattice norms in a similar w a y . 3. Strictl y convex dual norms on C ( K ) ∗ Eviden tly , Th eorem 2.8 relies on geometric a ss umptions, in the sense that only sets ha ving ( ∗ ) with slic es ar e considered. According to Example 2.9, it is not alwa ys p ossible to remov e the reliance on slices and deal instead with op en sets ha ving no sp ecial geometric prop erties . Ho w ev er, w e can liv e without slices in an imp ortant sp ecial case. W e dev ote this section to pro ving the next result. 12 Theorem 3.1. L et K b e a sc atter e d c omp a c t sp ac e. Then C ( K ) ∗ admits a strictly c onve x dual (lattic e) norm i f and only if K has ( ∗ ). Recall that any compact space K em b eds naturally into ( C ( K ) ∗ , w ∗ ) b y iden tifying p oin ts t ∈ K with their Dir ac measures δ t . It follows therefore from Theorem 3.1 that if K is scattered and ( C ( K ) ∗ , w ∗ ) has ( ∗ ) (without slices), the n ( C ( K ) ∗ , w ∗ ) has ( ∗ ) with slices. One implication of Theorem 3.1 ma y b e pro v ed easily . Prop osition 3.2. If C ( K ) ∗ admits a strictly c on v e x dual norm then K h as ( ∗ ). Pr o of. By Theorem 2.8, if C ( K ) ∗ admits a strictly con vex dua l norm then ( C ( K ) ∗ , w ∗ ) has ( ∗ ), whence K has ( ∗ ) b y the natural em b edding.  In order to pro v e the con v erse implication, we need to r efine our ( ∗ )-sequences so that they satisfy some additio nal prop erties. Assume that a top ological space X admits a ( ∗ )-sequence ( U n ) ∞ n =1 . Giv en any finite sequence o f nat ura l num b ers σ = ( n 1 , . . . , n k ), w e define the family U σ = ( k \ i =1 U i : U i ∈ U n i for all i ≤ k ) . Let us also set C n = S U n and C σ = S U σ . Lemma 3.3. Assume that F ⊆ X i s a finite subset such that for al l n , either F ∩ C n = ∅ or F ⊆ C n . Th e n ther e exists σ = ( n 1 , . . . , n k ) such that F ⊆ C σ and, mor e over, F ∩ V is at most a sing l e ton for al l V ∈ U σ . Pr o of. En umerate the set of doubletons { x, y } ⊆ F as { x 1 , y 1 } , . . . , { x k , y k } . F or ev ery i , there ex ists n i suc h that { x i , y i } ∩ C n i is non-empt y and { x i , y i } ∩ V is at most a singleton for all V ∈ U n i . By h ypothesis, w e ha v e F ⊆ C n i for all i . Put σ = ( n 1 , . . . , n k ). If x ∈ F , since F ⊆ C n i for all i , let U i ∈ U n i so that x ∈ T k i =1 U i ∈ U σ . Therefore F ⊆ C σ . Giv en V = T k i =1 V i ∈ U σ and distinct x, y ∈ F , w e hav e some i suc h that { x, y } ∩ W is at most a singleton for all W ∈ U n i . In particular, { x, y } ∩ V ⊆ { x, y } ∩ V i is at most a singleton. This prov es that F ∩ V is at most a singleton for any V ∈ U σ .  Bearing in mind the U σ , Lemma 3.3, and b y adding new singleton families if necessary , if X has ( ∗ ) then we can assume that there exists a ( ∗ )-sequence with additional prop erties, whic h w e list in the next lemma. Lemma 3 .4. If X ha s ( ∗ ) then it adm its a ( ∗ )-se quenc e ( U n ) ∞ n =1 with the fol lo w ing pr op erties. (1) X = C 1 ; (2) given n 1 , . . . , n k ∈ N , ther e exi s ts m ∈ N such that U m = ( k \ i =1 U i : U i ∈ U n i for al l i ≤ k ) ; (3) if F is a finite subset of X such that for e ach n ∈ N , either F ⊆ C n or F ∩ C n is empty, then ther e exists m ∈ N with two pr op erties: 13 (a) F ⊆ C m ; (b) F ∩ V is at most a sing l e ton for al l V ∈ U m . Armed with these enhanced ( ∗ ) - seq uences, w e can deliv er the pro of of Theorem 3.1. W e a sk that our compact spaces b e scattered b ecause the pro of relies o n the assumption that all measures in C ( K ) ∗ are atomic. Pr o of of The or em 3.1. One implication w as prov ed in Prop osition 3.2. No w assu me that K is scattered and let ( U n ) ∞ n =1 b e a ( ∗ )-sequence fo r K satisfying the prop erties of Lemma 3.4. Giv en n ≥ 1, k ≥ 0 and finite L ⊆ N , define the seminorm k µ k n,k ,L = sup ( | µ | [ i ∈ L C i ∪ [ F ! : F ⊆ U n and card F = k ) . W e sho w that these seminorms satisfy the requiremen ts of Prop osition 2.1. T o this end, supp ose that µ and ν are p ositiv e, and t hat (7) k µ k n,k ,L = k ν k n,k ,L = 1 2 k µ + ν k n,k ,L for all n , k and L . F or a con tradiction, w e shall supp ose also that µ 6 = ν . Since k µ k 1 , 0 , { n } = µ ( C n ) w e ha v e µ ( C n ) = ν ( C n ) = 1 2 ( µ + ν ) ( C n ) for all n , b y (7). By Lemma 3.4 (1) and (2), and the inclusion-exclusion principle, if I ⊆ N then we kno w that µ ( C I , n ) = ν ( C I , n ) = 1 2 ( µ + ν )( C I , n ) where C I , n = \ i ≤ n,i ∈ I C i \ [ i ≤ n,i / ∈ I C i . By monotone con v ergence, it follo ws that µ ( C I ) = ν ( C I ) = 1 2 ( µ + ν )( C I ) where C I = \ i ∈ I C i \ [ i ∈ N \ I C i . No w K is the disjoin t union of t he C I , where I ranges ov er non-empt y subsets of N , and since µ 6 = ν are atomic, we can find non-empty I ⊆ N suc h that µ ↾ I 6 = ν ↾ I . W e fix this I from no w on. T ak e a coun table set A ⊆ C I suc h that w e can write µ ↾ I = X t ∈ A a t δ t and ν ↾ I = X t ∈ A b t δ t for some n um b ers a t , b t ≥ 0. Let p = max { max { a t , b t } : t ∈ A, a t 6 = b t } q = max( { a t : a t < p } ∪ { b t : b t < p } ) and define the finite, p ossibly empt y , set F = { t ∈ A : a t = b t ≥ p } 14 and let k = card F . T a ke finite G ⊆ A suc h that (8) X t ∈ A \ G a t , X t ∈ A \ G b t < 1 4 ( p − q ) and n large enough so that (9) µ ( C I , n \ C I ) , ν ( C I , n \ C I ) < 1 4 ( p − q ) . Let H = { 1 , . . . , n } ∩ I a nd L = { 1 , . . . , n } \ I . By Lemma 3.4 (3), w e can find m ∈ N suc h that G ⊆ C m and G ∩ V is at most a singleton for all V ∈ U m . Since C I ⊆ T i ∈ H C i , w e can and do a ss ume that C m ⊆ T i ∈ H C i , b y Lemma 3.4 (2 ). It is b y considering the seminorm k · k m,k +1 ,L that w e reac h our con tradiction. Let u ∈ A suc h that a u 6 = b u and max { a u , b u } = p . Clearly u / ∈ F . Also, F ∪ { u } ⊆ G . Indeed, if t ∈ A \ G then a t , b t < 1 4 ( p − q ) < p . Without loss of generalit y , assume that a u < b u = p . Sinc e F ∪ { u } ⊆ C m , it is p ossible to find G ⊆ U m of cardinalit y k + 1, suc h that F ∪ { u } ⊆ S G . By considering k · k 1 , 0 ,L and (7), w e kno w t ha t µ  S i ∈ L C i  = ν  S i ∈ L C i  . W e shall denote this common quantit y b y c . W e estimate k ν k m,k +1 ,L ≥ ν [ i ∈ L C i ∪ [ G ! (10) ≥ ν [ i ∈ L C i ! + X t ∈ F ∪{ u } b t as ( F ∪ { u } ) ∩ [ i ∈ L C i = ∅ ≥ c + p + X t ∈ F b t = c + p + X t ∈ F a t . By (7) and the definition of the seminorms, let H ⊆ U m of cardinalit y k + 1 b e c hosen in suc h a w ay that 1 2 ( µ + ν ) [ i ∈ L C i ∪ [ H ! > k ν k m,k +1 ,L − 1 4 ( p − q ) . W e claim that a t ≥ p whenev er t ∈ S H ∩ G . In order to see this, first of all w e claim that if J ⊆ A has cardinality at most k , then (11) X t ∈ J a t ≤ X t ∈ F a t . Indeed, w e ha v e card F \ J ≥ card J \ F , since card J ≤ k = card F . If t ∈ J \ F then either a t < p or a t 6 = b t , whic h means a t ≤ p b y maximalit y of p . The refore X t ∈ F a t − X t ∈ J a t = X t ∈ F \ J a t − X t ∈ J \ F a t ≥ p (card F \ J ) − p (card J \ F ) ≥ 0 . This finishes the pro of of the claim. 15 No w w e can show that a t ≥ p whenev er t ∈ S H ∩ G . If not , then a s < p fo r some s ∈ S H ∩ G , meaning a s ≤ q . Observ e that (12) [ i ∈ L C i ∪ [ H ⊆  [ H ∩ G  ∪  [ H ∩ C I \ G  ∪ ( C I , n \ C I ) ∪ [ i ∈ L C i . T o see this, it helps to note t ha t [ H \ [ i ∈ L C i ⊆ C m \ [ i ∈ L C i ⊆ \ i ∈ H C i \ [ i ∈ L C i = C I , n . By c hoice of m , card S H ∩ G ≤ k + 1. Hence µ [ i ∈ L C i ∪ [ H ! ≤ X t ∈ F a t + a s + 1 4 ( p − q ) + 1 4 ( p − q ) + c b y (8) , (9) , (11) , (12) ≤ X t ∈ F a t + q + 1 2 ( p − q ) + c since a s ≤ q ≤ k ν k m,k +1 ,L − 1 2 ( p − q ) b y (10) . Ho w ev er, this means 1 2 ( µ + ν ) [ i ∈ L C i ∪ [ H ! ≤ 1 2 k ν k m,k +1 ,L − 1 4 ( p − q ) + 1 2 k ν k m,k +1 ,L whic h con tradicts the c hoice of H . Therefore a t ≥ p whenev er t ∈ S H ∩ G . By a similar argumen t applied to the b t , w e hav e b t ≥ p whenev er t ∈ S H ∩ G . Hence w e kno w that a t = b t for t ∈ S H ∩ G , lest w e con tradict the maximality of p . It follo ws that S H ∩ G ⊆ F . How ev er, this forces µ [ i ∈ L C i ∪ [ H ! ≤ X t ∈ F a t + 1 4 ( p − q ) + 1 4 ( p − q ) + c b y (8) , (9) and (12) < k ν k m,k +1 ,L − 1 2 ( p − q ) . Just as ab o v e, this con tradicts the ch oice o f H .  Remark 3.5. Most of Theorem 3.1 follows from Theorem 2.8 . Starting with a ( ∗ )-sequence from Lemma 3.4, we can sho w directly that ( C ( K ) ∗ , w ∗ ) has ( ∗ ) with slices. F or n, k ∈ N , finite L ⊆ N and ratio nal q > 0, define V n,k ,L,q , + to b e the family of all w ∗ -op en sets ( µ ∈ C ( K ) ∗ : µ + [ i ∈ L C i ∪ [ F ! > q ) where F ⊆ U n has cardinalit y k . Define V n,k ,L,q , − accordingly . By using essen tially the same metho d as that presen ted ab o v e, it can b e sho wn that the V n,k ,L,q , ± form a ( ∗ )-sequ ence. Moreo v er, if V ∈ V n,k ,L,q , ± then C ( K ) ∗ \ V is con vex . By the Hahn- Banac h Theorem, eac h suc h V can b e written as a union of w ∗ -op en half- space s. Therefore, w e can write do wn a ( ∗ )-sequence f o r ( C ( K ) ∗ , w ∗ ), the elemen ts o f whic h b eing families of half - spaces . What we lose here is the fact that the norm in Theorem 3.1 is a lattice norm, whic h is why w e giv e the pro of as is. 16 4. Topological proper tie s of ( ∗ ) and example s In this section, w e e xplore the prop erties o f ( ∗ ) and see how it compares with related concepts in the lit erature. In particular, unde r the con tin uum h yp othesis (CH), w e pro vide an example of a compact scattered non-Gruenhage space K ha ving ( ∗ ). This means that Theorem 3 .1 do es not follo w f r om existing r esults suc h as Theorem 1.3. A top ological space X is said to ha v e a G δ -diagonal if its diagonal { ( x, x ) : x ∈ X } is a G δ set in X 2 . This c oncept has b een studied extensiv ely in general metrisation theory; s ee, for example [12, Sec tion 2]. It is easy to sho w that X has a G δ -diagonal if and only if the admits a sequence ( G n ) ∞ n =1 of o p en co v ers of X , suc h that give n x, y ∈ X , there exists n with the property that { x, y } ∩ U is at m ost a s ingleton for all U ∈ G n [12, Theorem 2.2]. Equ iv alently , if we conside r the ‘stars’ st( x, n ) = [ { U ∈ G n : x ∈ U } , then T ∞ n =1 st( x, n ) = { x } for eve ry x ∈ X . In k eeping with previous notatio n, we call suc h a sequence a G δ -diagonal se quenc e . Compact space s with G δ -diagonals are metrisable (cf. [12, Theorem 2.13]), so ( ∗ ) is eviden tly a strict generalisation of the G δ -diagonal prop ert y . In some cases, it is p ossible to reduce problems ab out ( ∗ ) to the G δ -diagonal case; see Theorem 4 .3 and Prop osition 4.12, and also the partitioning of K in to the C I in the pro of of Theorem 3.1. Next, w e compare ( ∗ ) with G ruenhage’s prop ert y . Prop osition 4.1. If X is Gruenhage then it has ( ∗ ). Pr o of. If X is Gruenhage then let ( U n ) ∞ n =1 and R n b e as in Definition 1.2. Let V n = { R n } fo r eac h n . Give n distinct x, y ∈ X , there exists n and U ∈ U n , suc h that { x, y } ∩ U is a singleton. If x ∈ R n then y / ∈ R n and it is true that { x, y } ∩ U = { x } for ev ery U ∈ V n , b ecause V n is a singleton. Lik ewise if y ∈ R n . So w e assume no w that x, y / ∈ R n . No w it is true that { x, y } ∩ V is at most a singleton for ev ery V ∈ U n , since if y ∈ V then V 6 = U , and if x ∈ V then x ∈ U ∩ V = R n .  There are an abundance of compact spaces which are Gruenhage, but non-descriptiv e and so quite far from b eing metrisable; see [3 4 , Corolla ry 17] or Theorem 4.6 and subseque n t remarks, b elo w. In Ex ample 4 .10, w e show that under CH there exists a compact, scattered non-Gruenhage space that has ( ∗ ). No w w e see that ( ∗ ) implies fragmen tabilit y . Prop osition 4.2. If X has ( ∗ ) then X is fr agmen tabl e . Pr o of. Let X ha v e a ( ∗ )- se quence ( U n ) ∞ n =1 . W e w ell order eac h U n as ( U n ξ ) ξ <λ n . No w define V n α = S ξ ≤ α U n ξ , for α < λ n . W e claim that given distinct x, y ∈ X , there exists n and α < λ n suc h that { x, y } ∩ V n α is a singleton. As explained in t he In tro duction, this is enough to g ive fragmen tability . Indeed, tak e n ∈ N with the 17 prop erties giv en in Definition 2.6, and pic k the least α < λ n suc h that { x, y } ∩ U n α is a singleton. Then { x, y } ∩ U n ξ m ust b e empt y for all ξ < α , thus { x, y } ∩ V n α = { x, y } ∩ U n α is a singleton.  Theorem 4.3 b elo w is a generalisation of a result of Chab er ( cf. [12, Theorem 2.14]), whic h states that coun tably compact spaces with G δ -diagonals are compact (and th us metrisable). It allo ws us to glean a few more top olog ical consequences of the ( ∗ ) prop erty . As preparation, fix an op en co v er V of a coun tably compact (non-empt y) space X . Supp ose that X has a ( ∗ )- seq uence ( U n ) ∞ n =1 , with C n = S U n for eac h n . Define A X = ( I ⊆ N : X \ [ n ∈ I C n 6 = ∅ ) . Clearly , A X is a hereditary family of subsets of N . Moreo v er, it is compact in t he p oin t wise top ology . Indeed, if J / ∈ A X , t hen b y the coun table compactness of X , w e can find finite G ⊆ J suc h that G / ∈ A X . It follows that P ( N ) \ A X is op en. F urthermore, ∅ ∈ A X b ecause X is non-empt y , so A X is also non-empt y . F rom these facts, we deduce that A X admits an elemen t that is maximal with resp ect to inclusion. Theorem 4.3. If X is c ountably c om p act and has ( ∗ ) then X is c omp act. Pr o of. Fix an op en co v er V of X and ( ∗ )-sequence ( U n ) ∞ n =1 as ab o v e. W e define a decreasing transfinite sequence of coun tably compact subspaces X α of X , together with maximal M α ∈ A X α and finite F α ⊆ V , suc h that (1) X α = X \ S ξ <α S F ξ ; (2) M ξ / ∈ A X α whenev er ξ < α . T o b egin, set X 0 = X . G iv en X α , w e take some maximal M α ∈ A X α and set Y = X α \ S n ∈ M α C n . W e claim that ( U n ) n ∈ N \ M α is a G δ -diagonal sequence for Y . Indeed, the maximalit y of M α implies that Y ⊆ C n whenev er n ∈ N \ M α . If x, y ∈ Y then b y ( ∗ ), there exists n s uc h that { x, y } ∩ C n is non-empt y , and { x, y } ∩ U is at most a singleton for all U ∈ U n . By definition, Y ∩ C k is empt y whenev er k ∈ M α , so necessarily n ∈ N \ M α . Our claim is pro v ed. By Chab er’s result, Y is compact. Therefore there exists a finite set F α ⊆ V , suc h that X α \ [ n ∈ M α C n = Y ⊆ [ F α . Define X α +1 = X ′ α = X α \ S F α . W e hav e (1) imm ediately and (2) follows because M α / ∈ A X α +1 and A X α +1 ⊆ A X α . If X α +1 is empt y then w e stop the recursion. If λ is a countable limit ordinal and X α is non-empt y for all α < λ , set X λ = T α<λ X α . (1) and (2) follow. By coun table compactness, X λ is also non-empt y . This pro cess has to stop at a coun table (succe ssor) stage, b ecause ( A X α ) is a strictly decreasing family of closed subsets of the separable metric space P ( N ). Th us, 18 X α +1 is empt y for some α < ω 1 . By (1 ) , we get X ⊆ [ ξ ≤ α [ F ξ and so X is co v ered b y S ξ ≤ α F ξ . By a final application of countable compactness, w e extract from this a finite sub co v er.  The next result generalises [26, Corolla ry 4.3 ] from descriptiv e spaces t o spaces with ( ∗ ). Corollary 4.4. If L is lo c al ly c omp act and has ( ∗ ) then L ∪ {∞} is c ountably tight and se quential l y close d subsets of L ∪ {∞} ar e close d. Pr o of. The first assertion follow s directly from Theorem 4.3 and the second follows from Prop osition 4.2 and the fact that compact fragmen table spaces ar e sequen tially compact (see [29, C orollary 2 .7] a nd [10, Lemma 2.1.1]). Notice that if L is any lo cally compact space with ( ∗ ) then its 1- point compactification L ∪ {∞} ha s ( ∗ ) also. All w e need t o do is adjoin to any ( ∗ ) - seq uence f or L the sin gleton family { L } , whic h separates all p oin ts in L from ∞ .  Concerning stabilit y pro perties of ( ∗ ) under mappings, w e hav e t he nex t result. Prop osition 4.5. If K is a sc atter e d c om p act sp ac e with ( ∗ ) and π : K − → M is a c ontinuous, surje ctive map then M has ( ∗ ). Pr o of. If K has ( ∗ ) then by Theorem 3.1, C ( K ) ∗ admits a strictly conv ex dual norm k · k . If w e define T : C ( M ) − → C ( K ) b y T ( f ) = f ◦ π , it is standard to c hec k that | | | ν | | | = inf {k µ k : T ∗ ( µ ) = ν } defines a strictly con v ex dual norm on C ( M ) ∗ . Therefore M has ( ∗ ), again by Theorem 3.1.  The proo f ab o v e is concise and straightforw a rd, but also utterly opaque, as it lea v es the reader with no idea of how to construct a ( ∗ ) - seq uence on M in terms of a ( ∗ ) - seq uence on K . W e o utline a second approach to pro ving Prop osition 4.5 , whic h w e include b ecaus e w e believe it giv es the reader more idea of what is going on. The dual map S = T ∗ ab o v e is a natural extension of π if w e iden tify p oints in K a nd M with their Dirac measures in C ( K ) ∗ and C ( M ) ∗ , respectiv ely . Set Σ = { µ ∈ C ( K ) ∗ : µ is p ositiv e and k µ k 1 = 1 } . If t ∈ M and µ ∈ Σ then S ( µ ) = t if and only if supp µ ⊆ π − 1 ( t ). Giv en a ( ∗ )- sequence ( U n ) ∞ n =1 on K with t he prop erties of Lemma 3.4, to gether with the unions C n , define the w ∗ -compact and con v ex sets D n,q ,L = ( µ ∈ Σ : µ [ i ∈ L C i ∪ U ! ≤ q for all U ∈ U n ) where n ∈ N , q ∈ (0 , 1) ∩ Q and L ⊆ N is finite. The D n,q ,L should b e compared to the seminorms k · k n,k ,L in the pro of of Theorem 3.1. Giv en distinct s, t ∈ M and µ, ν ∈ Σ in S − 1 ( s ) and S − 1 ( t ) respectiv ely , by follo wing the pro of of Theorem 3.1, 19 w e can find n and q and L suc h that 1 2 ( µ + ν ) ∈ D n,q ,L , but { µ , ν } ∩ D n,q ,L is at most a s ingleton. There is less to consider in this case b ecause as the supports of µ and ν are neces sarily disjoint, the set F in the proo f of Theorem 3.1 is empt y . This is wh y w e only need to consider individual elemen ts of U n in the definition o f the D n,q ,L , rather than finite subsets o f U n as in the definition of the k · k n,k ,L . By app ealing to compac tness and con vex ity , it is p ossible to s elect a finite set G of triples ( n, q , L ) with the prop ert y that if w e conside r the in tersection D G = S ( n,q ,L ) ∈ G D n,q ,L , then D G ∩ S − 1 ( 1 2 ( s + t )) is non-empty , but either D G ∩ S − 1 ( s ) is empt y , or D G ∩ S − 1 ( t ) is empt y . Equiv alen tly , 1 2 ( s + t ) ∈ S ( D G ), but { s, t } ∩ S ( D G ) is at most a singleton. T he set S ( D G ) is w ∗ -compact and conv ex, so the complemen t C ( M ) ∗ \ S ( D G ) can b e written as the union of a family V G of w ∗ -op en halfspaces of C ( M ) ∗ . F rom what we kno w, it can b e easily v erified that the families V G , as G ranges o v er all finite subsets of triples ( n, q , L ), induce a ( ∗ )-sequence on M . No w w e mo v e on to examples. W e are c hiefly in tereste d in exploring ( ∗ ) , G r u- enhage’s pro perty and the gap betw een them. Giv en that descriptiv e space s are Gruenhage and spaces with ( ∗ ) are fragmen table, w e shall confine our atten tion t o spaces that are fragmentable but non-descriptiv e. The first thing to p oin t o ut is that ( ∗ ) is not equiv alen t to fragmen tabilit y , b e- cause ω 1 is scattered (hence fragmentable), but do es not ha v e ( ∗ ). That ω 1 do es not ha v e ( ∗ ) is clear, either directly from Corollar y 4.4, or fro m Theorem 3.1 and [38, Th ´ eor ` eme 3], whic h we men tioned in the In tro duction. An y lo cally compact space having ( ∗ ) necessarily has a coun tably tight 1-p oin t compactification, but this condition is not sufficie n t. Hereafter, all of our examples of lo cally compact spaces without ( ∗ ) ha v e countably tigh t 1-p oin t compactifications. Next, w e consider trees. A tr e e ( T , ≤ ) is a partially ordered set with the prop ert y that giv en any t ∈ T , its set of predecessors { s ∈ T : s ≤ t } is we ll ordered. The tree order induces a natural lo cally compact, scattered interval top olo gy . T o render this top ology Hausdorff, w e sh all only consider t r ee s T with t he property tha t eve ry non-empt y totally ordered subset of T has at most one minimal upp er b ound. An an tic hain is a subset of T , no tw o distinct elemen ts of whic h ar e comparable. F or further definitions and discussions ab out trees, and their ro le in renorming theory , w e refer the reader to [15, 16, 33 , 34, 36, 39]. If P and Q a re partially ordered sets then w e sa y tha t a map ρ : P − → Q is strictly i n cr e asing if ρ ( x ) < ρ ( y ) whenev er x < y . If s uc h a map exists then w e write P 4 Q . In [33, D efinition 5], the second-named author in tro duced a to tally ordered set Y to a ddres s the problem of when C 0 ( T ) ∗ admits a strictly conv ex dual norm. W e remark of Y that R 4 Y , Y α 4 Y for all α < ω 1 , where Y α is ordered lexicographically , and finally Y con tains no uncountable, w ell ordered subsets [3 3, Section 4]. By com bining Theorem 3.1 with [34, Corollary 17], w e obtain the next result. See also [36, Theorem 26]. Theorem 4.6. If T is a tr e e then the fol lowing ar e e quivale n t. (1) T is Gruenhage; (2) T has ( ∗ ); (3) C 0 ( T ) ∗ admits a strictly c on v e x dual norm; 20 (4) T 4 Y . Note that t he 1-p oint compactification T ∪ {∞} of a tree T is coun ta bly t ig h t if and only if T admits no uncoun table branche s. Indeed, suppo se that T admits no uncoun table branc hes. Since eac h t ∈ T admits a coun table neigh b ourho o d, the only p oin t w e need to test is ∞ . If ∞ ∈ A for some uncoun t a ble A ⊆ T , then by a standard resu lt of Ramsey theory , either A con tains an unc ountable tot a lly ordered set or a coun tably infinite a n tichain E . O nly the second p ossibilit y is v alid, whenc e ∞ ∈ E . The conv erse implication follows immediately from the fact that ω 1 + 1 is not countably tight. Th us we r estrict our atten tion to trees with no uncountable branc hes. Giv en a partially o r dered set P , w e set σ P = { A ⊆ P : A is w ell- ordered } . Kurepa intro duced this notion a nd pro v ed the follo wing fact: for all P , w e ha v e σ P 6 4 P . On the other hand, it is straigh tforw ard to sho w t ha t σ R α 4 R α × { 0 , 1 } [33, Prop osition 23]. Moreov er, it is kno wn that T is descriptiv e if and only if T 4 Q [33, Theorem 4]. Therefore, w e conclude that σ Q and σ R α , α < ω 1 , are all Gruenhage, non-descriptiv e spaces (see [3 4, p. 752] or [36 , p. 405]). Instead, if w e consider an y to t a l order W satisfying Y 4 W , then σ W 6 4 Y and so σ W do es not ha v e ( ∗ ) . In addition, if W do es n’t con tain any uncoun table w ell ordered subsets, then σ W is free of uncoun table bra nc hes. There is another t yp e of tree without uncoun table branc hes and without ( ∗ ). A subset E of a tree is a fin al p art if u ∈ E whenev er t ∈ E and t ≤ u . If E is a final pa rt then w e sa y that E is dense if ev ery elemen t of T is comparable with some elemen t o f E , and T is called Bai r e if ev ery coun table in tersection of dense final parts (whic h is itself a final part) is again dense. A subset E is called ev er-br anching if, giv en an y t ∈ E , there exist incomparable elemen ts u , v ∈ E satisfying t < u, v . If T admits an ev er-branc hing Baire subtree then C 0 ( T ) do es not admit a G ˆ ateaux norm [15, Theorem 2.1]. Therefore, no suc h tree can hav e ( ∗ ). An ev er-branc hing Ba ir e tree without uncoun table branc hes exists; see [39 , Lemma 9.12] and [15, Prop osition 3.1]. Recall that a tree T is called Suslin if it con tains no uncoun table branc hes or an tic hains. The existence of Suslin trees is indep enden t of Z FC; see, for example, [39, Section 6]. Ev ery Suslin t r ee con tains an ev er-branch ing Ba ire subtree [3 9, p. 246], so w e conclude that no Suslin tree has ( ∗ ) either. It is clear from T heorem 4.6 that in order to find ex amples of non-Gruenhage spaces with ( ∗ ), w e mus t searc h further afield. A top ological space X is said to b e her e ditarily sep ar a ble (HS) if ev ery subspace of X is separable. Clearly , the 1-p oin t compactification of a lo cally compact HS space is countably tight. Thes e spaces are in teresting f o r us b ecause if K is compact, HS and non-metrisable, then it is automatically non-descriptiv e. This fact is stated in [26, Propo sition 4.2] but no direct pro of is giv en, so an argumen t is sk etc hed here for completeness. If H is an isolated family of subsets of K then H mus t b e coun table, b ecause by hereditary separabilit y there is a coun table subset of S H whic h meets ev ery mem ber o f H . 21 Therefore, if K is a descriptiv e compact HS space then it admits a coun table net w ork, whence it is metrisable. Since w e w ant compact, non- metris able HS spaces that are also fragmentable, it is ne cessary to assume extra axioms. A space X is her e ditarily Lindel¨ o f (HL) if ev ery s ubspace of X is Lindel¨ of. If K is compact, f ragmen ta ble and HL then it is m etrisable (cf. [20, Corollary 9]). Th us, we w an t HS spaces that are not HL; suc h ob jects are called S-sp ac es . W e refer the reader to [31] f or an introduction to S -spaces and also the related L -spaces. It is k now n tha t unde r MA + ¬ CH (where MA stands fo r Martin’s axiom), there are no compact S -spaces (cf. [31, Theorem 6.4.1]), and in fact it is consisten t that there a re no S -spaces at all (cf. [31, Theorem 7.2.1]). Therefore, w e must assume extra a xioms if w e are to find an y animals in this particular zo o. Our treat ment of S -spaces pro ceeds a s follo ws. F irst, w e outline tw o approach es for constructing S -spaces by refining existing top ologies, and show that these yield Gruenhage spaces. Second, w e giv e an example under CH of a compact non- Gruenhage space of cardinalit y ℵ 1 with ( ∗ ) and sho w that, given a further mild assumption, no ob ject of this kind can exist under M A + ¬ CH. Finally , w e pre sen t a third metho d of c onstructing S -spaces and s how that no suc h s pace can ha v e ( ∗ ). The spaces dev eloped using the first approach are sometimes called ‘Kunen lines’, despite the fact that none of them are linearly ordered. Assumin g CH, the authors of [1 8] dev elop a mac hine whic h accepts as input a first countable HS space ( X , ρ ) of cardinalit y ℵ 1 , and generates a finer top ology ( X, τ ) whic h is lo cally compact, scattered, HS and non- Lindel¨ of. In applications, X is usually a subset of R and ρ is the induced metric top ology . Later, this process w as dev elop ed to ensure that ( X , τ ) n is HS for all n ∈ N ; [23, Section 7]. The resulting 1-p oin t compactification K is kno wn to Banach space theorists a s ‘Kunen’s compact space’. It is not explicitly stated in [23, Section 7] that the resulting top ology on X refines that of the real line, but the authors b eliev e that it is mean t to. If the top ology is suc h a refinemen t then neces sarily the Euclidean diameters of the B α k (whic h form the building blo c ks of neigh b ourho o ds of p oin ts, see (8) on [23, p. 1124 ]) hav e to tend to 0 as k → ∞ . It can b e c hec ke d that t his condition is also sufficien t to pro duce a refinemen t. W e not e further that an alternativ e a pproac h to [23, Section 7 ] is giv en in [7, Theorem 2.4], a nd there, the fact that the origina l t opolo gy is refined is explicitly stated. Of course, it is clear that an y refinemen t of a Gruenhage space is a gain Gruen- hage, b ecause w e can use exactly the same op en sets to separate points. Therefore, assuming the adjustment to the diameters of the B α k ab o v e, w e hav e the follow ing result. Prop osition 4.7. Th e Kunen lines ar e Gruenhage sp ac es. I n p articular, C ( K ) ∗ admits a strictly c onvex dual norm, the pr e dual of whi c h is ne c essarily Gˆ ate aux smo oth. The second approac h r efine s top ologies as ab o v e, but this time using the axiom b = ℵ 1 , where b is the minimal cardinality of a subset of N N whic h is un b ounded with resp ect to the or dering of ev en tual dominance. Under b = ℵ 1 , it is sho wn in 22 [40, Theorem 2.5] that the top ology o f an y set of reals of c ardinality ℵ 1 ma y b e refined to g iv e a lo cally compact, scattered , non-Lindel¨ of top ology whic h is HS in its finite p o w ers. Prop osition 4.8. The sp a c es of T o dor ˇ cevi´ c in [40, Theorem 2.5] ar e Gruenhage. Before presen ting our third approac h to construct S - spaces, w e giv e our example under CH of a compact, scattered non-Gruenhage space with ( ∗ ). W e shall a dopt the same basic approac h as [1 8] and [23 , Section 7], and use an idea from [1]. Ho w ever, the underlying motiv ation for the space should b e compared, at a distance, to the split in terv al, rather than the real line. In fact, w e construct a lo cally compact, scattered non-G ruenhage space with a G δ -diagonal. The 1-p oin t compactification of this space has ( ∗ ). F or o ur example, w e shall ma ke use of the follow ing observ ation ab out Gruenhage spaces o f cardinalit y no larger than the con tin uum. Prop osition 4.9. [36, Prop osition 2 ] L et X b e a top olo gic al sp ac e with card X ≤ c . Then X is Gruenhage if and o n ly if ther e is a se quenc e ( U n ) ∞ n =1 of op en subsets o f X wi th the pr op erty that if x, y ∈ X , then { x, y } ∩ U n is a single ton for some n . In that whic h fo llo ws, diam denotes Euclidean diameter. Example 4.10. (CH) There exists a lo cally compact, scattered, first coun table Hausdorff, non-Gruenhage space with a G δ -diagonal. Pr o of. Let ( x α ) α<ω 1 b e a set of distinct p oin ts in [0 , 1]. Define Y α = { x ξ : ξ < α } and X α = Y α × {± 1 } fo r α ≤ ω 1 , with Y = Y ω 1 and X = X ω 1 . Assuming CH, let ( A α ) α<ω 1 b e a n en umeration of all the countable subsets of Y . Let t : X − → X b e the map t ( x, i ) = ( x, − i ), and let q : X − → Y be the natural pro jection. W e obtain our top ology on X b y building increasing top ologies τ α on the X α , α < ω 1 , b y tra nsfin ite induction. The p oin ts ( x α , i ), i = ± 1, will hav e a coun table base of compact op en neigh b ourho ods U ( x α , i, n ), n ∈ N , suc h that (1) if β < α then X β is op en in τ α and τ β is the top ology on X β induced b y τ α ; (2) U ( x α , i, n ) \ { ( x α , i ) } ⊆ X α ; (3) diam ( q ( U ( x α , i, n ))) < 2 − n ; (4) U ( x α , − i, n ) = t ( U ( x α , i, n )); (5) q ↾ U ( x α ,i,n ) is injectiv e; (6) if ξ ≤ α , A ξ ⊆ Y α and x α ∈ A ξ R , then U ( x α , i, n ) ∩ ( A ξ × { − i } ) is non-empty for ev ery n . T o tak e care of limit stages α , w e set τ α = { U ⊆ X α : U ∩ X β ∈ τ β for all β < α } . No w ass ume that τ α has b een found. W e construct τ α +1 b y constructing neigh b our- ho o ds U ( x α , i, n ), n ∈ N , of the p oin ts ( x α , i ), i = ± 1. 23 If x α / ∈ Y α R then set U ( x α , i, n ) = { ( x α , i ) } for i = ± 1 and n ∈ N . Note that as Y is a separable subset of R , t his c an happ en for only countably many α . No w assume that x α ∈ Y α R . Define F α = n ξ ≤ α : A ξ ⊆ Y α and x α ∈ A ξ R o . Since F α is a t most countable, we can find an injectiv e sequence ( s n ) ∞ n =1 ⊆ Y α con v erging to x α , suc h that (i) diam ( { s m : m ≥ n } ) < 2 − n for eac h n (ii) { n ∈ N : s n ∈ A ξ } is infinite whenev er ξ ∈ F α . By considering (3) applied to β < α , and (i) ab o v e, for ev ery n w e can find k n suc h that (iii) q ( U ( s n , − 1 , k n )) ∩ q ( U ( s m , − 1 , k m )) = ∅ whenev er n 6 = m , and (iv) diam ( q  S m ≥ n U ( s m , − 1 , k m )  ) < 2 − n for ev ery n . Finally , define U ( x α , i, n ) = { ( x α , i ) } ∪ [ m ≥ n U ( s m , − i, k m ) . These neigh b ourho o ds are compact and op en. Extend τ α to τ α +1 in the ob vious wa y . It is clear that w e ha v e (1) and (2) , and then τ α +1 is lo cally compact. (3) f o llo ws from (iv) ab ov e. That τ α +1 is Hausdorff follows b y inductiv e h ypo thes is, (3), and the fact that U ( x α , 1 , n ) ∩ U ( x α , − 1 , m ) = ∅ . (4) and (5) follo w from the inductiv e h yp othesis , the definition of U ( x α , i, n ) and (iii) ab o v e. T o see (6), note that { s m : m ≥ n } × {− i } ⊆ U ( x α , i, n ) ∩ ( A ξ × { − i } ) so (6) no w follow s from (ii) ab o v e. This completes the induction. The to p olog y on X is giv en b y { U ⊆ X : U ∩ X α ∈ τ α for all α < ω 1 } W e sho w t ha t X is scattered. If E ⊆ X is non-empt y then let α be minimal, suc h that E ∩ { ( x α , ± 1) } is non- empt y . If ( x α , i ) ∈ E then by (1) and ( 2), w e hav e that U = X α ∪ { ( x α , i ) } is op en, and E ∩ U = { ( x α , i ) } . No w w e sho w that X has a G δ -diagonal. S et G n = { U ( x, i, n ) : ( x, i ) ∈ X } . Let ( x, i ) , ( y , j ) ∈ X . If x 6 = y t hen pic k n suc h that | x − y | ≥ 2 − n . W e cannot hav e ( y , j ) ∈ st(( x, i ) , n ) b ecause if so we w ould hav e ( x, i ) , ( y , j ) ∈ U ( z , k , n ) for some ( z , k ), giving | x − y | ≤ diam ( q ( U ( z , k , n ))) < 2 − n b y ( 3 ). If x = y and i 6 = j then b y (5), w e cannot ha v e ( x, i ) , ( y , j ) ∈ U ( z , k , n ) for an y ( z , k ) or n . Whatev er the case, ∞ \ n =1 st(( x, i ) , n ) = { ( x, i ) } . 24 This sho ws that ( G n ) ∞ n =1 is a G δ -diagonal sequenc e. Finally , w e pro v e that X is not Gruenhage. Bearing in mind Prop osition 4.9, w e supp ose for a con tradiction tha t there exists a seq uence of op en subs ets ( V n ) ∞ n =1 , with the prop ert y that giv en ( x, i ) , ( y , j ) ∈ X , w e can find n such that { ( x, i ) , ( y , j ) } ∩ V n is a singleton. Define J n,i = { x ∈ Y : ( x, i ) ∈ V n and ( x, − i ) / ∈ V n } . By assumption, Y = S n,i J n,i , so there exist n and i suc h that J = J n,i is uncount- able. Remem bering that R is HS, we can find a countable subse t A ξ suc h that A ξ ⊆ J ⊆ A ξ R . Because J is uncountable, w e can pic k α ≥ ξ suc h that A ξ ⊆ Y α and x α ∈ J ⊆ A ξ R . Since x α ∈ J , we ha ve ( x α , i ) ∈ V n , so tak e m s uc h that U ( x α , i, m ) ⊆ V n . F rom (6), w e kno w that U ( x α , i, m ) ∩ ( A ξ × {− i } ) ⊆ V n ∩ ( J × {− i } ) is non-empt y . How ev er, this violates the definition of J . This con tradiction estab- lishes that X is not Gruenhage.  T ogether with Theorem 3.1, this example sho ws that if C ( K ) ∗ admits a strictly con v ex dual norm then K is not necessarily Gruenhage. This giv es a cons isten t negativ e solution to [34 , Problem 14] and [36, Problem 4]. W e remark that the example ab o v e need not b e HS. Ho w ev er, it can easily b e made to b e HS b y c hanging (ii) ab ov e to r ead (ii) { n ∈ N : s 2 n , s 2 n +1 ∈ A ξ } is infinite whenev er ξ ∈ F α and setting U ( x α , i, n ) = { ( x α , i ) } ∪ [ m ≥ n U ( s m , ( − 1) m i, k m ) . T o see that this mak es X HS, w e let E ⊆ X and set E i = { x ∈ Y : ( x, i ) ∈ E } , i = ± 1. Then take ξ i < ω 1 suc h that A ξ i ⊆ E i ⊆ A ξ i R and c ho ose α ≥ ξ 1 , ξ − 1 large enough to satisfy A ξ 1 ∪ A ξ − 1 ⊆ Y α . It can now b e v erified that E is in the closure of E ∩ X α . There is no hop e of c onstructing s omething lik e Example 4.10 in ZFC . A space X is called lo c al ly c ountable if ev ery p oin t of X admits a coun table neighbourho o d. F or example, trees of heigh t at most ω 1 and ‘t hin-tall’ lo cally compact spaces are lo cally coun table. It is s traightforw ard to see that a lo cally compact, lo cally coun table space m ust b e scattered. Prop osition 4.11 (MA + ¬ CH) . S upp ose that L is a lo c al ly c o m p act, lo c al ly c ount- able sp ac e with ( ∗ ) a nd card L < c . Then L is σ -discr ete. Pr o of. This follo ws immediately fro m Coro llary 4.4 and [3, Theorem 2.1].  It is not p ossible use stronger a xioms to extend Prop osition 4.1 1 to include spaces of cardinality c : the tree σ Q is lo cally compact, lo cally coun table and Gruenhage, but is not σ -discrete. 25 W e end this section b y presen ting our third class of S - space s. W e s hall call a regular, uncoun table to polo g ical space X an O -sp a c e if ev ery op en subset of X is either coun table or co-coun table. Ostaszewski constructed a lo cally compact, scattered O -space using the clubsuit axiom ♣ [27, p. 506]. It is kno wn that ♣ is indep ende n t of CH and that ♣ + CH is equiv alen t to Jensen’s axiom ♦ (see [32] and [27, p. 506], resp ectiv ely). It is p ossible to obtain O -spaces b y assumin g principles strictly w eak er t ha n ♣ [19, Theorem 2.1]. Unlik e the previous constructions, these spaces are built from scratc h, r a ther than b y refining an initial space. Ev ery O -space contains an S - sub space. Indeed, if X is an O -space then notice that at most one p oin t of X can f a il to ha v e a countable op en neigh b ourho o d. Th us w e can construct b y induction an uncoun table subspace Y = { x α : α < ω 1 } suc h that { x ξ : ξ < α } is op en in Y for ev ery α < ω 1 . Th us Y is not Lindel¨ of. If, for a con tradiction, we supp ose that Z ⊆ Y is not separable, then by a no ther induction w e can construct an uncountable, relativ ely discrete subspace of Y . Ho w ev er, this cannot exis t by the O -space prop ert y . Therefore Y is an S -space. W e can ar g ue similarly to establish that ev ery lo cally compact O -space has a countably tight 1- p oin t compactification. Prop osition 4.12. If X is an O -sp ac e then it d o es not have ( ∗ ). Pr o of. Suppo se that ( U n ) ∞ n =1 is a ( ∗ )- seq uence fo r X , with C n = S U n for eac h n . Set J = { n ∈ I : C n is uncoun table } . If n ∈ J then X \ C n is coun table, so E = [ n ∈ J ( X \ C n ) ∪ [ n ∈ N \ J C n is a lso coun table. If w e let A = X \ E then w e see that A ⊆ C n for all n ∈ J , and A ∩ C n is empt y whenev er n / ∈ J . F or x ∈ A and n ∈ J , define st( x, n ) = [ { U ∈ U n : x ∈ U } . Since ( U n ) ∞ n =1 is assumed to b e a ( ∗ )-sequence for X , w e ha v e { x } = A ∩ \ n ∈ J st( x, n ) for all x ∈ A , i.e. ( U n ) ∞ n =1 induces a G δ -diagonal s equence on A . Giv en this, it follo ws that for each x ∈ A , there exists some n x ∈ J suc h that st( x, n x ) is coun table. Indeed, otherwise, E ∪ [ n ∈ J ( X \ st( x, n )) is coun table, giving { x } = A ∩ \ n ∈ J st( x, n ) 26 uncoun table. Since A is uncoun table, there exists n , which we fix fro m no w on, suc h that B = { x ∈ A : n x = n } is uncoun table. T ake an en umeration ( x α ) α<ω 1 of distinct p oin ts in B . W e find α 0 < α 1 < α 2 < . . . < ω 1 suc h that x α η / ∈ [ ξ <η st( x α ξ , n ) for a ll η < ω 1 . Observ e tha t b y the sym metry o f the sets st( x, n ), w e ha v e x α ξ / ∈ st( x α η , n ) whenev er ξ 6 = η . Therefore C =  x α ξ : ξ < ω 1  is a relative ly discrete subspace, whic h is not p ermitted by the O -space property .  Example 4.13. Ostaszew ski’s space [27, p. 5 06] is a lo cally compact, scattered HS O -space. Therefore, it do es not ha ve ( ∗ ). By refining Ostaszewski’s construction, it is p ossible to use ♣ to build a compact, scattered non-metrisable space K , such that K n is HS for all n [14, Theorem 4.36]. Moreo v er, it can b e che c k ed that this K is, in addition, an O -space. Therefore, unlik e C ( K ) ∗ , the space C ( K ) ∗ admits no strictly conv ex dual norm. W e mak e a remark ab out this C ( K ): the authors don’t know if it admits a Gˆ ateaux norm. Since K is separable, C ( K ) admits a b o unded linear, injectiv e map into c 0 . The authors don’t kno w of an y example of an Asplund space with an injectiv e map in to a c 0 (Γ), whic h do es not admit a Gˆ ateaux norm. 5. Problems T o finish, w e presen t a n um b er of related, unresolv ed problems. The first problem stems from Theorem 3.1. Problem 5.1. If K h as ( ∗ ) and is no t sc atter e d, then do es C ( K ) ∗ admit a strictly c onve x dual norm? In fact, w e don’t eve n kno w if C ( L ∪ {∞} ) ∗ admits a strictly conv ex norm when- ev er L is a lo cally compact space ha ving a G δ -diagonal. The next problem is prompted b y Example 4.1 0 . Problem 5.2. Is ther e in ZFC an example of a non-Gruenhage c om p act sp ac e with ( ∗ )? Prop osition 4.5 suggests the next problem. Problem 5.3. If K has ( ∗ ) and is not sc atter e d, and π : K − → M is a c ontinuous, surje ctive m ap, then do es M have ( ∗ )? Mor e gener al ly, if a top olo gic al sp ac e X has ( ∗ ) a nd f : X − → Y is a p e rf e ct, surje ctive map, do es Y have ( ∗ )? It is kno wn that the a nsw er to Problem 5.3 is po sitive in the Gruenhage case, including the more general p erfect ma p ass ertion [34, Theorem 23]. It is also k no wn that G δ -diagonals are not preserv ed under p erfect images. In [4, Example 2], an example is g iv en of a lo cally compact s cattered space L having a G δ -diagonal, and a p erfect surjectiv e map f : L − → M , where the diagonal of M is not a G δ . How ev er, L (2) is empt y , and the s ame will apply to an y p erfect image of L , so all suc h images 27 are σ - discrete and therefore ha v e ( ∗ ). If Problem 5.1 has a positiv e solution then so will the first part of Problem 5.3 , simply b y cop ying the pro of of Prop osition 4.5. F or our last problem, w e refer the reader to the end of Section 4. Problem 5.4. 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