Countable Choice and Compactness

COUNT ABLE CHOICE AN D COMP A CTNESS MARIANNE MORILLON Abstract. W e work in set-theory without c hoice ZF . Denoting by A C ( N ) the count able axiom o f choice, we show in ZF + A C ( N ) that the closed unit ball of a uniformly con vex Banac h space is compact in the conv ex top o lo gy (an a lternative to the weak top ol- ogy in ZF ). W e prove t hat this ball is (closely) con vex-compact in the conv ex top ology . Given a set I , a real num b er p ≥ 1 ( r e sp. p = 0), and some closed subset F o f [0 , 1] I which is a b ounded subset of ℓ p ( I ), we show that A C ( N ) ( r esp. DC , the a xiom of Dependent Choices ) implies the compactness o f F . ERMIT EQUIPE R ´ EUNIONNAISE DE MA TH ´ EMA TIQUES ET INF ORMA TIQUE TH ´ EORIQUE (ERMIT) h ttp://lab oratoires.univ-reunion.fr/ermit Contents 1. In tro duction 2 1.1. Presen tation of the results 2 1.2. Some we ak forms of A C 3 2. A criterion o f compactness 4 2.1. Filters 4 2.2. Complete metric spaces 5 2.3. A criterion of compactness in ZF + AC ( N ) 5 3. The con ve x to p ology on a normed space 7 Date : Nov ember 13, 2018 . 2000 Mathematics S ubje ct Classific ation. Pr imary 03E25 ; Seco ndary 46B26, 54D30. Key wor ds and phr ases. Bana ch space, weak compactness, Hahn-Banach, uni- formly conv ex, Axiom of Choice. 1 2 M. MORIL LON 3.1. Banac h spaces 7 3.2. W eak to p ologies on normed spaces 7 3.3. The con ve x t o p ology o n a normed space 8 4. W eak compactness in a uniformly conv ex Banach space 9 4.1. Uniform conv ex ity 9 4.2. V arious we ak forms of the Alaoglu theorem 10 4.3. Con v ex-compactness in ZF 11 5. A C ( N ) and compactness in [0 , 1 ] I 11 5.1. Spaces ℓ p ( I ), 1 ≤ p ≤ ∞ or p = 0 12 5.2. Closed subspaces of [0 , 1] I included in ℓ p ( I ), 1 ≤ p < + ∞ 13 6. DC and compactness in [0 , 1] I 14 6.1. Eb erlein’s criterion of compactness 14 6.2. Closed subsets of [0 , 1] I included in ℓ 0 ( I ) 16 References 17 1. Introduction 1.1. Presen tation of the results. W e w ork in ZF , Zermelo-F raenk el set-theory without the Axiom of Choice (fo r short A C ). Consider the c ountable Axiom of Choic e , whic h is not prov able in ZF , and whic h do es not imply AC : A C ( N ): If ( A n ) n ∈ N is a family of non-em pty sets, then ther e exists a mappi ng f : N → ∪ n ∈ N A n asso c i a ting t o every k ∈ N a n e l e ment f ( k ) ∈ A k . In this pap er, we first provide in ZF + A C ( N ) a criterion of compact- ness for top ological spaces coarser than some complete metric space, ha ving a sub-basis of closed sets satisfying “g o o d” prop erties with re- sp ect to the distance (see Theorem 1 in Section 2.3.4). W e then con- sider an alternativ e t o p ology for the we ak top ology on a normed space, namely the c onvex top olo gy (whic h is the w eak top olog y in ZF + HB ), and w e provide some prop erties o f this con ve x top ology: in particu- lar, using the ( choicele ss) Lusternik-Sc hnirelmann theorem, w e sho w (see Theorem 2 in Sec tion 3.3) that the closure of the unit sphere of E f or the con v ex topo logy is the closed unit ball of E . A pplying our criterion of compactness to the con v ex top ology , w e obtain some new results. First, in ZF + AC ( N ), “The closed unit ball of a uniformly con- v ex Banac h space is compact in the con v ex top ology .” (see Theorem 3 in Section 4.2): this extends a result obtained by F remlin for Hilb ert spaces (see [6, Chapter 56, Section 566P]) and this solves a question raised in [3 , Question 2]. W e then pro v e in ZF t hat “The closed unit COUNT ABLE CHOICE AND COMP ACTNESS 3 ball of a uniformly conv ex Banac h space is ( closely) c onvex-c omp act in the con v ex top olo gy .” (see Theorem 4 in Section 4.3). Giv en a set I , we apply our results to closed subsets of [0 , 1] I . In Section 5, we sho w that AC ( N ) implies the compactness of closed subsets o f [0 , 1] I whic h are b ounded subsets of some ℓ p ( I ), 1 ≤ p < + ∞ . In Section 6, w e pro v e that the Axiom o f Dep endent Cho ic es DC implie s that ev ery closed subset of [0 , 1] I whic h is con tained in ℓ 0 ( I ) is compact. 1.2. Some w eak forms of AC. W e no w review some w eak forms of the Axiom of Choice whic h will b e used in this pap er and the kno wn links b et w een them. F or detailed references a nd m uc h information on this sub ject, see [7]. 1.2.1. DC and AC ( N , fin ) . The axiom of Dep endent Choic es asserts that: DC : Giv en a non- empt y set X and a binary relation R o n X suc h that ∀ x ∈ X ∃ y ∈ X xRy , there exists a sequence ( x n ) n ∈ N of X suc h that for ev ery n ∈ N , x n Rx n +1 . The c ountable Axiom of C hoic e for fi nite sets says that: A C ( N , fin ): If ( A n ) n ∈ N is a family of finite no n-empty sets, then ther e exists a m a pping f : N → ∪ n ∈ N A n asso- ciating to e v e ry n ∈ N an element f ( n ) ∈ A n . Of course, AC ⇒ DC ⇒ A C ( N ) ⇒ AC ( N , fin ). Ho w ev er, the con v erse statements are not pro v able in ZF , and A C ( N , fin ) is not pro v a ble in ZF (see references in [7]) . 1.2.2. BPI an d HB . The Bo ole an Prime Ide al axiom sa ys tha t: BPI : Every non trivial b o ole an algebr a has a prime ide al. It is kno wn that BPI is not pro v able in ZF and that B P I does not imply AC . The f ollo wing well-kno wn statemen t s of functional analysis are equiv alen t to the axiom BP I ( see for example [13]): the T ychonov the or em for pro duct of compact Hausdorff spaces, the A lao glu the or em , the fact that for ev ery set I the pro duct space [0 , 1] I is compact. R emark 1 . If a set I is w ell-orderable, then the pro duct top olo gical space [0 , 1 ] I is compact in ZF . 4 M. MORIL LON 1.2.3. Hahn-Bana ch. Giv en a (real) v ector space E , a mapping p : E → R is sub-line ar if for ev ery x, y ∈ E , and every λ ∈ R + , p ( x + y ) ≤ p ( x ) + p ( y ) (sub-additivity ), and p ( λ.x ) = λp ( x ) (p ositiv e homogene- it y). Consider the “Hahn-Ba n ach” axiom, a well know n conseque nce of AC whic h is not pro v able in ZF : HB : L et E b e a (r e al) ve ctor sp ac e. If p : E → R is a s ub-l i n e ar mapping, if F is a ve c tor subsp ac e of E , if f : F → R is a line ar mapping such that f ≤ p ↾ F , then ther e exists a lin e ar ma p ping g : E → R e xtending f such that g ≤ p . Giv en a (real) to p ological v ector space E ( i.e. E is a v ector s pace suc h that the “sum” + : E × E → E and the exte rnal multiplicativ e la w . : R × E → E are con tinuous for the pro duct top olo gy), say that E satisfies the Continuous Hahn-Banach pr op erty (for short CHB prop- ert y) if “ F or every con tin uous sub-line ar mapping p : E → R , for every ve ctor subsp ac e F of E , if f : F → R is a line ar mapping such that f ≤ p ↾ F , then ther e exists a line ar mapping g : E → R extending f such that g ≤ p .” The statemen t HB is not pr ov able in ZF , ho w ev er, some real normed spaces satisfy (in ZF ) the CHB prop erty : for example normed spaces with a w ell-orderable dense subset (in particular sep- arable no rmed spaces), but also Hilb ert spaces, spaces ℓ 0 ( I ) (see [5]), uniformly conv ex Bana ch spaces with a Gˆ ateaux-differen tiable norm ([4]), uniformly smo oth Banac h spaces (see [1]). It is rather easy to pro ve that BPI implies HB and that HB is equiv alen t to most of its classical geometrical fo r ms (see [4, Section 6]). It is also easy t o see that AC ⇒ ( BPI + DC ) ⇒ BP I ⇒ HB . The con v erse statemen ts are not prov a ble in ZF and HB is not pro v able in ZF + DC (see [7]) . R emark 2 . There exist v arious ( ZF C -equiv alent) definitions of reflex- ivit y for Banac h spaces : most of them are equiv alent in ZF + HB + DC (see [11]), including James’ sup theorem (see [12]). 2. A criterion of comp actness 2.1. Filters. 2.1.1. Filters in lattic es of sets. Giv en a set X , a lattic e of subsets of X is a s ubset L of P ( X ) con taining ∅ and X , whic h is closed by finite in tersections a nd finite unions. A filter of the lattice L is a non-empt y prop er subset F of L suc h that for ev ery A, B ∈ L : (1) ( A, B ∈ F ) ⇒ A ∩ B ∈ F COUNT ABLE CHOICE AND COMP ACTNESS 5 (2) ( A ∈ F and A ⊆ B ) ⇒ B ∈ F A subset A o f L is contained in a filter of L if and o nly if A is c e n ter e d i.e. ev ery finite subset of A has a non-empt y in tersection; in this case, the in tersection of all filters of L containing A is called the filter gener a te d by A and w e denote it b y f il ( A ). 2.1.2. Stationary sets. Giv en a filter F of a lattice L o f subsets of a set X , an elemen t S ∈ L is F - s tationary if for ev ery A ∈ F , A ∩ S 6 = ∅ . The set S L ( F ) (also denoted b y S ( F )) of F -statio nary elemen ts of L satisfies the follo wing prop erties: (i) If A is a c hain of L a nd if A ⊆ S ( F ), then A ∪ F is cen tered. (ii) Let F 1 , . . . , F m ∈ L . If F 1 ∪ · · · ∪ F m ∈ S ( F ), then there exists some i 0 ∈ { 1 ..m } suc h that F i 0 is F -stationary . 2.2. Complete metric spaces. Giv en a metric space ( X , d ), some p oin t a ∈ X and real num b ers R , R ′ satisfying R ≤ R ′ , w e define lar ge d -b a l ls and lar ge d -cr owns as follo ws: B d ( a, R ) := { x ∈ X : d ( a, x ) ≤ R } D d ( a, R, R ′ ) := { x ∈ X : R ≤ d ( a, x ) ≤ R ′ } Moreo v er, if A is a subse t of X , w e define the d -diameter of A : diam d ( A ) := sup { d ( x, y ) : x, y ∈ A } ∈ [0 , + ∞ ] In particular, diam d ( ∅ ) = 0. A metric space ( X , d ) is said to b e c om - plete if ev ery Cauc hy filter of the lattice of closed subsets of X has a non-empt y in tersection. Here, a set A o f subsets of X is Cauchy if for ev ery ε > 0, there exists A ∈ A satisfying diam d ( A ) < ε . 2.3. A criterion of compa ct ness in ZF + AC ( N ) . 2.3.1. Comp actness. Definition 1 ( C -compactness, closed C - compactness) . Giv en a class C of subsets of a s et X , sa y that a subset A of X is C -c omp act if for ev ery family ( C i ) i ∈ I of C suc h that ( C i ∩ A ) i ∈ I is cen tered, A ∩ ∩ i ∈ I C i is non- empt y; sa y that A is clos e ly C -compact if there is a mapping asso ciating to ev ery family ( C i ) i ∈ I of C suc h that ( C i ∩ A ) i ∈ I is cen tered, an elemen t of A ∩ ∩ i ∈ I C i . Recall that a top olo gical space X is c omp ac t if X is C -compact, where C is the set of closed subsets of X . Equiv alen tly , ev ery filter o f the lattice of closed sets of X has a non-empty in tersection. 6 M. MORIL LON 2.3.2. Sub-b asis of close d sets. Definition 2 (basis, sub-basis of closed subsets) . A set B of closed subsets of a top o logical space X is a b asis of clo s e d sets if ev ery closed set o f X is an in tersection of elemen ts of B . A set S of closed subsets of X is a sub-b asis of close d sets if the set B of finite unions of eleme nts of S is a ba sis of closed sets of X . The fo llo wing result is easy . Prop osition 1. L et X b e a top olo gic al sp ac e, and L b e a la ttic e of close d subse ts of X . I f L i s a b asis of close d subsets of X , and if every filter of L h as a non-em pty interse ction, then X is c omp act. 2.3.3. Pr o p erty of smal lness. Giv en real n umbers a, b , we denote by ] a, b [ the op en interv al { x ∈ R : a < x < b } . Definition 3 (smallness in thin cro wns) . Let ( X, d ) be a metric space and let a ∈ X . Sa y that a set C of s ubsets of X satisfies the pr op erty of d -smal ln ess in thin cr owns c enter e d at a if for ev ery R ∈ R ∗ + , for ev ery ε > 0 there exists η ∈ ]0 , R [ suc h that for ev ery C ∈ C , C ⊆ D d ( a, R − η , R + η ) ⇒ diam d ( C ) < ε 2.3.4. Criterion of c omp actness. Theorem 1. L et ( X , d ) b e a c om plete metric sp ac e. L et T b e a top olo gy on X which is include d in the top olo gy T d of ( X, d ) . L et C b e a s ub- b asis of close d sets of ( X , T ) , which is cl o se d by finite interse ction. If a ∈ X , if C c ontains al l l a r ge d -b al ls c enter e d at a , and if C satisfies the pr op erty of d -s mal lness in thin cr owns c enter e d at a , then: (i) In ZF + A C ( N ) , every lar ge d - b al l with c enter a is T -c omp act (and thus, every d -b ounde d T -cl o se d subset of X is T -c om p act). (ii) In ZF , every lar ge d -b al l with c enter a is C -c om p act (an d thus, every d -b ounde d e lement of C is closely C -c omp a c t). Pr o of. Let L b e the sub-lattice of P ( X ) generated b y C . Let ρ > 0 and let B b e the large d -ball B d ( a, ρ ). (i) Let F b e a filter o f L con taining B . Let us prov e in ZF + A C ( N ) that ∩F is non-empt y (using Prop o sition 1, this will imply that B is T -compact). L et R := inf { r ∈ R + : B d ( a, r ) ∈ S ( F ) } . The set of balls { B d ( a, r ) : r > R } is a c hain of F -stationary sets of L , th us F ∪ { B d ( a, r ) : r > R } generates a filter G of L (see Section 2.1.2-(i)). If R = 0 then ∩G = { a } (because elemen ts of F are T d -closed) th us a ∈ ∩G ⊆ ∩F . If R > 0, for every ε > 0, there exists some elemen t of G whic h is included in the cro wn D d ( a, R − ε, R + ε ); with A C ( N ), choose COUNT ABLE CHOICE AND COMP ACTNESS 7 for e ve ry n ∈ N , a finite subset Z n of C suc h that ∪Z n ∈ G a nd ∪Z n ⊆ D d ( a, R − 1 n +1 , R + 1 n +1 ). With A C ( N , fin ), the set ∪ n ∈ N Z n is coun table. W e define b y induction a seq uence ( C n ) n ∈ N ∈ Q n ∈ N Z n suc h that for ev ery n ∈ N , G ∪ { C i : i < n } generates a filter G n and C n ∈ S ( G n ): giv en some n ∈ N , ∪Z n ∈ G ⊆ f il ( G , ( C i ) i R } . If R = 0 then { a } = ∩A ′ ⊆ ∩A . If R > 0, then for ev ery ε > 0, there exists some elemen t of A ′ whic h is included in the crown D d ( a, R − ε, R + ε ). Since C satisfies the prop ert y of d -smallness in thin d - cro wns ce ntered a t a , the cente red family A ′ is Cauc hy in the metric space ( X , d ); since this metric space ( X , d ) is complete, ∩A ′ is a singleton { b } , a nd { b } = ∩A ′ ⊆ ∩A ; mor eo v er, b is ZF -definable fro m ( X, d ) and A .  3. The convex topology on a normed s p ace In this p ap e r, al l ve ctor sp a c es that we c onsider ar e define d over the field R of real numb ers. 3.1. Banac h spaces. Giv en a normed space E endo wed with a norm k . k , w e denote b y B E the closed unit ball { x ∈ E : k x k ≤ 1 } , and by S E the unit sphere of E . The top ology on E asso ciated to t he norm is called the str ong top olo gy . A Ban a ch space is a normed space whic h is (Cauc hy)-complete for the metric asso ciated to the norm ( i.e. eve ry Cauc h y filter of closed sets has a non-empt y in tersection). 3.2. W eak top ologies on normed spa ces. 3.2.1. The c ontinuous dual E ′ of a norme d sp ac e E . W e endo w the v ector space E ′ of con tin uous linear mappings f : E → R with the dual norm k . k ∗ , and w e call this space the c ontinuous dual of the normed space E . W e also denote by can : E → E ′′ the c anonic al m apping asso ciating to ev ery x ∈ E the “ev aluat ing mapping” ˜ x : E ′ → R , satisfying for ev ery f ∈ E ′ the equalit y ˜ x ( f ) = f ( x ). 8 M. MORIL LON 3.2.2. The we ak top olo gy σ ( E , E ′ ) on E . It is the w eak est top ology T on E suc h tha t elemen ts f ∈ E ′ are T - con tin uous. The vec tor space E endo w ed the w eak top olo g y is a lo cally conv ex top olog ical v ector space. 3.2.3. The we ak* top olo gy σ ( E ′ , E ) on E ′ . It is the w eak est top ology T on E suc h that ev alua t ing mappings ˜ x : E ′ → R , x ∈ E are T - con tin uous. The v ector space E ′ endo w ed the w eak* top olog y is a Hausdorff lo cally con v ex to p ological vec tor space. R emark 3 . In a mo del of ZF where HB fails, there exists a non null (infinite dimensional) nor med space E suc h that E ′ = { 0 } (see [5 , Lemma 5] or [9 ]). In suc h a mo del o f ZF , the weak top o lo gy on E is trivial with only tw o op en sets. 3.3. The conv ex topology on a normed space. 3.3.1. Definition of the c onv ex top olo gy. Since the w eak top olog y on an infinite dimensional normed space E ma y b e trivial (in ZF ), w e define an alternativ e top ology on E , the c onv ex top olo g y (whic h w e intro duced in [11]): it is the w eak est top olog y T c for whic h strongly closed con v ex subsets of E are T c -closed. The lattice g enerated b y strongly closed con v ex subsets of E is called the c onvex la ttic e o f E . Elemen ts of this lattice are finite unions of strongly clos ed conv ex sets, so this la t t ice is a basis o f closed subsets of the conv ex topo lo gy . Thus the set C of strongly closed conv ex subsets of E is a sub-basis of closed sets of the con v ex top ology , whic h is closed b y finite inters ection. Prop osition 2. L et E b e a no rme d sp ac e. (i) “we ak top olo gy on E ” ⊆ “c on vex top olo gy on E ” ⊆ “str ong top ol- o gy o n E ”. (ii) If E satisfies the c ontinuous Hahn-Banach pr op e rty, t hen the we a k top olo gy and the c onvex top olo g y on E ar e e qual. Pr o of. (i) is trivial. (ii) follows from the fa ct that if a normed space E satisfies the CHB prop erty , then it satisfies sev eral classical geometric forms of the geometric Hahn-Banac h prop ert y (see [4]) and in particu- lar, ev ery closed conv ex set is w eakly closed.  In ZF + HB , the w eak top ology and the conv ex top ology on a no r med space are equal. 3.3.2. Convex top olo gy vs str ong top olo gy. Theorem ( Lusternik-Sc hnirelmann). L et n ∈ N ∗ , let N : R n → R b e a norm, a n d let S b e the unit spher e of N . L et a ∈ R n such that COUNT ABLE CHOICE AND COMP ACTNESS 9 N ( a ) < 1 . D enote by s a : S → S the “antip o dal mapping” a s s o ciating to every x ∈ S the p oint y ∈ S such that ( xa ) ∩ S = { x, y } , wher e ( xa ) is the line gene r ate d b y x and a . I f C 1 , . . . , C n ar e n close d subsets of R n such that S ⊆ ∪ 1 ≤ i ≤ n C i , then ther e exists i ∈ { 1 ..n } and x ∈ S such that { x, s a ( x ) } ⊆ C i . Pr o of. The pro of of this fa mous result is c hoiceless (see fo r example [10] for a = 0).  Theorem 2. L et E b e a norme d sp ac e which is not finite-dimensional. The closur e of the unit spher e S E for the c onvex top olo gy is the close d unit b al l B E . In p articular, the c onvex top olo gy on E is strictly c o n- taine d in the str ong top olo gy. Pr o of. Since B E is closed in the con v ex top o logy , the closure C of S E in the con v ex to p ology is con tained in B E . W e no w prov e that ev ery p oin t a ∈ E suc h that k a k < 1 b elong s to C . Consider some finite set { C i : 1 ≤ i ≤ n } of closed con v ex subsets of E suc h that F := ∪ 1 ≤ i ≤ n C i con tains S . W e ha v e to sho w that a ∈ F . Let V b e a v ector subspace of E , containing a , with dimension ≥ n . The Lusternik-Sc hnirelmann theorem implies that for some i 0 ∈ { 1 ..n } , C i 0 ∩ V contains tw o a - an tip o dal p oin ts of S E ; b y con v exity of C i 0 , a ∈ C i 0 .  Question 1. The con v ex top ology T c on a normed space E is T 1 ( i.e. ev ery singleton is closed). Is it Hausdorff ? Is the space ( E , T c ) a topo - logical v ector space? Is it lo cally conv ex? Is T c the top ology asso ciated to some f a mily of pseudo-metrics on E ? Is T c uniformizable? 4. Weak comp actness in a uniforml y convex Bana ch sp ace 4.1. Uniform con vexit y. 4.1.1. Strict c on vexity. A normed space ( E , k . k ) is strictly c onvex if ev ery segmen t contained in the unit sphere is a singleton: for eve ry x, y ∈ S E , x 6 = y ⇒ k x + y 2 k < 1. 4.1.2. Uniform c onvexity. This is a strong quantitativ e v ersion of the strict con v exity . A normed space ( E , k . k ) is uniformly c onvex if for ev ery real num b er ε > 0, there exists δ > 0 suc h that fo r ev ery x, y ∈ B E ,  k x − y k > ε ⇒ k x + y 2 k < 1 − δ  . Example 1 . Ev ery Hilb ert space is uniformly con vex (see [2, p. 1 90- 191]). Let p ∈ ]1 , + ∞ [. If B is a b o olean algebra of subsets of a set I , and if ν : B → [0 , + ∞ ] is non-null and finitely additiv e, then the normed space L p ( ν ) is uniformly con vex (see [4, Section 4]). In par- ticular, for ev ery set I , the normed space ℓ p ( I ) (see Section 5.1) is uniformly conv ex. 10 M. MORIL LON Prop osition 3. Given a uniformly c onvex norme d sp ac e E , an d de not- ing by d the metric on E giv en b y t he norm, the fam i ly of close d c on vex subsets of E satisfie s the pr op erty of d -smal l n ess in thin d -cr ow ns c en- ter e d at 0 E . Pr o of. This follo ws directly f r o m the definition of uniform con vex ity .  4.2. V arious wea k forms of the Alaoglu theorem. Consider the follo wing statemen ts (the first t w o w ere in tro duced in [3] and [11] and are consequences of BPI -or rather the Alaoglu theorem-): • A1 : The closed unit ball (and th us ev ery b ounded subset whic h is closed in the con v ex top olo g y) of a uniformly con vex Banac h space is compact in the conv ex top ology . • A2 : (Hilb ert) The closed unit ball (and thus ev ery b ounded w eakly closed subset) of a Hilb ert space is weak ly compact. • A3 : (Hilb ert with hilb ertian basis) F or ev ery set I , the closed unit ball of ℓ 2 ( I ) is weak ly compact. • A4 : F o r ev ery s equence ( F n ) n ∈ N of finite sets, the closed unit ball of ℓ 2 ( ∪ n ∈ N F n ) is weakly compact. Of course, A1 ⇒ A2 ⇒ A3 ⇒ A4 . R emark 4 . If a Hilb ert space H has a w ell orderable dense subset, then H has a well orderable hilb ertian basis, th us H is isometrically isomorphic with some ℓ 2 ( I ) where I is w ell or derable, and in this case, the closed ball B H , whic h is homeomorphic with a closed subset of [ − 1 , 1] I , is w eakly compact (use Remark 1). In particular, g iven an ordinal α , the c losed unit ball of ℓ 2 ( α ) (for example the c losed unit ball of ℓ 2 ( N )) is w eakly compact. Theorem 3. (i) AC ( N ) ⇒ A1 . (ii) A1 6⇒ AC ( N ) . (iii) A4 ⇔ AC ( N , fin ) . Pr o of. (i) Let E b e a uniformly conv ex Banac h space. Denote by C the class of (strongly) closed con v ex subsets of E . Denoting b y T the con v ex top olo g y on E , the class C is a sub-basis of closed subsets of the top olo gical space ( E , T ). Now, consider the distance d asso ciated to the norm on E : using Prop osition 3 , C satisfies the property o f d - smallness in thin d -cro wns cen tered at 0 E . Moreo v er, the metric space ( E , d ) is complete, closed d -balls b elong to C and T is included in the top ology asso ciated to d . Applying Theorem 1-(i) , it follows from A C ( N ) that the unit closed ball of E (a nd also ev ery b ounded subset of E whic h is closed in the con v ex top olog y) is compact in the conv ex COUNT ABLE CHOICE AND COMP ACTNESS 11 top ology of E . (ii) A1 6⇒ AC ( N ) b ecause BPI ⇒ A1 and BPI 6⇒ A C ( N ). (iii) The idea of the implication A4 ⇒ AC ( N , fin ) is in [5, th. 9 p. 16]: w e sk etc h it for sake of completeness. Let ( F n ) n ∈ N b e a disjoint sequence of non-empty finite sets. Let us show that Q n ∈ N F n is non-empty . Let I := ∪ n ∈ N F n . Then the Hilb ert spaces H := ℓ 2 ( I ) a nd ⊕ ℓ 2 ( N ) ℓ 2 ( F n ) are isometrically isomorph. Let ( ε n ) n ∈ N b e a sequence of ]0 , 1 [ suc h that P n ∈ N ε 2 n = 1 . F or ev ery n ∈ N , let ˜ F n := { ε n 1 { x } : x ∈ F n } where f or each x ∈ F n , 1 { x } : F n → { 0 , 1 } is the indicato r of { x } ; let Z n := { x = ( x k ) k ∈ N ∈ H : x n ↾ F n ∈ ˜ F n } . Eac h Z n is a w eakly closed subset of the ball B H ( Z n is a finite union of closed con v ex se ts). Moreo v er, the sequence ( Z n ) n ∈ N is cen tered. The w eak compactness of B H implies that Z := ∩ n ∈ N Z n is non-empty . An elemen t of Z defines an elemen t of Q n ∈ N ˜ F n , and thus an elemen t of Q n ∈ N F n (b ecause e ach ε n is > 0). F or the conv erse statemen t, if ( F n ) n ∈ N is a seq uence of finite sets, then the set I := ∪ n ∈ N F n is finite or coun table and in b oth cases, the closed unit ball of the Hilb ert space ℓ 2 ( I ) is we akly compact (see Remark 4).  R emark 5 . Theorem 3-(i) enhances our previous result DC ⇒ A1 whic h w e prov ed in [3], where w e left op en the t wo questions: D o es A C ( N ) imply A1 ? Do es A C ( N ) imply A2 ? A pro of of AC ( N ) ⇒ A2 has b een found by F remlin (see [6, chap. 56, Section 566P]). Question 2. Do es A2 imply A1 ? Do es A3 imply A2 ? Do es A C ( N , fin ) imply A3 ? 4.3. Con vex -compactness in ZF. Giv en a v ector space E , e ndow ed with a to p ology T , sa y that a subset A of E is c onv e x -c om p act if, denot- ing by C the set of T -closed conv ex subsets of E , A is C -compact; more- o v er, if A is closely C -compact, sa y that A is closely c onvex-c omp act . Theorem 4. The clo se d unit b al l o f a uniformly c onvex Banach sp ac e is closely c onvex-c omp act in the c onvex top olo gy. Pr o of. The pro of is analog to t he pro of of Theorem 3-(i), applying Theorem 1-(ii) instead of Theorem 1- (i).  5. A C ( N ) and comp actness in [0 , 1 ] I Giv en a set I , w e endo w the v ector sp ace R I with the the pro duct top ology , whic h w e denote b y T I . 12 M. MORIL LON A C   BPI + DC x x q q q q q q q q q q ' ' O O O O O O O O O O O BPI { { x x x x x x x x x   ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; DC   HB A C ( N ) w w p p p p p p p p p p p A1   A2   A3   A C ( N , fin ) Figure 1. Some w eak forms of A C 5.1. Spaces ℓ p ( I ) , 1 ≤ p ≤ ∞ or p = 0 . D enote b y ℓ ∞ ( I ) t he fo llowing v ector space endo w ed with the “sup” norm N ∞ : ℓ ∞ ( I ) := { x = ( x i ) i ∈ I : sup i ∈ I | x i | < + ∞} Denote b y ℓ 0 ( I ) the follow ing closed v ector space of ℓ ∞ ( I ) endow ed with the norm N ∞ : ℓ 0 ( I ) := { x = ( x i ) i ∈ I ∈ ℓ ∞ ( I ) : ∀ ε > 0 ∃ F ∈ P f ( I ) ∀ i ∈ I \ F | x i | ≤ ε } F or every p ∈ [1 , + ∞ [, denote by ℓ p ( I ) the followin g vec tor s pace en- do w ed with the N p -norm: ℓ p ( I ) := { x = ( x i ) i ∈ I ∈ R I : X i ∈ I | x i | p < + ∞} Recall that the con tinuous dual of ℓ 0 ( I ) is (canonically isometrically isomorphic with) ℓ 1 ( I ). Given some p ∈ ]1 , + ∞ [, the con tinuous dual of ℓ p ( I ) is (canonically isometrically isomorphic with) ℓ q ( I ) where q is the conjuguate of p . The follow ing Lemma is easy: COUNT ABLE CHOICE AND COMP ACTNESS 13 Lemma 1. L et I b e a set. (i) The top olo gy induc e d by T I on the subset ℓ 1 ( I ) ( resp. ℓ ∞ ( I ) ) is include d in the we ak* top olo gy σ ( ℓ 1 ( I ) , ℓ 0 ( I )) of ℓ 1 ( I ) ( resp. the w e ak* top olo gy σ ( ℓ ∞ ( I ) , ℓ 1 ( I )) of ℓ ∞ ( I ) ). Mor e over, the two top olo gies induc e the same top olo gy on b ounde d subsets of ℓ 1 ( I ) ( resp. ℓ ∞ ( I ) ). (ii) The top olo gy induc e d by T I on the subset ℓ 0 ( I ) is include d in the we ak top ol o gy σ ( ℓ 0 ( I ) , ℓ 1 ( I )) . Mor e over, the two top olo gies induc e the same top olo gy on b ounde d subsets of ℓ 0 ( I ) . (iii) Given some p ∈ ]1 , + ∞ [ , the top olo gy induc e d by T I on the subset ℓ p ( I ) is i n clude d in the w e ak top ol o gy σ ( ℓ p ( I ) , ℓ q ( I )) wher e q is the c onjuguate of p . Mor e over, the two top olo gies induc e the same top olo gy on b ounde d subsets of ℓ p ( I ) . 5.2. Closed subspaces of [0 , 1] I included in ℓ p ( I ) , 1 ≤ p < + ∞ . Giv en a set I , denote b y [0 , 1] I σ the set of elemen ts x = ( x i ) i ∈ I ∈ [0 , 1] I suc h tha t the supp ort { i ∈ I : x i 6 = 0 } of x is coun table. Say that a closed subset F of [0 , 1] I is Corson if F ⊆ [0 , 1] I σ . Prop osition 4. L et I b e a set. L et F b e a clos e d subset of [0 , 1] I . (i) If F ⊆ ℓ 0 ( I ) , then A C ( N , fin ) im p lies that F is Co rs o n. (ii) If F is Corson, then AC ( N ) im p l i es that F is se q uential ly c om - p act. (iii) If F ⊆ ℓ 0 ( I ) , then AC ( N ) i m plies that F is se quential ly c omp act. Pr o of. (i) assume that F ⊆ ℓ 0 ( I ). Giv en some x = ( x i ) i ∈ I ∈ F , the supp ort J := { i ∈ I : x i 6 = 0 } of x is a countable union of finite sets. Using AC ( N , fin ), J is coun table. Th us F is Corson. (ii) L et ( x n ) n ∈ N b e a sequen ce of F . F or ev ery n ∈ N , denote by J n the supp ort of x n : J n := { i ∈ I : x n i 6 = 0 } . Using AC ( N ), the set J := ∪ n ∈ N J n is coun table. So K := [0 , 1] J × { 0 } I \ J is compact and metrizable so K is sequen tially compact: extract fr o m ( x n ) n ∈ N a con v ergen t subsequence ( x n ) n ∈ A where A is some infinite subset of N . (iii) Use (i) and (ii).  Corollary 1. L et F b e a close d subset of [0 , 1] I . If ther e exists p ∈ [1 , + ∞ [ such that F is a b ounde d subset of ℓ p ( I ) , then A C ( N ) implies that F is c omp act in [0 , 1] I . Pr o of. Let r ∈ ] p, + ∞ [. Then N r ≤ N p , so F is a bounded subset of ℓ r ( I ). Since F is w eakly closed and b ounded in ℓ r ( I ), Theorem 3 - (i) implies that, using AC ( N ), F is compact in the w eak to p ology σ ( ℓ r ( I ) , ℓ r ′ ( I )) where r ′ is the conjuguate of r . It follo ws fr om Lemma 1- (iii) that F is compact for the top olog y T I .  14 M. MORIL LON Question 3. What is the p ow er of the statemen t “The close d unit b al l of ℓ 2 ( R ) is we akly c omp a c t” ? This stat ement is a consequence of A C ( N ). Are t here mo dels of ZF whic h do not satisfy this statemen t? 6. DC and c omp actness in [0 , 1] I 6.1. Eb erlein’s c r iterion of compactness. 6.1.1. ϑ -se q uenc es. Let E b e a normed space, and denote by d the metric giv en b y the norm on E . Give n a subset F of E , and some real n um b er ϑ > 0, a ϑ -se quenc e of F is a sequence ( a n ) n ∈ N of F satisfying for ev ery n ∈ N : d  span { a i : i < n } , conv { a i : i ≥ n }  ≥ ϑ In [11], w e prov ed in ( ZF + DC ) the follo wing result: Theorem ( DC ) . L et E b e a Banach sp ac e. Denote by T the c onvex top olo gy on E . L et F b e a con v ex subset o f E w h ich is d -b ounde d and T -close d. If F is not T -c omp act, then t her e exists some r e al numb er ϑ > 0 and a ϑ -se quenc e of F . If w e delete the h yp o t hesis “ F is conv ex”, our next result allo ws us to build in ( ZF + DC ) “pseudo ϑ -sequences”. Sa y that a sequence ( a n ) n ∈ N of F is a pseudo ϑ -se quenc e if for ev ery n ∈ N : d  span { a i : i < n } , ( F ∩ conv { a i : i ≥ n } )  ≥ ϑ 6.1.2. Building pseudo-se quenc es with D C . W e first r ecall the following result for saturating filters w.r.t. some numeric constrain t: Prop osition ( DC ) . L e t E b e a s e t, let L b e a la ttic e of subsets of E , with smal le s t element ∅ and gr e atest element E . L et ρ : L → R + b e some mapp ing. let ˜ ρ : P ( L ) → R + b e the mappin g asso ci a ting to every subset A of L the r e al numb er inf { ρ ( A ) : A ∈ A} . L et F b e a filter of L . The n ther e exi s ts a filter G of L in c luding F such that ˜ ρ ( G ) = ˜ ρ ( S L ( G )) Pr o of. See [11].  R emark 6 . The previous Prop osition is easy to prov e in ZF C : consider a maximal filter of L including F . Notation 1. Giv en a metric space ( X , d ) and s ome subset A of X , w e denote b y d A the (1-Lipsc hitzian henc e) con tin uous mapping d A : X → R asso ciating to ev ery x ∈ X the real num b er d ( x, A ) := inf { d ( x, a ) : COUNT ABLE CHOICE AND COMP ACTNESS 15 a ∈ A } . Moreo v er, given some real n um b er ϑ > 0, w e denote b y A ϑ the following closed subset of X : A ϑ := { x ∈ X : d ( x, A ) ≤ ϑ } Notice that if A is a conv ex subset of a normed space, then for ev ery ϑ > 0 the set A ϑ is conv ex (b ecause t he mapping d A is conv ex). Theorem 5 ( DC ) . L et E b e a Ba nach sp ac e. L et d b e the distanc e given by the n o rm on E . L et T b e the c onvex top olo gy on E . L et F b e a d -b ounde d subset of E , wh i ch is T -clos e d (thus d -close d, thus d - c omp l e te). If F is not T -c omp act, then ther e exists some r e a l numb er ϑ > 0 , and a se quenc e ( a n ) n ∈ N of F , such that for every n ∈ N : d  span { a k : k ≤ n } , ( con v T { a k : k > n } ∩ F )  ≥ ϑ Pr o of. Let C b e the set o f T -closed ( i.e. d -closed) con v ex subs ets of E . Let L c b e the lattice generated b y C . Let L 1 b e the lattice induced b y L c on F : L 1 = { A ∩ F : A ∈ L c } . Since F is not T - compact, let F be a filter of L 1 con taining F suc h that ∩F = ∅ . Let ρ b e the “diameter” function ( w.r.t. the distance d ), whic h is defined for d - b ounded subsets of E , and in particular on L 1 . Using the previous Prop osition for the “diameter” function ρ , DC implies the existence of a filter G of L 1 including F suc h that r := ˜ ρ ( S L 1 ( G )) = ˜ ρ ( G ) If r = 0, then, since the metric space ( F , d ) is complete, ∩G is a single- ton { a } , and a ∈ ∩F : contradictory! Th us r > 0. Let 0 < ϑ < r . W e will no w build a sequence ( K n ) n ∈ N of C , a nd a sequence ( a n ) n ∈ N of F , suc h that for eve ry n ∈ N , (3) a n ∈ ∩ i ≤ n K i and (span { a i : i < n } ) ϑ ∩ ( K n ∩ F ) = ∅ It will follo w that for every n ∈ N , d  span { a k : k < n } , ( con v T { a k : k ≥ n } ∩ F )  ≥ ϑ • B (0 , ϑ ) / ∈ S L 1 ( G ): th us there exists G ∈ G satisfying B (0 , ϑ ) ∩ G = ∅ ; since G ∈ L 1 , G is of the form ∪ i ∈ I ( C i ∩ F ) where I is finite and each C i b elongs to C ; using Section 2.1.2-(ii), let i ∈ I suc h that ( C i ∩ F ) ∈ S ( G ). let K 0 b e the con ve x s et C i . Let G 0 b e the filter of L 1 generated b y G and K 0 . Let a 0 ∈ K 0 ∩ F . • ( R a 0 ) ϑ / ∈ S L 1 ( G 0 ): let G ∈ G 0 suc h that (span { a 0 } ) ϑ ∩ G = ∅ . since G ∈ L 1 , G is of the form ∪ i ∈ I ( C i ∩ F ) where I is finite and eac h C i b elongs to C ; let i ∈ I suc h tha t ( C i ∩ F ) ∈ S ( G 0 ). Let K 1 b e the set C i . Let G 1 b e the filter of L 1 generated b y G 0 and K 1 . Let a 1 ∈ ( K 0 ∩ K 1 ∩ F ). 16 M. MORIL LON • ( span { a 0 , a 1 } ) ϑ / ∈ S L 1 ( G 1 ): let K 2 ∈ C suc h that K 2 ∈ S ( G 1 ) a nd (span { a 0 , a 1 } ) ϑ ∩ ( K 2 ∩ F ) = ∅ . Let G 2 b e the filter of L 1 generated b y G 1 and K 2 . Let a 2 ∈ ( K 0 ∩ K 1 ∩ K 2 ) ∩ F . • . . . Using DC , we construct a sequen ce ( a n , C n ) n ∈ N of F × C satisfying (3).  Corollary 2. L et E b e a Banach sp ac e. L et d b e the metric give n by the no rm on E . L et F b e a d -b ounde d subset of E whic h is clos e d for the c onvex top olo gy T on E . Consid er the thr e e fol low ing statements: (i) F is c omp act for the c onvex top o l o gy T . (ii) F is se quential ly c omp act for T . (iii) F or eve ry ϑ > 0 , F do es not c ontain any ps eudo ϑ -se quenc e. Then (i) ⇒ (ii) ⇒ (iii) . Mor e ov er, in ( ZF + DC ) , (iii) ⇒ (i) . Pr o of. (i) ⇒ (ii) It is sufficien t to prov e this implication when E is a sep ar able Banac h space. In this case, the no r med space satisfies the CHB pro p ert y (see Section 1.2.3), th us the con vex topolo gy T and the w eak top o logy on E are equal. Moreov er, there is a norm N on E whic h induces on the closed unit ball of E a top ology whic h is included in the w eak top ology of E ( see for example [8, Lemme I.4 p. 2]). This implies that the top ology giv en b y the norm N and the t o p ology T are equal on K ; thus the T -compact space K is metrisable whence K is sequen tially compact. (ii) ⇒ (iii) Assume that the subset K is seque ntially compact in the top ology T . Let ϑ > 0. Seeking for a contradiction, assume that F has a pseudo ϑ -sequence ( a n ) n ∈ N . Extract some sequence ( a n ) n ∈ A whic h con v erges to some l ∈ F in the top ology T . Then, for eve ry n ∈ N , l ∈ con v T { a i : i ≥ n } . Let V := span  { a i : i ∈ N } ∪ { l }  . Let ( u n ) n b e a con vex blo c k-sequence of ( a n ) n ∈ A whic h strong ly con v erges to l . Since the normed space V is separable, for eac h n ∈ N , c ho o se some f n in the unit sphere of V ′ suc h that f n is n ull on { a i : i < n } and f n ( l ) ≥ ϑ . Let n 0 ∈ N such that d ( u n 0 , l ) < ϑ 2 . Let N ≥ n 0 suc h that u n 0 ∈ span { a i : i < N } ; then f N ( u n 0 ) = 0 and f N ( l ) ≥ ϑ th us, since k f N k = 1, d ( l , u N 0 ) ≥ ϑ : this is contradictory! The implication (iii) ⇒ (i) holds in ZF + DC thanks to Theorem 5.  6.2. Closed subsets of [0 , 1] I included in ℓ 0 ( I ) . Corollary 3. L et F b e a close d subset of [0 , 1] I . Assume that F is a (b ounde d) subset of ℓ 0 ( I ) . Th en DC implies that F is c o mp ac t. COUNT ABLE CHOICE AND COMP ACTNESS 17 Pr o of. The normed space ℓ 0 ( I ) satisfies the CHB prop erty (see Sec- tion 1.2.3), th us the w eak top ology σ ( ℓ 0 ( I ) , ℓ 1 ( I )) and t he conv ex top ol- ogy T on ℓ 0 ( I ) are equal on ℓ 0 ( I ). Since F is a b o unded subse t of ℓ 0 ( I ), the top o logy T a nd the pro duct top ology T I induce the same topo l- ogy on F (see Lemma 1-(ii)). The subset F of ℓ 0 ( I ) is b ounded and T -closed; using Prop osition 4-(iii), F is sequ entially compact for the top ology T I i.e. for T . Using DC , Corollar y 2 implies that F is com- pact for T i.e. for T I .  Question 4. Let I b e a set and F b e some closed subset of [0 , 1] I . If F ⊆ ℓ 0 ( I ), do es AC ( N ) imply that F is compact? Question 5. Let F b e a closed subset of [0 , 1] I whic h is con tained in R ( I ) (the v ector subspace of elemen ts x ∈ R I whic h hav e a fi nite supp ort). Then F ⊆ ℓ 0 ( I ) th us, using Corolla ry 3, DC implies that F is compact. D o es AC ( N ) imply that F is compact? Question 6. 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