math.FA 2008-12-18 0

Uniform Eberlein spaces and the finite axiom of choice

We work in set-theory without choice $\ZF$. Given a closed subset $F$ of $[0,1]^I$ which is a bounded subset of $\ell^1(I)$ ({\em resp.} such that $F \subseteq \ell^0(I)$), we show that the countable axiom of choice for finite subsets of $I$, ({\em r…

Authors: Marianne Morillon (ERMIT)

UNIF ORM EBERLEI N SP A CES AND THE FINITE AXIOM OF CHOICE MARIANNE MORILLON Abstract. W e work in set- theo ry without ch oice ZF . Given a closed subset F of [0 , 1] I which is a b ounded subset of ℓ 1 ( I ) ( r e sp. such that F ⊆ ℓ 0 ( I )), w e show that the countable axiom of choice for finite subsets of I , ( r esp. the countable axiom of choice A C N ) implies that F is c ompact. This e nha nces previous re s ults where AC N ( r esp. the axiom of Dep endent Choices DC ) w as r equired. Moreov er, if I is linearly orderable (for e x ample I = R ), the closed unit ball of ℓ 2 ( I ) is weakly compa ct (in ZF ). ERMIT EQUIPE R ´ EUNIONNAISE DE MA TH ´ EMA TIQUES ET INF O RMA TIQUE TH ´ EORIQUE (ERMIT) Contents 1. In tr o duction 2 2. Some w eak forms of AC 3 2.1. Restricted axioms o f choice 3 2.2. W ell-orderable union of finite sets 4 2.3. Dep enden t Choices 4 2.4. The “ T ych onov ” a xiom 4 3. Some classe s of closed subsets of [0 , 1] I 4 3.1. Eb erlein closed subsets of [0 , 1] I 5 3.2. Uniform Eb erlein close d subsets of [0 , 1 ] I 5 3.3. Strongly Eb erlein closed subsets of [0 , 1] I 7 4. Compactness (in ZF ) 7 4.1. Lattices and filters 7 4.2. Con tin uo us imag e of a compact space 7 4.3. Sequen tial compactness 8 4.4. A C N and coun table pro ducts of compact spaces 8 5. One-p oint compactifications and related spaces 9 Date : Novem b er 21 , 2 0 18. 2000 Mathematics S ubje ct Classific ation. Primary 0 3E25 ; Secondary 54B 1 0, 54 D30, 4 6 B26. Key wor ds and phr ases. Axiom of Choice, pro duct top ology , compactness, Eb er lein spaces, uniform Eb er- lein spaces. 1 5.1. The o ne-p oin t compactification of a set 9 5.2. V arious notions of compactness for ˆ X α , α o rdinal 9 5.3. Spaces ˆ X α , α o rdinal 10 5.4. Spaces σ n ( X ), n in teger ≥ 1 10 6. A C f in ( I ) and closed compactness of B 1 ( I ) 12 6.1. Dy adic represen tations 12 6.2. Another equiv a len t of AC f in ( I ) 12 6.3. Consequence s 13 7. A C N and Eberlein spaces 14 7.1. Sequen tial compactness of I -Eb erlein spaces 14 7.2. Coun t able pro duct of finitely restricted spaces 14 7.3. Coun t able pro ducts of strong Eb erlein spaces 14 7.4. A C N and Eberlein closed subsets of [0 , 1] I 15 7.5. Con v ex-compactness a nd the Hahn-Ba na c h prop erty 15 References 16 1. Intro duction W e work in the set-theory without the Axiom of Choice ZF . It is a well kno wn theo- rem of Kelley (see [1 1 ]) that, in ZF , the Axiom of Choice (for short A C ) is equiv alen t to the Tyc hono v axiom T : “Eve ry fa m ily ( X i ) i ∈ I of c omp act top olo gic a l sp ac es has a c omp ac t pr o duct.” Here, a top o lo gical space X is c omp act if ev ery family ( F i ) i ∈ I of closed subsets of X satisfying the finite in tersection prop ert y ( F IP ) has a non-empt y in tersection. How ev er, some pa r ticular cases of the T yc hono v axiom are pro v able in ZF , for example: R emark 1 . A finite pro duct of compact spaces is compact (in ZF ). Sa y that the top ological space X is closely-c omp act if there is a mapping Φ asso ciating to ev ery family ( F i ) i ∈ I of closed subsets of X satisfying the FI P an elemen t Φ(( F i ) i ∈ I ) of ∩ i ∈ I F i : the mapping Φ is a witness of close d-c omp actness on X . Notice that a compact top ological space X is closely-compact if and only if there exists a mapping Ψ asso ciating to ev ery non-empty closed subset F of X a n elemen t of F . Example 1 . G iv en a linear order ( X, ≤ ) which is complete (ev ery non-empty subset of X has a least upper b ound), then the order top olo gy on X is closely compact. In particular, the closed b ounded in terv al [0 , 1] of R is closely compact. Pr o of. The space X is compact (the classic al pro of is v alid in ZF ). Moreo v er X is closely compact since o ne can consider the choice function asso ciating to ev ery non-empt y closed subset its first elemen t.  The fo llo wing Theorem is prov able in ZF : Theorem ([8 ]) . L et α b e an or dinal. If ( X i , Φ i ) i ∈ α is a family of witnesse d closely-c omp act sp ac es, then Q i ∈ α X i is close l y-c omp act, an d has a witness of close d-c omp actness which is definable fr om ( X i , Φ i ) i ∈ α . Example 2 . F or ev ery ordinal α , the pro duct top ological space [0 , 1 ] α is closely compact in ZF . 2 Giv en a set I , denote b y B 1 ( I ) the set o f x = ( x i ) i ∈ I ∈ R I suc h that P i ∈ I | x i | ≤ 1: then B 1 ( I ) is a closed subset of [ − 1 , 1 ] I . In this pap er, w e shall prov e that B 1 ( I ) is compact using the c ountable axiom of ch o ic e for finite subsets of I ( see Theorem 2 in Section 6.2). This enhances Corollary 1 of [14] a nd par t ially solves Question 2 in [1 4]. W e shall deduce (see Corollary 3) that, if I is linearly orderable, ev ery closed subset o f [0 , 1] I whic h is contained in B 1 ( I ) is closely compact. In particular, the closed unit ball o f the Hilb ert space ℓ 2 ( R ) is compact in ZF , and this solv es Question 3 of [14]. Notice that { 0 , 1 } R (and [0 , 1 ] R ) is not compact in ZF (see [12]). W e shall also pro v e that Eb erlein closed subsets of [0 , 1] I are compact using the c ountable axiom of choic e for subsets of I (see Corollary 4) of Section 7.4 . This enhances Coro llary 3 in [14] where the same result w as prov ed using the axiom of Dep endent C h oic es DC . This also solv es Questions 4 and 5 thereof. The pap er is organized as follows: in Section 2 w e review v arious consequences of AC (in particular the c ountable axiom of choic e A C N and the axiom of c hoice r e s tricte d to fi n ite subsets AC fin N ) and the know n links b et w een them. In Section 3 w e presen t definitions o f uniform Eb erlein spaces, strong Eb erlein spaces a nd Eb erlein spaces. In Section 4 we giv e some to ols for compactness or sequen tial compactness in ZF . In Section 5, we recall the one-p oint compactification ˆ X of a discrete space X , and we sho w that for eve ry ordinal α ≥ 1 , the closed-compactness of ˆ X α is equiv alen t to the axiom of c hoice restricted to finite subsets of X . Finally , in Section 6 ( r esp. 7) we prov e tha t the coun ta ble axiom o f c hoice for finite sets ( r esp. the countable axiom of c hoice) implies that uniform Eberlein spaces ( r esp. E b erlein spaces) are closely compact ( r esp. compact.) A basic to ol fo r these tw o last Sections is a “dy adic represen tation” of elemen ts of p o w ers of [0 , 1] (see the The orem in Section 6.1) whic h w e f ound in [3 , Lemma 1.1], and for whic h the autho rs cite [15]. 2. Some weak forms of AC In this Section, we review some we ak forms of the Axiom of Choice whic h will b e used in this paper and some know n links b etw een them. F or detailed references a nd m uc h informa- tion on this sub ject, see [10]. 2.1. Restricted axioms of choice. Giv en a formula φ o f set-theory with one free v ariable x , consider the following consequence of AC , denoted b y A C ( φ ): “F or every non-empty family A = ( A i ) i ∈ I of non-empty sets such that φ [ x/ A ] holds, then Q i ∈ I A ( i ) is non-empty.” Notation 1. In the particular case where the formula φ say s t ha t “ x is a mapping with domain I with v alues in some ZF - definable class C ”, the statement A C ( φ ) is denoted b y A C C I . The statemen t ∀ I AC C I is denoted b y A C C . The statemen t A C C I where C is the collection of all sets is denoted by AC I . Notation 2. F or ev ery set X , w e denote b y f in ( X ) the set o f finite subsets of X . W e denote b y f in the (definable) class of finite sets. So, giv en a set X , AC f in ( X ) is the follo wing stat ement: “F or eve ry non-empty fam ily ( F i ) i ∈ I of n o n-empty finite subse ts of X , Q i ∈ I F i is n o n-empty.” , and AC f in is the following statemen t: “F or every non - empty family ( F i ) i ∈ I of non-empty finite sets, Q i ∈ I F i is non- empty.” The c ountable Axiom of Choic e sa ys that: 3 A C N : If ( A n ) n ∈ N is a fa m ily of non-empty sets, then ther e exists a m apping f : N → ∪ n ∈ N A n asso ciating to every n ∈ N an element f ( n ) ∈ A n . And the c ountable Axiom of Choic e for fini te sets says that: A C fin N : If ( A n ) n ∈ N is a family of fin i te non-empty sets, then ther e exists a mapping f : N → ∪ n ∈ N A n asso ciating to ev ery n ∈ N an element f ( n ) ∈ A n . 2.2. W ell-orderable union of finite sets. G iven an infinite ordinal α , and a class C of sets, w e consider the fo llo wing conseq uence of A C C : Uw o C α : F or every family ( F i ) i ∈ α of elements of C , the set ∪ i ∈ α F i is wel l- or der able. R emark 2 . AC f in ( X ) implies Uw o f in ( X ) α . 2.3. Dep enden t Choices. The axiom of Dep endent Choic es sa ys that: DC : Given a non-empty set X and a binary r elation R on X such that ∀ x ∈ X ∃ y ∈ X xRy , then ther e e xists a se quenc e ( x n ) n ∈ N of X such that for every n ∈ N , x n Rx n +1 . Of course, A C ⇒ DC ⇒ AC N ⇒ A C fin N . Ho w ev er, the con v erse statements are not pro v able in ZF , and A C fin N is not prov able in ZF (see references in [10]). 2.4. The “T ychono v” axiom. Giv en a class C of compact top ological spaces and a set I , w e consider the follow ing conse quence of the T yc hono v axiom: T C I : Every fa mily ( X i ) i ∈ I of sp ac es b elonging to the class C has a c omp act pr o duct. F or example T f in ( X ) N is the statement “Every se quenc e of fini te discr ete subse ts of X has a c om p act pr o duct.” R emark 3 . (i) Given a set X , for ev ery ordinal α , A C f in ( X ) ⇒ Uw o f in ( X ) α ⇒ T f in ( X ) α ⇒ AC f in ( X ) α (ii) F or ev ery ordinal α , Uw o f in α ⇔ T f in α ⇔ A C f in α . Pr o of. (i) Uw o f in ( X ) α ⇒ T f in ( X ) α : Giv en a family ( F i ) i ∈ α of finite subsets of X , the statemen t Uw o f in ( X ) α implies the existence o f a family (Φ i ) i ∈ α suc h that for eac h i ∈ α , the discrete space is closely compact with witness Φ i . Using the Theorem of Section 1, it follo ws tha t Q i ∈ α F i is (closely) compact. T f in ( X ) α ⇒ A C f in ( X ) α : one can use Kelley’s argumen t (see [1 1 ]). (ii) F or AC f in α ⇒ Uw o f in α : g iv en some family ( F i ) i ∈ α of finite non-empty sets, then, fo r eac h i ∈ α , denote b y c i := { 0 ..c i − 1 } the (finite) cardinal of F i ; th us set G i of one-to-one mappings from F i to c i is finitel, and, b y A C f in α , the set Q i ∈ α G i is non-empt y . This implies a well-order on the set ∪ i ∈ α F i .  3. Some class es of closed sub s ets of [0 , 1] I Notation 3. Let I b e a set. Giv en some elemen t x = ( x i ) i ∈ I ∈ R I , denote b y su pp ( x ) the supp ort { i ∈ I : x i 6 = 0 } . Given some subset A of R containing 0, denote b y A ( I ) the set of elemen ts o f A I with finite supp ort. W e endo w the space R I with the pro duct top ology , whic h w e denote b y T I . 4 3.1. Eb erlein closed subsets of [0 , 1] I . Giv en a set I , w e denote b y ℓ ∞ ( I ) the Banac h space of b ounded mappings f : I → R , endo w ed with the “sup” no rm. If I is infinite, w e denote b y c 0 ( I ) the closed subspace of ℓ ∞ ( I ) consisting o f f ∈ ℓ ∞ ( I ) suc h that f con verges to 0 according to the F r ´ ec het filter on I ( i.e. the set of cofinite subsets of I ). Th us ℓ 0 ( I ) := { x = ( x i ) i ∈ I : ∀ ε > 0 ∃ F 0 ∈ P f ( I ) ∀ i ∈ I \ F 0 | x i | ≤ ε } If I is finite, then w e define c 0 ( I ) := ℓ ∞ ( I ) = R I . Definition 1. A t o p ological space F is I -Eb erlein if F is a closed subset of [0 , 1] I and if F ⊆ c 0 ( I ). A top ological space X is Eb erlei n if X is homeomorphic with some I -Eb erlein space. R emark 4 . Amir and Lindenstrauss ([1]) prov ed in ZFC that ev ery w eakly compact subset of a normed space is an Eb erlein space. This result relies on the existence of a Markhus hevic h basis in ev ery w eakly compactly generated Ba na c h space, and the pro of of the exis tence of suc h a basis (see [7]) relies on (m uc h) Axiom of Choice. R emark 5 . Conside r the compact top ological space X := [0 , 1] N . Then, the closed subset X of [0 , 1 ] N is not N -Eb erlein. How ev er, the mapping f : X → [0 , 1] N ∩ c 0 ( N ) asso ciat ing to each x = ( x n ) n ∈ N ∈ X the elemen t ( x n n +1 ) n ∈ N is con tinuous and one-to-one, so X is homeomorphic with the compact (hence closed) subset f [ X ] of [0 , 1] N ∩ c 0 ( N ). It follo ws tha t X is homeomorphic with some N -Eb erlein space. Prop osition 1. (i) Every close d subset of a I -Eb erlein ( resp. Eb erlein) s p ac e is I -Eb erlein ( resp. Eb erlein). (ii) L et ( I n ) n ∈ N b e a se quenc e of p airwise disjoint sets, and denote by I the set ⊔ n ∈ N I n . L et ( F n ) n ∈ N b e a se quenc e of top olo gic al sp ac es such that e ach F n is I n -Eb erlein . The n the close d subset Q n ∈ N F n of [0 , 1 ] I is hom e omorphic with a I -Eb erle in sp ac e. Pr o of. (i) is trivial. W e prov e (ii). F or ev ery n ∈ N , let f n : F n → [0 , 1] I n b e the mapping asso ciating to each x ∈ F n the elemen t 1 n +1 f n ( x ) of [0 , 1] I n . Let f := Q n ∈ N f n : Q n ∈ N F n → [0 , 1] I . Then f is one-t o -one and con tinuous. Moreov er, the subse t F := I m ( f ) of [0 , 1] I is closed since F is the pro duct Q n ∈ N ˜ F n where for eac h n ∈ N , ˜ F n is the closed subset 1 n +1 .F n of [0 , 1] I n . Finally , it can be easily c hec k ed that F ⊆ c 0 ( I ).  3.2. Uniform Eb erlein closed subsets of [0 , 1] I . 3.2.1. The b al l B p ( I ) , for 1 ≤ p < + ∞ . F or ev ery real n umber p ≥ 1, define as usual the normed space ℓ p ( I ) := { ( x i ) i ∈ I : P i | x i | p < + ∞} endow ed with the norm N p : x = ( x i ) i ∈ I 7→ ( P i | x i | p ) 1 /p . W e denote b y B p ( I ) t he large unit ball { x ∈ R I : P i | x i | p ≤ 1 } of ℓ p ( I ). Notice that for p = 1 ( r esp. 1 < p < + ∞ ) the top olog y induced b y T I on B p ( I ) is the top ology induced b y the weak * top ology σ ( ℓ 1 ( I ) , ℓ 0 ( I )) ( r esp. the top ology induced b y the w eak top ology σ ( ℓ p ( I ) , ℓ q ( I )) where q = p p − 1 is the conjuguate of p ). Also notice that for 1 ≤ p < + ∞ , B p ( I ) is a closed subset of [0 , 1] I . Prop osition 2. If 1 ≤ p < + ∞ , then B p ( I ) is home om orphic with B 1 ( I ) . Pr o of. Consider the mapping h p : B 1 ( I ) → B p ( I ) asso ciating to ev ery x = ( x i ) i ∈ B 1 ( I ) the family (sgn( x i ) | x i | 1 /p ) i ∈ I .  5 It follows that fo r ev ery p, q ∈ [1 , + ∞ [, spaces B p ( I ) and B q ( I ) are homeomorphic via h p,q := h q ◦ h − 1 p : B p ( I ) → B q ( I ). 3.2.2. Uniform Eb erlein sp ac es. Give n a set I , and some real n um b e r p ∈ [1 , + ∞ [, we denote b y B + p ( I ) the p ositiv e ball of ℓ p ( I ): B + p ( I ) := { x = ( x i ) i ∈ I ∈ [0 , 1] I : X i ∈ I x p i ≤ 1 } Definition 2. A top o lo gical space F is I -uniform Eb erlein if there exists a real n um b er p ∈ [1 , + ∞ [ suc h that F is a closed subset of B + p ( I ). A top ological space X is uniform Eb erlein if X is homeomorphic with some I -uniform Eb erlein space. Of course, eve ry I -unifor m Eb erlein space is I -Eb e rlein. Moreov er, using Prop osition 2 , ev ery I -uniform Eb erlein space is homeomorphic with a closed subset of B + 1 ( I ). Prop osition 3. (i) F or every set I , every close d subset of a I -uniform Eb erlein sp ac e is I -unifo rm Eb erlein. (ii) L et ( I n ) n ∈ N b e a se quenc e of p airwise disjoint sets, and denote by I the set ⊔ n ∈ N I n . L et ( F n ) n ∈ N b e a se quenc e of top olo gic al sp ac es s uch that e ach F n is a I n -uniform Eb erlein sp ac e. Then the close d subset F := Q n ∈ N F n of [0 , 1 ] I is I -uniform Eb erlein. Pr o of. (i) is easy . The pro of of (ii) is similar to the pro of of Prop osition 1-(ii).  In particular, the compact space [0 , 1] N (and th us ev ery metrisable compact space) is N -uniform Eberlein. F or ev ery set I , B + 1 ( I ) N is ( I × N )-uniform Eb erlein. R emark 6 . Let Z := ∩ i ∈ I { ( x, y ) ∈ B + 1 ( I ) × B + 1 ( I ) : x i .y i = 0 } : then Z is a closed subse t of B + 1 ( I ) × B + 1 ( I ), and the mapping − : Z → B 1 ( I ) is an homeomorphism; it follo ws that B 1 ( I ) is homeomorphic with a ( I × { 0 , 1 } )-uniform Eberlein space. 3.2.3. We akly close d b ounde d subse ts of a Hilb ert sp ac e. R emark 7 . Giv en a Hilbert space H with a Hilb ert basis ( e i ) i ∈ I , then its closed unit ball (and th us ev ery b ounded w eakly closed subset of H ) is ( linearly) homeomorphic with the uniform Eb erlein space B 2 ( I ). Consider the following statemen ts (the first t w o ones w ere in tro duc ed in [5] and [13] and are conseque nces of the Alaoglu theorem): • A1 : The closed unit ball (and thus ev ery b ounded subset which is close d in the conv ex top ology) of a uniformly con vex Banac h space is compact in the con ve x top ology . • A2 : (Hilb ert) The closed unit ball (and thus eve ry b ounded we akly closed subset) of a Hilb ert space is w eakly compact. • A3 : (Hilb ert with hilb ertian basis) F or ev ery set I , the closed unit ball of ℓ 2 ( I ) is w eakly compact. • A4 : F or eve ry seque nce ( F n ) n ∈ N of finite sets, the closed unit ball of ℓ 2 ( ∪ n ∈ N F n ) is w eakly compact. Of course, A1 ⇒ A2 ⇒ A 3 ⇒ A4 . Theorem ([8], [14]) . ( i ) A C N ⇒ A1 . (ii) A 1 6⇒ AC N . 6 (iii) A 4 ⇒ AC fin N . In this pap er, we will pro v e that the follow ing statemen ts are equiv alen t: A 3 , A4 , AC f in ω (see Corollary 2). Question 1. Do es A 2 imply A1 ? D o es A3 imply A2 ? R emark 8 . If a Hilb ert space H has a w ell orderable dense subset, then H has a w ell orderable hilb ertian ba sis, thus H is isometrically isomorphic with some ℓ 2 ( α ) where α is an ordinal. In this case, the closed unit ball of H endo wed with the we ak top ology is homeomorphic with a closed subset of [ − 1 , 1] α , so this ball is we akly compact. 3.3. Strongly Eb erlein closed subsets of [0 , 1] I . Definition 3. A top o logical space F is I -str ong Eb erlein if F is a closed subset of [0 , 1] I whic h is con tained in { 0 , 1 } ( I ) . A top o logical space X is str ong Eb erlein if X is homeomorphic with some I -stro ng Eb erlein space. Of course, ev ery I - strong Eberlein set is I - Eb erle in. R emark 9 . F or ev ery set I , ev ery closed subset of a I - stro ng Eb erlein space is I -strong Eb erlein. 4. Comp a ctness (in ZF ) 4.1. Lattices and filters. G iv en a lattice L of subsets of a set X , sa y that a non-empty prop er subset F of L is a filter if it satisfies the tw o following conditions: (i) ∀ A, B ∈ F , A ∩ B ∈ F (ii) ∀ A ∈ F , ∀ B ∈ L , ( A ⊆ B ⇒ B ∈ F ) Sa y that a n elemen t A ∈ L is F -stationa r if for ev ery F ∈ F , A ∩ F 6 = ∅ . R emark 10 . Let X b e a top olo gical space, let L b e a lattice of closed subsets of X , and let F b e a filter of L . Let K ∈ L . If K is a compact subse t of X and if K is F -stationar, then ∩F is non-empt y . Definition 4. Giv en a f a mily ( X i ) i ∈ I of top ological spaces, and denoting by X the top olog- ical pro duct o f this family , a closed subset F o f X is eleme ntary if F is a finite unio n of sets of the form Q i 6 = i 0 X i × C where i 0 ∈ I and C is a closed subset of X i 0 . Giv en a f amily ( X i ) i ∈ I of top ological spaces with pro duct X , the set of elemen tary closed subsets of X is a la t t ice of subsets of X that w e denote b y L X . Notice that giv en a elemen tary closed subset F of X , and some subset J of I , the pro jection p J [ F ] is a closed subset of Q j ∈ J X j . 4.2. Con tin uous image of a compact space. The follo wing Prop osition is easy: Prop osition 4. L et X , Y b e top o lo gic al sp ac es and let f : X ։ Y b e a c ontinuous on to mapping. If X is c omp act ( r esp. closely-c omp act), then Y is also c omp a c t ( resp. closely c omp act). If Φ is a witness of close d-c omp actness on X , then Y is closely-c o mp act, and has a witness of close d-c omp actness which is de fi nable fr om f and Φ . 7 4.3. Sequen tial compactness. W e denote b y [ N ] ω the set o f infinite subsets of N . Definition 5. A top olog ical space X is se quential ly c omp act if ev ery seq uence ( x n ) n ∈ N of X has an infinite subs equence which con v erges in X . A witness of se quential c omp actness on X is a mapping Φ : X N → [ ω ] ω × X asso ciating to eac h sequen ce ( x n ) n ∈ N of X an elemen t ( A, l ) ∈ [ ω ] ω × X suc h that ( x n ) n ∈ A con verges to l . Example 3 . If ( X , ≤ ) is a complete linear or der, then X is seq uen t ia lly compact, with a witness definable from ( X , ≤ ): giv en a sequence ( x n ) n ∈ N , build some infinite subset A o f N suc h that ( x n ) n ∈ A is monoto ne; then if ( x n ) n ∈ A is ascending ( r esp. descending), then ( x n ) n ∈ A con verges to sup n ∈ A x n ( r esp. inf n ∈ A x n ). Example 4 . Giv en an infinite set X , and some set ∞ / ∈ X , consider the top ology on ˜ X := X ∪ {∞} generated b y cofinite subsets of ˜ X and {∞} . This to p ology is compact and T 1 but it is not T 2 . This top ology is sequen tially compact, and, give n a p oint a ∈ X , there is a witness of sequen tial compactness whic h is definable from X and a : giv en a sequence ( x n ) n ∈ N of ˜ X , either the set of terms { x n : n ∈ N } is finite, and then o ne can define by induction an infinite subset A of N suc h that { x n : n ∈ A } is constant; else one can define by induction an infinite subset A of N suc h that { x n : n ∈ A } is one-to-one, th us it con ve rges to a (and also to ev ery p oint in X ). Notice that the top ology in Example 4 is the one used b y Kelley (see [11]) to prov e that “T ychono v implies AC ”. The follo wing Lemma is easy: Lemma 1. L e t X , Y b e two top olo gic al sp ac es and let f : X ։ Y b e an onto c ontinuous mapping w hich has a se ction j (for example if f is one-to-one). If X is se quential ly c omp act, then Y is also se quential ly c o mp act. Mor e over, if ther e is a witness Φ of se quential c om- p actness on X , ther e also exists a witness of se q uen tial c omp actness on Y which is d e finable fr om f , Φ and j . Lemma 2. L et ( X n , φ n ) n ∈ N b e a se quenc e of witnesse d se quential ly c om p act sp ac es. The sp ac e Q n ∈ N X n is se quential ly c omp act, and has a witness defin a ble fr om ( X n , φ n ) n ∈ N . Pr o of. Usual diagonalizatio n.  Example 5 . If D is a coun table set, then the top olo gical space [0 , 1] D is sequen tially compact, a witness of sequen tial compactness b e eing definable from ev ery w ell o r der on D . Sa y that a sequen tially compact top o lo gical space X is witnessa ble if there exists a witness of sequen tial compactness on X . It follo ws from Lemma 2, tha t with A C N , ev ery sequence ( K n ) n ∈ N of witnessable sequen tially compact spaces has a pro duct whic h is sequen tially compact. 4.4. AC N and coun table pro ducts of compact spaces. Denote b y T comp ω the follo wing statemen t: “Every se quenc e of c omp act sp ac es has a c omp act pr o duct.” Then Kelley’s ar- gumen t shows t ha t T comp ω ⇒ A C N . How ev er, it is an op en question ( see [4], [9]) to kno w whether A C N implies T comp ω . Definition 6. A top ological space X is ω -c om p act if ev ery descending sequenc e ( F n ) n ∈ N of non-empt y closed subsets of X has a no n- empt y in t ersection. Say that the space X is cluster- c omp act if ev ery seque nce ( x n ) ∈ N of X has a cluster p o int i.e. the set ∩ n ∈ N { x k : k ≥ n } is non-empt y . 8 R emark 11 . (i) Notice that sequ en tially compact ⇒ “ cluster-compact” . Also notice that “ ω -compact” ⇒ “clus ter-compact” and that the con ve rse holds with AC N (see [9, Lemma 1]). (ii) Giv en a sequence ( K n ) n ∈ N of compact spaces, then, denoting b y K the pro duc t of this family , K is compact iff K is ω -compact (see [9, Theorem 6]). (iii) If the pro duc t K of a sequence ( K n ) n ∈ N of compact spaces is sequen tia lly compact, then A C N implies that K is compact. Prop osition 5. AC N is e quivalent to the fol lowing statement: “Ev ery sequence ( K n ) n ∈ N of witnessable seque n tially compact spaces whic h are also compact has a compact pro duct.” Pr o of. ⇒ : Giv en a sequenc e ( K n ) n ∈ N of witnessable seque n tially compact spaces whic h are also compact, then, using AC N , one can c ho ose a witness of sequen tial compactness on ev ery space K n . It follows by L emma 2 that K is sequen tially compact, whence K is compact b y Remark 11-(iii). ⇐ : W e use Kelley’s a rgumen t (see [11]). Let ( A n ) n ∈ N b e a sequence of non-empt y sets. Consider some eleme n t ∞ / ∈ ∪ n ∈ N A n , and for every n ∈ N , denote b y K n the set A n ∪ {∞} endo wed with the top ology generated b y {∞} and cofinite subsets of K n (see Example 4). Then eac h K n is compact and sequen tially compact; moreov er, giv en an elemen t a ∈ A n , there is a witness of sequen tial compactness on K n whic h is definable from A n , ∞ and a . So eac h K n is a witnessable sequen tially compact space. It follo ws f rom the hypothesis that the pro duct K := Q n ∈ N K n is compact. W e end as in Kelley’s pro of: for ev ery n ∈ N , let F n b e the closed set A n × Q i 6 = n K i . By compactness of K , the set ∩ n ∈ N F n is non- empt y . This yields an elemen t of Q n ∈ N A n .  5. One -point comp a ct ifica tions and rela t e d sp aces 5.1. The one-point compactification of a set . Giv en a set X , w e denote b y ˆ X the Alexandro v compactification of the (Hausdorff lo cally compact) discrete space X : ˆ X := X ∪ {∞ } where ∞ is some set / ∈ X (for example ∞ := { x ∈ X : x / ∈ x } ; if X is finite, then ˆ X is discrete else op en subsets of the space ˆ X are subsets of X or cofinite subsets of ˆ X con t a ining ∞ . Notice t ha t the space ˆ X is compact and Ha usdorff in ZF . Example 6 . Giv en a discre te top olo gical space X , the one-p oin t compactification ˆ X of X is X - uniform Eberlein: consider the Hilb ert space ℓ 2 ( X ); and denote b y ( e i ) i ∈ X the canonical basis of the v ector space R ( X ) ; then the subspace X = { e i : i ∈ X } o f R ( X ) is discrete and the w eakly closed and bo unded subset X ∪ 0 R X is the one-p oin t compactification ˆ X of X . 5.2. V arious notions of compactness for ˆ X α , α ordinal. 5.2.1. ˆ X N is se q uential ly c omp act. Prop osition 6. L et X b e an infinite se t. (i) The sp ac e ˆ X is se q uen tial ly c omp act and has a wi tnes s of se quential c omp actness, de- finable fr om X . (ii) Th e sp ac e ˆ X N is se q uential ly c omp act with a witness d e fi nable fr om X . Pr o of. (i) W e define a witness Φ of sequen tial compactness on X as follows : given a sequenc e x = ( x n ) n ∈ N of ˆ X , if the set T := { x k : k ∈ N } is infinite, w e build (b y induction) some 9 infinite subset A o f N suc h that { x k : k ∈ A } is one-to -one, and w e define Φ( x ) := ( A, ∞ ); else the set T is finite, so w e build b y induction some infinite subset A of N suc h that the sequence { x k : k ∈ A } is a singleton { l } , and w e define Φ( x ) := ( A, l ) . (ii) W e apply (i) and Lemma 2 in Section 4.3.  5.2.2. A C f in and close d- c omp actness. Prop osition 7. L et X b e a set. (i) The r e is a mapping asso ciating to every n on-empty close d subset F of ˆ X , a finite non- empty close d subset ˜ F of F . (ii) AC f in ( X ) ⇔ The sp ac e ˆ X is closely c om p act. Pr o of. W e may assume that X is infinite. (i) G iv en a non-empt y closed subs et F of ˆ X , define ˜ F := { ∞ } if ∞ ∈ F and ˜ F := F if F is finite and ∞ / ∈ F . (ii) Use (i).  5.3. Spaces ˆ X α , α ordinal. R emark 12 . F or ev ery set X , the space ˆ X is X -unifor m Eb erlein, so, give n an ordinal α , ˆ X α is X × α -uniform Eberlein (see Prop osition 3-(ii)). Prop osition 8. L et X b e a set. L et α b e an or dinal ≥ 1 . (i) T f in ( X ) α ⇔ “ ˆ X α is c o m p act”. (ii) AC f in ( X ) ⇔ “ ˆ X α is closely c om p act”. Pr o of. (i) ⇒ : Let P b e the top ological pro duct space ˆ X α . Let F be a filter of the lattice L X of elemen tary closed subsets of P . W e are going to define b y transfinite recursion a family ( G n ) n ∈ α of finite subsets of ˆ X suc h that, denoting for ev ery n ∈ α by Z n the elemen ta r y closed subse t G n × ˆ X α \{ n } of P , the set F ∪ { Z i : i < n } satisfies the finite intersec tion prop ert y . Give n some n ∈ α , w e define G n in function of ( G i ) i

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