On Borel almost disjoint families

There is a close correspondence between uncountable almost disjoint families of subsets of ω and Aleksandrov–Urysohn compacta (in short, AU -compacta)— separable, uncountable compact spaces whose second derived set is a singleton. We shall show in particular, that AU -compacta embeddable in the space of ﬁrst Baire class functions on the Cantor set 2 ω , with the pointwise topology, are exactly the ones determined by almost disjoint families that are Borel sets in 2 ω , and they are also dis-tinguished among AU -compacta by the property that the cylindrical σ -algebras of their function spaces are standard measurable spaces. Although the ﬁrst condition implies the third one for arbitrary separable compact space, it is an open problem, whether the reverse implication is always true.


Introduction
Let B 1 (2 ω ) be the space of real-valued first Baire class functions on the Cantor set, equipped with the topology of pointwise convergence. Given a compact set K , we shall denote by C(K ) the Banach space of real-valued continuous functions on K . For a Banach space X , we denote by Cyl(X ) the cylindrical σ -algebra in X , i.e., the smallest σ -algebra for which all functionals from the dual space X * are measurable.
One of the main results of this paper is the following.
Theorem 1.1 For any compact separable space K whose set of accumulation points is the one-point compactification of an uncountable discrete space, the following conditions are equivalent: (i) K embeds in B 1 (2 ω ), (ii) (C(K ), Cyl(C(K ))) is a standard Borel space, (iii) K is determined by a Borel almost disjoint family of subsets of ω.
Compact spaces K that are the subject of this theorem were considered first by Aleksandrov and Urysohn [1] and we shall call them AU-compacta, cf. [12].
There is a close correspondence between AU-compacta and uncountable almost disjoint families of subsets of natural numbers ω, explained in Sect. 2.2, and condition (iii) in Theorem 1.1 refers to this correspondence. Moreover, identifying subsets of ω with the characteristic functions, we consider any almost disjoint family on ω as a subset of the Cantor set 2 ω .
Condition (ii) in Theorem 1.1 means that there is a bijection of C(K ) onto 2 ω taking the cylindrical sets in C(K ) to Borel sets in the Cantor set and vice versa. The compact space K being separable, this is equivalent to the condition that for any countable D dense in K , the restrictions of functions from C(K ) to D form a Borel set in the countable product R D of the real line, cf. Sect. 2.1.
Using a theorem of Dodos [5] (based on a deep result of Debs [3]) we shall show that the implication (i) ⇒ (ii) in Theorem 1.1 holds true for arbitrary compact separable spaces, cf. Theorem 3.2. It is an open problem whether the reverse implication is always true (this is the case for separable linearly ordered compact spaces), cf. [13].
We shall show in Sect. 7 that even for compact (as subsets of 2 ω ) almost disjoint families, there are continuum many topological types of AU-compacta determined by these families.
However, we have the following Theorem 1.2 There are Borel almost disjoint families A α ⊂ 2 ω , α < ω 1 , such that for any Borel almost disjoint family A ⊂ 2 ω , the compact space determined by A embeds as a retract in the compact space determined by some A α .
A boundedness theorem for a transfinite index introduced in [11], proved in Sect. 6, shows that the uncountable collection {A α : α < ω 1 } in Theorem 1.2 cannot be replaced by any countable collection of Borel almost disjoint families, even if the requirement concerning retractions is dropped.
In Sect. 8 we show that any AU-compactum embeddable in B 1 (2 ω ) can be embedded in fact in the unit ball of the double dual to a separable Banach space not containing any copy of 1 , equipped with the weak * topology (it is not known if this is true for any separable compact subspace of B 1 (2 ω )).
The last section of this paper deals with important Johnson-Lindenstrauss spaces. Following [8], one can associate with each AU-compactum K a Banach space JL(K )a twisted sum of c 0 and the Hilbert space of uncountable density, cf. [17]. We shall show for the measurable spaces (JL(K ), Cyl(JL(K ))) a counterpart of Theorem 1.1.

Terminology and some background
Our topological terminology follows [6,10] and the terminology related to the descriptive set theory follows [9,10]. We denote by |E| the cardinality of a set E.

The spaces B 1 (S) and C D (K )
Given a separable metrizable space S, B 1 (S) is the space of real-valued first Baire class functions on S, equipped with the topology of pointwise convergence.
Rosenthal compacta are compact spaces which can be embedded in B 1 (ω ω ), where ω ω is the space of the irrationals, cf. [7]. Let us recall an important characterization of separable Rosenthal compacta due to Godefroy, introducing first some notation.
Let K be a separable compact space. For each countable set D dense in K , we consider i.e., the space of restrictions of continuous functions on K to D, which is a subspace of the countable product of the real line. The space C D (K ) can be identified with the topological space (C(K ), τ D ), where τ D is the topology of pointwise convergence on D.
Now, a separable compact space K is a Rosenthal compactum if, and only if, for any countable set D dense in K , the set C D (K ) is analytic, cf. [7,Theorem 4]. We shall also use a fundamental fact that Rosenthal compacta are Fréchet spaces, cf. [2].
Following Godefroy [7], one can check that, for a separable compact space K , the measurable space (C(K ), Cyl(C(K ))) is standard if and only if, for each countable set D dense in K , C D (K ) is a Borel set in R D , cf. [13, Section 2].

The Aleksandrov-Urysohn compacta
As was agreed in Sect. 1, by an AU-compactum we mean an uncountable separable compact space K whose set K of accumulation points has exactly one non-isolated point. It will be also convenient to use the following description of AU-compacta.
Let D be a countable set and let A be an uncountable almost disjoint family of infinite subsets of D, i.e., the intersection of any two distinct members of A is finite.
Let A → p A be a one-to-one correspondence between members of A and points in some fixed set disjoint from D, an let ∞ be a point distinct from points in D and any point p A . In the set we introduce a topology declaring that points of D are isolated, basic neighborhoods p A are of the form { p A } ∪ (A\F), where F ⊂ A is finite, and ∞ is the point at infinity of the locally compact space D ∪ {p A : A ∈ A}. Then K A is an AU-compactum. Conversely, given an AU-compactum K with the countable set of isolated points D, for any x ∈ K distinct from the accumulation point of K , pick an open compact The collection A is almost disjoint and the identity on D extends to a homeomorphism of K onto the compactum K A .
The space 2 D = {u ∈ R D : u(D) ⊂ {0, 1}} is the Cantor set. We shall view points of 2 D interchangeably as function from D to {0, 1} or subsets of D, identifying a set A ⊂ D with its characteristic function χ A .
Let us notice that if A is an almost disjoint family of subsets of D which is analytic in 2 D , then K A is a Rosenthal compactum, cf. Lemma 4.4. It is not clear, however, if every AU-compactum K which is Rosenthal compact is homeomorphic to the compactum K A for some analytic almost disjoint family A.
We shall also show in Remark 4.7 (answering a query of the referee) that, for any family A, the space K A embeds in B 1 (C D (K A )).
Compacta K A , for Borel and analytic families A were investigated in [11,15]. In Sect. 6, we shall use a transfinite stratification of AU-compacta which are Rosenthal compacta, introduced in [11].

The Johnson-Lindenstrauss spaces JL(K )
Let K be an AU-compactum and let K be the set of accumulation points of K .
We denote by · ∞ the supremum norm in C(K ), and let 2 (K ) be the Hilbert space of square-summable functions u : K → R with the norm The Johnson-Lindenstrauss space associated with K is the space equipped with the norm cf. [8]. Let us recall that the functions in JL(K ) vanishing on K form a copy of c 0 such that JL(K )/c 0 can be identified with 2 (K ). An illuminating account of basic properties of JL(K ) can be found in [17,Example 3].

Function spaces on separable compacta in B 1 (2 ω )
A basis for this section is the following theorem of Dodos [5]. We shall derive from the Dodos theorem the following result, more general than the implication (i) ⇒ (ii) in Theorem 1.1. Proof Let us fix a countable set D dense in K , and let C and B be as in Theorem 3.1. Let The set A is a Borel subset of 2 D . Given an A ∈ C we denote by φ A the unique accumulation point of A. We also define Let us notice that u : D → R belongs to C D (K ) if and only if the oscillation of u at any point of K is zero, i.e., for any f ∈ K and ε > 0, there is a neighborhood V of f in K such that diam u(V ∩ D) ≤ ε. Since K is Fréchet, cf. Sect. 2, for any u ∈ R D and f ∈ K the oscillation of u at f is positive if and only if, for some n ≥ 1, either there exist A, B ∈ B such that φ A = φ B = f , and |u(g) − u(h)| ≥ 1/n, whenever g ∈ A, h ∈ B, or f ∈ D and there exists an Eq. (3.2), and |u(g) − u( f )| ≥ 1/n for g ∈ A. Therefore, setting we have proj (H(n)), (3.4) where proj : R D × B × A → R D is the projection onto the first factor. We shall show that each To that end, we shall verify first that, cf. Eqs. (3.1), (3.2), is σ -compact, as the the functions φ A , φ B are of the first Baire class, cf. Eqs. (3.6) and (3.2). Therefore, the projection of D parallel to 2 ω is Borel [9, 18.18], and so is the , and since C D (K ) is also analytic, cf. [7], it is Borel.

Remark 3.3
Any metrizable σ -compact space S is a continuous image of the space ω × 2 ω , which, in turn, embeds as an open subset in 2 ω . Using this observation, one can easily verify that the space B 1 (S) embeds into B 1 (2 ω ). Therefore Theorem 3.2 holds true for any sparable compact subspace K of B 1 (S), where S is metrizable and σ -compact.

Simple compacta in B 1 (2 ω ) and a proof of Theorem 1.1
We shall show that, among AU-compacta, the compacta K A associated with Borel families A are the ones that can be embedded in B 1 (ω ω ) in a particularly simple way. This will be used in our proof of Theorem 1.1 and also, in the next section, in the proof of Theorem 1.2. Definition 4.1 Let S be a separable metrizable space. We say that a compactum K ⊂ B 1 (S) is simple if K is separable, elements of K are characteristic functions of closed sets in S, and whenever f ∈ K is non-isolated in K , | f −1 (1)| ≤ 1.
In particular, a simple uncountable compactum in B 1 (S) is an AU-compactum.
is a simple compactum and S is absolutely Borel, then K is homeomorphic to a simple compactum in B 1 (ω ω ).
Indeed, let ϕ : F → S be a continuous injective map onto S defined on a closed subset F of the irrationals ω ω , cf. [9, 13.7]. We define : The mapping is an embedding and the compactum (K ) is simple.

Lemma 4.3 Let S be absolutely Borel. Then any simple compactum in B 1 (S) can be embedded in B 1 (2 ω ).
Proof Let Q n = {t ∈ 2 ω : |t −1 (1)| ≤ n} and P = 2 ω \ ∪ n Q n . Since P is homeomorphic to ω ω , by Remark 4.2 it is enough to show that any simple compactum K ⊂ B 1 (P) can be embedded in B 1 (2 ω ). To that end, let extend each function f ∈ K to a function f ∈ B 1 (2 ω ) in the following way. We shall list (without repetitions) isolated points in K as f 1 , f 2 , . . . and let f n be the characteristic function of the set f −1 n (1)\Q n , the closure being taken in 2 ω . If f ∈ K is non-isolated, we extend f to 2 ω letting f = 0 for t / ∈ P (recall that | f −1 (1)| ≤ 1). One readily checks that the map f → f is an embedding of K into B 1 (2 ω ).
The next lemma provides the implication (iii) ⇒ (i) in Theorem 1.1.

Lemma 4.4 Let
is the the characteristic function of the singleton {A}, and finally, ϕ(∞) is the zero function on S.
Let us check that ϕ : If the family A is Borel in 2 ω , so is the space S, therefore the second assertion of the lemma follows from Lemma 4.3.
We shall now address the implication (ii) ⇒ (iii) in Theorem 1.1.

Lemma 4.5 Let K be an AU-compactum and let D be the set of isolated points of K .
If the function space C D (K ) is Borel then K is homeomorphic to a compactum K A associated with a Borel almost disjoint family A ⊂ 2 ω . Proof Let K = K \D be the set of accumulation points of K and let p be the unique non-isolated point of the space K . For any point x ∈ K \{ p}, there is a sequence d n ∈ D converging to x. Therefore the mapping ϕ x : , is of the first Baire class. Since p is the limit of the sequence of the points from K \{ p}, the same argument shows that the mapping f |D → f ( p) is of the second Baire class on C D (K ). Therefore, the set where the closure is taken in K , is a Borel set in 2 D . Let us check that Indeed, the set is Borel in 2 D × 2 D × 2 D . Moreover, for any A ∈ S, since A ∩ K is finite, there are only countably many elements of S contained in A, cf. Eq. (4.1), and hence, for any The Borel family A is almost disjoint and it remains to make sure that the compactum This, however, follows readily from (4.2) and (4.7): identifying the isolated points of K A and K , sending each p A to the unique point in A ∩ K and ∞ to p, one defines a homeomorphism of K A onto K .
Since the implication (i) ⇒ (ii) in Theorem 1.1 was already established in Theorem 3.2, summarizing the results of this, and the preceding section, we obtain a Proof of Theorem 1.1.

Remark 4.6
Our reasoning shows also that one can add to the equivalent conditions in Theorem 1.1 yet another condition: (iv) K is homeomorphic to a simple compactum in B 1 (ω ω ).
Remark 4. 7 We shall show that any AU-compactum K embeds in B 1 (C D (K )), confirming a suggestion made by the referee. Let ∞ being the only non-isolated point of K .
Identifying points in K with corresponding evaluation maps on E, one embeds K into B 1 (E). To embed K into B 1 (C D (K )), it is enough to find a closed set F ⊂ C D (K ) and a continuous surjection ϕ : F → E. Indeed, given such ϕ, one can define an embedding : B 1 (E) → B 1 (C D (K )), assigning to u ∈ B 1 (E) the function (u) which coincides with u • ϕ on F and is zero outside of F. Now, the beginning of the Proof of Lemma 4.5 shows that E is a Borel set in C D (K ) ⊂ R D . Let E * be a Borel set in R D whose trace on C D (K ) is E, and let G ⊂ R D × ω ω be a closed set projecting onto E * . Then F = G ∩ (C D (K ) × ω ω ) is a closed set in C D (K )×ω ω and let ϕ : F → E be the restriction to F of the projection.
One can complete the reasoning noticing that C D (K ) × ω ω embeds in C D (K ) as a closed subspace. To this end, one should use the well-known fact that ω ω can be embedded as a closed subset into a subspace c 0 of R ω , consisting of sequences converging to zero, and the fact C D (K ) is homeomorphic to C D (K ) × c 0 , cf. the proof of Proposition 5.1 in [4].

Proof of Theorem 1.2
We shall use a representation of AU-compacta associated with Borel almost disjoint families, described in Sect. 4.
Having checked that a countable collection C α satisfies (5.1) and (5.2), we shall consider the corresponding simple compactum defined in (5.3), cf. Eq. (5.4), We shall show that for any uncountable Borel almost disjoint family A ⊂ 2 ω , the associated compactum K A embeds in some K α as a retract.
To that end, let us replace K A by its homeomorphic copy K ⊂ B 1 (P) which is a simple compactum, cf. Lemma 4.4. Let d 1 , d 2 , . . . be isolated points of K , enumerated without repetitions, and let C n = d −1 n (1). Since ϕ defined in (5.5) is a surjection, there is s ∈ 2 ω such that C n = ϕ n (s)∩P for n = 1, 2, . . .. Since the compactum K is simple, by the remark opening this section we have s ∈ S, cf. Eq. (5.7). Let us fix an α with s ∈ S α , cf. Eq. (5.10), let σ n be the restriction of s to {0, . . . , n − 1}, and let f n be the characteristic function of the set C(σ n ) ∩ T α , cf. Eqs. (5.6) and (5.11).
We have f n (s, t) = d n (t) for any t ∈ P, and if u = s and t ∈ P, f n (u, t) = 0 for all but finitely many n. It follows that the correspondence d n → f n extends to a homeomorphism of K onto the closure L = { f n : n ∈ ω} ⊂ K α .
Moreover, denoting by χ ∅ the zero function on T α , we see that L\{χ ∅ } is open in K α \{χ ∅ }, as the functions in L\{χ ∅ } do not vanish on {s} × P, while any function from K α \L is zero on {s} × P. Therefore, sending points from K α \L to χ ∅ and not moving the points in L, one defines a retraction of K α onto L. Now, to complete the proof it is enough to use Remark 4.6 to get, for each α < ω 1 , a Borel almost disjoint family A α ⊂ 2 ω such that the compactum K A α associated with this family is homeomorphic to the simple compactum K α .

A boundedness theorem
We shall show in this section that any collection of Borel almost disjoint families satisfying the assertion of Theorem 1.2 (even without the condition concerning retractions) must be uncountable.
We shall derive this result from a certain boundedness theorem for the following transfinite index η(K ) associated to any separable Rosenthal compact space K in [11]: The result stated at the beginning of this section is an immediate consequence of the following two theorems. For a Proof of Theorem 6.2 we will need the following auxiliary fact.
Recall that, for subsets A, B of ω, we write A ⊆ * B if A\B is finite, and A = * B if A ⊆ * B and B ⊆ * A. For a set X , by [X ] <ω ([X ] ω ) we denote the family of all finite subsets of X (all subsets of X of the cardinality ω). We also denote by ≺ the order on P(ω) corresponding to the lexicographic order on 2 ω .
Proof of Theorem 6.2 For n = 1, 2, . . . we define Letting π n : A n × [ω] ω → [ω] ω be the projection onto last axis, we put for n = 1, 2, . . . The set B n is Borel. Since C n+1 = B n+1 \ ∪ n i=1 π −1 n+1 (D i ) and the map p n is injective, one can show by induction that C n and D n are also Borel and the inverse function p −1 n : D n → C n is a Borel map. Let β be the upper bound for Borel classes of C n , D n and p −1 n , n = 1, 2, . . .. Let r n : D n → P(ω) be defined by Notice that the sets A 1 , . . . , A n in (6.4) are uniquely determined by B ∈ D n . Since r n (B) is the union of the first n coordinates of p −1 n (B), the map r n is also of the Borel class ≤ β.
Let h : L → K A be a homeomorphic embedding of an AU-compactum L into K A . Denote the image h(L\L ) of the set of isolated points of L by T . It may happen that T \ω = ∅, but clearly ∞ / ∈ T . By Lemma 6.3 it is enough to show that the space C T (h(L)) ∩ 2 T is Borel of additive class ≤ β + 2.
For n = 1, 2, . . . we put Since * is an F σ -subset of P(ω) × P(ω), the set E n = s −1 n ( * ) is Borel of the class ≤ β + 1, and the union E = ∪ ∞ n=1 E n is Borel of additive class ≤ β + 2. Observe that Since the space C T (h(L)) ∩ 2 T can be identified with the family G, the above formula shows that C T (h(L)) ∩ 2 T is a Borel subset of 2 T of additive class ≤ β + 2.
In view of Proposition 6.2 it is natural to state the following Problem 6.4 Let K be a separable compact subspace of B 1 (2 ω ). Does there exist a countable ordinal α such that η(L) ≤ α for every separable compact L ⊂ K ? 7 Continuum many topological types of K A determined by compact families A Example 7.1 There exists a collection {A α : α < 2 ω } of compact almost disjoint families such that the spaces K A α and K A β are not homeomorphic for α = β.
To that end we will construct a collection of almost disjoint families B of subsets of the Cantor tree T = 2 <ω , and we shall use the notation Given x ∈ 2 ω , by B x we denote the branch {x|n : n ∈ ω} in T determined by x. For a real number t ≥ 2, is the integer part of a real number x. Let A t = {x ∈ 2 ω : x n = 0 for n / ∈ N t } and A t = {B x : x ∈ A t }. Clearly, the family A t is almost disjoint and compact (the mapping x → B x is a homeomorphism of the compact set A t onto A t .
We will prove the following.

Proposition 7.2
For every 2 ≤ t < s, the AU-compacta K A s and K A t are not homeomorphic.
Proof Assume, towards a contradiction, that ϕ : Then, it is obvious that ϕ(T ) = T , ϕ(∞) = ∞, and there exists a bijection ψ : A s → A t such that ϕ( p B x ) = p B ψ(x) for x ∈ A s . By continuity of ϕ, for every x ∈ A s , there exists a k x ∈ ω such that Applying the Baire Category theorem, one can find k such that the set C k is dense in some open set in A t . For a sequence σ ∈ 2 i , i ∈ ω, we denote the basic clopen set {y ∈ A t : y n t and σ ∈ 2 i such that the set C k is dense in U σ . One can easily verify that for some j ≥ 1 we have Indeed, since (s/t) j tends to ∞, we can find j ≥ 1 such that therefore s j−1 > t i+ j + k + 1 and it remains to recall the definition of the numbers n s l , n t l : For a sequence τ ∈ 2 j , let σˆτ denote the sequence (σ 0 , . . . , σ i−1 , τ 0 , . . . , τ j−1 ) ∈ 2 i+ j . Since C k is dense in U σ , for every τ ∈ 2 j , we can pick a point y τ ∈ C k ∩ U σˆτ . For distinct τ, τ we have y τ |n t i+ j = y τ |n t i+ j , hence |B y τ ∩ B y τ | < n t i+ j for τ, τ ∈ 2 j , τ = τ . (7.3) Observe that the cardinality of the set {x|n s j−1 : x ∈ A s } equals 2 j−1 and therefore there exist τ, τ ∈ 2 j , τ = τ such that ψ −1 (y τ )|n s j−1 = ψ −1 (y τ )|n s j−1 . It follows that |B ψ −1 (y τ ) ∩ B ψ −1 (y τ ) | ≥ n s j−1 . Therefore, by inequality (7.2), for the set A = On the other hand, for every ρ ∈ A, condition (7.1) implies that ϕ(ρ) ∈ B y τ ∩ B y τ , a contradiction with inequality (7.3).

Embeddings in B 1 (2 ω ) related to the Odell-Rosenthal theorem
Important examples of separable Rosenthal compact spaces are provided by the following theorem of Odell and Rosenthal [14], where (B X * * , w * ) is the unit ball of the second dual X * * of a Banach space X , equipped with the weak * topology: Actually, in this case the restriction of an x * * ∈ B X * * to B X * gives an embedding of (B X * * , w * ) into B 1 ((B X * , w * )), hence (B X * * , w * ) embeds into B 1 (2 ω ). Observe that (B X * * , w * ) has a dense subset B X of elements continuous on (B X * , w * ), therefore η((B X * * , w * )) = 2, see [11,Thm. 2.3] and, cf. Sect. 6.
It is not clear which separable compact subspaces of B 1 (2 ω ) can be embedded into (B X * * , w * ) for some separable Banach space X not containing any isomorphic copy of 1 . The following result provides a useful criterium to that effect.
Theorem 8.2 [11] Let L ⊂ B 1 (2 ω ) be a compact set with L ∩ C(2 ω ) dense in L. Then there exists a separable Banach space X , such that X does not contain any isomorphic copy of 1 , and L embeds into (B X * * , w * ).
This result was proved in [11,Sec. 6.2] for the some special family of subspaces L of B 1 (2 ω ), but the proof used only the properties of L described in the assumptions of the above theorem. Let us notice that the assumptions of Theorem 8.2 yield the separability of L and this theorem suggests the following problem.
We shall show that this is the case for Aleksandrov-Urysohn compacta.

Theorem 8.4 Let A be a Borel almost disjoint family of subsets of ω.
There exists a compact subspace L of B 1 (2 ω ) such that the intersection L ∩ C(2 ω ) is dense in L and the space K A embeds in L.
Proof As shown in the proofs of Lemma 4.4 and Remark 4.2, there is a closed subset F of ω ω such that K A is homeomorphic to a simple compactum K in B 1 (F) whose isolated points are characteristic functions of closed-and-open sets in F. let T be the set of all isolated points of K . Take any embedding ϕ of F into 2 ω and let ψ be the embedding of F into 2 ω × 2 T defined by The image ψ(F) is a G δ -subset of 2 ω × 2 T , and therefore we can write 2 ω × 2 T = ∪ n∈ω A n , where A n are closed in 2 ω × 2 T . Let { f n : n ∈ ω} be an enumeration without repetitions of T . For every n ∈ ω, we define the following open subset of 2 ω × 2 T : Let g n be the characteristic function of G n . Observe that g n is an extension of of the function f n • ψ −1 : ψ(F) → {0, 1} over 2 ω × 2 T . Therefore, one can easily verify that the closure M of {g n : n ∈ ω} in B 1 (2 ω × 2 T ) is homeomorphic to K (cf. the Proof of Lemma 4.4) and, for accumulation points g of M, we have |g −1 (1)| ≤ 1. For each n ∈ ω, if g n is continuous, we define g k n = g n for k ∈ ω. Otherwise, we take a sequence (g k n ) k∈ω of continuous functions from 2 ω × 2 T into {0, 1} converging pointwise to g n and such that g k n ≤ g n for k ∈ ω. Let L be the closure of {g k n : k, n ∈ ω} in 2 2 ω ×2 T . We will show that L ⊂ B 1 (2 ω × 2 T ).
It is enough to verify that if h is an accumulation point of L and |h −1 (1)| ≥ 2, then h = g n for some n ∈ ω. Fix such h and x, y ∈ h −1 (1), x = y, and consider the closed-and-open neighborhood U = { f ∈ 2 2 ω ×2 T : f (x) = f (y) = 1} of h in 2 2 ω ×2 T . Since no function in U is an accumulation point of M, there is a finite set S ⊂ ω such that g n / ∈ U for n ∈ ω\S. Since g k n ≤ g n , we also have g k n / ∈ U for n ∈ ω\S, k ∈ ω. Therefore h = g n for some n ∈ S.

Johnson-Lindenstrauss spaces with standard cylindrical σ -algebras
In this section we prove a counterpart of Theorem 1.1 for twisted sums of c 0 and the Hilbert space of uncountable density, defined by Johnson and Lindenstrauss [8], cf. Sect. 2.3. In the other direction, let us assume that K = K A is determined by a Borel almost disjoint family A ⊂ 2 D , cf. Sect. 2.2, and let We shall check that also each set proj C M m,n, p is Borel in R D , (9.6) proj C being the projection onto the first coordinate. This will follow from the Lusin theorem and (9.5), once we make sure that the restriction proj C |M m,n, p is countable-to-one. (9.7) Let u ∈ C and u = f |D, f ∈ C(K ). Since f vanishes at ∞, cf. Eq. we get (9.2) from (9.4) and (9.6). Let us consider, for each d ∈ D, the evaluation functional e d ∈ JL(K ) * , and let Cyl D (JL(K )) be the σ -algebra in JL(K ) generated by the functionals e d .
The restriction map f → f |D is an isomorphism between the measurable space (JL(K ), Cyl D (JL(K ))) and the space JL D (K ) equipped with the σ -algebra of Borel sets, and therefore, by Eq. (9.2), (JL(K ), Cyl D (JL(K ))) is standard. (9.10) It is enough to make sure that Cyl D (JL(K )) = Cyl(JL(K )). (9.11) For any A ∈ A, the evaluation functional e A ( f ) = f ( p A ), f ∈ JL(K ), is the pointwise limit e A = lim d∈A e d , cf. Eq. (9.8) and hence e A is Cyl D (JL(K ))-measurable.
Since any functional ϕ ∈ JL(K ) * can be represented as ϕ = d∈D x(d)e d + A∈A y(A)e A , where x : D → R is summable and y : A → R is square-summable, cf. [17], the functional ϕ is Cyl D (JL(K ))-measurable. This demonstrates (9.11) and ends the proof.