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HEREDITARILY HUREWICZ SPACES AND ARHANGEL’SKI˘I SHEAF AMALGAMATIONS 2 BOAZ TSABAN ANDLYUBOMYRZDOMSKYY 1 0 2 Abstract. A classical theorem of Hurewicz characterizes spaces with n the Hurewicz covering property as those having bounded continuous a images intheBaire space. Wegiveasimilar characterization forspaces J X which havetheHurewicz property hereditarily. 3 WeproceedtoconsidertheclassofArhangel’ski˘ıα1spaces,forwhich everysheafatapointcanbeamalgamatedinanaturalway. LetCp(X) ] denote the space of continuous real-valued functions on X with the N topology of pointwise convergence. Our main result is that Cp(X) is G an α1 space if, and only if, each Borel image of X in the Baire space . is bounded. Using this characterization, we solve a variety of problems h posed in theliterature concerning spaces of continuousfunctions. t a m [ 2 v 1 1 Contents 2 0 1. Introduction 2 . 2. A characterization of hereditarily Hurewicz spaces 3 4 0 3. Bounded Borel images 6 0 4. Applications 11 1 4.1. QN spaces 11 : v 4.2. Convergent sequences of Borel functions 12 i X 4.3. Almost continuous functions 13 r 4.4. wQN spaces and the Scheepers Conjecture 13 a 4.5. QN spaces and M spaces 14 4.6. wQN spaces 15 ∗ 4.7. Bounded-ideal convergence spaces 15 4.8. Bounded Baire-class α images 17 5. Closing the circle: Continuous bounded images again 17 Acknowledgments 19 References 19 1991 Mathematics Subject Classification. Primary: 37F20; Secondary 26A03, 03E75. Keywordsandphrases. pointwiseconvergence,point-cofinitecovers,α1,eventualdom- inance, Hurewicz property,selection principles, QN sets, ideal convergence. Partially supported by theKoshland Center for Basic Research. 1 2 BOAZTSABANANDLYUBOMYRZDOMSKYY 1. Introduction We are mainly concerned with spaces X which are (homeomorphic to) sets of irrational numbers, and we recommend adopting this restriction for clarity. Our results (and proofs) apply to all topological spaces X in which each open set is a union of countably many clopen sets, and the spaces considered are assumed to have this property.1 Fix a topological space X. Let A,B be families of covers of X. The space X may or may not have the following property [35]. U (A,B): Whenever U ,U ,··· ∈ A and none contains a finite sub- fin 1 2 cover, there exist finite sets F ⊆ U , n ∈ N, such that { F : n ∈ n n n N} ∈ B. S Let O denote the collection of all countable open covers of X.2 A cover U of X is point-cofinite if U is infinite and each x ∈ X, is a member of all but finitely many members of U.3 Let Γ denote the collection of all open point-cofinite covers of X. Motivated by studies of Menger [26], Hurewicz [19] introduced the Hurewicz property U (O,Γ). fin Hurewicz [19] essentially obtained the following combinatorial characterization of U (O,Γ) (see Rec law [29]). For f,g ∈ NN, f ≤∗ g means fin f(n) ≤ g(n) for all but finitely many n. A subset Y of NN is bounded if there is g ∈ NN such that f ≤∗ g for all f ∈ Y. Theorem 1 (Hurewicz). X satisfies U (O,Γ) if, and only if, every con- fin tinuous image of X in NN is bounded. This characterization has found numerous applications—see [38, 24, 41] and references therein. We give a similar characterization for hereditarily Hurewicz spaces, that is, spaces X such that each subspace of X satisfies U (O,Γ). fin The property of being hereditarily Hurewicz was studied in, e.g., [15, 28, 27]. Rubinintroducedapropertyof subsetsof Rsuchthattheexistence ofa setwiththispropertyisequivalenttothepossibilityofacertainconstruction ofbooleanalgebras[31]. Miller[27]provedthattheRubin spaces areexactly the hereditarily Hurewicz spaces. The property of being hereditarily Hurewicz also manifests itself as follows: A set X ⊆ R is a σ′ space [32] if for each F set E, there is an F set σ σ F such that E∩F = ∅ and X ⊆ E∪F. This property was effectively used in studies of generalized metric spaces [13]. Recently, Sakai proved that X is a σ′ space if, and only if, X is hereditarily Hurewicz (Theorem 6 below). There exist additional classes of hereditarily Hurewicz spaces in the literature. We describe some of them. 1Every perfectly normal space (open sets are Fσ) with upper inductive dimension 0 (disjoint closed sets can be separated by a clopen set) has the required property. Thus, thespaces considered in thereferences also havethe required property. 2IfX is Lindelöf, we can consider arbitrary open covers of X. 3Traditionally, point-cofinite covers were called γ-covers [17]. HUREWICZ AND SHEAF AMALGAMATIONS 3 A topological space is Fréchet if each point in the closure of a subset of thespace is a limit of aconvergent sequence of points from that subset. The following concepts, dueto Arhangel’ski˘ı [1, 2], are important in determining when a product of Fréchet spaces is Fréchet. Let Y be a general topological space (not necessarily Lindelöf or zero-dimensional). A sheaf at a point y ∈ Y is a family of sequences, each converging to y. To avoid trivialities, we consider only sequences of distinct elements. We say that a countable set A converges to y if some (equivalently, each) bijective enumeration of A converges to y. Thespace Y is an α space if for each y ∈Y, each countable 1 sheaf {A : n ∈ N} at y can be amalgamated as follows: There are cofinite n subsets B ⊆ A , n ∈ N, such that the set B = B converges to y. The n n n n references dealing withα spaces are too numeroustobelisted here; see [40] 1 S and the references therein for a partial list. Fix a space X. The space C (X) is the family of all continuous real- p valued functions on X, viewed as a subspace of the Tychonoff product RX. A sequence of results by Bukovsky´–Rec law–Repicky´ [10], Rec law [30], Sakai [33], and Bukovsky´–Haleˇs [9], culminated in the result that if C (X) is an p α space, then X is hereditarily Hurewicz. Our main result is that if C (X) 1 p is an α space, then each Borel image of X in NN is bounded. It is easy 1 to see that the converse implication also holds, and we obtain a powerful characterization of spaces X such that C (X) is an α space. p 1 Historically, the realization that if C (X) is an α space then X is hered- p 1 itarily Hurewicz goes through QN spaces [10]: Let Y be a metric space. A function f : X → Y is a quasi-normal limit of functions f : X → Y if n there are positive reals ǫ , n ∈ N, converging to 0 such that for each x ∈ X, n d(f (x),f(x)) < ǫ for all but finitely many n. A topological space X is n n a QN space if whenever 0 is a pointwise limit of a sequence of continuous real-valued functions on X, we have that 0 is a quasi-normal limit of the same sequence. QN spaces are studied in, e.g., [10, 30, 36, 28, 11, 33, 9]. In [33, 9] it was shown that X is a QN space if, and only if, C (X) is an p α space. Thus, QN spaces are also characterized by having bounded Borel 1 images in NN. We use our main theorem to show that quite a few additional properties studied in the literature are equivalent to having bounded Borel images in NN, and consequently solve a variety of problems posed in the literature. To make the paper self-contained and accessible to a wide audience, we supply proofs for all needed results. Often, our proofs of known results are slightly simpler than those available in the literature. 2. A characterization of hereditarily Hurewicz spaces N Let N = N∪{∞} be the one-point compactification of N, and endow N N withtheTychonoffproducttopology. Anelementf ∈ N iseventually finite if there is m such that f(n) < ∞ for all n ≥ m. Let EF be the subspace of 4 BOAZTSABANANDLYUBOMYRZDOMSKYY N N N consisting of all eventually finite elements of N . The partial order ≤∗ extends to EF in the natural way. Theorem 2. X is hereditarily U (O,Γ) if, and only if, every continuous fin image of X in EF is bounded. Proof. (⇒) Assume that Ψ : X → EF is continuous. For each n, define the following (G ) subset of Ψ[X]: δ G = {f ∈ Ψ[X] : (∀m ≥n) f(m)< ∞}. n Let Tn : G → NN be the shift transformation defined by Tn(f)(m) = n f(m+n) for all m. For each n, X = Ψ−1[G ] ⊆ X, and therefore satisfies U (O,Γ). By n n fin Theorem 1, Tn[Ψ[X ]] = Tn[G ] is a bounded subset of NN. Thus, G is a n n n bounded subset of EF, and therefore so is Ψ[X] = G . n n (⇐) First, note that Hurewicz’s Theorem 1 and our assumption on X imply that X satisfies U (O,Γ). S fin Lemma 3. If each G subset of X satisfies U (O,Γ), then X is hereditarily δ fin U (O,Γ). fin Proof. Let Y ⊆ X. Assume that U , n ∈ N, are covers of Y by open subsets n of X, which do not contain finite subcovers. Each U is an open cover of n G = U ⊇ Y, and has no finite subcover of G. As G is a G subset of n n δ X, it satisfies U (O,Γ). Thus, there are finite F ⊆ U , n ∈ N, such that fin n n { FT:nS∈ N} is a point-cofinite cover of G, and therefore of Y. (cid:3) n SAssume that G is a Gδ subset of X. Lemma 4 (Sakai [33]). For each G subset G of X, there is an open point- δ cofinite cover {U :n ∈ N} of X such that G = U . n n n Proof. Gc = nCn with each Cn closed. If A isTclosed and B is open, then B is a union of countably many disjoint clopen sets, and therefore A∩B is a S union of countably many disjoint closed sets. Thus, each of the disjoint sets C \(C ∪...,C ), n ∈ N, is a union of countably many disjoint closed n 1 n−1 sets. Hence, Gc = C˜ where the sets C˜ are closed and disjoint, and n n n therefore G = C˜c, where {C˜c : n ∈ N} is an open point-cofinite cover of n nS n X. (cid:3) T So, let {U : n ∈ N} be an open point-cofinite cover of X such that n G = U . For each n, let U = Cn, a union of disjoint clopen sets. n n n m m Define Ψ : X → EF by T S m m ∈ N,x ∈ Cn Ψ(x)(n) = m (∞ x ∈/ Un As {U : n ∈ N} is a point-cofinite cover of X, Ψ(x) is eventually finite for n each x ∈ X. HUREWICZ AND SHEAF AMALGAMATIONS 5 The function Ψ is continuous: A basic open set in EF has the form V n n such that there are finite I ,I ⊆ N and elements m , n ∈ I ∪I , for which: 0 1 n 0 1 Q For each n ∈ I , V = {m }, for each n ∈ I , V = {m ,m +1,...}∪{∞}, 0 n n 1 n n n and for each n ∈/ I ∪I , V = N∪{∞}. Now, 0 1 n Ψ−1 V = Cn ∩ X \ Cn n mn   k "nY∈N # n\∈I0 n\∈I1 k<[mn is open.    Thus, Ψ[X] is bounded by some g ∈ NN. Now, G = {x ∈ X : (∀n) Ψ(x)(n) < ∞}= Ψ−1[{f ∈ NN :f ≤∗ g}]. The set {f ∈ NN : f ≤∗ g} is an F subset of EF. Indeed, let {g : n ∈ σ n N} enumerate all elements of NN which are eventually equal to g. Then {f ∈ NN : f ≤∗ g} = {f ∈ EF : f ≤ g }. Thus, G is an F subset of n n σ X. As U (O,Γ) is hereditary for closed subsets and preserved by countable fin unions, G satisfies U (SO,Γ). (cid:3) fin Recall that a topological space X is a σ space if each G subset of X is δ an F subset of X. The proof of Theorem 2 actually shows that (2 ⇒ 3), σ (2 ⇒ 1), (3 ⇒ 4), and (4 ⇒ 2) in the following theorem (and therefore establishes it). Theorem 5. The following are equivalent: (1) X is hereditarily U (O,Γ). fin (2) Each G subset of X satisfies U (O,Γ). δ fin (3) Every continuous image of X in EF is bounded. (4) X satisfies U (O,Γ) and is a σ space. (cid:3) fin The implication (1 ⇒ 4) in Theorem 5 was first proved by Fremlin and Miller [15]. The implication (4 ⇒ 1) can be alternatively deduced from Theorem 3.12 of [11] and Corollary 10 of [8]. An additional equivalent formulation was discovered by Sakai. Recall the definition of σ′ space from the introduction (page 2). Theorem 6 (Sakai). Let X ⊆ R. X is a σ′ space if, and only if, X is hereditarily U (O,Γ). fin Proof. This follows from Theorem 5.7 of [22]: X satisfies U (O,Γ) if, and fin only if, for each G set G ⊆ R containing X, there is an F set F ⊆ R such δ σ that X ⊆ F ⊆ G. (⇒) As being a σ′ space is hereditary, it suffices to show that X satisfies U (O,Γ). Indeed, for each G set G ⊆ R containing X, let E = R \ G, fin δ and take an F set F ⊆ R disjoint from E such that X ⊆ E ∪ F. Then σ X ⊆ F ⊆ G. (⇐) Let E ⊆ R be F . As X\E satisfies U (O,Γ) and is a subset of the σ fin G set R\E, there is an F set F ⊆ R such that X\E ⊆ F ⊆R\E. Then δ σ E∩F = ∅ and X ⊆ E ∪F. (cid:3) 6 BOAZTSABANANDLYUBOMYRZDOMSKYY ToindicatethepotentialusefulnessofTheorem2,weuseittogiveslightly moredirectproofsoftwoknowntheorems. RecallthedefinitionofQNspaces from the introduction (page 3). Theorem 7 (Rec law [30]). If X is a QN space, then X is hereditarily U (O,Γ). fin Proof. Let Y ⊆ EF be a continuous image of X. Then Y is a QN space. By Theorem 2, it suffices to show that Y is bounded. For each n, and each y ∈ Y, define 1 f (y) = , n miny−1(n) using the natural conventions that min∅= ∞ and 1/∞ = 0. lim f (y) = 0 n n for all y ∈ Y. As Y is a QN space, there are positive ǫ , n ∈N, dominating n this convergence. For each k, let Y = {y ∈ Y :(∀n ≥ k) f (y) < ǫ }. k n n Y = Y . We will show that each Y is bounded. k k k Fix k. Take an increasing g ∈ NN such that g(1) = k and for each n, S ǫ < 1/n for all m ≥ g(n). Then Y is bounded by g: Let y ∈ Y . Fix m k k n such that y(n) < ∞. If y(n) ≤ k, then y(n) ≤ g(1) ≤ g(n). Otherwise, y(n)> k, and since y ∈ Y , f (y) <ǫ . Thus, k y(n) y(n) 1 1 ≤ = f (y) < ǫ , n miny−1(y(n)) y(n) y(n) and therefore y(n) cannot be greater than g(n). (cid:3) Theorem 8 (Rec law [30]). If X is QN space, then X is a σ space. Proof. Theorems 5 and 7. (cid:3) The following sections give a deeper reason for the last two theorems. 3. Bounded Borel images Our main goal in this section is to establish the equivalence in the following Theorem 9. The implication (2 ⇒ 1) in this theorem is Proposition 9 of Scheepers [36]. The implication (1 ⇒ 2) is the more difficult one, and will be proved in a sequence of related results. Theorem 9. The following are equivalent: (1) C (X) is an α space. p 1 (2) Each Borel image of X in NN is bounded. Proof. (2 ⇒ 1) Consider a sheaf {A : n ∈ N} at f ∈ C (X). For each n p n, enumerate A = {fn : m ∈ N} bijectively. Define a Borel function n m Ψ :X → NN by Ψ(x)(n) = min{k :(∀m ≥ k) |fn(x)−f(x)|≤ 1/n}. m HUREWICZ AND SHEAF AMALGAMATIONS 7 Let g ∈ NN bound Ψ[X], and take the amalgamation B = {fn : m ≥ n m g(n)}. Then B converges to f. S (1 ⇒ 2)AssumethatC (X)isanα space. ThenthesubspaceC (X,{0,1}) p 1 p of C (X), consisting of all continuous functions f : X → {0,1}, is an α p 1 space.4 Each element of C (X,{0,1}) has the form χ , the characteristic p U functionof aclopensetU ⊆ X. Immediately fromthedefinition,asequence χ of elements of C (X,{0,1}) converges pointwise to the constant func- Un p tion 1 if, and only if, {U : n ∈ N} is a clopen point-cofinite cover of X. n This gives the following, which is due to Bukovsky´–Haleˇs (cf. [9, Theorem 17]), and independently Sakai (cf. [33, Theorem 3.7]). Lemma 10. The following are equivalent: (1) C (X,{0,1}) is an α space; p 1 (2) For each family {U : n ∈ N} of pairwise disjoint clopen point- n cofinite covers of X, there are cofinite V ⊆ U , n ∈ N, such that n n V is a point-cofinite cover of X. (cid:3) n n A funSction f with domain X is a discrete limit of functions fn, n ∈ N, if for each x ∈X, f (x) = f(x) for all but finitely many n. n Each bijectively enumerated family U = {U : n ∈ N} of subsets of a set n X induces a Marczewski map U :X → P(N) defined by U(x) = {n ∈ N :x ∈U } n for each x ∈X. The main step in our proof is the following. Lemma 11. Assume that C (X,{0,1}) is an α space, and U = {U : p 1 n n ∈ N} is a bijectively enumerated family of open subsets of X. Then the Marczewski map U :X → P(N) is a discrete limit of continuous functions. Proof. First, consider the case where for each n, U is not clopen. n For each n, write U as a union Cn of nonempty disjoint clopen n m m sets. We may assume that the partitions are disjoint: Inductively, for each n = 2,3,..., consider the elements CSn, m ∈ N, of the nth partition. For m each m, if Cn appears in the partition of U for some k < n, merge (in the m k nth partition) Cn with some other element of the nth partition. Continue m inthismanneruntilthenthpartition is disjointfromall previouspartitions. Thus, the families U = {(Cn)c : m ∈ N} are disjoint clopen point- n m cofinite covers of X. By Lemma 10, there are k , n ∈ N, and subsets n V = {(Cn)c : m ≥ k } ⊆ U , n ∈ N, such that V is a point-cofinite n m n n n n cover of X. In other words, S ∞ V = (Cn)c : n∈ N  m  m\=kn  is a point-cofinite cover of X.   4Infact,bythemethodsofGerlits–Nagy[18],theconverseimplicationalsoholds. This fact will not beused in our proof. 8 BOAZTSABANANDLYUBOMYRZDOMSKYY For each n,m, let max{m,kn} Un = Cn. m i i=1 [ For each m, define Ψ :X → P(N) by m Ψ (x) ={n :x ∈ Un}. m m As each Un is clopen, Ψ is continuous. It remains to prove that, viewed m m as a Marczewski map, U is a discrete limit of the maps Ψ , m ∈ N. m Fix x ∈ X. Let N be such that x ∈ ∞ (Cn)c for all n ≥ N. For each m=kn m n < N with x ∈ U , let m be such that x ∈ Un . Set M = max{m :n < n n T mn n N}. Fix m ≥ M. We show that x ∈ Un if, and only if, x ∈ U . One m n direction follows from Un ⊆ U . To prove the other direction, assume that m n x ∈ U , and consider the two possible cases: If n < N, then x ∈ Un n m because m ≥ M ≥ m , and we are done. Thus, assume that n ≥ N. Then n x ∈ ∞ (Cn)c. As x ∈U = Cn, it follows that x ∈ kn−1Cn ⊆ Un. i=kn i n m m i=1 i m Thus, for each x ∈X there is M such that Ψ (x)= U(x) for all m ≥ M. m T S S This completes the proof in the case that no U is clopen. n For the remaining case, let I ⊆ N be the set of all n such that U is n not clopen. The previous case shows that U = {U : n ∈ I}, viewed as I n a Marczewski function from X to P(I), is a discrete limit of continuous functions Ψ : X → P(I). m For each m, define Φ : X → P(N) by m Φ (x) = {n :(n ∈I and n∈ Ψ (x)) or (n ∈/ I and x ∈U )}. m m n Then U is a discrete limit of the continuous functions Φ , m ∈N. (cid:3) m As X satisfies item (2) of Lemma 10, it satisfies U (O,Γ): Refine each fin given cover to a clopen cover, turn it to a clopen point-cofinite cover by taking finite unions, and make the point-cofinite covers disjoint. Assume that U is a countable family of open subset of X. By Lemma 11, the Marczewski map U : X → P(N) is a discrete limit of continuous functions Ψ . n Clearly, every discrete limit is a quasi-normal limit. The proof of [10, Theorem 4.8] actually establishes the following. Lemma 12. Assume that P is a property of topological spaces, which is preserved by taking closed subsets, continuous images and countable unions. IfX hastheproperty P andΨ : X → Y isaquasi-normallimitofcontinuous functions into a metric space Y, then Ψ[X] has the property P. Proof. Let Ψ , n ∈ N, be continuous functions as in the premise of the n lemma,andletǫ ,n ∈ N,beasinthedefinitionofquasi-normalconvergence. n For each k, X ={x ∈ X :(∀n,m ≥ k) d(Ψ (x),Ψ (x)) ≤ ǫ +ǫ } k n m n m HUREWICZ AND SHEAF AMALGAMATIONS 9 is a closed subset of X, and the functions Ψ converge to Ψ uniformly on n X . Thus, Ψ is continuous on X , and therefore Ψ[X ] has the property P. k k k Now, X = X , and therefore Ψ[X] = Ψ[X ] has the property k k k k P. (cid:3) S S It follows that for each countable family U of open subsets of X, U[X] satisfies U (O,Γ). fin Let F,B denote the families of all countable closed and all countable Borel covers of X, respectively. Similarly, let F ,B denote the families of Γ Γ all countable closed point-cofinite covers of X and all Borel point-cofinite covers of X. Following isastrikingresultof Bukovsky´, Rec law, andRepicky´ [10]. In their terminology, it tells that the family of closed subsets of X is weakly distributive if, and only if, the same holds for the family of Borel subsets of X. In the language of selection principles, this result has the following compact form. Lemma 13 (Bukovsky´–Rec law–Repicky´ [10]). U (F,F )= U (B,B ). fin Γ fin Γ Proof. Assume that X satisfies U (F,F ). We first show that X is a σ fin Γ space [10, Theorem 5.2]. Assume that G = U where for each n, U ⊇ U are open subsets n n n n+1 of X. Write, for each n, T U = Cn, n m m∈N [ where for each m, Cn ⊆ Cn are closed subsets of X. We may assume m m+1 that the closed cover {Cn ∪(X\U ): m ∈ N} of X has no finite subcover.5 m n As X satisfies U (F,F ) and each given cover is monotone, there are m , fin Γ n n ∈ N, such that {Cn ∪(X \U ) : n ∈ N} is a closed point-cofinite cover mn n of X. For each k define ∞ Z = Cn . k mn n=k \ Then each Z is a closed subset of X, and G = Z is F . This shows k k k σ that X is a σ space. Now, assume that U ∈ B, n∈ N. Then for eachSn, each element of U is n n F and can therefore be replaced by countably many closed sets. Applying σ U (F,F ) to the thus modified covers, we obtain a cover in F . For each fin Γ Γ n, extend each of the finitely many chosen elements of the nth cover to an F set from the original cover U , to obtain an element of B chosen in σ n Γ accordance with the definition of U (B,B ).6 (cid:3) fin Γ Lemma 14. The following are equivalent: (1) X satisfies U (B,B ); fin Γ 5If there are infinitely many n for which there is some mn with Cmnn = Un, then G = T Cn is closed and we are done. Otherwise, we can ignore finitely many n and n mn assume that there are no n,m such that Cmn =Un contains G. 6This argument, in more general form, appears in [11, Theorem 2.1]. 10 BOAZTSABANANDLYUBOMYRZDOMSKYY (2) For each countable family U of open subsets of X, U[X] satisfies U (O,Γ); fin (3) For each countable family C of closed subsets of X, C[X] satisfies U (O,Γ). fin Proof. (2 ⇔ 3) Use the auto-homeomorphism of P(N) defined by mapping a set to its complement. (1 ⇒ 2) The Marczewski map U : X → P(N) is Borel. It is easy to see that U (B,B ) is preserved by Borel images [39]. Thus, U[X] satisfies fin Γ U (B,B ), and in particular U (O,Γ). fin Γ fin (3 ⇒ 1) By Lemma 13, it suffices to show that X satisfies U (F,F ). fin Γ For each C = {C : n ∈ N} ∈ F which does not contain a finite subcover, n { C : n ∈ N} ∈ F . Thus, U (F,F ) = U (F ,F ),7 and we prove m≤n m Γ fin Γ fin Γ Γ the latter property. SLet C = {Cn : m ∈ N}, n ∈ N, be bijectively enumerated closed point- n m cofinite covers of X which do not contain finite subcovers. We may assume that these covers are pairwise disjoint [35]. Let C = C , and consider the Marczewski map C : X → P(N × N) n n defined by S C(x) = {(n,m) : x ∈ Cn} m forallx ∈ X. Foreach(n,m),O = {A⊆ N×N :(n,m) ∈ A}isanopen (n,m) subset of P(N×N), and for each n, U = {O :m ∈ N} is an open cover n (n,m) of C[X] that does not contain a finite subcover. As C[X] satisfies U (O,Γ), fin there are k , n ∈ N, such that { O : n ∈ N} is a point-cofinite n m<kn (n,m) cover of C[X]. Then { Cn : n ∈ N} is a point-cofinite cover of X (it m<kn mS is infinite because X does not appear there as an element).8 (cid:3) S ByLemma14,X satisfiesU (B,B ). Itremainstoobservethefollowing. fin Γ For the reader’s convenience, we reproduce the proof of the implication needed in the present proof. Lemma 15 (Bartoszyn´ski–Scheepers [4]). X satisfies U (B,B ) if, and fin Γ only if, each Borel image of X in NN is bounded. Proof. (⇒) Assume that Y ⊆ NN is a Borel image of X. Then Y satisfies U (B,B ). BytakingtheimageofY underthecontinuousmappingf(n)7→ fin Γ f(1)+···+f(n) defined on NN , we may assume that all elements in Y are nondecreasing. We first consider the trivial case: There is an infinite I ⊆ N such that for each n ∈ I, F ={f(n) :f ∈ Y} is finite. For each n, let m ∈ I be minimal n such that n≤ m, and define g(n) = maxF . Then Y is bounded by g. m Thus, assume that there is N such that for each n ≥ N, {f(n) : f ∈ Y} is infinite. For all n,m, consider the open set Un = {f ∈ Y : f(n) ≤ m}. m 7This statement holds in a more general form [22]. 8The argument is standard: For each finite family of proper subsets of X, there is a finitesubset of X not contained in any memberof thisfamily.