Null sets and combinatorial covering properties
aa r X i v : . [ m a t h . GN ] J un NULL SETS AND COMBINATORIAL COVERING PROPERTIES
PIOTR SZEWCZAK AND TOMASZ WEISS
Abstract.
A subset of the Cantor cube is null-additive if its algebraic sum with any nullset is null. We construct a set of cardinality continuum such that: all continuous images ofthe set into the Cantor cube are null-additive, it contains a homeomorphic copy of a set thatis not null-additive, and it has the property γ , a strong combinatorial covering property. Wealso construct a nontrivial subset of the Cantor cube with the property γ that is not nulladditive. Set-theoretic assumptions used in our constructions are far milder than used earlierby Galvin–Miller and Bartoszy´nski–Rec law, to obtain sets with analogous properties. Wealso consider products of Sierpi´nski sets in the context of combinatorial covering properties. Introduction
Let N be the set of natural numbers and P( N ) be the power set of N . We identify eachset in P( N ) with its characteristic function, an element of the Cantor cube { , } N ; in thatway we introduce topology in P( N ). The Cantor space P( N ) with the symmetric differenceoperation ⊕ is a topological group; this operation coincides with the addition modulo 2 in { , } N . A set X ⊆ P( N ) is null-additive in P( N ) if for any null set Y ⊆ P( N ) the set X ⊕ Y := { x ⊕ y : x ∈ X, y ∈ Y } is null. In an analogous way, define null-additive subsetsof the real line with the addition + as a group operation. As we see in the forthcomingTheorem 2.2, it is relatively consistent with ZFC that null-additive subsets of P( N ) are notpreserved by homeomorphisms into P( N ). Subsets of the real line whose all continuous imagesinto the real line are null-additive were considered by Galvin and Miller [4]; to this end theyused combinatorial covering properties.By space we mean a Tychonoff topological space. A cover of a space is a family of propersubsets of the space whose union is the entire space. An open cover of a space is a cover whosemembers are open subsets of the space. A cover of a space is an ω -cover if each finite subset ofthe space is contained in a set from the cover and it is a γ -cover if it is infinite and each pointof the space belongs to all but finitely many sets from the cover. A space has the property γ ifevery open ω -cover of the space contains a γ -cover. This property was introduced by Gerlitsand Nagy in the context of local properties of functions spaces [5]. They proved that a space X has the property γ if and only if the space C p ( X ) of all continuous real-valued functionsdefined on X with the pointwise convergence topology is Fr´echet–Urysohn , i.e., each pointin the closure of a subset of C p ( X ) is a limit of a sequence from the set [5, Theorem 2].Galvin and Miller observed that for a subset of the real line X with the property γ and ameager subset of the real line Y , the set X + Y := { x + y : x ∈ X, y ∈ Y } is meager. Theypointed out that they were unable to prove an analogous statement for null sets, and thusthey introduced a formally stronger property than γ [4, p. 152]. For a natural number n , anopen cover of a space is an n -cover if each n -elements subset of the space is contained in amember of the cover. A space X has the property strongly γ if there is an increasing sequence Mathematics Subject Classification.
Key words and phrases. null sets, null-additive sets, selection principles, γ -property. f ∈ N N such that for each sequence U , U , . . . where U n is an f ( n )-cover of X , there are sets U ∈ U , U ∈ U , . . . such that the family { U n : n ∈ N } is a γ -cover of X .Galvin and Miller proved that any subset of the real line with the property strongly γ isnull-additive. The property strongly γ is preserved by continuous mappings, and thus anycontinuous image of a set with the property strongly γ , into the real line, is null-additive [4,Theorem 7]. Under Martin Axiom, Galvin and Miller constructed a subset of P( N ) of car-dinality continuum with the property strong- γ [4, Theorem 8]. In 1996, under some set-theoretic assumption, Bartoszy´nski and Rec law constructed a subset of P( N ) of cardinalitycontinuum with the property γ that is not null-additive (in particular, they separated theproperties γ and strongly γ ).Assuming only an equality between some cardinal characteristics of the continuum, weconstruct a subset of P( N ) of cardinality continuum with the property γ whose all continuousimages into P( N ) are null-additive and it contains a homeomorphic copy of a set that isnot null-additive. We also weaken a set-theoretic assumption in the result of Bartoszy´nskiand Rec law. In the both cases we use combinatorial methods of construction of subsetsof P( N ) with the property γ invented by Tsaban ([10, Theorem 3.6], [11, Theorem 6]) anddeveloped by W ludecka and the first named author [16]. We also use a combinatorial coveringcharacterization of null-additive subsets of P( N ) given by Zindulka [20].2. Null-additive sets with the property γ Let [ N ] ∞ be the family of infinite subsets of N . Each set in [ N ] ∞ we identify with anincreasing function from N N . Depending on the interpretation, points of [ N ] ∞ are referred toas sets or functions. For natural numbers n, m with n < m , define [ n, m ) := { i ∈ N : n ≤ i < m } . For sets a and b , we write a ⊆ ∗ b , if the set a \ b is finite. A pseudointersection ofa family of infinite sets is an infinite set a with a ⊆ ∗ b for all sets b in the family. A familyof infinite sets is centered if the finite intersections of its elements, are infinite. Let p be theminimal cardinality of a subfamily of [ N ] ∞ that is centered and has no pseudointersection.Let Fin be the family of finite subsets of N . The following notion plays a crucial role in ourconstructions. Definition 2.1 (Szewczak, W ludecka [16]) . A set X ⊆ [ N ] ∞ with | X | ≥ p is a p -generalizedtower if for each function a ∈ [ N ] ∞ , there are sets b ∈ [ N ] ∞ and S ⊆ X with | S | < p suchthat x ∩ [ n ∈ b [ a ( n ) , a ( n + 1)) ∈ Finfor all sets x ∈ X \ S .For functions f, g ∈ [ N ] ∞ , we write f ≤ ∗ g , if the set { n : f ( n ) > g ( n ) } is finite. Asubset of [ N ] ∞ is unbounded if for any function g ∈ [ N ] ∞ , there is a function f in the set with f ∗ g . Let b be the minimal cardinality of an unbounded subset of [ N ] ∞ . The existenceof a p -generalized tower in [ N ] ∞ is independent of ZFC, i.e., it is equivalent to the equality p = b [10, Lemma 3.3]. Let non( N add ) be the minimal cardinality of a subset of P( N ) that isnot null-additive. Theorem 2.2.
Assume that p = non( N add ) = c . There is a set X ⊆ [ N ] ∞ such that (1) The set X is a p -generalized tower. (2) The set X ∪ Fin has the property γ . (3) All continuous images of the set X ∪ Fin into P( N ) are null-additive. ULL SETS AND COMBINATORIAL COVERING PROPERTIES 3 (4)
The set X is homeomorphic to a subset of P( N ) that is not null-additive. We need the following notions and auxiliary results.For a class A of covers of spaces and a space X , let A ( X ) be the family of all covers of X from the class A . Let A , A , . . . and B be classes of covers of spaces. A space X satisfies S ( {A n } n ∈ N , B )if for each sequence U , U , . . . with U n ∈ A n ( X ) for each n , there are sets U ∈ U , U ∈ U ,. . . such that { U n : n ∈ N } ∈ B ( X ). Let O n be the class of all open n -covers of spaces foreach n and Γ be the class of all open γ -covers of spaces. Using the above notion, the strongly γ property is the property S ( { O n } n ∈ N ) , Γ). Let d be a metric in P( N ) that coincides withthe standard metric in { , } N , that is, for points x, y ∈ P( N ), let d ( x, y ) := ( − min(( x ∪ y ) \ ( x ∩ y )) , if x = y, , if x = y. Let ǫ be a positive real number. For a point x ∈ P( N ), let B ( x, ǫ ) := { y ∈ P( N ) : d ( x, y ) < ǫ } and for a set F ⊆ P( N ), let B ( F, ǫ ) := S { B ( x, ǫ ) : x ∈ F } . Fix a natural number n . Anopen cover of a subspace of P( N ) is a uniform n -cover if there is a positive real number ǫ such that for each n -elements subset F of the space, the set B ( F, ǫ ) is contained in a memberof the cover. Let O unif n be the family of all uniform n -covers of subspaces of P( N ). In 1995,Shelah characterized null-additive subsets of P( N ) [12]. Using the Shelah result, Zindulkacharacterized null-additive subsets of P( N ), in terms of combinatorial covering properties. Theorem 2.3 (Zindulka [20]) . A subset of P( N ) is null-additive if and only if it satisfies S ( { O unif n } n ∈ N , Γ) . For a set X and a natural number n , let P n ( X ) be the family of all n -elements subsets of X . By a similar argument as in the proof of a result of Tsaban [18, Theorem 2.1], we havethe following result. Lemma 2.4.
For a set X ⊆ P( N ) the following assertions are equivalent: (1) The set X satisfies S ( { O unif n } n ∈ N , Γ) . (2) There is a function f ∈ [ N ] ∞ such that the set X satisfies S ( { O unif f ( n ) } n ∈ N , Γ) . (3) For each function f ∈ [ N ] ∞ , the set X satisfies S ( { O unif f ( n ) } n ∈ N , Γ) . (cid:3) Proposition 2.5.
Let X ⊆ P( N ) be a set and assume that there is a function f ∈ [ N ] ∞ such that for each sequence U , U , . . . with U n ∈ O f ( n ) ( X ) for each n , there are sets U ∈ U ,U ∈ U , . . . and a set X ′ ⊆ X with | X ′ | < non( N add ) such that { U n : n ∈ N } ∈ Γ( X \ X ′ ) .Then all continuous images of the set X into P( N ) are null-additive.Proof. Let U , U , . . . be a sequence such that U n ∈ O unif f ( n ) ( X ) for each n . Fix a naturalnumber n . We may assume that there is a positive real number ǫ n such that U n = { B ( A, ǫ n ) : A ∈ P f ( n ) ( X ) } . We have U ′ n := { B ( F, ǫ n ) : F ∈ P f ( n ) ( X ) } ∈ O unif f ( n ) ( X ). By the assumption, there are sets F ∈ P f (1) ( X ) , F ∈ P f (2) ( X ) , . . . and a set X ′ ⊆ X with | X ′ | < non( N add ) such that { B ( F n , ǫ n ) : n ∈ N } ∈ Γ( X \ X ′ ) . Since | X ′ | < non( N add ), there are sets F ′ ∈ P f (1) ( X ′ ) , F ′ ∈ P f (2) ( X ′ ) , . . . such that { B ( F ′ n , ǫ n ) : n ∈ N } ∈ Γ( X ′ ) . P. SZEWCZAK AND T. WEISS
For each natural number n , there is a set A n ∈ P f ( n ) ( X ) such that F n ∪ F ′ n ⊆ A n and B ( F n , ǫ n ) ∪ B ( F ′ n , ǫ n ) ⊆ B ( A n , ǫ n ) ∈ U n . We have { B ( A n , ǫ n ) : n ∈ N } ∈ Γ( X )By Lemma 2.4 and Theorem 2.3, the set X is null-additive. (cid:3) By Theorem 2.3, we have the following result.
Proposition 2.6.
Null-additivity in P( N ) is preserved by uniformly continuous functions. (cid:3) Lemma 2.7.
Let X be a space and n, k be natural numbers. If U ∈ O n + k ( X ) , then each n -elements subset of X is contained in at least k pairwise different sets from U . (cid:3) For the remaining part of this section, let f ∈ [ N ] ∞ be a sequence such that f ( n + 1) > f ( n )+2 n + n for all natural numbers n . For a set d ∈ [ N ] ∞ , let [ d ] ∞ be the family of all infinitesubsets of d . Lemma 2.8.
Let U , U , . . . be a sequence of families of open sets in P( N ) such that U n ∈ O f ( n ) (Fin) for all natural numbers n and d ∈ [ N ] ∞ . There are an element x ∈ [ d ] ∞ andpairwise different sets U ∈ U , U ∈ U , . . . such that for each natural number n and eachset y ∈ P( N ) :If y ∩ [ f ( n ) , x ( n + 1)) ⊆ { x (1) , x (1) + 1 , . . . , x ( n ) , x ( n ) + 1 } , then y ∈ U n +1 . In particular, { U n : n ∈ N } ∈ Γ( { y ∈ P( N ) : y ⊆ ∗ x ∪ ( x + 1) } ) . Proof.
Let x (1) := a (1) and U ∈ U . Fix a natural number n and assume that naturalnumbers x (1) , . . . , x ( n ) and sets U ∈ U , . . . , U n ∈ U n have already been defined. We have | P([1 , f ( n )) ∪ { x (1) , x (1) + 1 , . . . , x ( n ) , x ( n ) + 1 } ) | ≤ f ( n )+2 n and U n +1 ∈ O f ( n +1) (Fin). By Lemma 2.7 there is a set U n +1 ∈ U n +1 \ { U , . . . , U n } such thatP([1 , f ( n )) ∪ { x (1) , x (1) + 1 , . . . , x ( n ) , x ( n ) + 1 } ) ⊆ U n +1 . For each set s ∈ P([1 , f ( n )) ∪ { x (1) , x (1) + 1 , . . . , x ( n ) , x ( n ) + 1 } ) , there is a natural number m s ∈ a with m s > max { f ( n ) , x ( n ) + 1 } such that for each set y ∈ P( N ): If y ∩ [1 , m s ) = s, then y ∈ U n +1 . Define x ( n + 1) := max { m s : s ∈ P([1 , f ( n )) ∪ { x (1) , x (1) + 1 , . . . , x ( n ) , x ( n ) + 1 } ) } . Fix a natural number n . Let y ∈ P( N ) be a set such that y ∩ [ f ( n ) , x ( n + 1)) ⊆ { x (1) , x (1) + 1 , . . . , x ( n ) , x ( n ) + 1 } . Then the set s := y ∩ [1 , x ( n + 1)) belongs to P([1 , f ( n )) ∪ { x (1) , x (1) + 1 , . . . , x ( n ) , x ( n ) + 1 } ),and y ∩ [1 , m s ) = y ∩ [1 , x ( n + 1)) = s. Thus, y ∈ U n +1 . (cid:3) ULL SETS AND COMBINATORIAL COVERING PROPERTIES 5
Lemma 2.9 (Folklore [17, Lemma 2.13]) . A set X ⊆ [ N ] ∞ is unbounded if and only if foreach function a ∈ [ N ] ∞ , there are a set b ∈ [ N ] ∞ and an element x ∈ X such that x ∩ [ n ∈ b [ a ( n ) , a ( n + 1)) = ∅ for all x ∈ X . For elements b ∈ [ N ] ∞ and c ∈ { , } N , let 2 b , b + 1 and b + c be elements in [ N ] ∞ such that(2 b )( n ) := 2 b ( n ), ( b + 1)( n ) := b ( n ) + 1 and ( b + c )( n ) := b ( n ) + c ( n ) for all natural numbers n . Proof of Theorem 2.2.
Construction of a set X : Let { ( U ( α )1 , U ( α )2 , . . . ) : α < c } be an enu-meration of all sequences of families of open sets in P( N ) such that U ( α ) n ∈ O f ( n ) (Fin) for allordinal numbers α < c and natural numbers n . Let [ N ] ∞ = { d α : α < c } .Apply Lemma 2.8 to the sequence U (0)1 , U (0)2 , . . . and to a set d ∈ [2 N ] ∞ with d ≤ ∗ d . Thenthere are sets U (0)1 ∈ U (0)1 , U (0)2 ∈ U (0)2 , . . . and an element x ∈ [2 N ] ∞ such that { U (0) n : n ∈ N } ∈ Γ( { y ∈ P( N ) : y ⊆ ∗ x ∪ ( x + 1) } ) . Fix an ordinal number α < c . Assume that elements x β ∈ [2 N ] ∞ with d β ≤ ∗ x β and sets U ( β )1 ∈ U ( β )1 , U ( β )2 ∈ U ( β )2 , . . . with { U ( β ) n : n ∈ N } ∈ Γ( { y ∈ P( N ) : y ⊆ ∗ x β ∪ ( x β + 1) } have already been defined for all ordinal numbers β < α such that for β, β ′ < α if β < β ′ ,then x β ∗ ⊇ x β ′ . Let d ∈ [2 N ] ∞ be a pseudointersection of the family { x β : β < α } with d α ≤ ∗ d . Apply Lemma 2.8 to the sequence U ( α )1 , U ( α )2 , . . . and to the element d . Then thereare an element x α ∈ [ d ] ∞ and sets U ( α )1 ∈ U ( α )1 , U ( α )2 ∈ U ( α )2 , . . . such that(2.9.1) { U αn : n ∈ N } ∈ Γ( { y ∈ P( N ) : y ⊆ ∗ x α ∪ ( x α + 1) } ) . Let { , } N = { c α : α < c } . Define X := { x α + c α : α < c } By the construction, for all ordinal numbers α, β with α ≤ β < c , we have(2.9.2) x β + c β ⊆ x β ∪ ( x β + 1) ⊆ ∗ x α ∪ ( x α + 1) . (1) Let a ∈ [ N ] ∞ . Since the set { x α : α < c } is unbounded, the set { x α ∪ ( x α + 1) : α < c } is unbounded, too. By Lemma 2.9, there are an ordinal number α < c and a set b ∈ [ N ] ∞ such that ( x α ∪ ( x α + 1)) ∩ [ n ∈ b [ a ( n ) , a ( n + 1)) = ∅ . By (2.9.2), we have( x β + c β ) ∩ [ n ∈ b [ a ( n ) , a ( n + 1)) ⊆ ∗ ( x α ∪ ( x α + 1)) ∩ [ n ∈ b [ a ( n ) , a ( n + 1)) = ∅ for all ordinal numbers β with α ≤ β < c . Thus, the set X is a p -generalized tower.(2) By (1), the set X is a p -generalized tower, and thus the set X ∪ Fin has the property γ [16, Theorem 4.1(1)].(3) The set X ∪ Fin satisfies the property from Proposition 2.5: Let U , U , . . . be a sequenceof open families in P( N ) such that U n ∈ O f ( n ) ( X ∪ Fin) for all natural numbers n . There P. SZEWCZAK AND T. WEISS is an ordinal number α < c such that this sequence is equal to the sequence U ( α )1 , U ( α )2 , . . . .By (2.9.1) and (2.9.2), we have { U ( α ) n : n ∈ N } ∈ Γ( { x β + c β : α ≤ β < c } ) . (4) In our proof we use Rothberger’s trick [7, Theorem 9.4]. The map Φ : P( N ) → { , } N unifying each subset of N with its characteristic function is a homeomorphism and an isometrywith respect to the standard metric in { , } N and the metric d on P( N ) defined in Section 2.The set G := (cid:8) ((Φ( x ))(1) , x (1) mod 2 , (Φ( x ))(2) , x (2) mod 2 , . . . ) : x ∈ X (cid:9) , is a homeomorphic copy of the set X . Since the projection π : G → { , } N is onto anduniformly continuous, the map Φ − ◦ π ◦ Φ : Φ − [ G ] → P( N )is onto and uniformly continuous, too. By Proposition 2.6, the set Φ − [ G ] is not null additive.The set Φ − [ G ] is homeomorphic to G . (cid:3) A γ -set that is not null-additive Let non( N ) be the minimal cardinality of a subset of P( N ) that is not null. A set { x α : α < p } ⊆ [ N ] ∞ is a p -unbounded tower if it is unbounded and for all ordinal numbers α, β < p with α < β , we have x α ∗ ⊇ x β . A p -unbounded tower exists if and only if p = b [10,Lemma 3.3]. Theorem 3.1.
Assume that p = b ≥ non( N ) . There is a p -generalized tower X ⊆ [ N ] ∞ thatis not null-additive. In particular, the set X ∪ Fin is a nontrivial set with the property γ thatis not null-additive.Proof. Let f ∈ [ N ] ∞ be a function such that f ( n ) := P ni =0 i for all natural numbers n .Define Y n := { x ∈ P( N ) : x ∩ [ f ( n ) , f ( n + 1)) = ∅ } for all natural numbers n . Each set Y n is clopen and has measure less or equal than 2 − ( n +1) ,and thus the set Y := T m S n>m Y n is null. Let T = { t α : α < p } be a p -unbounded towerin [ N ] ∞ and Z = { z α : α < p } be a nonnull set in P( N ).Define x α := z α ∩ [ n ∈ N [ f ( t α (2 n )) , f ( t α (2 n ) + 1) ∪ { f ( t α (2 n + 1)) : n ∈ N } for all ordinal numbers α < p and X := { x α : α < p } .The set X is a p -generalized tower: For each ordinal number α with α < p , we have { f ( t α (2 n + 1)) : n ∈ N } ⊆ x α , and thus X ⊆ [ N ] ∞ . Let a ∈ [ N ] ∞ . There is a set c ∈ [ N ] ∞ such that | [ f ( c ( k )) , f ( c ( k + 1))) ∩ a | ≥ k . By Lemma 2.9, there are a set d ∈ [ N ] ∞ and an ordinal number α < p such that(3.1.1) t α ∩ [ k ∈ d [ c ( k ) , c ( k + 1)) = ∅ . ULL SETS AND COMBINATORIAL COVERING PROPERTIES 7
Then there is a set b ∈ [ N ] ∞ such that [ n ∈ b [ a ( n ) , a ( n + 1)) ⊆ [ k ∈ d [ f ( c ( k )) , f ( c ( k + 1))) . Fix an ordinal number β with α ≤ β < p . We have x β ∩ [ n ∈ b [ a ( n ) , a ( n + 1)) ⊆ [ n ∈ t β [ f ( n ) , f ( n + 1)) ∩ [ n ∈ b [ a ( n ) , a ( n + 1)) ⊆ ∗ [ n ∈ t α [ f ( n ) , f ( n + 1)) ∩ [ k ∈ d [ f ( c ( k )) , f ( c ( k + 1))) . By (3.1.1), the latter intersection is empty.We have Z ⊆ X ⊕ Y : Fix an ordinal number α < p . Since x α ∩ [ n ∈ N [ f ( t α (2 n )) , f ( t α (2 n ) + 1) = z α ∩ [ n ∈ N [ f ( t α (2 n )) , f ( t α (2 n ) + 1) , we have ( x α ⊕ z α ) ∩ [ n ∈ N [ f ( t α (2 n )) , f ( t α (2 n ) + 1) = ∅ , and thus x α ⊕ z α ∈ Y . Then there is an element y α ∈ Y such that x α ⊕ z α = y α . Thus, z α = x α ⊕ y α .Since X is a p -generalized tower, the set X ∪ Fin has the property γ [16, Theorem 4.1(1)].Since Z ⊆ ( X ∪ Fin) ⊕ Y , the set X ∪ Fin is not null-additive. (cid:3)
Remark . The assumption of Theorem 3.1 is valid assuming the Continuum Hypothesisor Martin Axiom and in the following model. Let P ℵ be an ℵ -iteration with finite supportof a measure algebra and G be a generic filter in P ℵ . Let M be a model of ZFC and c = ℵ .In the model M [ G ], the assumption p = b ≥ non( N ) from Theorem 3.1 is true: Let non( M )be the minimal cardinality of a subset of P( N ) that is not meager. Adding ℵ Cohen reals,we have non( M ) = ℵ and adding ℵ Solovay reals, we have non( N ) = ℵ . Thus, p = ℵ .Since non( M ) = ℵ , we have b = ℵ . Theorem 3.3.
Assume that p = b ≥ non( N ) . There is a nontrivial subset of the real linewith the property γ that is not null-additive.Proof. We use notions from the paper of the second named author [19, Corollary 11]. Let X ⊆ [ N ] ∞ be a set from Theorem 3.1. Let p : P( N ) → A be a homeomorphism. Since themap f ◦ g − ◦ p is continuous, the image Y of the set X ∪ Fin under the map f ◦ g − ◦ p is asubset of the real line with the property γ . Suppose that the set Y is null-additive. Then theset g ◦ f − [ Y ] is null-additive [19, Theorem 12]. Since the map p − is unifromly continuousand the set X ∪ Fin is not null-additive, the set p [ X ] = ( g ◦ f − [ Y ]) is not null-additive, too,a contradiction. (cid:3) Theorem 3.4.
It is consistent with ZFC that there is a p -unbounded tower in [ N ] ∞ and non( N add ) < p .Proof. The dual Borel conjecture is the statement that for any set X ⊆ P( N ) with cardinality ℵ , there is a null set N ⊆ P( N ) such that X ⊕ N = P( N ). By the result of Judah andShelah [1, Theorem 8.5.23] there is a model for ZFC satisfying Martin Axiom for σ -centered P. SZEWCZAK AND T. WEISS sets and the dual Borel conjecture. In that model, we have p = ℵ = b , an thus a p -unbounded tower exists. On the other hand, since the dual Borel conjecture holds, we havenon( N add ) = ℵ . (cid:3) Nonproductivity of Sierpi´nski-type sets
Let O be the class of open covers of spaces. A space X satisfies Menger’s property S fin (O , O)if for each sequence U , U , . . . ∈ O( X ), there are finite sets F ⊆ U , F ⊆ U , . . . such that S n F n ∈ O( X ). A space X satisfies Hurewicz’s property U fin ( O , Γ) if for each sequence U , U , . . . ∈ O( X ), there are finite sets F ⊆ U , F ⊆ U , . . . such that { S F n : n ∈ N } ∈ Γ( X ).The property U fin ( O , Γ) implies S fin (O , O) and it generalizes σ -compactness. An uncountablesubset of P( N ) is a Sierpi´nski set if its intersection with any null set is at most countable;its existence is independent of ZFC. Any Sierpi´nski set satisfies U fin ( O , Γ) but it is not σ -compact. Assuming that the Continuum Hypothesis holds, there are two Sierpi´nski setswhose product space does not even satisfy S fin (O , O) [6, p. 250].A category theoretic counterpart to a Sierpi´nski set is a
Luzin set , i.e., an uncountablesubset of P( N ) whose intersection with any meager set is at most countable. Each Luzin setsatisfies S fin (O , O) but no U fin ( O , Γ). A set of reals is a space homeomorphic to a subset of thereal line. Let P be a property of spaces. A space is productively P if its product space withany space satisfying P , satisfies P . In the class of sets of reals, no Luzin set is productively S fin (O , O). There are open problems ([14, Problem 7.5], [15, Problem 5.5]), whether in theclass of sets of reals (or in the class of general topological spaces) for any Sierpi´nski set X ,there is a space Y satisfying U fin ( O , Γ) such that the product space X × Y does not satisfy U fin ( O , Γ), and what if we assume the Continuum Hypothesis? We consider an analogousproblem with respect to combinatorial covering properties, stronger than U fin ( O , Γ).Let A , B be classes of covers of spaces. A space X satisfies S ( A , B ) if for each sequence U , U , . . . ∈ A ( X ), there are sets U ∈ U , U ∈ U , . . . such that { U n : n ∈ N } ∈ B ( X ).Let Ω be the class of all open ω -covers of spaces. A Borel cover of a space is a cover whosemembers are Borel subsets of the space. Let Γ B and Ω B be classes of all countable Borel γ -covers and countable Borel ω -covers of spaces, respectively. The property γ , considered inthe previous sections, is one of the classic properties in the selection principles theory, it isequivalent to the property S (Ω , Γ) [5, Theorem 2]. A set of reals is totally imperfect if it doesnot contain an uncountable compact set and it is perfectly meager if its intersection with aperfect set is meager in this perfect set. The following diagram presents relations betweenconsidered properties [13], [6]. perfectlymeager U fin ( O , Γ) / / S fin (O , O) γ / / S (Γ , Γ) O O / / e e ❑❑❑❑❑❑❑❑❑❑ S (Γ , Ω) / / S (Γ , O) O O / / totallyimperfect S (Γ B , Γ B ) O O / / S (Γ B , Ω B ) O O Let cov( N ) be the minimal cardinality of a family of null subsets of P( N ) whose union isP( N ) and cof( N ) be the minimal cardinality of a family of null subsets of P( N ) such that ULL SETS AND COMBINATORIAL COVERING PROPERTIES 9 any null subset of P( N ) is contained in a set from the family. For an uncountable ordinalnumber κ , a set X ⊆ P( N ) is a κ -Sierpi´nski set if | X | ≥ κ and for any null set Y ⊆ P( N ),we have | X ∩ Y | < κ . Every b -Sierpi´nski set satisfies S (Γ B , Γ B ) ([6, Theorem 2.9], [17,Theorem 2.4]). Theorem 4.1. (1)
Assume that cov( N ) = cof( N ) = b . For every b -Sierpi´nski set S , there is a b -Sierpi´nski set S ′ such that the product space S × S ′ does not satisfy S (Γ , Γ) . (2) Assume that cov( N ) = cof( N ) = d = c and the cardinal number c is regular. Forevery c -Sierpi´nski set S , there is a c -Sierpi´nski set S ′ such that the product space S × S ′ does not satisfy S (Γ , O) . In order to prove Theorem 4.1, we need the following Lemma.
Lemma 4.2.
Assume that cov( N ) = cof( N ) and the cardinal number cov( N ) is regular.For every cov( N ) -Sierpi´nski set S and every set Y of cardinality at most cov( N ) , there is a cov( N ) -Sierpi´nski set S ′ such that Y ⊆ S ⊕ S ′ .Proof. Let { N α : α < cov( N ) } be a cofinal family of null sets in P( N ) and Y = { y α : α < cov( N ) } ⊆ P( N ). Fix an ordinal number α < cov( N ) and assume that elements x β ∈ P( N )have already been defined for all ordinal numbers β with β < α . The set y α ⊕ S is a cov( N )-Sierpi´nski. Since the cardinal number cov( N ) is regular, the union S β<α N β ∪ { x β : β < α } cannot cover any cov( N )-Sierpi´nski set. Then there is an element x α ∈ ( y α ⊕ S ) \ [ β<α N β ∪ { x β : β < α } . By the construction the set S ′ := { x α : α < cov( N ) } is a cov( N )-Sierpi´nski set and foreach ordinal number α < cov( N ), there is an element s α ∈ S such that y α = x α ⊕ s α . Thus, Y ⊆ S ⊕ S ′ . (cid:3) Let non( M ) be the minimal cardinality of a nonmeager subset of P( N ). Proof of Theorem 4.1. (1) By the Cicho´n diagram, we have cov( N ) ≤ non( M ) ≤ cof( N ),and thus non( M ) = b . Let Y ⊆ P( N ) be a nonmeager set of cardinality b . By Lemma 4.2,there is a b -Sierpi´nski set S ′ such that Y ⊆ S ⊕ S ′ . Every set with the property S (Γ , Γ)is meager and the property S (Γ , Γ) is preserved by continuous functions. Since S ⊕ S ′ is anonmeager continuous image of the product space S × S ′ , the product S × S ′ does not satisfy S (Γ , Γ).(2) Every set satisfying S (Γ , O) is totally imperfect. Let Y = [ N ] ∞ . Proceed analogouslyas in (1). (cid:3) Remark . The assumptions of Theorem 4.1 are valid assuming the Continuum Hypothesisor Martin Axiom. In a model obtained by ℵ -iteration of Sacks forcing with countablesupports, we have cov( N ) = non( M ) = cof( N ) = b = ℵ . In a model obtained by ℵ -iteration of amoeba forcing with finite supports, we have cov( N ) = cof( N ) = d = c .It is not known whether, in the class of totally imperfect sets, the properties S (Γ , O) and S fin (O , O) are different. Theorem 4.1 can be useful to solve this problem.
Remark . Assume that the Continuum Hypothesis holds. If there is a Sierpi´nski set whoseproduct space with any Sierpi´nski set satisfies S fin (O , O), then, in the class of totally imperfectsets, the properties S fin (O , O) and S (Γ , O) are different.
For a finite set F ⊆ [ N ] ∞ and a set X ⊆ [ N ] ∞ let max[ F ] := { max f ∈ F f ( n ) : n ∈ N } ,an element of [ N ] ∞ , and maxfin[ X ] := { max[ F ] : F is a finite subset of X } . For elements f, g ∈ [ N ] ∞ we write f ≤ ∞ g if the set { n : f ( n ) ≤ g ( n ) } is infinite.Using similar ideas as in the proof that any Sierpi´nski set satisfies S (Γ , Γ) [6, Theorem 2.9,Theorem 2.10], we obtain the following results.
Proposition 4.5.
Every d -Sierpi´nski set satisfies S (Γ B , Ω B ) .Proof. Let S be a d -Sierpi´nski set. There is a positive real number p such that the outermeasure of S is equal to p . Let B be a Borel set containing S with µ ( B ) = p . Let U , U ,. . . ∈ Γ Bor ( S ) be a sequence of families of Borel subsets of the set B . Let U n = { U nm : m ∈ N } for all natural numbers n , and we may assume that each such a family is increasing (if notconsider the family { T i ≥ j U ni : j ∈ N } , an increasing countable Borel cover of S ). Fix anatural number k . There is a function f k ∈ N N such that µ ( U nf k ( n ) ) ≥ (1 − n + k ) p . For a set A k := T n U nf k ( n ) , we have µ ( A k ) ≥ (1 − k ) p . The set A = S k A k has measure p , and thus | S \ A | < d . For each element x ∈ S \ A , there is a function f x ∈ N N such that x ∈ T n U nf x ( n ) .There is a function g ∈ N N such thatmaxfin[ { f k : k ∈ N } ∪ { f x : x ∈ S \ A } ] ≤ ∞ g. We have { U ng ( n ) : n ∈ N } ∈ Ω( S ): Let F be a finite subset of S . There is a set a ∈ Fin suchthat F ∩ A ⊆ S k ∈ a A k . Let f := maxfin[ { f k : k ∈ a } ∪ { f x : x ∈ S \ A } ] . Then F ⊆ U nf ( n ) for all but finitely many natural numbers n . Since f ≤ ∞ g , and families U n are increasing, we have F ⊆ U ng ( n ) for infinitely many natural numbers n . (cid:3) The following corollary is a straightforward consequence of Theorem 4.1.
Corollary 4.6. (1)
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