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On Some New Ideals on the Cantor and Baire Spaces 0 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 5, May 1998, Pages 1549{1555 S 0002-9939(98)04378-0 ON SOME NEW IDEALS ON THE CANTOR AND BAIRE SPACES JACEK CICHON´ AND JAN KRASZEWSKI (Communicated by Andreas R. Blass) Abstract. We define and investigate some new ideals of subsets of the Cantor space and the Baire space. We show that combinatorial properties of these ideals can be described by the splitting and reaping cardinal numbers. We show that there exist perfect Luzin sets for these ideals on the Baire space. 0. Introduction For each infinite subset T of the set ! of all natural numbers let us denote by (T )theσ-ideal of meagre subsets of the space 2T with the canonical product topology.K By (T )wedenotetheσ-ideal of Lebesgue measure zero subsets of 2T with respect toL the canonical product measure. T ! T Notice that if T is a subset of ! then we can indentify the spaces 2 2 \ and ! × the Cantor space 2 using the canonical homeomorphism πT defined by πT (x)= (xT;x (! T)). Directly from the definition of meagre sets it follows that if A | | \ ! T ∈ (T ), then A 2 \ (!). The same observation is also true for the ideal (!), Kbut it is evidently× false∈K for the σ-ideal of all countable subsets of the Cantor space.L We call this property of the ideals (!)and (!)productivity. There are other natural productiveK σ-idealsL of subsets of the Cantor space, e.g. the σ-ideal (!) (!). It is interesting that among them there exists the least K ∩L productive σ-ideal which contains all points. We call this ideal S2.Thereexists ! also the least productive ideal of subsets of 2 , and we call it I2. These ideals have Borel bases but they do not satisfy the countable chain condition|there exists a family of continuum many pairwise disjoint Borel sets outside the ideal S2.The ideal I2 is not σ-additive, and the ideal S2 is precisely σ-additive. The minimum cardinality of bases of these ideals is continuum. The covering number cov of both ideals is equal to the reaping cardinal r, and the last basic combinatorial invariants (cardinal numbers non) are described in terms of splitting cardinal numbers. These results show that both splitting and reaping cardinals are closely connected with natural mathematical objects on the classical Cantor space. We also consider the minimal productive σ-ideal S! of subsets of the Baire space Received by the editors December 18, 1995 and, in revised form, October 16, 1996. 1991 Mathematics Subject Classification. Primary 04A20, 28A05. Key words and phrases. Set theory, ideals, Borel sets, Cantor space, Baire space, cardinal functions. Research of the second author supported by a grant 2149/W/IM/96 from the University of Wroclaw. c 1998 American Mathematical Society 1549 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1550 JACEK CICHON´ AND JAN KRASZEWSKI !!. We show that there exists an uncountable closed subset P of !! which is a Luzin set for S!, i.e. the intersection of the set P with any set from S! is countable. This fact completely determines the basic combinatorial invariants of this ideal. 1. Notation, definitions and basic observations In this paper we use the standard set theoretical notation. For example, ! denotes the first infinite cardinal number, which we shall identify with the set of all natural numbers. The cardinality of the set of all real numbers is denoted by c.Ifκ is a cardinal number then [X]κ denotes the family of all subsets of X of cardinality κ and [X]<κ denotes the family of all subsets of X of cardinality strictly less then κ. X<! denotes the set of all finite sequences of elements of the set X.Thepower set of a set X is denoted by P (X). For A; B ! we put A ∗ B if and only if card (A B) <!. ⊆ ⊆ If X is\ a discrete topological space then we endow X! with the standard product topology. In particular, for X =2andX=!we get the Cantor space and the Baire space, respectively. For an ideal of subsets of X we consider the following cardinal numbers: J add( )=min card ( ): and ; J { A A⊆J A6∈J} cov( )=min card ( ): and [ = X ; J { A A⊆J A } non( )=min card (B):B X and B[ ; J { ⊆ 6∈ J } cof( )=min card ( ): and ( A )( B )(A B) : J { A A⊆J ∀ ∈J ∃ ∈A ⊆ } Note that if is a proper ideal and = X, then the following relations hold: J J add( ) cov( ); add( ) non(S ); cov( ) cof( ); non( ) cof( ): J ≤ J J ≤ J J ≤ J J ≤ J Suppose that is an ideal of subsets of X and P(X). We say that has an -base if forJ each A there exists B A⊆such that A B. Hence,J in particular,A if X is a topological∈J space, then ∈A∩Jhas an F -base if each⊆ element from J σ can be covered by some Fσ subset of X from . J Let and denote the σ-ideals of meagre subsetsJ and of Lebesgue measure zero subsetsK of theL Cantor space 2!, respectively. The ideal has an F -base and the K σ ideal has a Gδ-base. FromL now on let us assume that X has at least two elements. We define Pif(X)= ': 'is a function and dom(') [!]! and rng(') X : { ∈ ⊆ } If ' Pif(X), then we put ∈ ['] = x X! : ' x ; X { ∈ ⊆ } ! [']∗ = x X : ' ∗ x : X { ∈ ⊆ } If we treat X as a discrete topological space, then [']X is a closed and [']X∗ is an F subset of X! for each ' Pif(X). σ ∈ Now we are able to define the ideals we are going to deal with. Let IX and IX∗ denote the ideals generated by the families [']X : ' Pif(X) and [']X∗ : ' <! { ∈ } { ∈ Pif(X) , respectively. Then [X] IX IX∗ and IX∗ is a proper ideal of subsets ! } ⊆ ⊆ of X . The first ideal has a closed base and the second has an Fσ-base. Similarly, let S and S∗ denote the σ-ideals generated by families ['] : ' X X { X ∈ Pif(X) , [']∗ : ' Pif(X) , respectively. It is easy to see that S = S∗ . } { X ∈ } X X License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON SOME NEW IDEALS ON THE CANTOR AND BAIRE SPACES 1551 HencewehaveIX IX∗ SX. The ideal SX has an Fσ -base and is proper. Directly from the definition⊆ ⊆ of the ideals we can deduce that cov(IX )=cov(IX∗)=cov(SX): If j : ! ! is an injection and A 2!, then we define → ⊆ j A = x 2! : x j A : ∗ { ∈ ◦ ∈ } We say that an ideal of subsets of 2! is productive if j A for each A and any injection j : ! J !. As we have mentioned in the∗ introduction,∈J the ideals∈J and are productive.→ K L Let ' Pif(2) and let j : ! dom(') be any bijection. Then j ' j =[']2. ∈ → ! 1 ∗{ ◦ } Conversely, if j : ! ! is an injection and x 2 ,thenx j− Pif(2) and → ∈ ◦ ∈ 1 j x =[x j− ] : ∗{ } ◦ 2 These observations imply that I2 and S2 are the least productive ideal and σ-ideal of subsets of 2! (containing all points), respectively. Since and are productive, K L we see that S2 . Let us recall⊆K∩L that σ-ideals of subsets of a Polish space with Borel bases have plenty of interesting properties (see e.g. [2]). The next result shows the main difference between the ideal S and the ideals and . 2 K L Theorem 1.1. There exists a family of pairwise disjoint Borel subsets of 2! such that card ( )=cand none of elementsF of belongs to S . F F 2 Proof. For each real number α [0; 1] we put ∈ n ! 1 Aα = x 2 : lim x(i)=α : { ∈ n n } →∞ i=1 X It is easy to check that A : α [0; 1] is the required family. { α ∈ } Let A; S be two infinite subsets of !.WesaythatSsplitsAif card (A S)= card (A S)=!. Let us recall the following three cardinal numbers (see e.g.∩ [3]): \ s =min card ( ): [!]! and ( A [!]!)( S )(S splits A) ; { S S⊆ ∀ ∈ ∃ ∈S } -s =min card ( ): [!]! and ( [[!]!]!)( S )( A )(S splits A) ; ℵ0 { S S⊆ ∀A ∈ ∃ ∈S ∀ ∈A } r =min card ( ): [!]! and ( S [!]!)( R )(S does not split R) : { R R⊆ ∀ ∈ ∃ ∈R } The cardinal numbers s and r are called splitting and reaping, respectively. It is easy to check that ! s -s cand ! r c. Itisanopenproblem 1 ≤ ≤ℵ0 ≤ 1 ≤ ≤ if s = 0-s can be proved in ZFC. An easy reformulation of the definition of the reapingℵ number r gives us the following description: r =min card ( ): [!]! and ( S [!]!)( R )(R S R ! S) : { R R⊆ ∀ ∈ ∃ ∈R ⊆ ∨ ⊆ \ } Let us introduce an auxiliary cardinal number. Namely, we define fin-s =min card ( ): [!]! { S S⊆ and ( [[!]!]<!)( S )( A )(S splits A) : ∀A ∈ ∃ ∈S ∀ ∈A } Lemma 1.2. s = fin-s : License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1552 JACEK CICHON´ AND JAN KRASZEWSKI Proof. It is clear that s fin-s. Suppose now that [!]! is such that card ( )= sand every infinite subset≤ A ! is split by someS⊆ element of . We may assumeS that is a field of subsets of !⊆containing all finite sets. One canS show by an easy inductionS on n ! that ∈ ( [[!]!]n)( S )( A )(S splits A) : ∀A ∈ ∃ ∈S ∀ ∈A } 2. Basic properties Assume that X Y; card (X) 2. Then it is easy to see that ⊆ ≥ ! ! ! I = I P (X );I∗=I∗ P(X)andS =S P(X): X Y ∩ X Y∩ X Y∩ Let us recall a well-known fact.
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