MAKING DOUGHNUTS OF COHEN REALS

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1 MAKING DUGHNUTS F CHEN REALS Lrenz Halbeisen Department f Pure Mathematics Queen s University Belfast Belfast BT7 1NN, Nrthern Ireland halbeis@qub.ac.uk Abstract Fr a b ω with b \ a infinite, the set D = {x [ω] ω : a x b} is called a dughnut. A set S [ω] ω has the dughnut prperty D, if it cntains r is disjint frm a dughnut. It is knwn that nt every set S [ω] ω has the dughnut prperty, but S has the dughnut prperty if it has the Baire prperty B r the Ramsey prperty R. In this paper it is shwn that a finite supprt iteratin f length ω 1 f Chen frcing, starting frm L, yields a mdel fr CH + Σ 1 2 (D ) + Σ1 2 (B ) + Σ1 2 (R ). 0. Intrductin Investigating arrw partitin prperties, Carls DiPrisc and James Henle intrduced in [DH00] the s-called dughnut prperty: Fr a set x, let x dente its cardinality and let [ω] ω := {x ω : x = ω}. Then, fr a b ω with b \ a [ω] ω, the set D = {x [ω] ω : a x b} is called a dughnut, r mre precisely, the (a, b)-dughnut. A set S [ω] ω has the dughnut prperty, dented by D, if it cntains r is disjint frm sme dughnut. A set S [ω] ω has the Ramsey prperty, dented by R, if it cntains r is disjint frm sme (, b)-dughnut. Hence, it is bvius that if S has the Ramsey prperty, then it has the dughnut prperty as well. Like fr the Ramsey prperty, it is easy t shw using the Axim f Chice that nt every set S [ω] ω has the dughnut prperty. Mrever, in the cnstructible universe L nt even every 1 2-set has the dughnut prperty. Indeed, let A = { y [ω] ω : z [ω] ω( z < L y ( y M z is infinite) )}, where M dentes the symmetric difference, and let S = { x [ω] ω : y A( x M y is infinite r dd) }. Since < L is a 1 2 -relatin (cf. [Je78, Therem 97]), the set A is a Π1 2-set and by cnstructin, fr every x [ω] ω there is a unique y A such that x My is finite. Thus, fr every x [ω] ω we either have x S r y A( x M y is infinite r even), and since dd and even are arithmetical relatins, S is a 1 2 -set. Nw, if x [ω]ω and n x, then x S if and nly if x \ {n} / S, which implies that S des nt have the dughnut prperty. n the ther hand, ne can shw that if S has the Baire prperty B (which means there is an pen set such that S M is meager), then S has als the dughnut prperty (cf. [DH00, Prpsitin 2.2] r [MS80]) Mathematics Subject Classificatin: 03E35 05D10 05E05 03E40 Key-wrds: dughnut prperty, Ramsey prperty, Baire prperty, Chen frcing 1

2 2 Let P be any prperty f subsets f [ω] ω. We write Σ 1 2 (P ) if every bldface Σ1 2 -set S [ω] ω has the prperty P. By the facts mentined abve we have Σ 1 2 (B ) = Σ1 2 (D ) = Σ1 2 (R ). This is similar t the fact that if all Σ 1 2-sets S have the Baire prperty r the Ramsey prperty, then all Σ 1 2 -sets are K -regular, dented by K σ. (The prperty K σ will be defined later.) Thus, we als have Σ 1 2 (B ) = Σ1 2 (K σ) = Σ 1 2 (R ). The aim f this paper is t shw that Cn(ZFC) Cn ( ZFC+CH+Σ 1 2 (D )+ Σ1 2 (K σ) ). In particular we get Cn(ZFC) Cn ( ZFC + CH + Σ 1 2 (D ) + Σ1 2 (B ) + Σ1 2 (R )) Acknwledgement: I like t thank Carls DiPrisc fr many fruitful and inspiring discussins. 1. Cmpletely Dughnut Sets In this sectin we intrduce a pseud tplgy n [ω] ω, called the dughnut tplgy, which is related t the dughnut prperty and shw that this pseud tplgy has the same features as the Ellentuck tplgy, which was intrduced by Erik Ellentuck in [El74] t prve that analytic sets are cmpletely Ramsey. In ur terminlgy, the nn-empty basic pen sets f the Ellentuck tplgy are the (, b)-dughnuts, where b [ω] ω. In fact, Ellentuck prved that a set S [ω] ω is cmpletely Ramsey if and nly if it has the Baire prperty with respect t the Ellentuck tplgy, and that S is cmpletely Ramsey null if and nly if it is meager (r equivalently, nwhere dense) with respect t the Ellentuck tplgy. In the fllwing we will see that a set is cmpletely dughnut (defined belw) if and nly if it has the Baire prperty with respect t the dughnut tplgy and it is cmpletely dughnut null if and nly if it is meager (r equivalently, nwhere dense) with respect t the dughnut tplgy. Let us start by defining the dughnut tplgy: Fr a, b ω let (a, b) ω := { x [ω] ω : a x b x \ a = b \ x = ω }. The sets (a, b) ω tgether with the sets [ω] ω and are the basic pen sets. Since the intersectin f tw basic pen sets might cntain just ne element, the basic pen sets d nt frm a basis fr a tplgy n [ω] ω. Hwever, since we use in the fllwing just unins f basic pen sets and the intersectin f unins f basic pen dense sets, let us say that a set S is pen with respect t the dughnut tplgy if S is the unin f basic pen sets, and similarly we define nwhere dense and meager with respect t the dughnut tplgy. Fr a, b ω let [a, b] ω := { x [ω] ω : a x b ( x \ a = ω b \ x = ω )}. Thus, a set [a, b] ω is either an (a, b)-dughnut r empty. In ur terminlgy, the nn-empty basic pen sets f the Ellentuck tplgy are the dughnuts [, b] ω, where b [ω] ω.

3 3 Fact 1.1. Fr every dughnut [a, b] ω there are a, b [a, b] ω such that (a, b ) ω [a, b ] ω (a, b) ω [a, b] ω. Prf. Take a, b [a, b] ω such that a a b b and each f the sets a \ a, b \ a and b \ b is infinite. A set S [ω] ω is cmpletely dughnut, dented by D c, if fr each dughnut [a, b] ω there is a dughnut [a, b ] ω [a, b] ω such that [a, b ] ω S r [a, b ] ω S =. If we are always in the latter case, then S is called cmpletely dughnut null. bviusly, the cmplement f a cmpletely dughnut set is als cmpletely dughnut (cf. [El74, Lemma 5]). Mrever, every pen set (w.r.t. the dughnut tplgy) is cmpletely dughnut (cf. [El74, Lemma 4]): Fact 1.2. If S [ω] ω is an pen set with respect t the dughnut tplgy, then S is cmpletely dughnut. Prf. If S [ω] ω is pen (w.r.t. the dughnut tplgy), then S is the unin f basic pen sets. Thus, fr any dughnut [a, b] ω, S (a, b) ω is pen. Hence, S (a, b) ω is either empty r cntains a basic pen set. Thus, by Fact 1.1, S [a, b] ω cntains r is disjint frm sme dughnut, and since [a, b] ω was arbitrary, this implies that S is cmpletely dughnut. The fllwing fact characterizes cmpletely dughnut null sets in terms f nwhere dense sets (cf. [El74, Lemma 6]). Fact 1.3. A set S [ω] ω is cmpletely dughnut null if and nly if it is nwhere dense with respect t the dughnut tplgy. Prf. Take any set S [ω] ω. By definitin, S is cmpletely dughnut null if fr each dughnut [a, b] ω there is a dughnut [a, b ] ω [a, b] ω \ S, and hence, by Fact 1.1, [ω] ω \ S cntains an pen dense set (w.r.t. the dughnut tplgy). n the ther hand, if [ω] ω \ S cntains an pen dense set (w.r.t. the dughnut tplgy), then fr each dughnut [a, b] ω there is a basic pen set (a, b ) ω (a, b) ω such that (a, b ) ω S =. Nw, fr any dughnut [a, b ] ω (a, b ) ω we have [a, b ] ω S =, and since [a, b] ω was arbitrary, this implies that S is cmpletely dughnut null. Befre we prceed, let us first define cmpletely dughnut sets in terms f trees. Let {0, 1} <ω be the set f all finite sequences f 0 s and 1 s. Fr s, t {0, 1} <ω we write s t if s is a prper initial segment f t, and we write s 4 t if s t r s = t. A set T {0, 1} <ω is called a tree, if s T and t s implies t T. If T {0, 1} <ω is a tree and s T, then T s = {t T : s 4 t t s}. If s 0,..., s n {0, 1} <ω and u {0, 1}, then s u := s 0,..., s n, u. A tree T {0, 1} <ω is called unifrm, if fr all s, t T f the same length we have s 0 T t 0 T and s 1 T t 1 T. If T is a tree, then a set ξ T is called a branch thrugh T, if fr any s, t ξ we have s 4 t r t s and ξ is maximal with respect t this prperty. The set f all branches thrugh T is dented by [T ]. Ntice that all branches thrugh a finite unifrm tree are f the same length. A tree T {0, 1} <ω is called perfect, if fr every s T there is

4 4 a t T with s 4 t such that bth t 0 and t 1 belng t T ; such a sequence t is called a splitting nde f T. The set f all splitting ndes f T is dented by split(t ) and fr n ω, split n (T ) := { s split(t ) : {t split(t ) : t s} = n }. Finally, the set splev(t ) := { t : t split(t )} dentes the set f all split levels f T. Let ω 2 be the set f all functins frm ω t {0, 1}. Fr a set a ω, let χ a ω 2 be such that χ a (n) = 1 iff n a, and fr ξ ω 2 let χ 1 (ξ) = {n ω : ξ(n) = 1}. Fact 1.4. Each unifrm perfect tree T {0, 1} <ω crrespnds in a unique way t a dughnut, and vice versa. Prf. Let T {0, 1} <ω be a unifrm perfect tree. Let D = {χ 1 (ξ) : ξ [T ] χ 1 (ξ) [ω] ω }, then D is equal t the dughnut [a, b] ω, where a = ξ [T ]{n ω : ξ(n) = 1} and b \ a = splev(t ). n the ther hand, if a b ω with b \ a [ω] ω, then let T {0, 1} <ω be the tree with [T ] = {ξ ω 2 : a χ 1 (ξ) b}. It is easy t see that T is a unifrm perfect tree. If T is a unifrm perfect tree, then let dnut(t ) dente the dughnut which by Fact 1.4 crrespnds t T. The fllwing lemma is similar t [El74, Lemma 7]. Lemma 1.5. With respect t the dughnut tplgy, a set S [ω] ω is meager if and nly if it is nwhere dense. Prf. Let S [ω] ω be any meager set. By Fact 1.3 it is enugh t shw that S is cmpletely dughnut null. Since S is meager, there are cuntably many nwhere dense sets W n such that S = n ω W n. Fr each n ω, let n be an pen dense set with n W n =. Let [a, b] ω be any dughnut and let T 0 be the crrespnding unifrm perfect tree. Assume we already have cnstructed a unifrm perfect tree T n fr sme n ω. Fr each t split n (T n ) cnsider the trees T n t and T n 0 t. By a successive amalgamatin we can 1 cnstruct a unifrm perfect tree T n+1 such that split n (T n+1 ) = split n (T n ), [T n+1 ] [T n ] and dnut ( T n+1) n. Finally, let T = n ω T n, which is by cnstructin a unifrm perfect tree. Nw, dnut(t ) = [a, b ] ω fr sme a b ω with b \ a [ω] ω, and by cnstructin we have [a, b ] ω [a, b] ω and [a, b ] ω n ω n, and since S n ω n = we get [a, b ] ω S =. The fllwing is a cnsequence f the preceding bservatins (cf. [El74, Therem 9]). Prpsitin 1.6. A set S [ω] ω is cmpletely dughnut if and nly if S has the Baire prperty with respect t the dughnut tplgy. Prf. Let S [ω] ω be cmpletely dughnut and let = { (a, b) ω : [a, b] ω S }. The set, as a unin f pen sets, is pen. Since S is cmpletely dughnut, fr every dughnut [a, b] ω there is a dughnut [a, b ] ω [a, b] ω, such that either [a, b ] ω S r [a, b ] ω S =. Hence, fr every nn-empty basic pen set (a, b) ω there is a nn-empty basic pen set (a, b ) ω (a, b) ω such that (a, b ) ω S \ =, which implies that the set S \ is nwhere dense. Thus, S is the unin f an pen set and a nwhere dense set, and therefre has the Baire prperty.

5 n the ther hand, let us assume that S [ω] ω has the Baire prperty. Thus, there is an pen set such that S M is meager. Since meager sets are nwhere dense (by Lemma 1.5) and nwhere dense sets are cmpletely dughnut null (by Fact 1.3), and since pen sets are cmpletely dughnut (by Fact 1.2), S is cmpletely dughnut. Remark. The set f cmpletely dughnut null sets frms an ideal n [ω] ω. This ideal, dented by v 0, is related t Silver frcing and was studied by Jörg Brendle in [Br95]. Fr example he shws that cv(v 0 ) r (where r dentes the reaping number) and that ω 1 = cv(v 0 ) < cv(p 2 ) = ω 2 = c is cnsistent with ZFC (where P 2 dentes the Mycielski ideal). 2. Making Dughnuts f Perfect Sets f Chen Reals In the fllwing we shw hw t make dughnuts f Chen reals. First we cnsider the case when ne Chen real is added and then we shw that a finite supprt iteratin f length ω 1 f Chen frcing, starting frm L, yields a mdel in which every Σ 1 2-set is cmpletely dughnut. Let C := C, be the Chen partial rdering, where C := {0, 1} <ω and fr s, t C we stipulate s t iff t 4 s. Remember that the algebra determined by C is the unique atmless cmplete Blean algebra which has a cuntable dense subset (cf. [BJ95, Therem 3.3.1]). T define the frcing ntin P, which will be used later, we have t give sme definitins. Let T {0, 1} <ω be a finite unifrm tree. As mentined in Sectin 1, all branches thrugh a finite unifrm tree are f the same length. Fr a finite unifrm tree T, let ht(t ) dente the length f a branch thrugh T. Finally, fr a tree T {0, 1} <ω let T n = {t T : t n}. Let P := P, be the partial rdering, where and fr T 1, T 2 P we stipulate P = { T {0, 1} <ω : T is a finite unifrm tree }, T 1 T 2 ht(t 1 ) ht(t 2 ) and T 1 ht(t2 ) = T 2. Since P = ℵ 0, the algebra determined by P is ismrphic t the algebra determined by C, hence, the frcing ntin P is ismrphic t C. If G is a P-generic filter ver sme mdel V, then T G := G is a unifrm perfect tree. Mrever, by genericity, every ξ [T G ] is Chen generic ver V (cf. [BJ95, Lemma 3.3.2]). In the sequel we will nt distinguish between the sets [ω] ω and ω 2, and all tplgical terms will refer t the usual tplgy n [ω] ω and ω 2, respectively. Let T {0, 1} <ω be any unifrm perfect tree and let σ T : ω splev(t ) n t fr sme t split n (T ) be the functin which enumerates the split levels f T. Further, we define a bijectin Θ T between [T ] and ω 2, by stipulating Θ T (ξ)(n) := ξ ( σ T (n) ) fr ξ [T ]. Fr any set S ω 2, let Θ T (S) := {Θ T (ξ) : ξ S [T ]}. It is easy t see that if S ω 2 is a Σ 1 2-set, then als Θ T (S) is a Σ 1 2 -set. Nw we are prepared t prve the fllwing: 5

6 6 Lemma 2.1. Let G be P-generic ver V and let a, b V be such that a b ω and b \ a [ω] ω. Then fr every Σ 1 2-set S V [G] with parameters in V there is a dughnut [a, b ] ω [a, b] ω with a, b V [G], such that either [a, b ] ω S r [a, b ] ω S =. Prf. Take any dughnut [a, b] ω with a and b in V and let T be the unifrm perfect tree which by Fact 1.4 crrespnds t [a, b] ω. Further, take any Σ 1 2 -set S [ω]ω with parameters in V. Put S = Θ T (S), then, since S is a Σ 1 2 -set, S = ι ω 1 B ι, where fr each ι ω 1, B ι is a Brel set (cf. [Je78, Therem 95, p. 520]) with Brel cde in V. The fllwing tw cases are pssible: Case 1: There is an α ω 1 such that B α is nn-meager. Case 2: Fr each ι ω 1, B ι is meager. We first cnsider case 1: S, let us assume that B α is nn-meager fr sme α ω 1. Since every Brel set has the Baire prperty, there is a nn-empty basic pen set such that \ B α is meager. Thus, there are cuntably many pen dense sets D n, n ω, such that n ω D n B α. Fllwing the prf f Lemma 1.5 (see als the prf f [DH00, Prpsitin 2.2]), by a successive amalgamatin we can grw a unifrm perfect tree T such that [ T ] B α. Let T {0, 1} <ω be the tree with [T ] = Θ 1 ( ) T [ T ], then T is a unifrm perfect subtree f T. Let [a, b ] ω = dnut(t ), then [a, b ] ω [a, b] ω and [a, b ] ω Θ 1 T (B α) Θ 1 T (S ) S. S far, we just wrked in V, but since the ntins like meager and nwhere dense are abslute fr Brel cdes (cf. [Je78, Lemma 42.4]), [a, b ] ω S is als valid in V[G], which cmpletes case 1. Let us nw cnsider case 2: S, assume that fr every ι ω 1, B ι is meager. By genericity, T G = G is a unifrm perfect tree where [T G ] avids every meager set with Brel cde in V. Hence, [T G ] ι ω 1 B ι =, which implies that Θ 1 ( T [TG ] ) S =. Let T {0, 1} <ω be the tree such that [T ] = Θ 1 ( T [TG ] ), then T is a unifrm perfect subtree f T. Let [a, b ] ω = dnut(t ), then [a, b ] ω [a, b] ω and [a, b ] ω S =, which cmpletes case 2 and the prf as well. Lemma 2.2. Let C ω1 be the finite supprt iteratin f length ω 1 f Chen frcing, starting frm L, and let G ω1 be C ω1 -generic ver L. Then L[G ω1 ] = Σ 1 2 (D c). Prf. Let c ι : ι < ω 1 be the generic sequence f Chen reals which crrespnds t G ω1. Let S [ω] ω be any Σ 1 2 -set with parameter r and let [a, b]ω be any dughnut. By [Ku83, Chapter VIII, Lemma 5.14], there is a λ ω 1 such that a, b, r V[G λ ], where G λ = c ι : ι < λ. Since the frcing ntins P and C are ismrphic, by Lemma 2.1 there is a dughnut [a, b ] ω in V[G λ ][c λ ] such that [a, b ] ω [a, b] ω and either V[G λ ][c λ ] = [a, b ] ω S r V[G λ ][c λ ] = [a, b ] ω S =. If we are in the frmer case (which crrespnds t case 1 in the prf f Lemma 2.1), then [a, b ] ω B fr sme Brel set B cntained in S with Brel cde in V[G λ ] and by absluteness we get V[G ω1 ] = [a, b ] ω S. n the ther hand, x(x [a, b ] ω x / S) is a Π 1 2-sentence with parameters in V[G λ ][c λ ] which hlds in V[G λ ][c λ ], thus, by Shenfield s Absluteness Therem (cf. [Je78, Therem 98, p. 530]) we get V[G ω1 ] = [a, b ] ω S =. Since the Σ 1 2 -set S and the dughnut [a, b]ω were arbitrary, this cmpletes the prf.

7 ;3 +# SK kc 7 3. Cnclusin Befre we can prve the main result f this paper, we have t give sme definitins. Let ω ω be the set f all functins frm ω t ω. Fr f, g ω ω we write f g, if f(n) g(n) fr all but finitely many n ω. A family F ω ω is called bunded, if there is a functin g ω ω such that fr every f F we have f g ; therwise, we call F unbunded. Using the techniques given in [BJ95, Chapter 6, Sectin 5] (see als [G93, Sectin 8]), ne can shw the fllwing: Lemma 3.1. A finite supprt iteratin f Chen frcing preserves unbunded families. In particular, a finite supprt iteratin f length ω 1 f Chen frcing, starting frm L, yields a mdel in which ω ω L is unbunded. A tree T ω <ω is called superperfect, if fr every s T there is a t T such that s t and {n ω : t n T } is infinite. A set F ω ω is K -regular, dented by K σ, if F is either bunded r there is a superperfect tree T such that [T ] F. Cncerning Σ 1 2 (K σ) we have the fllwing tw lemmata: Lemma 3.2 ([Ju88, Therem 1.1]). V = Σ 1 2 (K σ) V = r ω ω ( ω ω L[r] is bunded). Lemma 3.3 ([Ju88, 3]). Σ 1 2 (B ) = Σ1 2 (K σ) = Σ 1 2 (R ) are the nly implicatins between the three prperties B, R and K σ. Nw we are ready t prve the main result: Therem 3.4. Cn(ZFC) Cn ( ZFC + CH + Σ 1 2 (D c) + Σ 1 2 (K σ) ). In particular, it is cnsistent with ZFC that 2 ℵ 0 = ℵ 1 and that there is a Σ 1 2-set which is cmpletely dughnut, but which has neither the Baire prperty nr the Ramsey prperty. Prf. Let C ω1 dente the finite supprt iteratin f length ω 1 f Chen frcing, and let G ω1 be C ω1 -generic ver L. bviusly we have L[G ω1 ] = 2 ℵ 0 = ℵ 1 and by Lemma 2.2 we als have L[G ω1 ] = Σ 1 2 (D c). n the ther hand, by Lemma 3.1, L[G ω1 ] = ω ω L is unbunded, and hence, by Lemma 3.2 we get L[G ω1 ] = Σ 1 2 (K σ). In particular, by Lemma 3.3, there is a Σ 1 2 -set in L[G ω 1 ] which is cmpletely dughnut, but which has neither the Baire prperty nr the Ramsey prperty. Putting the previus results tgether, we get the fllwing diagram: Σ 1 2 (B ) Cnsidering this diagram, we get Σ 1 2 (K σ) P PPP PPP P PP PP Σ 1 2 (R ) Σ 1 2 (D c) n nn n nnn n nnn s{n nn

8 8 Questin 1. Σ 1 2 (K σ) = Σ 1 2 (D c)? Haim Judah and Saharn Shelah have shwn that 1 2 (R ) Σ1 2 (R ) (see [JS89, Therem 2.10]). n the ther hand, it is well-knwn that 1 2 (B ) des nt imply Σ1 2 (B ). This leads t Questin (D c) Σ 1 2 (D c)? Further, they have als shwn that 1 2 (B ) r ω ω ( Chen (L[r]) ), where Chen (L[r]) dentes the set f Chen reals ver L[r] (see [JS89, Therem 3.1 (iii)]). Thus, in the mdel L[G ω1 ] cnstructed abve, every 1 2-set has the Baire prperty, which mtivates Questin 3. Σ 1 2 (D c) = 1 2 (B )? Accrding t [Br95], let r 0 and v 0 dente the ideals n [ω] ω f cmpletely Ramsey null and cmpletely dughnut null sets, respectively. Szymn Plewik has shwn in [Pl86] that add(r 0 ) = cv(r 0 ). Thus, we like t mentin als Questin 4. add(v 0 ) = cv(v 0 )? References [BJ95] Tmek Bartszyński, Haim Judah: Set Thery: n the structure f the real line, A.K. Peters, Wellesley (1995). [Br95] Jörg Brendle: Strlling thrugh paradise, Fundamenta Mathematicae, vl. 148 (1995), [DH00] Carls A. DiPrisc and James M. Henle: Dughnuts, flating rdinals, square brackets, and ultraflitters, Jurnal f Symblic Lgic, vl. 65 (2000), [El74] Erik Ellentuck: A new prf that analytic sets are Ramsey, The Jurnal f Symblic Lgic, vl. 39 (1974), [G93] Martin Gldstern: Tls fr yur frcing cnstructin, Israel Mathematical Cnference Prceedings, H. Judah, Ed., Bar-Ilan University, Israel, vl. 6, (1993), [Je78] Thmas Jech: Set Thery, Academic Press [Pure and Applied Mathematics], Lndn (1978). [Ju88] Haim Judah: Σ 1 2-sets f reals, Jurnal f Symblic Lgic, vl. 53 (1988), [JS89] Haim Judah and Saharn Shelah: 1 2-sets f reals, Annals f Pure and Applied Lgic, vl. 42 (1989), [Ku83] Kenneth Kunen: Set Thery, An Intrductin t Independence Prfs, Nrth Hlland [Studies in lgic and the fundatins f mathematics; v. 102, Amsterdam (1983). [MS80] Gadi Mran and Dna Strauss: Cuntable partitins f prducts, Mathematika, vl. 27 (1980), [Pl86] Szymn Plewik: n cmpletely Ramsey sets, Fundamenta Mathematicae, vl. 127 (1986),