PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 7, Pages 2141{2150 S 0002-9939(04)07360-5 Article electronically published on January 23, 2004 MYCIELSKI IDEAL AND THE PERFECT SET THEOREM MIROSLAV REPICKY´ (Communicated by Alan Dow) Abstract. We make several observations on the Mycielski ideal and prove a version of the perfect set theorem concerning this ideal for analytic sets: If A ⊆ !2 is an analytic set all projections of which are uncountable, then there is a perfect set B ⊆ A a projection of which is the whole space. We also prove that (a modification of) an infinite game of Mycielski is determined for analytic sets. 1. Introduction For a 2 [!]! and A ⊆ !2letΓ(A; a) be the infinite game for players I and II in which both players choose consecutive elements of a sequence x 2 !2. Player II chooses x(n)forn 2 a, and the other moves are done by player I. Player I wins if x 2 A, and otherwise player II wins. This game was used by J. Mycielski [6] for introducing new invariant ideals on !2 that are orthogonal to the ideals of sets of Lebesgue measure zero and of meager sets. A. Ros lanowski [8] introduced the next two ideals related to this game: ! ! C2 = fA ⊆ 2:(8a 2 [!] ) player II has a winning strategy in Γ(A; a)g; ! ! a P2 = fA ⊆ 2:(8a 2 [!] ) Aa =6 2g; where Aa = fxa : x 2 Ag. The ideals C2, P2 are translation-invariant σ-ideals, P2 ⊆ C2,everysetinC2 having the Baire property is meager, and every measurable set in C2 has measure zero. In this paper we make several observations on these two ideals and prove variants of the perfect set theorem for them (Theorems 2.2 and 3.2). In the case of the ideal C2, this perfect set theorem is the determinacy of a modified game for analytic sets. Namely, let Γ(A) be the following modification of the above game: At the beginning of the game, player I chooses a 2 [!]!, and then the players follow the rules of the game Γ(A; a). Clearly, ! C2 = fA ⊆ 2 : player II has a winning strategy in Γ(A)g: Theorem 3.2 says that the game Γ(A) is determined for analytic sets A ⊆ !2. We use the standard notation. If p ⊆ <!2(orp ⊆ <!!)isatreeands 2 p,then <! we denote [p]=fx :(8n) xn 2 pg and ps = ft 2 p : s ⊆ t _ t ⊆ sg.Fors 2 2or s 2 <!!,[s] denotes the corresponding clopen basic set fx : s ⊂ xg in !2or!!. Received by the editors August 21, 2002 and, in revised form, March 27, 2003. 2000 Mathematics Subject Classification. Primary 03E15; Secondary 03E17, 91A44. Key words and phrases. Mycielski ideal, analytic sets, perfect set theorem. This work was supported by a grant of Slovak Grant Agency VEGA 2/7555/20. c 2004 American Mathematical Society 2141 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2142 MIROSLAV REPICKY´ Let us recall that s is the minimal cardinality of a set A ⊆ [!]! such that for every a 2 [!]! there is b 2 A such that ja n bj = ja \ bj = !; h is the least cardinal κ such that P(!)=fin is not κ-distributive (see [5]). 2. The ideal P2 The ideal P2 is called the Mycielski ideal (see [8] and [3]). We consider the following subideal I0 of P2, defined by ! ! ! I0 = fA ⊆ 2:(8a 2 [!] )(9b 2 [a] ) jAb|≤!g: This is the σ-ideal IX defined in [7] for the sequence of open sets X = hUn : n 2 !i with U = fx 2 !2:x(n)=0g where n S T S T ! ! ! ! IX = fA ⊆ 2:(8a 2 [!] )(9b 2 [a] ) A ⊆ Un [ ( 2 n Un)g: m n2bnm m n2bnm Theorem 2.1. (i) For every cardinal number κ with !<κ≤ s, ! ! ! I0 = fA ⊆ 2:(8a 2 [!] )(9b 2 [a] ) jAbj <κg ( P2: ! (ii) s =minfjAj : A ⊆ 2 and A 2 P2 nI0g. (iii) add(I0) ≥ h. Proof. (i) The inclusions from left to right and the first equality are consequences of definitions. The only nontrivial part is the inequality (. Consider a base matrix B in [!]! (constructed in [1]). Every projection Ba again is a base matrix in [a]! and has the size of the continuum. Therefore B 2 P2 nI0. (ii) Every set of size < s is in both P2 and I0.Ifs < c, then any splitting family of size s is in P2 nI0.Ifs = c, then the set of the size of the continuum (base matrix) used in the proof of (i) belongs to P2 nI0. (iii) The inequality add(I0) ≥ h is due to the fact that A 2I0 if and only if the set fa 2 [!]! :(8x 2 A)((81n 2 a) x(n)=0_ (81n 2 a) x(n)=1)g is open dense in P(!)=fin. Another example of a set belonging to P2 nI0 is constructed in Theorem 4.3. Now we prove a version of a perfect set theorem, which in a particular case says that if every projection of an analytic set in !2 is uncountable, then it contains a perfect set a projection of which is the whole space. ! Theorem 2.2. If an analytic subset of 2 is not in I0, then it contains a perfect set that is not in P2. Hence there is no analytic set in P2 nI0. ! Proof. Let A ⊆ 2 be analytic, and let A=2I0.Letf be a continuous mapping ! <! from ! onto A.Letfun : n 2 !g be an enumeration of !, and by induction on n 2 ! let us construct a sequence of infinite sets an ⊆ ! such that ! (i) (8b 2 [a0] ) jAbj >!, ! (ii) (8n 2 !)(jf([un])an|≤! _ (8b 2 [an] ) jf([un])bj >!), and (iii) (8n 2 !) an+1 ⊆ an. Condition (i) is due to the assumption that A=2I0, and condition (ii) is due to the an−1 ! fact that either f([un])an−1 belongs to the ideal I0 (in 2) or not. Let a 2 [!] <! be a pseudo-intersection of fan : n 2 !g and let T = fu 2 ! : jf([u])aj >!g. For every u 2 T and every b 2 [a]!, S <! f([Tu])b ⊇ f([u])b n ff([v])b : v 2 ! n T g; License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use MYCIELSKI IDEAL AND THE PERFECT SET THEOREM 2143 and therefore ! (2.1) (8u 2 T )(8b 2 [a] ) jf([Tu])bj >!: (u) <! (u) For u 2 T let p = fs 2 2:[s] \ f([Tu]) =6 ;g, i.e., f([Tu]) = [p ]. Then 1 (u) n+1 (2.2) (8u 2 T )(8 n 2 a)(9s1;s2 2 p \ 2) s1(n) =6 s2(n); since otherwise there exists an infinite set b ⊆ a such that j[p(u)]bj =1,which contradicts (2.1). Condition (2.2) enables us to construct, by induction on n = jsj <! for s 2 2, an infinite set b = fkn : n 2 !}⊆a with kn+1 >kn,andsystems <! <! <! X = fus : s 2 2}⊆T , Y = fts : s 2 2}⊆ 2sothatu; = ; = t;, (us) kn+1 ts_0;ts_1 2 p \ 2aresuchthatts_i(kn)=i for i =0,1,andus_0;us_1 2 n ⊆ _ 2 ⊆ _ Tus are such that f([Tus_i ]) [ts i]fors 2andi = 0, 1. Clearly, ts ts i. The systems X and Y determine perfect trees in <!! and <!2, respectively, namely the trees <! T0 = fu 2 T :(9v 2 X) u ⊆ vg;p0 = fs 2 2:(9t 2 Y ) s ⊆ tg: Hence the sets B =[T0]andC =[p0] are perfect compact, and clearly C ⊆ A,since C = f(B). From the construction of Y it follows that Cb = b2. Corollary 2.3. The union of less than h analytic sets from P2 is in P2. Let P be the set of all perfect trees in <!2. For n 2 ! and p 2 P we define n n p ≤n q if p ⊆ q and p \ 2=q \ 2. A set D ⊆ P is said to be !-dense if for every tree p 2 P and every n 2 ! there is q 2 D such that q ≤n p. In [7] it was said that an ideal I on !2 is tall if it contains all singletons and every perfect set A ⊆ !2 has a perfect subset in I. Let us recall that the condition in this definition concerning perfect sets is the so-called property (P) introduced by M. Balcerzak and studied in a number of papers; see, e.g., [2] for references. In [7] the question arose whether the ideals IX defined for various sequences of open sets are tall. The referee has kindly informed us that this is not true, because W. D¸ebski, J. Kleszcz, and Sz. Plewik in [4] have proved that the ideal P2 (and hence also I0) does not have property (P). Anyway, we can make several observations in this direction. Let us note that if I is a tall ideal on !2, then for every p 2 P there exists q ≤n p such that [q] 2I, i.e., the family of perfect trees DI = fp 2 P :[p] 2Igis !-dense in P . Theorem 2.4. (i) The ideal I0 contains perfect sets. In fact, for every set B 2 P2 there is a perfect set A 2I0 disjoint from B. <! ! (ii) For every perfect tree p ⊆ 2 there are a perfect tree q ≤n p and c 2 [!] such that for every a 2 [c]! there is b 2 [a]! such that [q]b is countable c (hence [q]c 2I0 inside 2).