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Products of Perfectly Meagre Sets proceedings of the american mathematical society Volume 112, Number 4, August 1991 PRODUCTS OF PERFECTLY MEAGRE SETS IRENEUSZ RECLAW (Communicated by Andrew M. Bruckner) Abstract. We show that there exists a perfect set D Ç 2W x 2m such that for every Luzin set in D both projections of it are perfectly meagre. It follows (under CH) that the product of two perfectly meagre sets need not be perfectly meagre (or even have the Baire property in the restricted sense). This provides an answer to a 55-year-old question of Marczewski. A subset L of a completely separable metric space is a (generalized) Luzin set iff L n K is countable (less than the continuum) for every meagre set K and \L\> œ (\L\ = c). A subset X of a completely separable metric space is perfectly meagre iff, for every perfect set D, X C\D is meagre in D. X has the Baire property in the restricted sense if, for every perfect set D, X n D has the Baire property (symmetric difference of an open set and a first-category set) relative to D. non(K) = {\X\ : X ç R and X is not meagre}. In 1934, Sierpinski [Si2] showed (under CH) that the product of a set with the Baire property in the restricted sense and a complete metric space need not have the Baire property in the restricted sense. Sierpinski used the so-called function of Luzin [L] in his construction. In 1935, E. Marczewski (Szpilrajn) asked a question: Is the product of two perfectly meagre sets perfectly meagre? (See [Mi] or [S].) If there exists a Luzin set, then the answer to this question is "No." Theorem. There exists a perfect set D ç 2W x 2m such that, for every perfect set E c 2W, there exists G ç E dense Gö-setin E suchthat ((Gx2w)n(2w xG))í)D is nowhere dense in D. Corollary 1. Let D be a perfect set as above. ( 1) If L C D is a Luzin set in D, or (2) if L ç D is a generalized Luzin set in D and non(K) = c, then nx(L) and n2(L) are perfectly meagre. Received by the editors February 9, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 28A05; Secondary 04A15. Key words and phrases. Perfectly meagre set, Luzin set. ©1991 American Mathematical Society 0002-9939/91 $1.00 +$.25 per page 1029 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1030 IRENEUSZ RECLAW Proof. Let E and G ç E have the properties from the thesis of the theorem. If L is a Luzin set in D, then \Lf)(Gx2co)\ < œ. So K^^Z.) n G)| < œ. Thus G \ nx(L) is residual in E . The proof for the generalized Luzin set is similar. Corollary 2. (1) If there exists a Luzin set, or (2) if there exists a generalized Luzin set and non(K) = c, then there exist X, Y c 2W perfectly meagre such that X x Y is not perfectly meagre. Proof. We can assume that L is a Luzin set in D. Let X = nx(L) and Y = n2(L). Then Lçlxv,so X x Y is second-category in D. Proof of Theorem. Let {An : n e co] be a family of infinite pairwise dis- joint subsets of co. We will define D ç 2W x 2W such that (x, y) £ D •«• 3n3k,i€Ax(k) - y(l) = ° or> equivalent^, (x,y) e D <# V„x\An = 1 or y \An = 1. Since it is easier to investigate properties of D in (2W)Wx (2W)W, we will define D in this space: ((xk), (yk)) e D <&Vkxk = 1 or yk = 1, where 1 is the constant function equal to 1. Let E ç (2W)Wbe a perfect set and Un = {(xk) e (2œ)w : xn ¿ 1}. Define G = f)neJE n Un) U (E \ (EnUn) .Let IV, VC (2œf be open sets such that W= W0xWxx- ■■xWn_xx2wx2bJ x- ■■ and V = VQxVxx-■ ■xVn_xx2wx2ax-■ ■ and (W x V) n D ¿ 0. We will find open sets W*, V* such that W* ç W, V ç V, and (W x V") n D ¿ 0 and (W x V*) n(Cx (2T) n O = 0. Consider the following cases: (1) Supposethat ((W\(ËJTLT))x V)nD ¿ 0. Then W* = (W\(E nUn))n Un and V = V. It is obvious that (W7*x V") n (G x (2a')w) = 0. We will show that (JF* x F"*) n D ^ 0. By the assumption, there exists an open set H = H0xHxx--xHnx---xHkx2wx2wx-- suchthat (HxV)C\D¿0 and H ç W \ (E n Un). It means that there exist x = (xk) e H and y = (yk) 6 F such that for every i, x¡ = 1 or v, = 1. Let x be an arbitrary element of //n different from 1. Then let x* = (x*) such that , _ ( Xj ifi£n, ' \x if / = n and y* = (y*) such that * = ( y¡ if i 9e», y' \ 1 if i = «. Then x* eHnUn and y* e V, so x* e H>" and y* e V and (x*, y") e D because for every i, x* = 1 or y* = 1. (2) Suppose that ((W \ (£nl/,)) x K) n D = 0. Then let W* = W and V* = V r\Un. There exist x = (x(), y = (y,-) such that x e ^, y e V, and (x,y)eD. Let * « * f x if i' ^ n, x = (x ), where x¡ = < ' 1 if¡ = fl License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use PRODUCTS OF PERFECTLY MEAGRE SETS 1031 and * , •*. u * ( y¡ if i¿n, y = (y. , where y, = <! ... I.y if í = «, where y is an arbitrary element of 2W different from 1. Then x* 6 W = W*, y* e V n U„ = V*, and (x*, y*) e D. We show that (W* x V*)n(Gx (2(ü)C0)f)D= 0. We know that GC(EnUn) U(E\ (EnUn)). By the assumption, we have that ( W* x V*) n ( W \ ËlTiT) x (2a')cor\D = 0. Since (Un x Un)nD = 0, we have (EnUn) x (Vf)Un)nD = 0. Thus ((E n Un) x (2ü)f) n (W* xV*)nD = 0. D Note. After seeing the solution to the problem given in this paper, Pawlikowski constructed an alternative solution which is more similar to the construction of Sierpinski [Si2] and uses the "function of Luzin" [L]. Pawlikowski's solution has appeared in [P]. ACKNOWLEDGMENT The author would like to thank J. Pawlikowski for helpful conversations. References [L] N. Luzin, Ensembles de l-re catégorie, Fund. Math. 21 (1933), 114-126. [Mi] A. W. Miller, Special subsets of the real line, Handbook of Set-theoretic Topology (K. Kunen and J. E. Vaughan, eds.), Chapter 5, 1984, pp. 201-233. [P] J. Pawlikowski, Products of perfectly meagre sets and Luzin function, Proc Amer. Math. Soc. 107(1989), 811-815. [Si] W. Sierpiñski, Hypothèse du continu, Chelsea, New York, 1956. [Si2] _, Sur un problème de M. Kuratowski concernant la propriété de Baire des ensembles, Fund. Math. 22 (1934), 262-266. [S] E. Szpilrajn (Marczewski),Problème 68 , Fund. Math. 25 (1935), 579. Institute of Mathematics, University of Gdansk, Wita Stwosza 57, 80-952 Gdansk, Poland License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use.
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