Andryszczak, A.

Recław, I.

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A note on strong measure zero sets

Institute of Mathematics AS CR, v. v. i.

Acta Universitatis Carolinae. Mathematica et PhysicaA. Andryszczak; Ireneusz RecƚawA note on strong measure zero setsActa Universitatis Carolinae. Mathematica et Physica, Vol. 34 (1993), No. 2, 7--9Persistent URL: http://dml.cz/dmlcz/701988Terms of use:© Univerzita Karlova v Praze, 1993Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.cz1993 ACTA UNTVERSITATIS CAROUNAE-MATHEMATICA ET PHYSICA VOL. 34. NO. 2 A Note on Strong Measure Zero Sets A. ANDRYSZCZAK, I. RECLAW Gdansk*) Received 14 April 1993 We say that a metric space (X> Q) is strong measure zero if for every sequence of positive numbers {en}n€ m there is a sequence of sets {Lrt}n6e> such that X « (jLn and diam(Ln) < en for each n. n In [GMS] the autors proved a theorem which characterizes strong measure zero sets. Theorem. X c R is strong measure zero iffVF<ziLimeagreX + -F ^ R We strenghten it to the following result. Theorem 0. X c R is strong measure zero iff VD<zR2,F0se,\ V / ) , « meagre =» |J Dx * R] \xeR xeX ) Observe that if F is meagre F«,-set then X + F - (J Dx where D - \}{x}x x*X *eR (F + x). So we see that =» in Theorem 0 implies => in Theorem. It will be shown in Theorem 1. The <= in both theorems are very simple but in a case we do not need the algebraic structure in the proof. It will be shown in Theorem 2. Theorem 1. Let Y be a o-compact metric space, Z a locally compact space or completely metrizable space. Then if X c Y is a strong measure zero set then VDCYXZ,F0 set\ V Dx is meagre =» U Dx± Z Proof. The proof is very similar to the proof of Theorem in [M]. Let D be a Fa-set in y X Z with meager vertical sections. Then D — |J Fn are closed with nowhere dense vertical sections. We may assume also that FnczFn+l and Fna KnX Z, where Kn are compact, Kn c Kn+1 and Y- (J Kn. *) Institute of Mathematics, University of Gdaňsk, Wita Stwosza 57, 80-952 Gdaňsk, Poland Lemma. Let C c K x Z be a closed set with nowhere dense vertical sections, where K c Yis a compact set. Let Be: Z be a closed ball. Then there exists e > 0 and a finite family G of closed balls contained in B such that: VLC***(L><. 3 ( L X G ) n C « 0 G*G This follow strictly from the compactness of K. Indeed, if x is any point from K then there exist ex and a closed ball Gx in Z such that B(x9 ex) X G n C =» 0. We can find a finite family {xl9..., xk} of elements of K such that K c (J 5(^„ cx/3). It is easy to see, that the family G « {GX(: i -• 1, . . . , A:} and £ — min{ex/3: i •- 1, . . . , £} has the required property. Using Lemma we will construct a finitely branched tree T c o><6> and a sequen-ces: (Bs)seT of closed balls from Z and (£s)56r of positive real numbers with the following properties: (1) -V* c Bs (2) V56ft/,n rVLC Ydiam(L) < es =» 3(L X 5,**) n F„ - 0 For every « we define 6n — min {e5: |s| — n & 5 € 7] We know that 5n > 0. Now let X c y be a strong measure zero. From the definition of the property of strong measure zero we know that there exists a sequence of balls {Ln}nea) contained in Y such that X c p| [) Ln and m n>m diam(Ln) < dn. Now we construct a function/: co -+ a) such that (Ln X BAn+1) n F,, - 0. Let JC e H^/ln- F r o m this w e o b t a i n that (fl U Lm X {*}) n |jFn) - 0. n n m>/t n Theorem 2. Let y 6e a separable metric space, Z Hausdorff, second-countable dense in itself space, X c Y. Then if (*) VDC yxz> closedset V Dx is meagre •> |J Dx ^ Z then X is strong measure zero. Proof. Let us assume that we have a sequence (£„)-,««, of positive real numbers. We will construct a cover of X (Kn)n6a) with open balls such that diam(Kn) < en. Let (Un)n€Q) be a countable base of Z, and for any n < co let (I/n,m)mec> be open disjoint sets in Un and let (Bnm)m6(0 be a cover of Y with open balls of diameter less than en. Let us fix n < co and put Wn — (J^«.« x ^ r ^ut al s o A. •" m yxZ\W„andD-- f]^-.-n 8 Clearly D is a closed set in Y X Z. Next we show that V Dx is meagre. For that x* ) if Un is any base set in Z and x e Y than let m < cy be such ihat xe Bnm. We have tf„,m c (|JS,,,m X Un>m)x = (W„), - Z\(Dn)x c Z \ ( f | A,)* - Z\ A so D is m m meager. From the (*) we obtain U (D)x * z so let z e Z\ U (o)*- Let n < a> be fixed. We have that U(->)*" U fl(A0-- U n ( y x Z ^ X - ^ n UTO, * e * *** n x e * n J t e * n SO( + )2 6 f) U(W„),. j r e * n Now we define sets (Kn)n€o): For any A* e a> let m e <y be such, that z e Un>m, if there exists any In this case we put Kn =* .Bnm. If there does not exist any m e w such that z € £/nm we will take as Kn any open ball in Ywith diameter less than en. * must check that Jf c |J #„. So let x £ X. From the statement ( + ) we know, n that :here exists n e <0 such that z e ( Wn)x, that means z e (J(_Bn m X Untfn)x, so we m must have for some m e cy: z e f/n m and JC e Bn m, and this implies that x £ Kn from the definition of Kn. This ends the proof. References [GMS] GALVIN F., MүaELSKi J. and SOLOVAY R. M., Strong measure zero sets, AMS Notices, 26, A-280. [M] MILLER A. W., Special subsets of the real line, Handbook of Set Theoretic Topology, (K. Kunen and J. Vaughan, Eds.), North-HoUand, Amsterdam, 1984. 

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A note on strong measure zero sets

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