ON POINT-PICKING GAMES

Transcription

1 Volue 10, 1985 Pges ON POINT-PICKING GAMES by I. Juhász Topology Proceedigs Web: Mil: Topology Proceedigs Deprtet of Mthetics & Sttistics Aubur Uiversity, Alb 36849, USA E-il: ISSN: COPYRIGHT c by Topology Proceedigs. All rights reserved.

2 TOPOLOGY PROCEEDINGS Volue ON POINT-PICKING GAMES I. Juhsz 1. Itroductio Let X be topologicl spce, P property of subsets of X d ordil the the poit-pickig ge GP(X) is defied s follos: To plyers, I d II, tke turs plyig. A roud cosists of Plyer I choosig (o-epty) ope set U c X d the Plyer II choosig poit x E U. A roud is plyed for ech ordil less th. Plyer I is the ge if the set of poits picked by Plyer II hs property 'P, otherise Plyer II is. This ge s itroduced d ivestigted for the cses P = D (= dese) d P = SD (= soehere dese) i [BJ], here, og other thigs, resoble criteri ere forulted d prove for Plyer I to hve iig strtegy. The i here is to try to do the se for Plyer II. As e shll see, the situtio is ore coplex i this cse. Let us put for y spce X o(x) = sup{d(y): Y c X dese i X}. It is strightforrd to sho tht if is y ordil d o(x) > the II t GD(X). I fct, s A. Berer hs sho i [B], this is exctly he Plyer II hs so-clled sttiory iig strtegy for GD(X). Our i is to ivestigte uder ht circustces is this obvious d turl sufficiet coditio for II to i lso ecessry. I fct, e shll oly do this for the cse i hich the legth of the ge is.

3 104 Juhsz Before e forulte our i result let us recll tht, for T 3 spce X, I t G~(X) if d oly if (X) =, see [BJ]. It ill lso be helpful to keep i id tht, gi for T spce X, oe lys hs o(x) < (X) < 2 o (X). 3 Our i result reds s follos: Theore 1. for y give crdil K ~ 2 : The folloig to stteets re equivlet (1) There exists (T ) spce X ith o(x) d 3 (X) = K such tht II t GD(X). (2) The rel lie R c be ritte s the uio of < K ohere dese subsets. Before e give the proof of this result let us dd fe coets. If K = the this is trivil sice both (1) d (2) re flse. If K = 2 (the xiu possible vlue of K) the (2) of course is vlid, hece so is (1), hece e get ZFC exple of T spce X ot stisfyig 3 O(X) > d such tht II t GD(X). Filly, if < K < 2 the theore 1 shos tht, t lest for (T 3 ) spces of -eight K, it is idepedet of ZFC hether the coditio o(x) > is lso ecessry for Plyer II to i the ge GD(X). 2. ProofofTheore 1 As is ell ko, see e.g. [K], the egtio of stteet (2) of theore 1, i.e. tht R is ot the uio of < K ohere dese sets, is equivlet to MACK' hich deotes Mrti's xio for coutble posets d ~ K dese sets. Hece the folloig result yields "oe hlf" of theore 1.

4 TOPOLOGY PROCEEDINGS Volue Theore 2. Assue MACK d let X be y spce ith 6(X) = d (X) < K. The Plyep II does ot hve iig strtegy for G D (X). Proof. Let s: l(x}< x lex) ~ X be strtegy for Plyer II, e shll sho tht s is o't iig. I order to do this e first defie, by iductio o E, for every sequece E coutble dese subset S s follos. of X To defie S e first cosider the set X = {s(u}: U E l(x)} hich is dese i X sice s is strtegy for Plyer II, hece e y, by 6(X) =, tke S to be coutble dese subset of X. No, let E d ssue k tht S hs bee defied for ll E u{ : k < }, ore over, for every such 0, So ope sets U O such tht x {x o : ill E } d e hve soe here = (0) is the legth (i.e. doi) of o. ~i ht follos, e shll rite U(o,i) isted of Uo~i).} l No, if E + X o the e cosider the set (I {s(u(o,o),u(o,l),,u(o,)"u): U E T(X)} hich is gi dese i X hece hs cout:ble dese subset S {xo: E}, d, of course, e y choose for ech ill E ope set UO such tht x O s(u(o,o},u(o,l},,u(o,),u ). This copletes the iductio. The (obviously coutble) prtil order tht e t to pply MACK to is < ith the extesio of sequeces s the prtil order. To get the dese sets e first recll tht X hs -bse B ith IBI ~ K d for every B E B e put

5 106 Juhsz < o ~i (3i < (0)) (xo(i) = x(o,i) E B)}. Sice every So is dese i X it is obvious tht DB is dese i < for ech B E 8, cosequetly by MAC geeric brch T E WW over the fily {DB: B E B}. K there is Hoever, it is iedite fro our iductive costructio tht «U(T,i),x(T,i»: i E ) is ply of the ge GD(X) i hich Plyer II hs folloed the strtegy sd still the set {x(t,i): i E } is dese i X (becuse it itersects every B E B), hece s is ot iig. I vie of our bove rerk, it ieditely follos tht (1) iplies (2) i theore 1. To see the coverse, e first prove the folloig result. Theore 3. Let S ith lsi = K be T 2 spce ith coutble bse B hich dits fuctio ~: B + R+ = (0,00) stisfyig the folloig to properties: (i) for every pes d every s > 0 there is BE B such tht p E B d ~(B) < s; (ii) there is sequece {s : E} c R+ such tht for every {B : E } c B if ~ (B ) < s holds for ech E the S t- U{B : E}. The 2 K hs coutble dese subspce X for hich SD D II t G (X) (hece II t G (X) s e ll). Note tht (X) = K d o(x) = IXI. Proof. Let us strt by otig tht, possibly by pssig to pproprite subsequece of {E: E}, e y ssue tht i (ii) for every {B : E} c B ith

6 TOPOLOGY PROCEEDINGS Volue (B ) < E e ctully hve tht S\U{B : E} is ifiite (or, i fct, ucoutble). The costructio of the spce X, for coveiece e S K shll defie it s subspce of 2 rther th 2, is stdrd, see e.g. [H] or [J]. Let D deote the set of ll fuctios d such tht do(d) is fiite d disjoit subset of Bd rge(d) c 2, d for every d E De defie f d E 2 S by puttig: fd(p) = {d(b), if P E B E do(d); 0, if p E S\U do(d). We the put X {f d : d ED}, it is ell-ko tht X is S (coutble) dese subset of 2. We o describe, iforlly, iig strtegy for Plyer II i the ge G~D(X). First, it clerly suffices to do this oly for oves of Plyer I hich re trces of eleetry ope sets i 2 8, i.e. hve the for [3] = {f E X: s c f}, here s is y 0-1 fuctio defied o soe fiite subset of S. Suppose tht Plyer lis first ~ove is [so] here do(so) = O = {p~: i < O}. I respose to this Plyer II 1 first picks disjoit collectio B O = {~i: i < O} c B such tht Pi E B d (B ) < E hold for ll i < O. i i i This is possible by (i). The II y pick the eleet f of [so], here do ED is give: by do(do) = B O d d do(b.) so(p.) for ech i < O. If i the ext roud the 1 1 l ope set [sl] is plyed by I, here do(sl) = 1 1 {Pi: i < }, the II first chooses disjoit collectio Bl = {B : < j < l } such tht p. E: B d j O+i

7 108 Juhsz (B 0.) < 0. 1 for i < d the picks the poit l fdl E [sl], here do(d ) = B l d dl(b 0.) = Sl(p~) l i <. +1 for Cotiuig i this y, he the ply is fiished sequece {B i : i E } is geerted ith the property tht (Bi) < f for ll i E, hece S\U{B : i E } is ifiite. i i But clerly every choice fdi of Plyer II i this ply is such tht fdi(p) = 0 for every p E S\ {B : i E }, hece i {fdi: i E } is ohere dese i 2 S d cosequetly i X s ell. This shos tht the strtegy e described is ideed iig. A iedite corollry of theore 3 is tht if there is set S E [R]K hich does ot hve strog esure 0 the there is coutble dese X c 2 K such tht II t GSD(X). Hoever, i order to fiish the proof of theore 1 e shll hve to look t soe other exples. Let us cosider the Bire spce WW ith its stdrd coutble bse B = {[s]: s E <}, the "Bire itervls." For y subset S c W e let B S = {S [s]: s E <} d defie s: B s ~ 1 21s/ R+ by It is obvious tht (s,bs' s > lys stisfies (i) of theore 3. No, the fil lik i the proof of Theore 1 is give by the folloig (so fr upublished) result of A. Miller d D. Freli [MF]: Propositio (Miller-Freli). The folloig to stteets re equivlet for y give crdil K:

8 TOPOLOGY PROCEEDINGS Volue (A) t-1ac K ; (B) For every S E [W]K d every {E : E} c R+ there re {B : E} c B such tht ~(B ) < for ll s d S U{B: E}. Hece if (2) of theore 1 holds, i.e~. MACK fils, the by the bove propositio there is soe S E [W]K for hich B d ~S stisfy coditios (i) d (ii) of theore 3, s d cosequetly there is coutble dese X c 2 K for hich II t GSD(X). This copletes the proof of theore 1. Before e coclude, let us etio the folloig esy cosequece of theore 1: If K > d I\1AC I< holds the the ges GD(X) d GSD(X) re udecided for every T spce X stis 3 fyig 8(X) = d (X) = K (i prticulr, for every coutble dese subspce of 2 K ). Moreover, it s sho i [BJ] tht siilr "udecided" spces lso exist if oe ssues O. This leds to the folloig turl questio. Proble. Does there exist, i ZFC, T spce X for 3 hich the ge G~(X) is udecided? Refereces [B] A. J. Berer, Types of strtegies i poit-pickig ges, Top. Proc. 9 (1984), [BJ] d I. Juh&sz, Poit-pickig ges d HFD's, I: Models d Sets, Proc. Logic Cll. '83, Spriger LNS 1103, [H] E. Heitt, A rerk o des1.:ty hreter, Bull. AMS 52 (1946), [J] I. Juh&sz, Crdil fuctios--te yers lter, Mth. Cetre Trct o. 123, Asterd, 1980.

9 110 Juhsz [K] K. Kue, Set theory, Asterd, North Holld, [MF] A. Miller, Orl couictio, fll of Mtheticl Istitute of Hugri Acdey BUdpest, Hugry