Selective screenability in topological groups

SELECTI VE SCREENABILITY IN TOPOLO GICAL GR OUPS LILJANA BABINKOSTO V A Abstra ct. W e examine the sel ective screenabilit y property in top o- logical groups. In the metrizable case we also give characterizatio ns of S c ( O nbd , O ) an d Smirnov - S c ( O nbd , O ) in terms of the Hav er property and finitary Ha ver prop erty respectively relative to left-inv ari ant met- rics. W e prov e theorems stating conditions under which S c ( O nbd , O ) is preserved by pro ducts. Among metrizable groups we chara cterize the countable dimensional ones by a natural game. 1. Def initions and not a tion Let G b e top ological sp ace. W e shall u se the notations: • O : The collection of op en co vers of G . An op en cov er U of a top ological space G is said to b e • an ω -co v er if G is not a mem b er of U , bu t for eac h finite subset F of G there is a U ∈ U su c h that F ⊂ U . The symbol Ω denotes the collect ion of ω co ve rs of G . • gr oup a ble if there is a partition U = ∪ n< ∞ U n , where eac h U n is finite, and for eac h x ∈ G the set { n : x 6∈ ∪U n } is fi nite. The symb ol O g p denotes the collection of groupable op en co v ers of the space. • lar ge if ea ch element of the space is con tained in infinitely many elemen ts of the co v er. Th e symb ol Λ denotes the collect ion of large co v ers of th e space. • c- gr oup able if there is a p artition U = ∪ n< ∞ U n , where eac h U n is pairwise disjoin t and eac h x is in all b ut fin itely many ∪U n . The sym b ol O cg p denotes the collection of c-groupable op en co v ers of the space. No w let ( G, ∗ ) b e a topological group w ith iden tit y elemen t e . W e will assume that G is not compact. F or A and B sub sets of G , A ∗ B denotes { a ∗ b : a ∈ A, b ∈ B } . W e use the notation A 2 to denote A ∗ A , and for n > 1, A n denotes A n − 1 ∗ A . F or a neighborh o o d U of e , and f or a finite subset F of G the set F ∗ U is a neigh b orho o d of the fin ite set F . Thus, Key words and phr ases: Hav er prop erty , selectiv e screenabilit y , H u rewicz prop erty , finitary H av er prop erty , countable dimensional, selection principle, c- groupable co ver. Sub ject Cl assification: Primary 54D20, 54D45, 55M10; Secondary 03E20. 1 the set { F ∗ U : F ⊂ G finite } is an ω -co v er of G , which is denoted b y the sym b ol Ω( U ). Th e s et Ω nbd = { Ω( U ) : U a n eigh b orh o o d of e } is the set of all such ω -co vers of G . The set O ( U ) = { x ∗ U : x ∈ G } is an op en cov er of G . The symb ol O nbd = {O ( U ) : U a n eigh b orh o o d of e } denotes the collect ion of all suc h op en co ve rs of G . Selectio n pr inciples u s- ing these op en co ve rs of top ologica l group s hav e b een consider ed in sev eral pap ers, including [4], [5], [16] and [23], where information relev an t to our topic can b e foun d. No w we describ e the relev an t selection principles for this pap er. Let S b e an infin ite s et, an d let A and B b e collections of families of subsets of S . The selection principle S c ( A , B ), introd uced in [2], is defined as follo ws: F or eac h sequence ( A n : n < ∞ ) of elemen ts of the family A there exists a sequence ( B n : n < ∞ ) suc h that for eac h n B n is a pairwise disjoin t family refining A n , and S n< ∞ B n is a mem b er of the family B . The selection principle Smirnov − S c ( A , B ) is defined as follo ws: F or eac h sequence ( A n : n < ∞ ) of elemen ts of the family A there exists a p ositiv e in teger k < ∞ and a sequence ( B n : n ≤ k ) where eac h B n is a pairwise disjoint family of op en sets refining A n , n ≤ k and S j ≤ k B j is a mem b er of the family B . The metrizable space X is said to b e Haver [12] with resp ect to a metric d if there is for eac h sequence ( ǫ n : n < ∞ ) of p ositiv e r eals a sequence ( V n : n < ∞ ) where eac h V n is a p airwise disjoint family of op en sets, eac h of d -diamet er less than ǫ n , such that S n< ∞ V n is a co v er of X . And it is said to b e finitary Haver [8] with resp ect to the metric d if there is for eac h sequence ( ǫ n : n < ∞ ) a p ositiv e in teger k and a sequence ( V n : n ≤ k ) where eac h V n is a pairwise disjoin t family of op en sets, eac h of diameter less than ǫ n , suc h that S n ≤ k V n is a cov er of X . 2. S elective scre enability and S c ( O nbd , O ) Recen t inv estigat ions in to the Ha v er prop erty and its relation to the se- lectiv e screenability prop erty S c ( O , O ) rev ealed that the Ha ve r prop erty is w eak er than selectiv e screenabilit y . E. and R. Po l h as r ep orted th e follo wing nice charac terizations of S c ( O , O ) in terms of the Ha ve r prop ert y: Theorem 1 ([20]) . L et (X,d) b e a metrizable sp ac e. The fol lowing ar e e quivalent: (1) X has pr op erty S c ( O , O ) . 2 (2) X has the Haver pr op erty in al l e quivalent metrics. F or a top ological group the prop ert y S c ( O , O ) is stronger than S c ( O nbd , O ). This is in part seen by comparing Theorem 1 with the follo wing result: Theorem 2. L et ( G, ∗ ) b e a metrizable gr oup . The fol lowing ar e e quivalent: (1) The gr oup has pr op erty S c ( O nbd , O ) . (2) The gr oup has the Haver pr op erty in al l e quivalent left invariant metrics. In the p ro of of Theorem 2 w e us e the follo wing result of Kakutani: Theorem 3 ([14]) . L et ( U k : k < ∞ ) b e a se q uenc e of subsets of the top olo g- ic al g r oup ( H , ∗ ) wher e { U k : k < ∞} is a neighb orh o o d b asis of the identity element e and e ach U k is symmetric 1 , and for e ach k also U 2 k +1 ⊆ U k . Then ther e is a left-invariant metric d on H such that (1) d is uniformly c ontinuous i n the left uniform structur e on H × H . (2) If y − 1 ∗ x ∈ U k then d ( x, y ) ≤ ( 1 2 ) k − 2 . (3) If d ( x, y ) < ( 1 2 ) k then y − 1 ∗ x ∈ U k . In the ab ov e theorem V 2 denotes { a ∗ b : a, b ∈ V } . F or n > 1 a p ositiv e in teger the symbol V n is d efined s im ilarly . And no w the pr o of of T heorem 2: Pro of: 1 ⇒ 2: Let d b e a left-in v arian t metric of G and let ( ǫ n : n < ∞ ) b e a sequence of p ositiv e real n umb ers. F or eac h n c ho ose an op en neighborh o o d U n of the iden tit y elemen t e of G with diam d ( U n ) < ǫ n and put U n = O ( U n ). Then {U n : n < ∞} is a sequence f rom O nbd ( U ). App ly S c ( O nbd , O ). F or eac h n there is a pairwise disjoin t family V n of op en sets refinin g U n suc h that S n< ∞ V n is an elemen t of O . Now for eac h n , for V ∈ V \ there is an x ∈ G with V ⊆ x ∗ U n . But then diam d ( V ) ≤ diam d ( x ∗ U n ) = diam d ( U n ) ≤ ǫ n . Th us the V n ’s witnesses Ha v er’s prop erty for the given sequence of ǫ n ’s. 2 ⇒ 1: Let U n = O ( U n ), n < ∞ b e giv en. F or eac h n c ho ose a neigh b orh o o d V n of the iden tit y element e in G such that: (1) F or all n , V n ⊂ U n . (2) F or all n , V n ∗ V n ⊂ V n − 1 . (3) { V n : n < ∞} is a n eigh b orh o o d basis of the iden tit y e . By Kakutani’s theorem choose a left in v arian t metric d so that for ev ery n : (1) If y − 1 ∗ x ∈ V n then d ( x, y ) ≤ ( 1 2 ) n − 2 . (2) If d ( x, y ) < ( 1 2 ) n then y − 1 ∗ x ∈ V n . F or eac h n , p ut ǫ n = ( 1 2 ) n . Since G has the Hav er pr op erty with r esp ect to d , c ho ose for eac h n a p airwice disj oin t family V n of op en sets suc h that: (1) F or eac h n and f or eac h V ∈ V n , diam d ( V ) < ǫ n . (2) S n< ∞ V n co v ers G . 1 U k is symmetric if U k = U − 1 k 3 Then f or ev ery n and f or ev ery V ∈ V n , there is and x V with V ⊆ x ∗ V n ⊆ x V ∗ U n ∈ U n and so V n refines U n . But then V n witness S c ( O nbd , O ) for {U n : n < ∞} . ♦ Using the similar ideas one can pro ve the follo wing: Theorem 4. L et ( G, ∗ ) b e a metrizable gr oup . The fol lowing ar e e quivalent: (1) The gr oup has pr op erty Smirnov − S c ( O nbd , O ) . (2) The gr oup has the finitary Haver pr op erty in al l e quivalent left in- variant metrics. One ma y ask when th e prop erties S c ( O , O ) and S c ( O nbd , O ) are equiv alen t in a topological group. W e do n ot ha v e a complete answer. The Hurewicz prop er ty giv es a condition: A top ological sp ace G has the H ur ewicz pr o p erty if for eac h sequence U n , n < ∞ of op en co v ers of X there is a sequence F n , n < ∞ of finite sets s u c h th at eac h F n ⊂ U n , and f or eac h x ∈ G , the set { n : x 6∈ ∪F n } is fi nite. Theorem 5. L et ( G, ∗ ) b e a top olo gic al gr oup with the Hur ewicz pr op erty. Then S c ( O nbd , O ) is e quivalent to S c ( O , O ) . Pro of: Let ( G, ∗ ) b e a top ologic al group. It is clear that S c ( O , O ) implies S c ( O nbd , O ). F or the conv erse implicat ion, assume the group has prop ert y S c ( O nbd , O ). Let ( U n : n < ∞ ) b e a sequen ce of op en co ve rs of G . F or eac h n , and eac h x ∈ G c ho ose a neigh b orho o d V ( x, n ) of the id en titit y e suc h that x ∗ V ( x, n ) 4 is a s ubset of some U in U n . Put H n = { x ∗ V ( x, n ) : x ∈ G } . Apply the Hurewicz prop erty to the sequence ( H n : n < ∞ ). F or eac h n c ho ose a fi nite F n ⊂ H n suc h that for eac h g ∈ G , the set { n : g 6∈ ∪F n } is finite. W rite F n = { x i n ∗ V ( x i n , n ) : i ∈ I n } and I n is finite. F or eac h n , define V n = T i ∈ I n V ( x i n , n ) a neigh b orho o d of the identit y e . Cho ose a partition N = S k < ∞ J k where eac h J k is in fi nite, and for l 6 = k , J l ∩ J k = ∅ . F or eac h k , apply S c ( O nbd , O ) to the sequ ence ( O ( V n ) : n ∈ J k ). F or eac h n ∈ J k find a pairwise disjoin t family S n of op en sets suc h that S n refines O ( V n ) and S n ∈ J k S n co v ers G . F or eac h n defin e V n = { S ∈ S n : ( ∃ U ∈ U n )( S ⊆ U ) } . S ince V n ⊂ S n , V n is pairwise disjoin t and refines U n . W e will sho w that S n< ∞ V n co v ers G . Pic k any g ∈ G . Fix N g so that for all n ≥ N g , g ∈ ∪F n . Pic k k g so large that min ( J k g ) ≥ N g . Pic k m ∈ J k g with g ∈ ∪S m . Pic k J ∈ S m with g ∈ J . W e will sho w that J ∈ V m . W e ha v e that g ∈ ∪ F m , so pic k i ∈ I m with g ∈ x i m ∗ V ( x i m , m ). Since J ∈ S m , also pic k h m so that J ⊆ h m ∗ V m = h m × ( T i ∈ I m V ( x i m , m ) ⊆ h m ∗ V ( x i m , m ). W e ha v e that g = x i m ∗ z g = h m ∗ t g for some z g , t g ∈ V ( x i m , m ). So h m = x i m ∗ z g ∗ t g − 1 . No w consider any y ∈ J . Ch o ose t y ∈ V ( x i m , m ) with y = h m ∗ t y . But then y = x i m ∗ ( z g ∗ t g − 1 ∗ t y ∗ e ) ∈ x i m ∗ V 4 ( x i m , m ) ⊆ U , for s ome U ∈ U m . So we ha ve that J ∈ V m and g ∈ J . ♦ The sym b ol S 1 ( A , B ) d enotes the statemen t that there is f or eac h sequen ce ( O n : n < ∞ ) of elemen ts of A a sequ en ce ( T n : n < ∞ ) suc h that for eac h n T n ∈ O n , and { T n : n < ∞} ∈ B . A top ological group ( G, ∗ ) is said to b e a Hur ewicz-b o unde d group if it satisfies the selection pr inciple S 1 (Ω nbd , O g p ). 4 In [2] was sho wn that S c ( O , O ) is equiv alen t to S c (Ω , O ). The analogous equiv ale nce d o esn’t hold in top ological group s: Theorem 6. S c (Ω nbd , O ) do es not imply S c ( O nbd , O ) . Pro of: Let ( C, ∗ ) b e the unit circl e in th e complex p lane with complex m ultiplication. It is a compact metrizable group em b edd ing th e unit in terv al [0 , 1] as a subsp ace. Since ( C N , ∗ ) is a compact group it h as the Hurewicz prop er ty , so is Hu r ewicz b oun ded. Also R , th e real line with addition, is a Hurewicz-b ound ed top ological group . Thus the pr o duct group R × C N is Hurewicz b ounded , so h as th e prop ert y S 1 (Ω nbd , O ), and so has S c (Ω nbd , O ). But [0 , 1] N em b ed s as closed sub space into R × C N , and [0 , 1] N do es not h a v e the p r op erty S c ( O , O ). Thus the top ologica l group R × C N do es not ha ve S c ( O , O ), and as it has the Hurewicz p rop erty , Theorem 5 implies it is n ot S c ( O nbd , O ). ♦ The symbol S f in ( A , B ) denotes the statemen t that there is for eac h se- quence ( O n : n < ∞ ) of ele ments of A a sequence ( T n : n < ∞ ) of finite sets suc h that for eac h n T n ⊆ O n , and ∪ { T n : n < ∞} ∈ B . It w as shown in [15] that S f in (Ω , O g p ) is equ iv ale nt to th e Hurewicz prop ert y . And it is w ell known that S f in ( O , O ) is the Menger prop er ty , whic h is equiv alen t to S f in (Ω , O ). A top ological group is said to b e a M e nger b ound e d group if it has th e p rop erty S 1 (Ω nbd , O ). By ho w muc h can th e requiremen t that ( G, ∗ ) has the Hurewicz prop erty b e we ak ened in Th eorem 5? Natural p ossibilities include the Menger prop- ert y , Menger b oundedn ess or Hurewicz b oundedn ess. In ligh t of in teresting recen t examples of E. and R. Po l - [19], [20] w e conjecture that n on e of these w eak enings is enough: Conjecture 1. Ther e is a metrizable Menger b ounde d top olo gic al gr oup which has the pr o p erty S c ( O nbd , O ) , but not the pr op erty S c ( O , O ) . Conjecture 2 . Ther e is a metrizable Hur ewicz b ounde d top olo gic al gr oup which has the pr o p erty S c ( O nbd , O ) , but not the pr op erty S c ( O , O ) . Conjecture 3. Ther e is a metrizable top olo gic al gr oup which has the Menger pr op erty and pr op erty S c ( O nbd , O ) , but not the pr op erty S c ( O , O ) . It is clear that Conjecture 3 ⇒ C onjecture 1 and C onjecture 2 ⇒ Conjec- ture 1. It ma y b e that C on j ecture 2 is in d ep end en t of the Zermelo-F raenk el axioms. Recen tly E. and R. P ol s ho we d that C H imp lies Conj ecture 3. 3. Products E. Pol sh o w ed in [17] that there exist a zerod imensional su bset Y of the real line and a separable metric sp ace X and such that X has the prop ert y S c ( O , O ) in all finite p o w ers, b ut X × Y do es not h a v e S c ( O , O ). This failure do es not h app en for the group analogue: Theorem 7. L et ( G, ∗ ) b e a gr oup satisfying S c ( O nbd , O ) . If ( H , ∗ ) is a gr oup with pr op erty S c ( O nbd , O cg p ) , then ( G × H , ∗ ) also has S c ( O nbd , O ) . 5 Pro of: F or eac h n let U n b e an elemen t of O nbd ( G × H ). Eac h U n is of the form U n = O ( U n ) where U n is a neighborho o d of the identi ty ( e G , e H ) of G × H . Pic k V n ⊂ G a neighb orh o o d of e G , and W n ⊂ H a neigh b orho o d of e H so that V n × W n ⊆ U n . Then W n = O ( V n × W n ) is a refin emen t of U n , for all n . Let H n = O ( W n ) ∈ O nbd . Ap ply S c ( O nbd , O cg p ) to the sequ ence ( H n : n < ∞ ). F or eac h n find a finite pairwise disjoin t refin emen t K n of H n so that eac h x is in all but fin itely man y of S K n . Next, for eac h n pu t G n = O ( V n ) ∈ O nbd . Apply S c ( O nbd , O ) to the sequence ( G n : n < ∞ ). F or eac h n c ho ose pairwise disjoin t J n that refines G n so that S J n is a la rge op en cov er of G . F or eac h n d efine V n = { J × K : J ∈ J n , K ∈ K n } . Claim 1: V n refines W n : F or J ∈ J n and K ∈ K n there is an elemen t g ∈ G an d h ∈ H suc h that J ⊆ g ∗ V n and K ⊆ g ∗ W n . Bu t then J × K ⊆ g ∗ V n × h ∗ W n ∈ W n . Claim 2: V n is p airw ise disjoint : Let J 1 × K 1 and J 2 × K 2 b e elements of V n with J 1 × K 1 6 = J 2 × K 2 . If J 1 6 = J 2 then J 1 ∩ J 2 = ∅ b ecause the J n is disjoin t. So ( J 1 × K 1 ) T ( J 2 × K 2 ) = ∅ . Similarly , ( J 1 × K 1 ) T ( J 2 × K 2 ) = ∅ if K 1 6 = K 2 . Claim 3: S V n co v ers G × H . Consid er ( g , h ) as an element of C × H . Since S J n is a large co v er of G the set S 1 = { n : ( ∃ J ∈ J n )( g ∈ J ) } is infinite and there is an N suc h that S 2 = { n : ( ∃ K ∈ K n )( h ∈ K ) } ⊇ { n : n ≥ N } . Pic k an n ∈ S 1 ∩ S 2 . Pick J ∈ J n with g ∈ J and K ∈ K n with h ∈ K . Then ( g , h ) ∈ J × K ∈ V n . ♦ Corollary 8. L e t ( G, ∗ 1 ) and ( H , ∗ 2 ) b e metrizable top o lo gic al gr oups such that ( G, ∗ 1 ) has S c ( O nbd , O ) and H is zer o-dimensio nal. Then ( G × H , ∗ ) is a gr oup with pr o p erty S c ( O nbd , O ) . Pro of: W e sho w that ( H , ∗ ) has S c ( O nbd , O cg p ). Th e reason for this is that since H is zero d imensional, eac h op en cov er of it has a r efinement by a disjoin t op en co ve r. Thus for a giv en sequence ( U n : n < ∞ ) from O nbd for H we can c ho ose for eac h n a disjoin t op en refin emen t V n whic h co v ers H . Clearly ∪ n< ∞ V n is c -groupable. ♦ T o illustrate: Let P denote the set of irrational n umb ers. E. Pol has sho wn under CH 2 that there is a metrizable s pace X with p rop erty S c ( O , O ) suc h that X × P do es not ha v e S c ( O , O ). Now P is homeomorphic to a closed subset of th e zero d imensional group ( Z N , +). T h us X × Z N also do es not hav e S c ( O , O ). But for an y topological group ( G, ∗ ) with prop erty S c ( O nbd , O ), the group G × Z N still has S c ( O nbd , O ). Hattori, Y amada and indep end en tly Rohm, hav e pro ve n the follo wing pro du ct theorem for S c ( O , O ): Theorem 9 (Hatto ri-Y amada, Rohm) . If X is σ -c omp act and if X and Y b oth have the pr op erty S c ( O , O ) , then X × Y has the pr op erty S c ( O , O ) . 2 F or a new pro of using a w eaker hyp othesis, see [19] and [20]. 6 W e shall prov e an analogous th eorem, Th eorem 11, for top ological groups. Since S c ( O nbd , O ) is we ak er than S c ( O , O ) (see the remarks f ollo wing Con- jecture 3), w e are able to use a w eak er restriction than σ -compact. W e u se the follo wing result in our pro of: Lemma 10 ([7]) . The fol lowing statements ar e e quivalent: (1) X has the Hur ewicz pr op erty and pr op erty S c ( O , O ) . (2) F or e ach se quenc e ( U n : n < ∞ ) of op en c overs of X ther e is a se quenc e ( V n : n < ∞ ) such that: (a) Each V n is a finite c ol le ction of op en sets; (b) Each V n is p airwise disjoint; (c) Each V n r efines U n ; (d) ther e i s a se quenc e n 1 < n 2 < · · · < n k < · · · of p o sitive inte gers such that e ach element of X i s in al l but finitely many of the sets ∪ ( ∪ n k ≤ j N with h ∈ ∪ R k . It follo ws that for a j with n k ≤ j < n k +1 w e hav e ( g , h ) in ∪ K j . This completes the pro of. ♦ Theorem 12. L et ( G, ∗ ) b e a metrizable top olo gic al gr oup with no isolate d p oints. Then S c ( O nbd , O ) is e quivalent to S c ( O nbd , Λ) . Pro of: Let ( O ( U n ) : n < ∞ ) b e a sequen ce in O nbd ( G ). Cho ose a sequence ǫ n : n < ∞ ) such th at ǫ i > ǫ i +1 for all i < ∞ and diam d ( U 1 ∩ U 2 ∩ · · · ∩ U n ) > ǫ n for all n . Define ( O ( V n ) : n < ∞ ) suc h that diam d ( V i ) = ǫ i for i = 1 , 2 , · · · , n . W r ite N = S m< ∞ I m where eac h I m is infinite, and for m 6 = k , I m ∩ I k = ∅ . Apply S c ( O nbd , O ) to the sequence ( O ( V n ) : n ∈ I m ) for all m . Let T n b e a pairwise d isjoin t family refinin g O ( V n ), n ∈ I m suc h that ∪{ T n : n ∈ I m } co v ers G . W e will show that ∪{ T n : n ∈ N } is a large co v er. T ak e an element x ∈ G and pick m 1 ∈ I 1 with x ∈ ∪ T m 1 . Next, pic k W 1 ∈ T m 1 with x ∈ W 1 and N 1 so large that for all n ≥ N 1 w e ha v e ǫ n < diam d ( W 1 ). Then pick i 2 so large th at the smallest elemen t of I i 2 is larger than N 1 . No w choose m 2 ∈ I i 2 with x ∈ S T m 2 . Pic k W 2 ∈ T m 2 with x ∈ W 2 . S ince m 2 ≥ N 1 , ǫ m 2 < diam d ( W 1 ), and by definition of O ( V m 2 ), diam d ( W 2 ) ≤ diam d ( V m 2 ) ≤ ǫ m 2 < diam d ( W 1 ). Next pic k N 2 so large that for all n ≥ N 2 w e hav e ǫ n < diam d ( W 2 ) and con tinue the same wa y as w e did with N 1 . Con tinuing lik e th is w e find W 1 , W 2 , W 3 , · · · infinitely man y distinct elements of ∪{ T n : n < ∞} cov ering x . ♦ Note in particular that if for eac h n V n is a disjoin t family of op en sets, and if ∪ n< ∞ V n is a large co v er of G , th en for eac h g ∈ G th e set { n : g ∈ ∪V n } is infinite. Th is is b ecause for eac h n there is at most one set in V n that migh t con tain g . Corollary 13. L et ( G, ∗ ) b e a gr oup which has pr op erty S c ( O nbd , O ) as wel l as the Hur ewicz pr op erty. Then for any metrizable to p olo gic al gr oup ( H , ∗ ) satisfying S c ( O nbd , O ) , G × H also satisfies S c ( O nbd , O ) . Pro of: Use Theorems 11 and 12. ♦ Corollary 14. L et ( G, ∗ ) b e a metrizable gr oup which has pr op erty S c ( O nbd , O ) as wel l as th e Hur ewicz pr op erty. Then al l finite p owers of ( G, ∗ ) have the pr op erty S c ( O nbd , O ) . Pro of: Use Corollary 13. ♦ It is not clear that the full Hurewicz prop ert y is needed in Theorem 11 or C orollaries 13 and 14: mayb e Hur ewicz-b oundedn ess is enough. Problem 4. In The or em 11, c an we r eplac e the r e quir ement that G has the Hur ewicz pr op erty with the we aker r e quir ement that ( G, ∗ ) has the pr op erty S 1 (Ω nbd , O g p ) ? 8 In ligh t of results of E. and R. Pol - [19] - w e conjecture that neither Menger b oun dedness, nor the Menger prop ert y is enough to obtain T heorem 11: Conjecture 5. Ther e is a metrizable Menger b ound e d gr oup ( G, ∗ ) with pr op erty S c ( O nbd , O ) , such that G 2 is Menger b ounde d but do es not have S c ( O nbd , O ) . Conjecture 6. Ther e is a metrizable gr oup ( G, ∗ ) which has the pr op erty S c ( O nbd , O ) , and G 2 has the Menger pr op erty but do es not have S c ( O nbd , O ) . 4. Gam es The follo wing game, denoted G c ( A , B ), is naturally associated with S c ( A , B ): Pla y ers O NE and TWO pla y as follo ws: They pla y an inning for eac h natu- ral num b er n . In the n -th inning ONE first chooses O n , a mem b er of A , and then TW O r esp ond s with T n refining O n . A play ( O 1 , T 1 , · · · , O n , T n , · · · ) is won by TW O if ∪ n< ∞ T n is a member of B ; else, ONE wins. V ersions of differen t length of this game can also b e considered: F or an ord inal num b er α let G α c ( A , B ) b e the game play ed as follo ws: in the β -th inning ( β < α ) ONE first chooses O β , a member of A , and then TW O r esp ond s with a pairwise disjoin t T β whic h refin es O β . A pla y O 0 , T 0 , · · · , O β , T β , · · · β < α is wo n b y TW O if ∪ β <α T β is a mem b er of B ; else , ONE wins. Thus the game G c ( A , B ) is G ω c ( A , B ). Theorem 15. L et ( G , ∗ ) b e a metrizable gr oup. Then the f ol lowing state- ments hold : (1) If dim ( G ) ≤ n then TWO has a winning str ate g y in G n +1 c ( O nbd , O ) . (2) If TWO has a winning str ate gy in G n +1 c ( O nbd , O ) , then the dim ( G ) ≤ n . (3) If G is c ountable dimensional, then TWO has a winning str ate gy in G ω c ( O nbd , O ) . (4) If TWO has a winning str at e gy in G ω c ( O nbd , O ) , then G is c ounta ble dimensional. Pro of: W e pr ov e 3 and 4. The pro ofs of 1 and 2 are similar. Pro of of 3 : Supp ose that G is countable dimensional. W e define the follo wing strategy for T W O: W rite G = ∪ n< ∞ G n where eac h G n is zero-dimensional. Let U b e an elemen t of O nbd . F or U = O ( U ) of G an d n < ∞ , consid er U as a co v er of G n . S in ce G n is zero-dimensional, fin d a pairwise disjoin t family V n of subsets of G n op en in G n suc h that V n co v ers G n and refines O ( U ). Cho ose a pairwise disjoint family σ ( U , n ) refin ing O ( U ) s u c h th at eac h elemen t V of V n is of the form U ∩ G n for some U ∈ σ ( U , n ). No w T W O pla ys as follo ws: In inn ing 1 ONE plays U 1 , and T W O resp ond s with σ ( U 1 , 1), thus co v ering G 1 . When ONE has pla y ed U 2 in the second inn ing TW O resp onds with 9 σ ( U 2 , 2), th us co v ering G 2 , and so on. And in the n -th inn ing, when ONE has c hosen the co v er U n of G TW O resp onds with σ ( U n , n ), co ve rin g G n . This strategy eviden tly is a winn ing strategy for TWO. Pro of of 4 : Let σ b e a winnin g strategy f or T W O. Ch o ose a neighb orh o o d basis ( U n : n < ∞ ) of the identit y element e of G so that diam d ( U ) < 1 n for all n . Consider the p la ys of the game in wh ic h in eac h innin g ONE c ho oses for s ome n a co ver U n of G of the form O ( U n ). Define a family ( C τ : τ ∈ <ω N ) of subsets of G as f ollo ws: (1) C ∅ = ∩{∪ σ ( U n ) : n < ∞} ; (2) F or τ = ( n 1 , · · · , n k ), C τ = ∩{∪ σ ( U n 1 , · · · , U n k , U n ) : n < ∞} Claim 1: G = ∪{ C τ : τ ∈ <ω N } : F or supp ose on the contrary that x 6∈ ∪{ C τ : τ ∈ <ω N } . Cho ose an n 1 suc h that x 6∈ σ ( U n 1 ). With n 1 , · · · , n k c hosen suc h that x 6∈ σ ( U n 1 , · · · , U n k ), c ho ose an n k +1 suc h that x 6∈ σ ( U n 1 , · · · , U n k +1 ), and so on. Then U n 1 , σ ( U n 1 ) , U n 2 , σ ( U n 1 , U n 2 ) , · · · is a σ -pla y lost b y TW O, con tradicting the fact that σ is a win ning strategy for T W O. Claim 2: Eac h C τ is zero-dimensional. F or consider an x ∈ C τ . Say τ = ( n 1 , · · · , n k ). Thus, x is a mem b er of ∩ {∪ σ ( B n 1 , · · · , U n k , U n ) : n < ∞} . F or eac h n c ho ose a neigh b orho o d V n ( x ) ∈ σ ( U n 1 , · · · , U n k , U n ). Since for ea c h n we ha v e diam d ( V n ( x )) < 1 n , the set { V n ( x ) ∩ C τ : n < ∞} is a n eighb orh o o d basis for x in C τ . Observ e also that eac h V n ( x ) is closed in C τ b ecause: The set V = ∪ σ ( U n 1 , · · · , U n k , U n ) \ V n ( x ) is op en in G and so C τ \ V n ( x ) = C τ ∩ V is op en in C τ . Th us eac h elemen t of C τ has a basis consisting of clop en sets. ♦ 5. R emarks and a cknowledgment. Regarding Theorem 2: F or a left in v arian t m etric d let U d b e a the family of sets U ǫ , ǫ > 0 where we defin e U ǫ = { ( x, y ) ∈ G × G : d ( x, y ) < ǫ } . Th e family U d generates the left-uniformity of the top ological group G . Refer to [9] C hapters I I I § 3 and IX § 3 and [10] Chapter 8.1 r egarding these f acts. Let O d denote the collection of op en co v ers of the form { U ǫ ( x ) : x ∈ G } wh ere U ǫ ( x ) = { y : ( x, y ) ∈ U ǫ } . T he r eferee p oin ted out that a third equiv alence can b e ad d ed in Theorem 2: For each left-i nvariant metric d , S c ( O d , O ) holds. Regarding Theorem 5: There is a more general theorem. L et U b e uniformity on X generati ng th e top ology τ U . F or V ⊂ X × X , define V ( x ) = { y ∈ X : ( x, y ) ∈ V } . W e say that an op en co ve r of ( X, τ U ) is uniform with resp ect to U if it is of the form { V ( x ) : x ∈ X } , for some V ∈ U . De fin e O U = {{ V ( x ) : x ∈ X } : V ∈ U } . Theorem 16. L et U b e a uniformity gene r ating the top olo gy τ U on the set G . A ssume that the top olo gic al sp ac e ( G, τ U ) has the Hur ewicz pr op erty. Then S c ( O U , O ) is e quivalent to S c ( O , O ) . 10 The pro of of this theorem is ve ry similar to th e pro of of T h eorem 5 . I thank th e referee for the useful remarks and E. Po l and R. P ol for comm unicating their resu lt on Conjecture 3 to me. Referen ces [1] D.F. Ad dis and J.H. 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Tsaban, o-Bounde d gr oups and other top olo gic al gr oups wi th str ong c ombinatorial pr op er ties , Pro ce edings of the A merican Mathematical Socie ty ,1 34 (2006), 881- 891. 11 Con tact Inf ormation: Liljana Babink osto v a Departmen t of Mathematics Boise State Univ ersit y Boise, ID 83725 e-mail: liljanab@diamond.b oisestate.edu fax: 208 - 426 - 1356 phone: 208 - 426 - 2896 12