Topological games and covering dimension

TOPOLOGICAL GAMES AND CO VERING DIMEN SIO N LILJANA BABINKOSTO V A Abstract. W e consider a natural wa y of extending the Leb esgue co v ering dimension to v arious classes of infinite dimensional separable metric spaces. All spaces in this pap er a re ass umed to be separable metric spaces. Infinite games can b e us ed in a natural way to define or dinal v a lued functions o n the class of separable metric s pa ces. One of our examples o f such an ordinal function coincides in an y finite dimensio na l metric spaces with the cov ering dimension o f the space, a nd may thus be though t of as an ex tension o f Leb esgue co v ering dimension to all sepa r able metric spaces. W e will call this particular ex tens io n of Leb esgue cov ering dimensio n the game dimension o f a space. Game dimension is defined using a game motiv ated by a s election principle. Several natural clas sical selection principles are rela ted to the one motiv ating game dimension, and their ass o ciated g ames can be used to compute upper b ounds on game dimension. These games, and the upp er b ounds they pr ovide, are interesting in their own r ig ht. W e develop tw o such games and use them then to o btain upp er bo unds on our ga me dimension. W e also compute the game dimension of a few sp ecific examples. 1. Selection principles, open covers an d games The s e le ction principle S 1 ( A , B ) s ta tes that there is for ea ch sequence ( A n : n ∈ N ) of elements of A a sequenc e ( B n : n ∈ N ) such that for each n we hav e B n ∈ A n , and { B n : n ∈ N } ∈ B . The s e le ction principle S c ( A , B ) s ta tes that there is for each s equence ( A n : n ∈ N ) of member s o f A a s equence ( B n : n ∈ N ) of s ets such that for ea ch n , B n is a pairwise disjoint family o f sets and refines A n and S n ∈ N B n ∈ B . F o r a co llection T of subsets of a top o logical s pace X we call an op en cover U of X a “ T -cover” if for e ach T ∈ T , ther e is a U ∈ U with T ⊆ U . The symbol O ( T ) denotes the collection of T -cov ers of X . A trivial situation, but o ne w e cannot ignore, aris es when X itself is a member o f T . W e don’t follow the usual pr actice of re quiring X 6∈ U for U ∈ O ( T ). The mo tiv atio n, as will b e seen b elow, is that allowing this trivial situation provides a unifor mity to the statements of so me of our results. A few special combinatorial pr op erties of the family T are impor tant for our consideratio ns. Here are some of them: T is said to be u p-dir e cte d if for all A and B in T there is a C in T with A S B ⊂ C . It is said to b e first-c ountable if there is for each T ∈ T a sequence ( U n : n ∈ N ) o f op en sets suc h that for each n , U n ⊃ U n +1 ⊃ T , and for ea ch op en s et V ⊃ T , ther e is a n n with V ⊇ U n . W e s ha ll say that X is T - first c ountable if ther e is fo r ea ch T ∈ T a sequence ( U n : n = 1 , 2 , . . . ) o f op en sets such that for all n , T ⊂ U n +1 ⊂ U n , and for each op en set U ⊃ T there is a n n with U n ⊂ U . Let hT i denote the subspa c es which 1 2 LILJANA BABINKOSTO V A are unions o f countably many elements o f T . When T is a colle c tion of co mpact sets in a metriz a ble s pace X , T is fir st c ountable a nd up-directed. Ca ll a subset C of T c ofin al if there is fo r each T ∈ T a C ∈ C with T ⊆ C . When T is the co llection of one- element subsets of X , then O ( T ) w ill be denoted simply O . With O fd we denote the collection of all finite dimensional subse ts of a separable metric spa c e and with O kfd we denote the collectio n o f all co mpa ct, finite dimensional subsets o f a separ able metric spa c e. When T is the collection of finite subsets of X then T -cov ers a re ca lled ω - cov ers, and Ω deno tes O ( T ). The collection of op en covers ha ving just tw o elemen ts is denoted O 2 . A topo log- ical space is sa id to b e weakly infinite dimensional if it has the prop erty S c ( O 2 , O ). The cla ss of spaces satisfying S c ( O , O ) was introduce d in [1]. If a space is a union of co untably many zero dimensio nal subsets it is s a id to b e co unt able dimensional. Hurewicz intro duced the latter notio n. As was shown in [1], countable dimensional implies S c ( O , O ), a nd S c ( O , O ) implies S c ( O 2 , O ). If a space is not weakly infinite dimensional, then it is said to b e str ongly infinite dimensional . The Hilb er t cub e H is an ex ample of s trongly infinite dimensiona l space. Let α b e an ordinal num ber. The game G α 1 ( A , B ) asso cia ted with the selec tio n principle S 1 ( A , B ) is as follows: The play ers play an inning p er γ < α . In the γ -th inning ONE first chooses an A γ ∈ A : TWO then resp o nds with a B γ ∈ A γ . A play A 0 , B 0 , · · · , A γ , B γ , · · · of leng th α is won by TW O if { B γ : γ < α } ∈ B . Else, ONE wins. When for a set S a nd families A a nd B there is an or dinal num ber α such that TW O has a winning stra tegy in the ga me G α 1 ( A , B ) play ed o n S, then we define: tp S 1 ( A , B ) ( S ) = min { α : TW O has a winning stra tegy in G α 1 ( A , B ) o n S } . W e adopt the co nven tion that tp S 1 ( A , B ) ( ∅ ) = − 1. The ordina l- v a lue d function tp S 1 ( A , B ) ( S ) has b een studied for a few sp ecific choices of the families A and B . Examples c losely affilia ted with what w e will examine here can b e found in [3], [5], [6], [7 ], [12] and [13]. The following monotonicity prop erties are eas y to chec k: Lemma 1 . F or e ach sep ar able met r ic sp ac e X , for e ach close d set Y ⊆ X and for families S ⊂ T and A ⊂ B the fol lowing hold: (1) tp S 1 ( O ( T ) , A ) ( X ) ≤ tp S 1 ( O ( S ) , A ) ( X ) . (2) tp S 1 ( O ( T ) , B ) ( X ) ≤ tp S 1 ( O ( T ) , A ) ( X ) . (3) tp S 1 ( O ( T ) , A ) ( Y ) ≤ tp S 1 ( O ( T ) , A ) ( X ) . In par ticular we find that tp S 1 ( O fd , A ) ( X ) ≤ tp S 1 ( O kfd , A ) ( X ). Some o f the results from our inv estigation in [3 ] have some applications in this pap er. W e first recall these . In order to compar e the ordina ls tp S ξ ( A , B ) ( X ) for v arious c hoices ξ , A and B , we extend Lemma 1 o f [3] as follows: Lemma 2. L et α b e an or dinal n umb er, and let F b e a stra te gy for TWO in t he game G α 1 ( O ( T ) , O ) . Then ther e is for e ach ν < α , for e ach s e qu enc e ( O β : β < ν ) fr om O ( T ) a set C ∈ T such that for e ach op en set U ⊇ C t her e is an O ∈ O ( T ) with F (( O β : β < ν ) ⌢ ( O )) = U . Theorem 3 ([3] Theor em 4) . Le t α b e an infin ite or dinal and let T b e up-dir e cte d. If F is any str ate gy for TWO in G α 1 ( O ( T ) , O ) and if X is T -first c ountable, t hen TOPOLOGICAL GAMES AND COVERING DIMENSION 3 ther e is for e ach set T ∈ hT i a set S ∈ hT i such that: T ⊆ S and for any close d set C ⊂ X \ S , ther e is an ω -length F -play O 1 , T 1 , · · · , O n , T n · · · such that T ⊆ S ∞ n =1 T n ⊆ X \ C . Theorem 4 ([3] Theor em 5) . Le t α b e an infin ite or dinal and let T b e up-dir e cte d. If F is any str ate gy for TWO in G α 1 ( O ( T ) , O ) and if X is C -first c ountable wher e C ⊂ T is c ofinal in T , then ther e is for e ach s et T ∈ hT i a set S ∈ hC i su ch that: T ⊆ S and for any close d set C ⊂ X \ S , t her e is an ω -length F -play O 1 , T 1 , · · · , O n , T n · · · such that T ⊆ S ∞ n =1 T n ⊆ X \ C . Lemma 5 ([3] Theore m 6) . If T has a c ofinal subset c onsist ing of G δ -sets, then TWO has a winning st r ate gy in G ω 1 ( O ( T ) , O ) if, and only if, X is a union of c ou n tably many element s of T . Lemma 6 ([3] Lemma 13) . L et X and T b e such that X 6∈ hT i and T is up-dir e cte d and first- c ountable. If α is the le ast or dinal for which ther e is a B ∈ hT i such that e ach close d set C ⊂ X \ B satisfies tp S 1 ( O ( T ) , O ) ( C ) ≤ α , then tp S 1 ( O ( T ) , O ) ( X ) = ω + α . The co nv erse is also true : Lemma 7. L et X and T b e su ch that X 6∈ hT i and T is up-dir e cte d and first- c ou n table. If tp S 1 ( O ( T ) , O ) ( X ) = ω + α then ther e is a B ∈ hT i such t hat e ach close d set C ⊂ X \ B satisfi es tp S 1 ( O ( T ) , O ) ( C ) ≤ α , and α is the minimal s u ch or dinal. Pro of: F or let F b e a winning s trategy for TW O in the game G ω + α 1 ( O ( T ) , O ). By Lemma 2 cho ose C ∅ ∈ T such that for each op en set U ⊇ C ∅ there is a U ∈ O ( T ) with U = F ( U ). Since T is first countable choose a sequence ( U n : n < ω ) o f o pen sets with: F or each op en s e t U ⊇ C ∅ there is an n with U ⊇ U n ⊇ C ∅ , and for all n , U n ⊃ U n +1 . F or each n cho ose O n ∈ O ( T ) with U n = F ( O n ). Applying Lemma 2 to each O n , fix sets C n ∈ T suc h that there is for each op en U ⊇ C n an O ∈ O ( T ) with U = F ( O n , O ). Since T is fir s t countable choose for ea ch n a sequence ( U n,m : m < ω ) of op en sets such that: F or ea ch op en set U ⊇ C n there is an m with U ⊇ U n,m ⊇ C n , a nd for a ll m , U n,m ⊃ U n,m +1 . Then choose for each m an O n,m ∈ O ( T ) with U n,m = F ( O n , O n,m ). Contin uing like this we find for each finite sequence ( n 0 , · · · , n j ) sets C n 0 , ··· ,n j ∈ T , and op en s ets U n 0 , ··· ,n j ,t , and O n 0 , ··· ,n j ,t ∈ O ( T ), t < ω , such that for each op en set U ⊇ C n 0 , ··· ,n j there is a t with U ⊇ U n 0 , ··· ,n j ,t ⊇ C n 0 , ··· ,n j and U n 0 , ··· ,n j ,t = F ( O n 0 , O n 0 ,n 1 , · · · , O n 0 , ··· ,n j ,t ), and for all t , U n 0 , ··· ,n j ,t ⊃ U n 0 , ··· ,n j ,t +1 . Put B = ∪ τ ∈ <ω ω C τ , an element of hT i . Consider a ny closed se t C ⊂ X \ B . Since U = X \ C ⊇ B is op en, ch o ose n 0 with C ∅ ⊆ U n 0 ⊂ U . Then choose n 1 with C n 0 ⊆ U n 0 ,n 1 ⊆ U , and then n 2 with C n 0 ,n 1 ⊆ U n 0 ,n 1 ,n 2 ⊆ U , and so on. In this wa y we obtain a n ω -sequence of mov es during whic h TW O used the strategy F , and the unio n of TWO’s r esp onses, say V , is a subs et of U . Since F is a winning strategy for T WO in the game G ω + α 1 ( O ( T ) , O ) it follows that tp S 1 ( O ( T , O ) ( C ) ≤ tp S 1 ( O ( T , O ) ( X \ U ) ≤ tp S 1 ( O ( T , O ) ( X \ V ) = α . The closed set X \ V witnesses that α is the minimal such ordinal.  4 LILJANA BABINKOSTO V A 2. The o rdinal tp S 1 ( O kfd , O ) ( X ) . The prop erties of op en p oint-cov ers of spaces ar e at the bas is of Lebesg ue’s notion o f covering dimension. The p oints hav e, in that theor y , dimension zero . W e will consider a more general situation of op en T -covers of spa c e s. In this ca se by analogy with p oint cov ers, w e require that the notion of dimension assign the v a lue 0 to the member s of T . W e feature here the sp e c ific case when T is the collection of finite dimensional co mpact spaces . 2.1 When tp S 1 ( O kfd , O ) ( X ) is countable Prop ositi o n 8. F or X a sep ar able metric sp ac e the fol lowing ar e e quivalent: (1) X is a c ountable union of c omp act finite dimensional su bset s. (2) 1 < tp S 1 ( O kfd , O ) ( X ) ≤ ω . W e shall see that (2) of P rop osition 8 cannot b e improved. 2.2 Pro ducts and unio n s . Lemma 9. F or X a metric sp ac e, and Y a c omp act fi nite dimensional metric sp ac e, tp S 1 ( O kfd , O ) ( X × Y ) = tp S 1 ( O kfd , O ) ( X ) . Pro of: First note that since X is a c losed subset of X × Y , we hav e tp S 1 ( O kfd , O ) ( X ) ≤ tp S 1 ( O kfd , O ) ( X × Y ). W e s how that tp S 1 ( O kfd , O ) ( X × Y ) ≤ tp S 1 ( O kfd , O ) ( X ). Let F b e a winning stra tegy for TWO in G α 1 ( O kfd , O ) played on X . TWO us es F as follows to play the game o n X × Y . Firs t obser ve that if O is in O kfd then there is an S ( O ) ∈ O kfd such that each element of S ( O ) is of the form U × Y , a nd such that S ( O ) is a refinement of O . Define a strategy G for TWO in the game G α 1 ( O kfd , O ) on X × Y as fo llows: In inning 1, when ONE plays O 1 , put B 1 = { U ⊂ X : U × Y ∈ S ( O 1 ) } , a mem ber of O kfd for X . Then compute W 1 = F ( B 1 ) ∈ B 1 , a nd choose T 1 ∈ O 1 with W 1 × Y ⊆ T 1 . Define G ( O 1 ) = T 1 . Let γ < α b e g iven, as well a s a sequence ( O ξ : ξ ≤ γ ) of members of O kfd for X × Y . F or e a ch ξ ≤ γ define B ξ = { U ⊂ X : U × Y ∈ S ( O ξ ) } , a member of O kfd on X . Co mpute W γ = F ( B ξ : ξ ≤ γ ) ∈ B γ and then choose T γ ∈ O γ with W γ × Y ⊆ T γ , and put G ( O ξ : ξ ≤ γ ) = T γ . Then G is a winning strategy for TWO in G α 1 ( O kfd , O ) o n X × Y .  Lemma 10. L et X b e a metric sp ac e which is not c omp act and finite dimensional. If tp S 1 ( O kfd , O ) ( X ) is finite then ther e is a nonempty finite dimensional c omp act set C such that for e ach op en set U ⊃ C we have tp S 1 ( O kfd , O ) ( X \ U ) ≤ tp S 1 ( O kfd , O ) ( X ) − 1 , and for some op en set U ⊃ C we have tp S 1 ( O kfd , O ) ( X \ U ) = tp S 1 ( O kfd , O ) ( X ) − 1 Pro of: Suppose no t. F or each compact finite dimensiona l C ⊂ X cho ose an op en set U ( C ) such that C ⊂ U ( C ), and tp S 1 ( O kfd , O ) ( X \ U ( C )) ≥ tp S 1 ( O kfd , O ) ( X ). Then O 1 = { U ( C ) : C ⊂ X finite dimensional compact } ∈ O kfd , and for each s tr ategy G of TW O, tp S 1 ( O kfd , O ) ( X \ G ( O 1 )) ≥ tp S 1 ( O kfd , O ) ( X ). But this contradicts the fact that tp S 1 ( O kfd , O ) ( X ) is finite. Suppo se that for each compact finite dimensional C ⊂ X , for each op en set U ⊃ C we hav e tp S 1 ( O kfd , O ) ( X \ U ) ≤ tp S 1 ( O kfd , O ) ( X ) − 2. Then we hav e the contradiction that tp S 1 ( O kfd , O ) ( X \ U ) ≤ tp S 1 ( O kfd , O ) ( X ) − 1.  TOPOLOGICAL GAMES AND COVERING DIMENSION 5 Lemma 11. F or X a metric sp ac e, and Y 1 , · · · , Y n subsp ac es such that for e ach n , tp S 1 ( O kfd , O ) ( Y n ) ≤ α , also tp S 1 ( O kfd , O ) ( S j ≤ n Y j ) ≤ α . Pro of: F o r each n we prov e this by induction on α . F or n = 1 there is nothing to prov e. Th us ass ume n > 1. When α = 0 there is nothing to prove since each Y j then is compact and finite dimensional, as is their union. Th us, assume tha t 0 < α and we have prov en this res ult for all β < α . Ther e are three cases to consider: α is finite, α is an infinite limit or dinal and α is an infinite succe ssor or dinal. First, the case when α is finite: Since we have already disp osed of the case when α = 0, consider now the case wher e α = m + 1 and the result is proved for α ≤ m . By Lemma 1 0 choo se for j ≤ n a compac t finite dimensional set C j ⊂ Y j such tha t for any op e n (in Y j ) set U containing C j , tp S 1 ( O kfd , O ) ( Y j \ U ) ≤ m . Then C = ∪ j ≤ n C j is co mpact and finite dimensional, and fo r every op en set U ⊃ C , for ea ch j , tp S 1 ( O kfd , O ) ( Y j \ U ) ≤ m . Th us TWO ca n play as follows: When ONE in inning 0 play an O 0 ∈ O kfd for ∪ j ≤ n Y j , TWO choo ses T 0 ∈ O 0 with C ⊆ T 0 . Since now for j ≤ n we hav e for Z j = Y j \ T 0 that tp S 1 ( O kfd , O ) ( Z j ) ≤ m , the induction hypothesis gives that tp S 1 ( O kfd , O ) ( ∪ j ≤ n Z j ) ≤ m = tp S 1 ( O kfd , O ) (( ∪ j ≤ n Y j ) \ T 0 ) ≤ m . It follows that tp S 1 ( O kfd , O ) ( ∪ j ≤ n Y j ) ≤ m + 1, completing the induction step, and the pro of for α finite. In the case when α is a limit ordinal we can wr ite α = ∪ j ≤ m A j where each A j has o rder type α , and in innings γ ∈ A j , we use a winning str ategy for TWO o n Y j to choo se a n element T γ ∈ O γ . This plan pr o duces a winning strateg y in the α -length ga me on ∪ j ≤ n Y j . In the case when α is an infinite succes sor or dina l, write α = ℓ ( α ) + n ( α ) where ℓ ( α ) is a limit ordinal and n ( α ) < ω . During the first ℓ ( α ) innings w e follow the plan ab ov e. Af ter these innings for each j the uncov ered part Z j of Y j has tp S 1 ( O kfd , O ) ( Z j ) ≤ n ( α ), and thus the uncov ered part Z of ∪ j ≤ n Y j has tp S 1 ( O kfd , O ) ( Z ) ≤ n ( α ), by the finite case. This completes the pro of.  Theorem 1 6 b elow shows that Lemma 11 fails for infinite unions . 2.3 When tp S 1 ( O kfd , O ) ( X ) is fini te and p o sitive Note that when X is a compact finite-dimensio nal space, tp S 1 ( O kfd , O ) ( X ) = 0. Next we describ e the structur e of those metriz a ble X with tp S 1 ( O kfd , O ) ( X ) finite. Smirnov’s compactum S ω is cons tructed as follows: F o r p os itive integer n define S n = I n . Then define S ω to be the one-p oint compactification of the top o logical sum ⊕ ∞ n =1 S n , say S ω = ( ⊕ ∞ n =1 S n ) S { p ω } . It is clear that S ω is a union of countably many co mpact finite dimensional spa ces. Lemma 12. F or e ach p ositive inte ger n , tp S 1 ( O kfd , O ) ( S n ω ) = n . Pro of: The pro of is by induction on n . n = 1: Since S ω is co mpa ct and not finite dimensional, O kfd 6 = ∅ , and so TWO do es not win in one inning. When ONE plays O 0 ∈ O kfd , then TWO chooses T 0 ∈ O 0 such that p ω ∈ T 0 . Then S ω \ T 0 is compact finite dimensional, and thus in the next inning TWO choos es T 1 ∈ O 1 containing this set. n = k + 1: Assume that we ha ve alre a dy prov en the r esult for n ≤ k . Consider n = k + 1: 6 LILJANA BABINKOSTO V A Play er ONE will play op en cov ers O with the prop erty that for any element U o f O 1 which contains the k+1 vector ( p ω , · · · , p ω ) there is so me m > k + 1 such that the pro jection of U in each of the k + 1 directions is disjoint fro m ⊕ j ≤ m S j . It fo llows that for these elements U of ONE ’s mov e, S k +1 ω \ U contains the closed subspace S k +1 j =1 X j , wher e X j is the pro duct Q k +1 i =1 Z i where Z i =  S ω if i 6 = j S k +1 otherwise Each X j is homeomor phic to S k +1 × S k ω . By L e mma 9, tp S 1 ( O kfd , O ) ( S k +1 × S k ω ) = tp S 1 ( O kfd , O ) ( S k ω ) = k . Then by Lemma 11 tp S 1 ( O kfd , O ) ( S k +1 ω \ U ) ≥ k . Th us we conclude that tp S 1 ( O kfd , O ) ( S k +1 ω ) ≥ k + 1. Proving that this inequality is in fact a n equality is left to the reader .  2.4 When tp S 1 ( O kfd , O ) ( X ) is ω + 1 The following Lemma is useful for computing lowerbounds on tp S 1 ( O kfd , O ) ( X ): Lemma 13. L et X and C b e metrizable sp ac es with X not c ountable dimensional. If E ⊂ X × C is c ountable dimensio nal, then ther e is a set Y ⊂ X with Y not c ou n table dimensional and E T ( Y × C ) = ∅ . Pro of: Supp o se not. Then the set Y := { x ∈ X : ( { x } × C ) \ E = ∅} is countable dimensio nal, a nd so Z = X \ Y is not countable dimensio nal. F or e a ch x ∈ Z choo s e φ ( x ) ∈ C with ( x, φ ( x )) ∈ E . Then the set T := { ( x, φ ( x )) : x ∈ Z } , being a subs et of E , is countable dimensional. But the function f : T − → Z defined by f ( x, y ) = x , the pro jection on the firs t co or dinate, b eing the restriction o f Π 1 to T , is contin uous, and o ne-to-one a nd o nto Z . Thus, the inv erse of f is a b o th op en and clo sed function fro m Z to T . Since T is countable dimensional, a theor em of Arhangel’ski ˇ i (see Theo rem 7.7 in [8]) implies that Z is countable dimensional, a contradiction. It follows that Y is not countable dimensional.  Pol’s c ompactum K is constructed as fo llows: Start with a c o mplete heredita rily strongly infinite dimensional totally disconnected metric space M and then com- pactify it such that the extensio n L is a union of countably many compact, finite dimensional spaces. Then K = M S L . F o r the rest of the pap er fix a repre- sentation of L as a union of co un tably many compact finite dimensio nal s ets, say L = S ∞ n =1 L n , wher e for m < n we hav e L m ⊂ L n and dim ( L m ) < dim ( L n ). Theorem 14 . F or e ach p ositive inte ger n , tp S 1 ( O kfd , O ) ( K n ) = ω · n + 1 Pro of: W e use induction o n n . n=1: tp S 1 ( O kfd , O ) ( K ) ≤ ω + 1: In inning n when ONE plays the open cov er O n ∈ O kfd , TW O chooses T n ∈ O n with L n ⊂ T n . After ω innings L ⊂ S ∞ n =1 T n and the part of K not yet cov ered by T WO is a compa c t set contained in the to ta lly disconnec ted space M , a nd thu s is a co mpact z e ro dimensional set. In inning ω + 1 TWO c ho oses T ω ∈ O ω containing this compact z ero dimensional se t. tp S 1 ( O kfd , O ) ( K ) ≥ ω + 1: K is not countable dimensional. By Lemma 5, tp S 1 ( O kfd , O ) ( K ) > ω . TOPOLOGICAL GAMES AND COVERING DIMENSION 7 n > 1: Assume tha t the statement is true for k < n . tp S 1 ( O kfd , O ) ( K n ) ≤ ω · n + 1: F or j < n and m < ω define K ( j, m ) = K j × L m × K n − j − 1 . F or each such ( j, m ): K ( j, m ) is ho meo morphic to L m × K n − 1 and b y Lemma 9 a nd the induction hypothes is tp S 1 ( O kfd , O ) ( K ( j, m )) = ω · ( n − 1) + 1. Let F j,m be TWO’s winning str ategy in G ω · ( n − 1)+1 1 ( O kfd , O ) o n K ( j, m ). Now write ω · ( n − 1) as a union S m<ω S j 1 and the theorem holds for all j < m : (1) tp S 1 ( O kfd , O ) ( K m × X ) ≤ ω · ( m + 1): By Lemma 9 and Theo rem 1 4, for each n TW O ha s a winning strategy F n in G ω · m +1 1 ( O kfd , O ) o n K m × X n . W rite ω · m = ∪ n< ∞ S n where each S n is infinite, and S m ∩ S n = ∅ fo r m 6 = n . F o r each n we enumerate S n in o r der as ( s n γ : γ < ω · m ). In the first ω · m innings, when O NE plays O γ in inning γ , TWO choo ses n with γ ∈ S n and then fixes δ with γ = s n δ , and think of O γ as O n δ , the δ -th mov e of ONE in the game on K m × X n . Then TWO choos es T γ ∈ O γ using F n ( O n 0 , · · · , O n δ ). Let U be the union of TWO’s mov es during the first ω · m innings. F or each n , since TWO w as using the winning strategy F n through ω · m innings on K m × X n , the set C n = ( K m × X n ) \ U is compact finite dimensional. TWO cov ers these countably many compact finite dimensio nal sets in the next ω innings. (2) tp S 1 ( O kfd , O ) ( K m × X ) ≥ ω · ( m + 1): Let F be a winning strategy fo r TWO in G α 1 ( O kfd , O ) pla yed on K m × X . The set T = L m × X is co mpa ctly count able dimensiona l. By Theor em 3 choo se a countable dimensional set S ⊂ K m × X such that T ⊆ S and for eac h closed set C ⊂ ( K m × X ) \ S there is an ω -leng th F -play O 0 , T 0 , · · · , O n , T n , · · · with T ⊂ ∪ n< ∞ T n ⊂ ( K m × X ) \ C . Then as K is not countable dimensional, L emma 13 implies that ther e is an x ∈ K with the clos e d set C = { x } × ( K m − 1 × X ) disjoint from S . Then choose a n ω - le ng th play O 0 , T 0 , · · · , O n , T n , · · · with T ⊂ ∪ n< ∞ T n ⊂ ( K m × X ) \ C . Since C = { x } × ( K m − 1 × X ) is disjoint fro m the set cov ered by TWO in these innings, a nd is homeo morphic to K m − 1 × X the induction hypothesis implies that at leas t ω · m additional inning a re needed for TWO to cover C . It follows that α ≥ ω + ω · m = ω · ( m + 1). This c o mpletes the induction step a nd the pr o of.  TOPOLOGICAL GAMES AND COVERING DIMENSION 9 Theorem 17 . L et X b e a metrizable sp ac e with tp S 1 ( O kfd , O ) ( X ) a suc c essor or dinal. L et Y b e a metric sp ac e with tp S 1 ( O kfd , O ) ( Y ) ≤ ω . Then: tp S 1 ( O kfd , O ) ( X × Y ) ≤ tp S 1 ( O kfd , O ) ( X ) + ω . Pro of: W rite α = β + m where β < α = tp S 1 ( O kfd , O ) ( X ) is a limit ordinal, and 0 < m < ω . W rite β = ∪ n< ∞ S n where ea ch S n is a subset of β o f order t ype β , and S m ∩ S n = ∅ for m 6 = n . F or each n we enu merate S n as ( s n γ : γ < β ). When O NE plays O γ in inning γ < β , TW O chooses n with γ ∈ S n and then fixes ν with γ = s n ν , and think of O γ as O n ν , the ν -th move of ONE in the game on X × Y n . Then TWO choos es T γ ∈ O γ using F n ( O n 0 , · · · , O n ν ). Let U b e the union of the mov es TWO made during these β innings. F o r each n , since TWO w as using the winning stra tegy F n through β innings o n X × Y n , we hav e tp S 1 ( O kfd , O ) (( X × Y n ) \ U ) ≤ m . But then, by Prop osition 8 , for each n the set C n = ( X × Y n ) \ U is a union of co unt ably many compact finite dimensio nal sets. During the next ω innings TWO covers these. T hus we have that tp S 1 ( O kfd , O ) ( X × Y ) ≤ α + ω .  Theorem 1 6 b elow shows that the v a lue of tp S 1 ( O kfd , O ) ( X × Y ) obtained in The- orem 17 cannot b e impr ov ed. W e susp ect that the upp er b ound in Theor em 1 7 is in fact achiev ed: Problem 1. If X b e a metrizable sp ac e with tp S 1 ( O kfd , O ) ( X ) a suc c essor or dinal and Y is a metric sp ac e with tp S 1 ( O kfd , O ) ( Y ) ≤ ω , do es it fol low that: tp S 1 ( O kfd , O ) ( X × Y ) = tp S 1 ( O kfd , O ) ( X ) + ω ? 2.4 When tp S 1 ( O kfd , O ) ( X ) is a limit ordinal Theorem 18 . L et X b e a metrizable sp ac e with tp S 1 ( O kfd , O ) ( X ) = α a limit or dina l, and let Y b e a met ric sp ac e with tp S 1 ( O kfd , O ) ( Y ) ≤ ω . Then: tp S 1 ( O kfd , O ) ( X × Y ) = α . Pro of: Since tp S 1 ( O kfd , O ) ( Y ) ≤ ω , by P rop osition 8 cho ose compac t finite dimen- sional sets Y n ⊂ Y with Y = ∪ n<ω Y n . By Lemma 9 fix for ea ch n a winning strategy F n for TWO in the ga me G α 1 ( O kfd , O ) played on X × Y n . By Lemma 1 (3), since X is homeomo r phic to a c losed subspace of X × Y , we hav e α ≤ tp S 1 ( O kfd , O ) ( X × Y ). W e m ust s how that tp S 1 ( O kfd , O ) ( X × Y ) ≤ α . W rite α = ∪ n< ∞ S n where each S n is a subset of α of or de r type α , and S m ∩ S n = ∅ for m 6 = n . F or ea ch n we en umerate S n in o r der as ( s n γ : γ < α ). When O NE plays O γ in inning γ < α , TW O chooses n with γ ∈ S n and then fixes ν with γ = s n ν , and think of O γ as O n ν , the ν -th move of ONE in the game on X × Y n . Then TWO choos es T γ ∈ O γ using F n ( O n 0 , · · · , O n ν ). Let U b e the union of the moves TWO made during these α innings. F or each n , since TWO w as using the winning strategy F n through α innings on X × Y n , we hav e X × Y n ⊆ U . Thus, TWO won in α inning s. This co mpletes the pro of.  Let X b e ⊕ ∞ n =1 K n , the top o logical sum of the finite p ow ers of the Pol co mpactum. ω 2 = tp S 1 ( O kfd , O ) ( X ). This is a lo cally compact space. Le t K ω = ( ⊕ ∞ n =1 K n ) ∪ { p ω } be the one-p o int co mpa ctification of X . It is ea sy to see that tp S 1 ( O kfd , O ) ( K ω ) = ω 2 . W e now show tha t tp S 1 ( O kfd , O ) ( X × K ) = ω 2 . 10 LILJANA BABINKOSTO V A Lemma 19. tp S 1 ( O kfd , O ) ( K ω × K ) = ω 2 . Thus, tp S 1 ( O kfd , O ) ( K ω × X ) = ω 2 . Pro of: During the fir st ω + 1 innings, cover { p ω } × K . The rema ining part of the space is a clo sed subset of X and th us req uir es at mos t ω 2 more innings. Thus ω + 1 + ω 2 ≥ tp S 1 ( O kfd , O ) ( K ω × K ). B ut K ω is a close d subset o f K ω × K , a nd so also ω 2 ≤ tp S 1 ( O kfd , O ) ( K ω × K ).  3. The o rdinal tp S c ( O , O ) ( S ) and the game dimension. W e now define an ordina l-v a lued function on the set of subspa ces of the Hilb ert cube such that this function • coincides with Leb es gue covering dimension in the case o f subspaces with finite cov ering dimensio n, and • capture s a n imp ortant a sp ect o f extending the covering dimension to sub- spaces tha t ar e not finite dimensional. Other imp o r tant ex a mples o f such or dinal-v alued functions were develop e d by P . Borst [4], and R. Pol [10]. Let α b e an ordinal num ber. The game G α c ( A , B ) asso cia ted with the selec tio n principle S c ( A , B ) is as follows: The play ers play an inning p er γ < α . In the γ -th inning ONE firs t cho o ses an A γ ∈ A : TWO then resp onds with a pairwise disjoint family of se ts B γ that refines A γ . A pla y A 0 , B 0 , · · · , A γ , B γ , · · · of leng th α is won by TWO if S n ∈ N B n ∈ B . Else, O NE wins. When for a set S a nd families A a nd B there is an or dinal num ber α such that TW O has a winning stra tegy in the ga me G α c ( A , B ) play ed o n S, then we define: tp S c ( A , B ) ( S ) = min { α : TW O has a winning stra tegy in G α c ( A , B ) on S } . F o r X a separable finite-dimensio na l metric space, dim( X ) denotes the Leb esgue cov ering dimensio n of X . The star ting p oint for this e xploration is the following game-theor etic result: Lemma 20 ([2]) . L et X b e a sep ar able m etric sp ac e and let n b e a nonne gative inte ger. The fol lowing ar e e quivalent: (1) dim (X) = n. (2) tp S c ( O , O ) ( X ) = n + 1 . W e define for separa ble metric space X the game dimension o f X , denoted dim G ( X ), by 1 + dim G ( X ) = tp S c ( O , O ) ( X ) . The for X a finite-dimensional separable metric s pa ce, dim G ( X ) = dim( X ). By a theorem of Nag a mi [9] and Smir nov [11] (indep endent ly), ea ch separ a ble metric space is the unio n o f ≤ ω 1 zero-dimensio nal subsets. This implies Theorem 21 . F or e ach sep ar able metric sp ac e X , dim G ( X ) ≤ ω 1 . The following three results are easy to prov e. Theorem 22. If X is a sep ar able metric sp ac e for which dim G ( X ) < ω 1 , then X has pr op erty S c ( O , O ) . Since the Hilber t cub e H = [0 , 1] N do es not hav e the prop er t y S c ( O , O ), we have dim G ( H ) = ω 1 . TOPOLOGICAL GAMES AND COVERING DIMENSION 11 Lemma 23 (Subspa ce Lemma) . L et Y b e a close d subsp ac e of X . Then dim G ( Y ) ≤ dim G ( X ) . Lemma 24 (Addition Lemma ) . L et Y ⊆ X b e such that dim G ( Y ) = β , and α is minimal such that for e ach close d set C ⊂ X \ Y we have dim G ( C ) ≤ α . Then dim G ( X ) = β + α . W e hav e the following particula rly s atisfying pr op erty of ga me dimension in the case of countable dimensional spa ces: Lemma 25 ([2]) . L et X b e a sep ar able m et ric sp ac e. The fol lowing ar e e quival ent: (1) X is c ountable dimensional. (2) dim G ( X ) = ω . Now co nsider the behavior of g ame dimension on space s X with ω < dim G ( X ) < ω 1 . As noted in T heo rem 22 these spac es are a mong the spaces with proper ty S c ( O , O ). A num ber of imp or tant sub class es of the spaces with prop er ty S c ( O , O ) hav e b een iden tified and play an imp orta nt ro le in developing ga me dimension. The following res ults p oint out s ome of thes e co nnections. Theorem 26 . S c ( O , O ) = S c ( O kfd , O ) = S c ( O fd , O ) . Pro of: W e show that S c ( O fd , O ) ⇒ S c ( O , O ). Let X s a tisfy S c ( O fd , O ). Let ( U n : n < ∞ ) be a sequence of op en cov ers of X . No w we use the tec hnique in [ ? ] to con vert this sequence of op en cov ers to a sequence of FD -cov ers: W r ite N = S k< ∞ Y k where ea ch Y k is infinite and the Y k ’s are pairwise disjoint. Fix a k , a finite dimensional s ubset C of the space, and a set I ⊂ Y k with | I | = dim ( C ) + 1. Then for each i ∈ I U i is a n op en cov er of C and we can find a pairwise disjoint refinement V i of U i , where V i consists of sets op en in the rela tive top ology of C , such tha t S i ∈ I V i is a n op en cover of C . Without loss of gener ality we may as sume that the sets in V i are op en in X , and pair wise disjoint in X . Define U ( C , I ) = [ ( [ i ∈ I V i ) . Finally , set V k = { U ( C, I ) : C ⊂ X finite dimensional and I ⊂ Y k with | I | = dim ( C ) + 1 } . Then each V k is in O fd . Applying S c ( O fd , O ) to the sequence ( V k : k < ∞ ) we find a sequence ( H k : k < ∞ ) s o that each H k is pairwise disjoint, refines V k , and S k< ∞ H k is an op en cover of X . F or each k , and for ea ch W ∈ H k , choo se a finite dimensio nal set C W and a finite set I W ⊂ Y k so that W ⊂ U ( C W , I W ). F or ea ch i ∈ I W , c ho ose V i ( W ) an op en pa irwise disjoint refinement of U i so that U ( C W , I W ) = S ( S i ∈ I W V i ( W )). Put I k = S W ∈H k I W , a subset of Y k . F o r each i ∈ I k define G i = { W ∩ V : W ∈ H i and V ∈ V i ( W ) } . Each G i is a pairwise disjoint op en refinement of the c o rresp onding U i , and the union of all the H i ’s is an o pe n cover of X .  The inequalities in the following theore m ca n b e proved also for the family O kfd substituting for O fd . How ev er, the result given here s eems optimal. Theorem 27 . L et X b e an infinite dimensional metrizable sp ac e. Then: dim G ( X ) ≤  tp S 1 ( O fd , O ) ( X ) if tp S 1 ( O fd , O ) ( X ) is a limit or dinal tp S 1 ( O fd , O ) ( X ) + ω otherwise. 12 LILJANA BABINKOSTO V A Pro of: Let α b e an ordinal suc h that TW O ha s a winning strateg y in the game G α 1 ( O fd , O ). Let F b e a winning strateg y for TW O. W e shall use F to define a winning strategy for TW O in G β c ( O , O ) for the a ppropria te β . W e shall rep eatedly use Lemma 2 and Lemma 20. W e may assume without los s of gener ality that the op en cov ers play ed by play er O NE of G β c ( O , O ) never include X as an element. T o b egin, choose by Lemma 2 a finite dimensio nal set C 0 ⊂ X such tha t there is for each prop er op en s e t U ⊇ C 0 an O ∈ O fd with F ( O ) = U . Put n 0 = dim ( C 0 ) + 1 . During the first n 0 innings of G β c ( O , O ), TW O choos es for each O j ∈ O a disjoint op en refinement T j such that C 0 ⊆ W 0 = S ( S j ≤ n 0 T j ) (this us es Lemma 20). Since X is infinite dimensional X 6 = W 0 . Then cho ose a B 0 ∈ O fd with W 0 = F ( B 0 ). F o r the second s tage of the ga me, choo se b y Lemma 2 a finite dimensio nal s et C 1 ⊂ X such that there is for each prop er op en set U ⊇ C 1 a U ∈ O fd with U = F ( B 1 , U ). Put n 1 = dim ( C 1 ) + 1. During the next n 1 innings o f G β c ( O , O ), TW O c ho oses for each O j ∈ O a dis jo in t op en refinement T j of O j such that C 1 ⊆ W 1 = S ( S n 0 + n 1 j = n 0 +1 T j ) (this uses Lemma 2 0). As b efore X 6 = W 1 : Cho os e B 1 ∈ O fd with F ( B 0 , B 1 ) = W 1 . Contin uing like this we find that after ω innings we hav e chosen finite dimens io nal subsets C 0 , · · · , C j , · · · of X , op en disjoint r efinements T j of op en cov ers O j , and nonnegative integers n 0 , · · · , n j , · · · , j < ω , and elements B j of O fd such that for all j ≥ 0: (1) n j = dim ( C j ) + 1; (2) C j ⊆ S ( S n j − 1 + n j i = n j − 1 +1) T i ) = W j 6 = X ; (3) F ( B 0 , · · · , B j ) = W j . Assume that ν < α is an infinite ordinal and that for ea ch ρ < ν we hav e selected a finite dimensio nal s et C ρ , an element B ρ of O fd , an element W ρ of B ρ , an op en cov er O ρ and a disjoint o pen refinemen t T ρ of O ρ such that the following hold: W rite ρ as ρ = ω β 1 · n 1 + · · · + ω β k · n k where now β 1 > · · · > β k ≥ 0, and ea ch n i is p o sitive. (1) C ρ is such that for each pro p er o p en s et U ⊇ C ρ there is a B ∈ O fd with U = F (( B σ : σ < ρ ) ⌢ ( B )); (2) If β k = 0, put µ = sup { σ < ρ : σ a limit ordinal } , s o that ρ = µ + n k . And set m = P µ ≤ j <µ + n k ( dim ( C j ) + 1). Then we alr eady have av ailable O µ , · · · , O µ + m + dim ( C ρ )+1 and T µ , · · · , T µ + m + dim ( C ρ )+1 , and S ρ ⊆ S ( S µ + m + dim ( C ρ )+1 i = µ + m +1 T i ) = W ρ ; (3) If β k > 0, then ρ is a limit ordinal. And set m = dim ( C ρ ) + 1. Then we a lready have av a ilable O ρ +1 , · · · , O ρ + m and T ρ +1 , · · · , T ρ + m , a nd S ρ ⊆ S ( S ρ + m i = ρ +1 T i ) = W ρ . (4) W ρ = F ( B σ : σ ≤ ρ ). W e now describ e wha t happ ens at ν . Case 1: ν is a limit or dinal: By Lemma 2 choose a finite dimensiona l subset C ν of X such that for each pro p e r op en set U ⊇ C ν there is a U ∈ O fd with U = F (( B ρ : ρ < ν ) ⌢ ( U )) and put m = dim ( C ν ). In innings j ∈ { ν, ν + 1 , · · · , ν + m } of G β c ( O , O ), choose op en disjoint refinements T j of O j such that C ν ⊆ S ( S ν + m j = ν T j ) = W ν (this uses Lemma 20). Since X is infinite dimensional, W ν 6 = X , so choo se by Lemma 2 TOPOLOGICAL GAMES AND COVERING DIMENSION 13 a B ν ∈ O fd with W ν = F ( B ρ : ρ ≤ ν ). Case 2: ν is a successo r ordinal: W rite ν = ρ + k + 1 for some nonnega tive integer k and limit ordinal ρ . By the induction hypo thesis we have finite dimensio nal sets C ρ , · · · , C ρ + k av ailable, and with ℓ = P ρ + k j = ρ ( dim ( C j ) + 1 ), we also hav e av a ilable the disjoint op en refinements T ρ , · · · , T ρ + ℓ of the op en cov ers O ρ , · · · , O ρ + ℓ of X . Now choose by Lemma 2 a finite dimensio nal subset C ν of X such that fo r each prop er op en set U ⊇ C ν there is a U ∈ O fd such that U = F (( B σ : σ < ν ) ⌢ ( U )). P ut m = dim ( C ν ) + 1 . During the next m innings of G β c ( O , O ) TWO cho o ses op en disjoint refinements T ρ + ℓ + i of O ρ + ℓ + i , 1 ≤ i ≤ m , so that C ν ⊆ S ( S ρ + ℓ + m i = ρ + ℓ +1 T i ) = W ν 6 = X (this uses Lemma 2 0), and then cho ose B ν ∈ O fd such that W ν = F ( B σ : σ ≤ ν ). Thu s, we see that the recurs ive conditions hold als o at ν . Th us for a play (( O σ , T σ ) : σ < β ) of G β c ( O , O ) play ed acco rding to this strategy we find a sequence (( B ν , W ν ) : ν < α ) which is an F - play of G α 1 ( O fd , O ), thus w on by TW O, for which the union of the set of W ν is covered by the unio n o f the set of T σ ’s. Moreov er, the strategy descr ib ed shows that if α is a limit or dinal, then β = α works, and if α is a succes s or ordina l, then TWO wins G β c ( O , O ) a t so me inning b efor e α + ω .  Note that since tp S 1 ( O fd , O ) ( X ) ≤ tp S 1 ( O kfd , O ) ( X ) we find that Corollary 28. F or X an infinite dimensional sep ar able m et ric sp ac e, dim G ( X ) ≤  tp S 1 ( O kfd , O ) ( X ) if tp S 1 ( O kfd , O ) ( X ) is a limit or dinal tp S 1 ( O kfd , O ) ( X ) + ω otherwise. W e now demonstrate a few features o f these results. Prop ositi o n 29. L et X and Y b e c omp act metrizable sp ac es. F or e ach op en c over U of X × Y ther e ar e op en c overs A of X and B of Y such that { A × B : A ∈ A and B ∈ B } re fines U . Pro of: Let d X be a compatible metric o n X and let d Y be a compatible metric o n Y . Define metr ic d on X × Y s o that d (( x 1 , y 1 ) , ( x 2 , y 2 )) = p d X ( x 1 , x 2 ) 2 + d Y ( y 1 , y 2 ) 2 . Then d is a compatible metric o n X × Y . Consider a n op en cov er U of X × Y . Since X × Y is compact choose by the Leb esgue Covering Lemma a p ositive real num ber δ suc h that for each subset S ⊂ X × Y with d -diameter less than δ ther e is a U ∈ U with S ⊂ U . Let V be a finite op en cov er of X consisting of sets o f d X -diameter less than δ 2 , and le t B b e a finite op en cover of Y consisting o f op en sets of d Y -diameter less than δ 2 . Then { A × B : A ∈ A a nd B ∈ B } is a n op en cov er of X × Y a nd consists of sets of d -diameter less than δ . Thus, this op en cov er refines U .  Since K is no t countable dimens ional, Le mma 25 implies that di m G ( K ) ≥ ω + 1. Lemma 30. F or e ach n , di m G ( K × I n ) ≤ ω + n + 1 . Pro of: W rite I n = A 0 ∪ A 1 ∪ · · · ∪ A n where each A i is zero- dimensional. B y Prop ositio n 29 we may assume the open cov ers U o f K × I n play ed by ONE ar e finite, of the for m { A × B : A ∈ A a nd B ∈ B } where A is an op en cover of K and B is an o p en cover of I n . Now consider the following s trategy for TWO: Let F be TWO’s winning stra tegy in G ω +1 c ( O , O ) on K . In even indexed innings (including index 0), when TW O con- siders the mov es ( O 0 , · · · , O 2 · n ) of ONE, TWO plays T 2 · n K = F ( O 0 K , O 2 K , · · · , O 2 · n K ) 14 LILJANA BABINKOSTO V A on K Since A 0 ⊂ I is zero dimensional TWO cho o ses an op en refinement T 2 · n I of O 2 · n I which covers A 0 . TWO plays T 2 · n = G ( O 0 , O 1 , · · · , O 2 · n − 1 , O 2 · n ) = { U × V : U ∈ T 2 · n K and V ∈ T 2 · n I } . In o dd innings the sa me plan is used on A 1 instead of A 0 . Consider the s tatus after ω innings have elapsed. The uncov ered part is contained in a set of the form C × I n where C is compact and zero -dimensional. Thus it takes at most n + 1 more innings. This shows that dim G ( K × I n ) ≤ ω + n + 1.  Using Lemma 25 for the Smirnov compactum S ω we hav e that tp S 1 ( O kfd , O ) ( S ω ) = 2 < dim G ( S ω ) = ω . Th us the second alter native o f Corollary 28 cannot b e im- prov ed. F o r the Pol compa ctum K we have tp S 1 ( O kfd , O ) ( K ) = ω + 1. Thus, using n = 0 , dim G ( K ) = ω + 1. By the second a lternative of Theorem 27 we conclude that dim G ( K ) ≤ ( ω + 1) + ω = ω · 2. Thus the second alternative of Theorem 27 do es not give optimal information in all cases. F or n > 1 we a lso hav e that tp S 1 ( O kfd , O ) ( K n ) = ω · n + 1 so that Co rollar y 28 implies that dim G ( K n ) ≤ ω · ( n + 1 ). Conjecture 1. dim G ( K n ) = ω · n + 1 . Let X b e ⊕ ∞ n =1 K n , the to p o logical sum of the finite pow ers of the P ol com- pactum. This is a loca lly compact space. Let K ω = ( ⊕ ∞ n =1 K n ) ∪ { p ω } b e the one-p oint compac tification of X . Then we have dim G ( X ) = ω 2 = tp S 1 ( O kfd , O ) ( X ) and dim G ( K ω ) = ω 2 = tp S 1 ( O kfd , O ) ( K ω ). Contin uing in vestigating g ame dimension’s pro duct theory we also note: Theorem 31. If X is a c omp act metrizable sp ac e and C is the Cantor set, t hen dim G ( X ) = dim G ( X × C ) Pro of: When U is an op en cover of X × C we may assume by P rop osition 29 that U is finite and that ther e a re finite op en cov ers U X of X and U C of the Cantor set such tha t U = { U × V : U ∈ U X and V ∈ U C } . Since C is zer o dimensional we may assume that U C is pa irwise disjoint. Let α denote dim G ( X ). W e may assume that α > ω . Let F b e T WO’s winning strategy in G α c ( O , O ) o n X . W e now define a winning strategy G for TW O in the game G α c ( O , O ) played on X × C . With γ < α and op en covers ( U 0 , · · · , U γ ) of X × C giv en, each finite, compute V γ X = F ( U 0 X , · · · , U γ X ), and compute U γ C , and then define G ( U 0 , · · · , U γ ) = { U × V : U ∈ V γ X and V ∈ U γ C } . This defines G for inning γ fo r each γ < α . T o see that G is a winning stra tegy , co nsider a G -play of length α , O NE’s mov es denoted U γ , γ < α , and TWO’s, T γ . Consider a n ( x, c ) ∈ X × C . F rom the definition of G we hav e an F -play U γ X , T γ X = F ( U δ X : δ ≤ γ ) , γ < α on X , won by TW O. Cho ose γ < α w ith x ∈ S T γ X . Pick a V ∈ T γ X with x ∈ V . Also, choo se a C ∈ U γ C with c ∈ C . Then V × C is in T γ . Thu s G is a winning strateg y for TW O. This shows that dim G ( X × C ) ≤ α . Since X is homeo morphic to a clos ed subspa ce of X × C , Le mma 23 implies that TOPOLOGICAL GAMES AND COVERING DIMENSION 15 α = dim G ( X ) ≤ dim G ( X × C ).  Theorem 32 . L et X and Y b e c omp act metrizable sp ac es. If dim G ( X ) is a limit or dinal and Y is c ountable dimensional, t hen dim G ( X ) = dim G ( X × Y ) Pro of: Put α = dim G ( X ). W r ite α = S n< ∞ S n where each S n has order type α , and for m < n , S m ∩ S n = ∅ . F or ea ch n enumerate S n in an o rder preserving way as ( s n γ : γ < α ). W rite Y = ∪ n< ∞ Y n where ea ch Y n is a zero dimensio nal set. Let F be a winning strategy for ONE in the ga me G α c ( O , O ) on X . W e define a winning stra tegy G for Y in the game G α c ( O , O ) on X × Y : First, when ONE plays an op en cov er U of X × Y , TW O will repla ce it, using Prop o sition 29, with a finite refinement of the form { U × V : U ∈ U X and V ∈ U Y } where U X is a n o p e n cover of X and U Y is an op e n cover of Y . Now we define G . Let a n inning γ < α b e g iven, as well as a mov e O γ of O NE. Cho ose n with γ ∈ S n , say γ = s n ν . Replace O γ with the refinement { U × V : U ∈ O γ X and V ∈ O γ Y } where O γ X is a finite o p en cover of X and O γ Y a finite op en cov er of Y . Let V γ be a finite disjoint refinement of O γ Y which cov ers Y n , and let T γ X = F ( O γ X,s n 0 , · · · , O γ X,s n ν ). Then define T γ = G ( O ν : ν ≤ γ ) := { U × V : U ∈ T γ X and V ∈ V γ } . It follows that G is a winning strategy for TWO in G α c ( O , O ), showing dim G ( X × Y ) ≤ α . But X is homeomor phic to a clo sed subspace of X × Y , implying by Lemma 23 that α ≤ dim G ( X × Y ).  References [1] D.F. Addis and J.H. Gresham, A class of infinite dimensional sp ac es. Part I: Dimension the ory and Alexandr off ’s pr oblem , F undamenta Mathematica e 101 (3) (1978), 195 - 205. [2] L. Babink osto v a, Selective scr e enability game and c overing dimension , T op ol ogy Pro ceed- ings 29:1 (2005), 13 - 17. [3] L. 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