Very I-favorable spaces

VER Y I-F A VORABLE S P A CES A. KUCHARSKI, SZ. PLEWIK, AND V. V A LOV Abstract. W e prov e that a Hausdorff space X is very I -fav orable if and only if X is the almost limit spa ce of a σ -co mplete inverse system co nsisting of (n ot necessarily Hausdorff ) second coun table spaces and surjectiv e d-op en b onding maps. It is also sho wn that the class of Tychonoff very I -favorable spaces with respect to the co-zero sets coincides with the d-op enly generated spaces. 1. Introduction The classes of I-fa v orable and v ery I-fav orable space s w ere in tro d uced b y P . Daniels, K. Kunen and H. Zhou [2]. Let us recall the corres p ond- ing definitions. T w o play ers are play ing the so called op en- o p en game in a space ( X , T X ) , a round consists of play er I c ho os ing a nonempt y op en set U ⊂ X and pla y er I I a nonempt y op en set V ⊂ U ; I wins if the union o f I I’s op en sets is dense in X , otherwise I I wins. A space X is called I- f a vor able if pla y er I has a winning strategy . Thi s means that there exists a function σ : S {T n X : n ≥ 0 } → T X suc h that for eac h game σ ( ∅ ) , B 0 , σ ( B 0 ) , B 1 , σ ( B 0 , B 1 ) , B 2 , . . . , B n , σ ( B 0 , . . . , B n ) , B n +1 , . . . the union S n ≥ 0 B n is dense in X , where ∅ 6 = σ ( ∅ ) ∈ T X and B k +1 ⊂ σ ( B 0 , B 1 , .., B k ) 6 = ∅ and ∅ 6 = B k ∈ T X for k ≥ 0 . A family C ⊂ [ T X ] ≤ ω is said to b e a club if: (i) C is closed under increasing ω -c hains, i.e., if C 1 ⊂ C 2 ⊂ ... is an increasing ω -c ha in from C , then S n ≥ 1 C n ∈ C ; ( ii) f o r an y B ∈ [ T X ] ≤ ω there exists C ∈ C with B ⊂ C . Let us recall [7, p. 218], that C ⊂ c T X means that for an y nonempt y V ∈ T X there exists W ∈ C suc h that if U ∈ C and U ⊂ W , then 2000 Mathematics Subje ct Classific ation. Primary: 54B 35, 9 1A44; Secondary : 54C10. Key wor ds and phr ases. In verse system; very I-fav or able space; sk eletal map, κ -metrizable compact space, d-op en map. The third author was supported b y NSER C Gran t 26 1 914-0 8. 1 U ∩ V 6 = ∅ . A space X is I -favo r able if and only if the family {P ∈ [ T X ] ≤ ω : P ⊂ c T X } con tains a club, see [2, Theorem 1.6]. A space X is called very I -favor able if the family {P ∈ [ T X ] ≤ ω : P ⊂ ! T X } con tains a club. Here, P ⊂ ! T X means that for an y S ⊂ P and x / ∈ cl X S S , there exists W ∈ P suc h that x ∈ W and W ∩ S S = ∅ . It is easily seen that P ⊂ ! T X implies P ⊂ c T X . It w as sh o wn b y the first tw o authors in [5] that a compact Hausdorff space is I-fa vorable if and only if it can b e represen ted as the limit of a σ -complete (in the sense of Shc hepin [10]) in ve rse system consisting of I-fav orable compact metrizable spaces and sk eletal b o nding maps, see also [4] a nd [6]. F or similar c haracterization of I-fav orable spaces with resp ect to co-zero sets , se e [14]. Recall that a con tinuous map f : X → Y is called skeletal if the set Int Y cl Y f ( U ) is non-empt y , for an y U ∈ T X , see [8]. In this pap er we sho w that there exists an analogy b etw een the re- lations I-fa v orable spaces - sk eletal maps and v ery I-fa vorable spaces - d-op en maps (see Section 2 for the definition of d-op en maps). The follo wing t w o theorems are our main results : Theorem 3.3. A r e gular sp ac e X is ve ry I -favor able if and only if X = a − lim ← − S , wher e S = { X A , q A B , C } is a σ -c omple te inverse system such that al l X A ar e ( not-ne c essarily Hausdorff ) sp ac es with c ountable weight and the b onding maps q A B ar e d-op en and onto. Theorem 4.1. A c ompletely r e gular sp ac e X is very I -favor able w i th r esp e ct to the c o-ze r o sets if and only if X is d -op enly gener ate d. Here, a completely regular space X is d- op enly gener ate d if there exists a σ -complete inv erse system S = { X σ , π σ  , Γ } consisting of sep- arable metric spaces X σ and d-op en surjectiv e b onding maps π σ  with X b eing embedded in lim ← − S suc h that π σ ( X ) = X σ for eac h σ ∈ Γ . Theorem 4.1 implies the following c haracterization o f κ - metrizable compacta (see Corollary 4.3), whic h pro vides an answ er of a question from [14]: A compact Hausdorff space is very I -fav orable with resp ect to the co-zero se ts if and only if X is κ -metrizable. 2 2. Ver y I-f a v ora ble sp a ces and d-open maps T. Byczk o wski and R. P ol [1] in tro duced nearly op en sets and nearly op en maps as follo ws. A subset of a top ological space is ne arly op en if it is in the inte rior of its closure. A map is ne arly op en if the image of ev ery op en subset is nearly op en. Contin uous nearly op en maps w ere called d- op en b y M. Tk a c henk o [12]. Ob viously , ev ery d-op en map is sk eletal. Prop osition 2.1. L et ( X , T X ) and ( Y , T Y ) b e top olo gi c al s p ac es and f : X → Y a c on tinuous function. Then the fol lowin g c on d itions ar e e quivalent: (1) f is d-op en; (2) cl X f − 1 ( V ) = f − 1 (cl Y V ) for any op en V ⊂ Y ; (3) f ( U ) ⊂ In t Y cl Y f ( U ) for every op en subset U ⊂ X ; (4) { f − 1 ( V ) : V ∈ T Y } ⊂ ! T X . Pr o of. The implication (1) ⇒ (2) w as establishe d in [12, Lemma 5]. Ob viously (3) ⇒ (1) . Let us pro v e the implication (2) ⇒ (3) . Supp ose U ⊂ X is open. Then w e ha v e X \ f − 1 (In t Y cl Y f ( U )) ⊂ X \ U . Indeed, Y \ In t Y cl Y f ( U ) = cl Y ( Y \ cl Y f ( U )) and b y ( 2 ) w e get f − 1 (cl Y ( Y \ cl Y f ( U ))) = cl X ( f − 1 ( Y \ cl Y f ( U ))) . But cl X ( f − 1 ( Y \ cl Y f ( U ))) = cl X ( X \ f − 1 (cl Y f ( U ))) and X \ f − 1 (cl Y f ( U )) ⊂ X \ cl X f − 1 ( f ( U )) ⊂ X \ cl X U ⊂ X \ U. Hence f ( U ) ∩ Y \ In t Y cl Y f ( U ) = ∅ and f ( U ) ⊂ In t Y cl Y f ( U ) . T o show (4) ⇒ (2) , assume that { f − 1 ( V ) : V ∈ T Y } ⊂ ! T X . Since f is contin uous w e get cl X f − 1 ( V ) ⊂ f − 1 (cl Y V ) for an y open set V ⊂ Y . W e shall show that f − 1 (cl Y V ) ⊂ cl X f − 1 ( V ) f o r an y op en V ⊂ Y . Supp ose there exists an op en set V ⊂ Y suc h that f − 1 (cl Y V ) \ cl X f − 1 ( V ) 6 = ∅ . Let x ∈ f − 1 (cl Y V ) \ cl X f − 1 ( V ) and S = { f − 1 ( V ) } . Since x 6∈ cl X S S = cl X f − 1 ( V ) , there is a n op en set U ∈ B Y suc h that x ∈ f − 1 ( U ) and f − 1 ( U ) ∩ f − 1 ( V ) = ∅ . Therefore, f ( x ) ∈ U ∩ cl Y V whic h con tradicts V ∩ U = ∅ . Finally , w e can sho w that (2) yields { f − 1 ( V ) : V ∈ T Y } ⊂ ! T X . Indeed, let S ⊂ { f − 1 ( V ) : V ∈ T Y } and x 6∈ cl X S S . The n there is U ∈ T Y suc h that S S = f − 1 ( U ) . Hence, cl X S S = f − 1 (cl Y U ) . Put W = f − 1 ( Y \ cl Y U ) . 3 W e ha v e x ∈ W and W ∩ cl X S S = ∅ .  Remark 2.2. If, under the hyp otheses of Pr op os ition 2.1 , ther e exists a b ase B Y ⊂ T Y with { f − 1 ( V ) : V ∈ B Y } ⊂ ! T X , then f is d-op en. Indeed, we can follow the pro of of the implication (4) ⇒ (2) from Prop osition 2.1. The only difference is the c hoice of the family S . If there exis ts x ∈ f − 1 (cl Y V ) \ cl X f − 1 ( V ) for some o p en V ⊂ Y , w e c ho o se S = { f − 1 ( W ) : W ∈ B Y and W ⊂ V } . Next lemma w as established in [12, Lemma 9 ]. Lemma 2.3. L et f : X → Y and g : Y → Z b e c ontinuous maps with f b ein g surje c tive . T hen g is d-op en pr ovide d so is g ◦ f .  Let X b e a top ological space equipped with a t o p ology T X and Q ⊂ T X . Supp ose that there exis ts a function σ : S {Q n : n ≥ 0 } → Q suc h that if B 0 , B 1 , . . . is a sequence of non-empt y elemen ts of Q with B 0 ⊂ σ ( ∅ ) a nd B n +1 ⊂ σ (( B 0 , B 1 , . . . , B n )) for all n ∈ ω , then { B n : n ∈ ω } ∪ { σ (( B 0 , B 1 , . . . , B n )) : n ∈ ω } ⊂ ! Q . The function σ is called a s tr ong winn ing str ate gy in Q . If Q = T X , σ is called a strong winning strategy . It is clear that if σ is strong winning strategy , then it is a winning strategy for play er I in the op en-op en game. Lemma 2.4. L et σ : S {Q n : n ≥ 0 } → Q b e a str ong winning str ate g y in Q , wher e Q is a family of op en subs e ts of X . Then P ⊂ ! Q for every family P ⊂ Q such that P is close d under σ and finite interse ctions. Pr o of. Let P ⊂ Q b e closed under σ and finite in t ersections. Fix a family S ⊂ P and x 6∈ cl S S . If σ ( ∅ ) ∩ S S 6 = ∅ , t hen ta k e an elemen t U ∈ S suc h that σ ( ∅ ) ∩ U 6 = ∅ and put V 0 = σ ( ∅ ) ∩ U ∈ P . If σ ( ∅ ) ∩ S S = ∅ , then put V 0 = σ ( ∅ ) ∈ P . Assume that sets V 0 , . . . , V n ∈ P are just defi ned. If σ ( V 0 , . . . , V n ) ∩ S S 6 = ∅ , then tak e an elemen t U ∈ S suc h that σ ( V 0 , . . . , V n ) ∩ U 6 = ∅ and pu t V n +1 = σ ( V 0 , . . . , V n ) ∩ U ∈ P . If σ ( V 0 , . . . , V n ) ∩ S S = ∅ , then put V n +1 = σ ( V 0 , . . . , V n ) ∈ P . T ak e a subfamily U = { V k : V k ∩ [ S 6 = ∅ and k ∈ ω } ⊂ Q . Since σ is strong strategy , then S { V n : n ∈ ω } is dense in X . He nce cl S U = cl S S . Since { V n : n ∈ ω } ∪{ σ (( V 0 , V 1 , . . . , V n )) : n ∈ ω } ⊂ ! Q there exists V ∈ { V n : n ∈ ω } ∪ { σ (( V 0 , V 1 , . . . , V n )) : n ∈ ω } ⊂ P suc h that x ∈ V and V ∩ S S = ∅ .  Prop osition 2.5. L et X b e a top olo gic al sp ac e and Q ⊂ T X b e a family close d under finite interse ction. Then ther e is a str ong winning str ate gy 4 σ : S {Q n : n ≥ 0 } → Q in Q if and only if the family {P ∈ [ Q ] ≤ ω : P ⊂ ! Q} c ontains a club C such that every A ∈ C is cl o se d under finite interse ctions. Pr o of. If there is a club C ⊂ {P ∈ [ Q ] ≤ ω : P ⊂ ! Q} , then followin g the argumen ts from [2, Theorem 1.6] one can construct a strong winning strategy in Q . Supp ose there exis ts a strong winni ng strategy σ : S {Q n : n ≥ 0 } → Q . Let C b e the family of all countable subfamilies A ⊂ Q suc h that A is closed under σ and finite in tersections. The family C ⊂ [ Q ] ≤ ω is a club. Obvious ly , C is closed under increasing ω -c hains. If B ∈ [ Q ] ≤ ω , there exists a countable family A B ⊂ Q whic h con tains B and is closed under σ and finite in tersections. So, A B ∈ C . A ccording to Lemma 2.4, A ⊂ ! Q for a ll A ∈ C .  Corollary 2.6. A Hausdorff sp ac e ( X , T ) is ve ry I -favo r able if and only if the family {P ∈ [ T ] ≤ ω : P ⊂ ! T } c on tains a club C with the fol lowing pr op erties: (i) every A ∈ C c ov e rs X and i t is close d under finite interse ctions; (ii) for an y two diffe r ent p oi n ts x, y ∈ X ther e exists A ∈ C c on tain - ing two disjoint elements U x , U y ∈ A with x ∈ U x and y ∈ U y ; (iii) S C = T .  The next prop osition show s that ev ery space X ha ving a base B X suc h that the f a mily {P ∈ [ B X ] ≤ ω : P ⊂ ! B X } con tains a club is ve ry I-fa v orable. Prop osition 2.7. If ther e exists a b ase B of X such that the family {P ∈ [ B ] ≤ ω : P ⊂ ! B } c ontains a club, then the family {P ∈ [ T X ] ≤ ω : P ⊂ ! T X } c ontains a club to o. Pr o of. If there exists a base B of X suc h that the family {P ∈ [ B ] ≤ ω : P ⊂ ! B } con tains a club, then there exists a strong winning strategy in B . Therefore, pla y er I has winning strategy in the open-o p en game G ( B ) (i.e., the op en-op en ga me when eac h play er c ho oses a set from B ). This implies that X satisfies the coun table c hain condition, oth- erwise the strategy fo r play er I I to c ho ose at eac h stage a nonempt y subset of a mem b er of a fixed uncoun table maximal disjoin t collection of elemen ts of B is winning (see [2, Theorem 1.1(ii)] for a similar situ- ation). Consequen tly , eve ry nonempt y op en subset G ⊂ X con tains a coun table disjoint op en family whose union is dense in G (just tak e a maximal disjoin t op en family in G ). Now, for each elemen t U ∈ T X \ B 5 w e assign a coun table family A U ⊂ B of pairwis e disjoin t op en subsets of U suc h that cl S A U = cl U . If U ∈ B , then w e assign A U = { U } . Let C ⊂ {P ∈ [ B ] ≤ ω : P ⊂ ! B } b e a club. Put C ′ = { A ∪ Q : Q ∈ C and A ∈ [ T X ] ≤ ω with A U ⊂ Q for a ll U ∈ A } . First, observ e that if A ∪ Q A ⊂ D ∪ Q D and A ∪ Q A , D ∪ Q D ∈ C ′ , then Q A ⊂ Q D . Indeed, if U ∈ Q A ⊂ B then U ∈ D ∪ Q D and U ∈ B . If U ∈ D , then we get { U } = A U ⊂ Q D (i.e. U ∈ Q D ). Therefore, if w e ha v e a c hain { A n ∪ Q A n : n ∈ ω } ⊂ C ′ , then [ { A n ∪ Q A n : n ∈ ω } = [ n ∈ ω A n ∪ [ n ∈ ω Q A n ∈ C ′ The absorbing prop ert y (i.e. for ev ery A ∈ [ T X ] ≤ ω there is an elemen t P ∈ C ′ suc h that A ⊂ P ) for C ′ is ob vious. So, C ′ ⊂ [ T X ] ≤ ω is a club. It remains to prov e that A ∪ Q ⊂ ! T X for ev ery A ∪ Q ∈ C ′ . Fix a subfamily S ⊂ A ∪ Q and x 6∈ cl S S . Define S ′ = { U ∈ S : U ∈ Q} ∪ [ {A U : U ∈ A } and note that cl S S = cl S S ′ . The last equalit y follows f r o m the inclusion S S ′ ⊂ S S and the fact that S A U is dense in U for ev ery U ∈ A . So, if x 6∈ cl S S then x 6∈ cl S S ′ . Since S ′ ⊂ Q ∈ C there is G ∈ Q suc h that x ∈ G a nd G ∩ cl S S ′ = ∅  If X is a completely regular sp ace, then Σ X denotes the collection of all co-zero sets in X . Corollary 2.8. L et X b e a c ompletely r e gular sp ac e and B ⊂ Σ X a b ase for X . If {P ∈ [ B ] ≤ ω : P ⊂ ! B } c ontains a club, then the fam ily {P ∈ [Σ X ] ≤ ω : P ⊂ ! Σ X } c ontains a club to o. Pr o of. The pro of of previous prop osition w orks in the presen t situation. The only mo dification is that for eac h U ∈ Σ X \ B w e assign a coun table family A U ⊂ B of pairwise disjoin t co-zero subsets of U s uc h that cl S A U = cl U . Suc h A U exists. F or example, an y maximal disjoin t family of elemen ts from B whic h are con tained in U can serv e as A U . The new club is the family C ′ = { A ∪ Q : Q ∈ C and A ∈ [Σ X ] ≤ ω with A U ⊂ Q for a ll U ∈ A } , where C ⊂ {P ∈ [ B ] ≤ ω : P ⊂ ! B } is a club.  6 3. In verse systems with d-open bounding maps Recall some facts from [5]. Let P b e an op en f a mily in a top ological space X and x, y ∈ X . W e say that x ∼ P y if and only if x ∈ V ⇔ y ∈ V for ev ery V ∈ P . The family o f all sets [ x ] P = { y : y ∼ P x } is denoted by X/ P . There exists a mapping q : X → X/ P d efined by q [ x ] = [ x ] P . The set X/ P is equipped with the top ology T P generated b y all images q ( V ) , V ∈ P . Lemma 3.1. [5, Lemma 1] The ma pping q : X → X/ P is c ontinuous pr ovide d P i s an op en family X which is close d under finite interse ction. Mor e over, if X = S P , then the family { q ( V ) : V ∈ P } is a b ase for the top ol o gy T P .  Lemma 3.2. L et a sp ac e X b e t he limit of a inverse system { X σ , π σ  , Σ } with surje ctive b onding maps π σ  . Then π σ  ar e d-op en if and only if e ach pr oje ction p σ : X → X σ is d-op en. Pr o of. Assume all π σ  are d-op en. It suffices to sho w that p ρ  ( p σ ) − 1 ( U )  is dense in some op en subset of X ρ for an y o p en U ⊂ X σ , where σ ≥ ρ . Since p σ ρ is d-op en and p ρ  ( p σ ) − 1 ( U )  = p σ ρ ( U ) , the pro of is completed. Con v ersely , if the limit pro jections are d-op en, then, b y Lemma 2.3, the b onding maps are also d-op en.  W e sa y that a space X is an almost limit o f the in v erse sy stem S = { X σ , π σ  , Γ } , if X can be em b edded in lim ← − S suc h that π σ ( X ) = X σ for eac h σ ∈ Γ . W e denote this by X = a − lim ← − S , and it implies that X is a dense subse t of lim ← − S . Theorem 3.3. A Hausdorff s p ac e X i s very I -fa v or able if and only if X = a − lim ← − S , wher e S = { X A , q A B , C } is a σ -c omple te inverse system such that al l X A ar e ( not-ne c essarily Hausdorff ) sp ac es with c ountable weight and the b onding maps q A B ar e d-op en and onto. Pr o of. Supp ose ( X, T ) is v ery I-fa v ora ble. By Corollary 2.6, there exist a club C ⊂ {P ∈ [ T ] ≤ ω : P ⊂ ! T } satisfying conditions (i)-(iii). F or ev ery A ∈ C cons ider the space X A = X/ A and the map q A : X → X A . Since eac h A is a co v er of X closed under finite inte rsections, b y Lemma 3.1, q A is a contin uous surjection and { q A ( U ) : U ∈ A } is a coun table base for X A . Moreov er, q − 1 A ( q A ( U )) = U for all U ∈ A , see [5]. This, according to Remark 2.2, implies that eac h q A is d-op en (recall that A ⊂ ! T ) ). If A, B ∈ C with B ⊂ A , then there exists a map q A B : X A → X B whic h is contin uous b ecause ( q A B ) − 1 ( q B ( U )) = q A ( U ) for 7 ev ery U ∈ B . The maps q A B are also d- o p en, s ee Lemma 3.2. In this w ay w e obtained the in v erse system S = { X A , q A B , C } consisting of spaces with coun table w eigh t and d- o p en b onding maps. Sinc e C is closed under increasing chains , S is σ -complete. It remains to sho w that the map h : X → lim ← − S , h ( x ) = ( q A ( x )) A ∈C , is a dense em b edding. Let π A : lim ← − S → X A , A ∈ C , b e the limit pro jections of S . The family { π − 1 A ( q A ( U )) : U ∈ A, A ∈ C } is a base f or the top ology of lim ← − S . Since h − 1  π − 1 A ( q A ( U ))  = U for an y U ∈ A ∈ C , h is con tinuous and h ( X ) is dense in lim ← − S . Because C satisfies c ondition (ii ) (se e C orollary 2.6 ), h is one-to-one. Finally , since h ( U ) = h ( X ) ∩ π − 1 A ( q A ( U )) for an y U ∈ A ∈ C (see [5, the pro o f of Theorem 11] and C con tains a base fo r T , h is a n em b edding. Supp ose no w that X = a − lim ← − S , where S = { X A , q A B , C } is a σ - complete inv erse system suc h that all X A are spaces with countable w eigh t and the b onding maps q A B are d-open and on to. Then, b y Lemma 3.2, all limit pro jections π A : lim ← − S → X A , A ∈ C , are d-op en. Since X is dense in lim ← − S , any restriction q A = π A | X : X → X A is also d-op en. Moreo ve r, all q A are surjectiv e (see the definition of a − lim ← − ). Then, according to Propo sition 2.1, { q − 1 A ( U ) : U ∈ T A } ⊂ ! T , where T A is the top ology of X A . Conse quen tly , if B A is a coun table base for T A , w e ha v e P A = { q − 1 A ( U ) : U ∈ B A } ⊂ ! T . The last relation implie s P A ⊂ ! B with B = S {P A : A ∈ C } b eing a base for T . Let us show that P = {P A : A ∈ C } is a club i n {Q ∈ [ B ] ≤ ω : Q ⊂ ! B } . Since S is σ - complete, the suprem um of an y increasing sequence from C is again in C . This implies tha t P is clos ed under increasing c hains. So, it remains to pro ve tha t for ev ery coun table family { U j : j = 1 , 2 , .. } ⊂ B there exists A ∈ C with U j ∈ P A for a ll j ≥ 1 . Because ev ery U j is of the form q − 1 A j ( V j ) for some A j ∈ C and V j ∈ B A j , the re e xists A ∈ C with A > A j for eac h j . It is easily see n that P A con tains the family { U j : j ≥ 1 } for an y suc h A . The refore, P is a club in {Q ∈ [ B ] ≤ ω : Q ⊂ ! B } . Finally , according to Prop osition 2.7, t he fa mily {Q ∈ [ T ] ≤ ω : Q ⊂ ! T } also con tains a club. Hence, X is v ery I-fa vorable .  It follo ws from Theorem 3.3 that ev ery dense subset of a space from eac h of the followi ng classes is v ery I-fa v o ra ble: pro ducts of first coun t- able spaces, κ -metrizable compacta. More g enerally , b y [13, Theorem 2.1 (iv)], ev ery space with a lattice of d-op en maps is v ery I-fa vorable. The next theorem pro vides another examples of v ery I-fav orable spaces. 8 Theorem 3.4. L et f : X onto − − → Y b e a p erfe ct map with X , Y b ei n g r e gular sp ac es . Then Y is very I -favor able, pr ovi d e d so is X . Pr o of. This theorem was established in [ 2] whe n X and Y are compact. The same pro of w orks in our more general situation.  Corollary 3.5. Every c on tinuous image under a p erfe ct map of a sp ac e p ossessing a l a ttic e of d-op en maps is v ery I-favor able.  4. ver y I-f a vorable sp a c es wi th respect to the c o-zer o sets W e say that a space X is very I- f a vor able with r esp e ct to the c o- zer o sets if there exists a strong winning strategy σ : S { Σ n X : n ≥ 0 } → Σ X , where Σ X denotes the collection o f all co-zero sets in X . By Prop osition 2.5, this is equiv alent to the existence of a club in the family {P ∈ [Σ X ] ≤ ω : P ⊂ ! Σ X } . A completely regular space X is d- op enly gener ate d if X is the almost limit of a σ -complete in v erse system S = { X σ , π σ  , Γ } consisting of separable metric spaces X σ and d-op en surjectiv e b onding maps π σ  . Theorem 4.1. A c ompletely r e gular sp ac e X is very I -favor able w i th r esp e ct to the c o-ze r o sets if and only if X is d -op enly gener ate d. Pr o of. Supp ose X is very I -fav orable with resp ect to the co-zero sets and σ : S { Σ n X : n ≥ 0 } → Σ X is a strong winning strategy in Σ X . W e place X as a C ∗ -em b edded subset o f a T yc honoff cub e I A . If B ⊂ A , let π B : I A → I B b e the natural pro j ection and p B b e restriction map π B | X . Let also X B = p B ( X ) . If U ⊂ X w e write B ∈ k ( U ) to denote that p − 1 B  p B ( U )  = U . Claim 1 . F or eve ry U ∈ Σ X ther e exists a c ountable B U ⊂ A such that B U ∈ k ( U ) w i th p B U ( U ) b ein g a c o-ze r o set in X B U . F or ev ery U ∈ Σ X there exists a con tin uous function f U : X → [0 , 1] with f − 1 U  (0 , 1]  = U . Next, ex tend f U to a con tin uous function g : I A → [0 , 1] (recall that X is C ∗ -em b edded in I A ). Then , the re ex ists a coun table set B U ⊂ A and a function h : I B U → [0 , 1] with g = h ◦ π B U . Ob viously , U = p − 1 B U  h − 1  (0 , 1]  ∩ p B U ( X )  , whic h comple tes the pro of of the claim. Let B = { U α : α < τ } b e a base for the top ology o f X consisting of co-zero sets suc h that for eac h α there exists a finite set H α ⊂ A with 9 H α ∈ k ( U α ) . F or an y finite set C ⊂ A let γ C b e a fixed coun table base for X C . Claim 2 . F or every c ountable B ⊂ A ther e exists a c ountable set Γ ⊂ A c ontaining B and a c ountable family U Γ ⊂ Σ X satisfying the fol lowing c onditions: (i) U Γ is close d under σ and finite interse ctions; (ii) Γ ∈ k ( U ) for al l U ∈ U Γ ; (iii) B Γ = { p Γ ( U ) : U ∈ U Γ } is a b ase for p Γ ( X ) . W e construct b y induction a sequence { C ( m ) } m ≥ 0 of coun table sub- sets of A , and a se quence {V m } m ≥ 0 of coun table subfamilies of Σ X suc h that: • C 0 = B and V 0 = { p − 1 B ( V ) : V ∈ B B } , where B B is a base fo r X B ; • C ( m + 1) = C ( m ) ∪ S { B U : U ∈ V m } ; • V 3 m +1 = V 3 m ∪ { σ ( U 1 , .., U n ) : U 1 , .., U n ∈ V 3 m , n ≥ 1 } ; • V 3 m +2 = V 3 m +1 ∪ S { p − 1 C ( γ C ) : C ⊂ C (3 m + 1 ) is finite } ; • V 3 m +3 = V 3 m +2 ∪ { T i = n i =1 U i : U 1 , .., U n ∈ V 3 m +2 , n ≥ 1 } . It is easily seen that the set Γ = S ∞ m =0 C m and the family U Γ = S ∞ m =0 V m satisfy the conditions (i)-(iii) from Claim 2. Claim 3 . The map p Γ : X → X Γ is a d-op en map. It follow s from (ii) that U Γ = { p − 1 Γ ( V ) : V ∈ B Γ } . A ccording to Lemma 2.4, U Γ ⊂ ! Σ X . Conse quen tly , U Γ ⊂ ! T X . There fore, w e can apply Prop osition 2.1 to conclude that p Γ is d-op en. No w, consider the family Λ of all Γ ∈ [ A ] ≤ ω suc h that there exists a coun table family U Γ ⊂ Σ X satisfying the c onditions (i)-(iii) from Claim 2. W e consider the in v erse system S = { X Γ , p Γ Θ , Λ } , where Θ ⊂ Γ ∈ Λ and p Γ Θ : X Γ → X Θ is the restriction of the pro jection π Γ Θ : I Γ → I Θ on the set X Γ . Since p Θ = p Γ Θ ◦ p Γ and both p Γ and p Θ are d-op en surjections, p Γ Θ is also d-op en (see Lemma 2.3). Moreo ver, the union of an y increasing c hain in Λ is again in Λ . So , Λ , equipped the inclusion order, is σ -complete. Finally , b y Claim 2 , Λ co v ers the set A . Therefore, the limit of S is a subset of I A con taining X as a dens e subset. Hence, X is d-op enly generated. Supp ose that X is d-op enly generated. So, X = a − lim ← − S , where S = { X σ , p σ  , Γ } is a σ -complete in v erse system consisting o f separa- ble metric spaces X σ and d-o p en surjectiv e b onding maps p σ  . Let p σ : lim ← − S → X σ , σ ∈ Γ , b e the limit pro jections and q σ = p σ | X . As in 10 the pro of of Theorem 3.3, we can sho w that P = {P σ : σ ∈ Γ } is a club in the family {Q ∈ [ B X ] ≤ ω : Q ⊂ ! B X } , where B X = S {P σ : σ ∈ Γ } and P σ = { q − 1 σ ( V ) : V ∈ B σ } with B σ b eing a coun table base for the top ology of X σ . Since B X consists of co-zero sets, b y Coro lla ry 2 .8 , the family {Q ∈ [Σ X ] ≤ ω : Q ⊂ ! Σ X } contains also a club. Hence , X is v ery I-fa v orable with respect to the co- zero sets.  W e sa y that a space X ⊂ Y is r egularly em b edded in Y is there exists a function e : T X → T Y satisfying the follo wing conditions for an y U, V ∈ T X : • e( ∅ ) = ∅ ; • e( U ) ∩ X = U ; • e( U ) ∩ e( V ) = ∅ pro vided U ∩ V = ∅ . Theorem 4.1 and [13, Theorem 2.1(ii)] yield t he fo llo wing external c haracterization o f v ery I-fav orable spaces with resp ect to the co-zero sets (I-fa v orable space s with resp ect to the co-zero sets hav e a similar external c haracterization, see [14, The orem 1.1]). Corollary 4.2. A c ompletely r e gular sp ac e is very I -favor able with r e- sp e ct to the c o-zer o sets if and on ly if every C ∗ -emb e dding of X in a n y T ychonoff sp ac e Y is r e gular. The nex t corollary pro vides an a nsw er of a question from [14 ] wh ether there exists a characterization of κ -metrizable compacta in t erms a game b et w een t wo pla ye rs. Corollary 4.3. A c omp act Hausdorff sp ac e is very I -favor able with r esp e ct to the c o-ze r o sets if and only if X is κ -metrizable. Pr o of. A compact Hausdorff space is κ -metrizable spaces iff X is the limit space of a σ -complete in v erse system consisting of compact metric spaces a nd op en surjectiv e b onding maps, se e [11] and [10]. Since ev ery d-op en surjec tiv e map b et we en compact Hausdorff space s is open, this corollary follo ws from The orem 4.1.  Recall that a normal space is called p erfectly normal if ev ery op en set is a co-zero set. So, any p erfectly normal spaces is ve ry I-fav orable if and o nly if it is v ery I-fav orable with resp ect to the co-zero sets. Th us, w e hav e the next corollary . Corollary 4.4. Eve ry p erfe ctly normal very I -f a vor able sp a c e is d- op enly gener ate d. 11 Lemma 4.5. L et ( X, T ) b e a c ompletely r e gular sp ac e. If ther e is a str ong winning s tr ate gy σ ′ : S {T n : n ≥ 0 } → T , then ther e is a str ong winning str ate gy σ : S {R n : n ≥ 0 } → R , wher e R c onsis ts of al l r e gular op en subset of X . Pr o of. Assume that σ ′ : S {T n : n ≥ 0 } → T is a strong winni ng strat- egy . W e define a strong winn ing strategy on R . Let σ ( ∅ ) = In t cl σ ′ ( ∅ ) . W e define b y induction σ (( V 0 , V 1 , . . . , V k )) , V k +1 ⊂ σ (( V 0 , V 1 , . . . , V k )) , b y σ (( V 0 , V 1 , . . . , V n +1 )) = In t cl σ ′ (( V ′ 0 , V ′ 1 , . . . , V ′ n +1 )) , where V ′ k +1 = V k +1 ∩ σ ′ (( V ′ 0 , V ′ 1 , . . . , V ′ k )) . Let us sho w that F = { V n : n ∈ ω } ∪ { σ (( V 0 , V 1 , . . . , V n +1 )) : n ∈ ω } ⊂ ! R . If S ⊂ F and x 6∈ cl S S , let F ′ = { V ′ n : n ∈ ω } ∪ { σ ′ (( V ′ 0 , V ′ 1 , . . . , V ′ n +1 )) : n ∈ ω } and S ′ = { W ′ ∈ F ′ : W ∈ S } . Note that S S ′ ⊂ S S , hen ce x 6∈ cl S S ′ . So, there is W ′ ∈ S ′ suc h that W ′ ∩ U ′ = ∅ for all U ′ ∈ F ′ . Assume that W ′ = V k +1 ∩ σ ′ (( V ′ 0 , V ′ 1 , . . . , V ′ k )) and U ′ = V i +1 ∩ σ ′ (( V ′ 0 , V ′ 1 , . . . , V ′ i )) . Then w e infer that V k +1 ∩ In t cl σ ′ (( V ′ 0 , V ′ 1 , . . . , V ′ k )) ∩ V i +1 ∩ In t cl σ ′ (( V ′ 0 , V ′ 1 , . . . , V ′ i )) = ∅ . Since V k +1 ⊂ σ (( V 0 , V 1 , . . . , V k )) = In t cl σ ′ (( V ′ 0 , V ′ 1 , . . . , V ′ k )) and V i +1 ⊂ σ (( V 0 , V 1 , . . . , V i )) = Int cl σ ′ (( V ′ 0 , V ′ 1 , . . . , V ′ i )) , w e get V k +1 ∩ V i +1 = ∅ . Supp ose W ′ = V k +1 ∩ σ ′ (( V ′ 0 , V ′ 1 , . . . , V ′ k )) and U ′ = σ ′ (( V ′ 0 , V ′ 1 , . . . , V ′ i )) . Then V k +1 ∩ In t cl σ ′ (( V ′ 0 , V ′ 1 , . . . , V ′ k )) ∩ In t cl σ ′ (( V ′ 0 , V ′ 1 , . . . , V ′ i )) = ∅ . So, W ∩ U = ∅ . Similarly , w e o bta in W ∩ U = ∅ if W ′ = σ ′ (( V ′ 0 , V ′ 1 , . . . , V ′ k )) and U ′ = σ ′ (( V ′ 0 , V ′ 1 , . . . , V ′ i )) . This completes the pro of.  W e sa y that a top ological space X is p erfe ctly κ -normal if for ev ery op en and disjoin t subset U, V there are op en F σ subset W U , W V with W U ∩ W V = ∅ and U ⊂ W U and V ⊂ W V . It is clear that a space X is p erfectly κ -normal if and only if that eac h regular open set in X is F σ . Prop osition 4.6. I f a normal p erfe ctly κ -n o rmal sp ac e is a c ontinuous image of a very I -favor able sp ac e under a p erfe ct map, then X is d- op enly gener ate d. 12 Pr o of. Ev ery op en F σ -subset of a normal space is a co- zero set, see [3]. So, ev ery regular op en subset of a normal and p erfectly κ -normal space is a co-zero s et. Consequen tly , if X is the image of v ery I-fa v orable s pace and X is normal and p erfectly κ - normal, the n X is v ery I-fav orable (see Theorem 3.4). Hence, according to Lemma 4.5, X is a v ery I-fa v orable with respect to the co-zero sets. Finally , Theorem 4.1 implies that X is d-op enly generated.  Corollary 4.7. If the image of a c om p act Hausdorff very I -favor able sp ac e under a c o ntinuous map is p erfe ctly κ - n ormal, then X is κ - metrizable. Corollary 4.7 implies the followin g result of Shc hepin [11, Theorem 18] whic h has b een prov ed by different metho ds: If the image of a κ -metrizable compact Hausdorff space X under a conti n uous map is p erfectly κ -normal, then X is κ -metrizable to o. Let us also me n tion that , according to Shapiro’s result [9], contin uous images of κ -metrizable compacta hav e sp ecial spectral rep resen tations. This result implies that an y suc h a n image is I - fa vorable. References [1] T. Byczk owski and R. P ol, O n the close d gr aph and op en mapping the or ems , Bull. A ca d. P olon. Sci. Sér. Sci. Math. Astronom. P h ys. 24 (1 976), no. 9, 723 - 726. [2] P . Daniels, K. Kunen and H. Zhou, O n the op en- op en game , F und. Math. 1 45 (1994), no. 3, 205–2 20. [3] R. Engelking, Gener al top olo gy , P olish Scientific Publishers, W arszawa (19 77). [4] A. Kucharski and Sz. Plewik, Game appr o ach to universal ly Kur atowski-Ulam sp ac es , T opolo gy Appl. 154 (2007), no. 2, 4 21–4 27. [5] A. Kuc ha rski and Sz. Plewik, Inverse sy stems and I -favor able sp ac es , T op olog y Appl. 156 (2008), no. 1, 110–1 16. [6] A. K ucharski and Sz. Plewik, Skeletal maps and I-favor able sp ac es , A cta Uni- versitatis Ca r olinae - Math. et Ph ys to a ppe a r (arXiv:math.GN/1003 230 8). [7] K . Kunen, Set the ory. An intr o duction to inde p endenc e pr o ofs , Studies in Logic and the F oundations o f Mathematics, 102. North-Ho lland Publishing Co., Am- sterdam, (1980). [8] J. Mioduszewsk i and L. Rudolf, H -close d and extr emal ly disc onne cte d Haus- dorff sp ac es , D issertationes Math. 66 (1969 ). [9] L. Shapiro, On a sp e ctr al r epr esentation of image s of κ -metrizable bic omp acta , Uspehi Mat. Nauk 37 ( 1982 ), no. 2(224 ), 245–246 (in Russian). [10] E. Sh chepin, T op olo gy of limi t sp ac es with unc ount able inverse sp e ctr a , Uspekhi Mat. Nauk 31 (1976), no. 5(191), 191–2 26. [11] E. Shc hepin, F unctors and un c ountable p owers of c omp acta , Usp ekhi Ma t. Nauk 36 (1981 ), no. 3(219), 3–62 (in Russian). 13 [12] M. Tk achenko, Some r esults on inverse sp etr a II , Comment. Math. Univ. Car ol. 22 (1981), no. 4, 819–8 41. [13] V. M. V alov, Some char acterizations of t he sp ac es wi th a lattic e of d -op en map- pings , C. R. A cad. Bulgare Sci. 39 (1986), no. 9, 9-12. [14] V. V a lov, Ext ernal char acterization of I- favor able sp ac es , arXiv:math.GN/100 50074. Institute of Ma thema tics, Un iversity of S ilesia, ul. Banko w a 14, 40-007 Ka towice E-mail ad dr ess : akuc har@u x2.ma th.us.edu.pl Institute of Ma thema tics, Un iversity of S ilesia, ul. Banko w a 14, 40-007 Ka towice E-mail ad dr ess : plew ik@ma th.us .edu.pl Dep ar tment of Computer Science and Ma thema tics, Nipissing U ni- versity, 100 Col lege Drive, P.O. Box 5 002, Nor th Ba y, ON, P1B 8L 7, Canada E-mail ad dr ess : vesk ov@ni pissi ngu.ca 14