Inverse Systems and I-Favorable Spaces

INVERSE SYSTEMS AND I-F A V ORABLE SP A CES ANDRZEJ KUCHARSKI AND SZYMON PLEWIK Abstra t. W e sho w that a ompat spae is I-fa v orable if, and only if it an b e represen ted as the limit of a σ -omplete in v erse system of ompat metrizable spaes with sk eletal b onding maps. W e also sho w that an y ompletely regular I-fa v orable spae an b e em b edded as a dense subset of the limit of a σ -omplete in v erse system of separable metrizable spaes with sk eletal b onding maps. 1. Intr odution W e in v estigate the lass of all limits of σ -omplete in v erse systems of ompat metrizable spaes with sk eletal b onding maps. Notations are used the same as in the monograph [5℄. F or example, a ompat spae is Hausdor, and a regular spae is T 1 . A direted set Σ is said to b e σ -omplete if an y oun table  hain of its elemen ts has least upp er b ound in Σ . An in v erse system { X σ , π σ  , Σ } is said to b e a σ -omplete, whenev er Σ is σ -omplete and for ev ery  hain { σ n : n ∈ ω } ⊆ Σ , su h that σ = sup { σ n : n ∈ ω } ∈ Σ , there holds X σ = lim ← − { X σ n , π σ n +1 σ n } , ompare [15℄. Ho w ev er, w e will onsider in v erse systems where b onding maps are surjetions. Another details ab out in v erse systems one an nd in [5℄ pages 135 - 144. F or basi fats ab out I-fa v orable spaes w e refer to [4℄, ompare also [10℄. Through the ourse of this note w e mo dify quotien t top ologies and quotien t maps, in tro duing Q P -top ologies and Q P -maps, where P is a family of subsets of X . Next, w e assign the family P seq (of all sets with some prop erties of ozero sets) to a giv en family P . F rink's theorem is used to sho w that the Q P -top ology is ompletely regular, whenev er P ⊆ P seq is a ring of subsets of X , see Theorem 5. Afterw ards, some 2000 Mathematis Subje t Classi ation. Primary: 54B35, 90D44; Seondary: 54B15, 90D05. Key wor ds and phr ases. In v erse system, Op en-op en game, sk eletal map. 1 sp eial lub lters are desrib ed as systems of oun table sk eletal fam- ilies. This yields that ea h family whi h b elongs to a su h lub lter is a oun table sk eletal family , whi h pro dues a sk eletal map on to a ompat metrizable spae. Theorem 12 is the main result: I-fa v orable ompat spaes oinides with limits of σ -omplete in v erse systems of ompat metrizable spaes with sk eletal b onding maps. E.V. Sh hepin has onsidered sev eral lasses of ompat spaes in a few pap ers, for example [13℄, [14℄ and [ 15 ℄. He in tro dued the lass of ompat op enly generated spaes. A ompat spae X is alled op enly gener ate d , whenev er X is the limit of a σ -omplete in v erse sys- tem of ompat metrizable spaes with op en b onding maps. Originally , Sh hepin used another name: op en-generated spaes; see [15 ℄. A. V. Iv ano v sho w ed that a ompat spae X is op enly generated if, and only if its sup erextension is a Dugundji spae, see [9℄. Then Sh hepin established that the lasses of op enly generated ompat spaes and of κ -metrizable spaes are the same, see Theorem 21 in [ 15℄. Something lik ewise is established for ompat I-fa v orable spaes in Theorem 12. A Bo olean algebra B is semi-Cohen (regularly ltered) if, and only if [ B ] ω has a losed un b ounded set of oun table regular subalgebras, in other w ords [ B ] ω on tains a lub lter. Hene, the Stone spae of a semi-Cohen algebras is I-fa v orable. T ranslating Corollary 5.5.5 of L. Heindorf and L. B. Shapiro [7℄ on top ologial notions, one an obtain our's main result in zero-dimensional ases, ompare also Theorem 4.3 of B. Balar, T. Je h and J. Zapletal [2 ℄. W e get Theorem 11 whi h sa ys that ea h ompletely regular I-fa v orable spae is homeomorphi to a dense subspae of the limit of an in v erse system { X/ R , q R P , C } , where spaes X/ R are metrizable and separable, b onding maps q R P are sk eletal and the direted set C is σ -omplete. 2. Q P -topologies Let P b e a family of subsets of X . W e sa y that y ∈ [ x ] P , whenev er x ∈ V if, and only if y ∈ V , for ea h V ∈ P . The family of all lasses [ x ] P is denoted X / P . Note that [ x ] P ⊆ V if, and only if [ x ] P ∩ V 6 = ∅ , for ea h V ∈ P . Put q ( x ) = [ x ] P . The funtion q : X → X/ P is alled an Q P -map . The oarser top ology on X / P whi h on tains all images q [ V ] = { [ x ] P : x ∈ V } , where V ∈ P , is alled an Q P -top olo gy . If V ∈ P , then q − 1 ( q [ V ]) = V . Indeed, w e ha v e V ⊆ q − 1 ( q [ V ]) , sine q : X → X / P is a surjetion. Supp ose x ∈ q − 1 ( q [ V ]) . Then q ( x ) ∈ q [ V ] , and [ x ] P ∩ V 6 = ∅ . W e get [ x ] P ⊆ V , sine V ∈ P . Therefore x ∈ V . 2 Lemma 1. L et P b e a family of op en subsets of a top olo gi al sp a e X . If P is a lose d under nite interse tions, then the Q P -map q : X → X / P is  ontinuous. Mor e over, if X = S P , then the family { q [ V ] : V ∈ P } is a b ase for the Q P -top olo gy. Pr o of. W e ha v e q [ V ∩ U ] = q [ V ] ∩ q [ U ] , for ev ery U, V ∈ P . Hene, the family { q [ V ] : V ∈ P } is losed under nite in tersetions. This family is a base for the Q P -top ology , sine X = S P implies that X / P is an union of basi sets. Ob viously , the Q P -map q is on tin uous.  A dditionally , if X is a ompat spae and X/ P is Hausdor, then the Q P - map q : X → X / P is a quotien t map. Also, the Q P -top ology oinides with the quotien t top ology , ompare [5℄ p. 124. Let R b e a family of subsets of X . Denote b y R seq the family of all sets W whi h satisfy the follo wing ondition: Ther e exist se quen es { U n : n ∈ ω } ⊆ R and { V n : n ∈ ω } ⊆ R suh that U k ⊆ ( X \ V k ) ⊆ U k +1 , for any k ∈ ω , and S { U n : n ∈ ω } = W . If R seq 6 = ∅ , then S R = X . Indeed, tak e W ∈ R seq . Whenev er U n and V n are elemen ts of sequenes witnessing W ∈ R seq , then X \ V k ⊆ U k +1 ⊆ W implies U k +1 ∪ V k = X . If X is a ompletely regular spae and T onsists of all ozero sets of X , then T = T seq . Indeed, for ea h W ∈ T , x a on tin uous funtion f : X → [0 , 1] su h that W = f − 1 ((0 , 1]) . Put U n = f − 1 (( 1 n , 1]) and X \ V n = f − 1 ([ 1 n , 1]) . Reall that, a family of sets is alled a ring of sets whenev er it is losed under nite in tersetions and nite unions. Lemma 2. If a ring of sets R is  ontaine d in R seq , then any  ountable union S { U n ∈ R : n ∈ ω } b elongs to R seq . Pr o of. Supp ose that sequenes { U n k : k ∈ ω } ⊆ R and { V n k : k ∈ ω } ⊆ R witnessing U n ∈ R seq , resp etiv ely . Then sets U 0 n ∪ U 1 n ∪ . . . ∪ U n n and V 0 n ∩ V 1 n ∩ . . . ∩ V n n are suessiv e elemen ts of sequenes whi h witnessing S { U n ∈ R : n ∈ ω } ∈ R seq .  Lemma 3. If a family of sets P is  ontaine d in P seq , then the Q P -top olo gy is Hausdor. Pr o of. T ak e [ x ] P 6 = [ y ] P and W ∈ P su h that x ∈ W and y 6∈ W . Fix sequenes { U n : n ∈ ω } and { V n : n ∈ ω } witnessing W ∈ P seq . Cho ose k ∈ ω su h that x ∈ U k and y ∈ V k . Hene [ x ] P ⊆ U k and [ y ] P ⊆ V k . 3 Therefore, sets q [ U k ] and q [ V k ] are disjoin t neigh b ourho o ds of [ x ] P and [ y ] P , resp etiv ely .  Lemma 4. If a non-empty family of sets P ⊆ P seq is lose d under nite interse tions, then Q P -top olo gy is r e gular. Pr o of. W e ha v e q [ A ] ∩ q [ B ] = q [ A ∩ B ] for ea h A, B ∈ P . The family { q [ A ] : A ∈ P } is a base of op en sets for the Q P -top ology . Fix x ∈ W ∈ P and sequenes { U n : n ∈ ω } ⊆ P and { V n : n ∈ ω } ⊆ P witnessing W ∈ P seq . T ak e an y U k ⊆ W su h that [ x ] P ⊆ U k ∈ P . W e get q ( x ) ∈ q [ U k ] ⊆ cl q [ U k ] ⊆ q [ X \ V k ] = X / P \ q [ V k ] ⊆ q [ W ] , where ∪P = X .  T o sho w whi h Q P -top ologies are ompletely regular, w e apply the F rink's theorem, ompare [6℄ or [5℄ p. 72. Theorem [O. F rink (1964)℄. A T 1 -sp a e X is  ompletely r e gular if, and only if ther e exists a b ase B satisfying : (1) If x ∈ U ∈ B , then ther e exists V ∈ B suh that x 6∈ V and U ∪ V = X ; (2) If U, V ∈ B and U ∪ V = X , then ther e exists disjoint sets M , N ∈ B suh that X \ U ⊆ M and X \ V ⊆ N .  Theorem 5. If P is a ring of subsets of X and P ⊆ P seq , then the Q P -top olo gy is  ompletely r e gular. Pr o of. The Q P -top ology is Hausdor b y Lemma 3. Let B b e the mini- mal family whi h on tains { q [ V ] : V ∈ P } and is losed under oun table unions. This family is a base for the Q P -top ology , b y Lemma 1 . W e should sho w that B fullls onditions (1) and (2) in F rink's theorem. Let [ x ] P ∈ q [ W ] ∈ B . Fix sequenes { U k : k ∈ ω } and { V k : k ∈ ω } witnessing W ∈ P seq and k ∈ ω su h that x ∈ X \ V k ⊆ W . W e ha v e W ∪ V k = X . Therefore [ x ] P 6∈ q [ V k ] and q [ W ] ∪ q [ V k ] = X/ P . Th us B fullls (1) . Fix sets S { U n : n ∈ ω } ∈ B and S { V n : n ∈ ω } ∈ B su h that X/ P = [ { q [ U n ] : n ∈ ω } ∪ [ { q [ V n ] : n ∈ ω } , where U n and V n b elong to P . Th us, U = S { U n : n ∈ ω } ∈ P seq and V = S { V n : n ∈ ω } ∈ P seq b y Lemma 2. Next, x sequenes { A n : n ∈ ω } , { B n : n ∈ ω } , { C n : n ∈ ω } and { D n : n ∈ ω } witnessing U ∈ P seq and V ∈ P seq , resp etiv ely . Therefore A k ⊆ ( X \ B k ) ⊆ A k +1 ⊆ U and C k ⊆ ( X \ D k ) ⊆ C k +1 ⊆ V , 4 for ev ery k ∈ ω . Put N n = A n ∩ D n and M n = C n ∩ B n . Let M = [ { M n : n ∈ ω } and N = [ { N n : n ∈ ω } . Sets q [ M ] and q [ N ] fulll (2) in F rink's theorem. Indeed, if k ≤ n , then A k ∩ D k ∩ C n ∩ B n ⊆ A n ∩ B n = ∅ and A n ∩ D n ∩ C k ∩ B k ⊆ C n ∩ D n = ∅ . Consequen tly M k ∩ N n = ∅ , for an y k , n ∈ ω . Hene sets q [ M ] and q [ N ] are disjoin t. Also, it is q [ V ] ∪ q [ N ] = X/ P . Indeed, supp ose that x 6∈ V , then x ∈ U and there is k su h that x ∈ A k . Sine x 6∈ V , then x ∈ D k for all k ∈ ω . W e ha v e x ∈ A k ∩ D k = N k ⊆ N . Therefore [ x ] P ∈ q [ N ] . Similarly , one gets q [ U ] ∪ q [ M ] = X/ P . Th us B fullls (2) .  If P ⊆ P seq is nite, then X/ P is disrete, b eing a nite Hausdor spae. Whenev er P ⊆ P seq is oun table and losed under nite in ter- setions, then X/ P is a regular spae with a oun table base. Therefore, X/ P is metrizable and separable. 3. Skelet al f amilies and skelet al funtions A on tin uous surjetion is alled skeletal whenev er for an y non-empt y op en sets U ⊆ X the losure of f [ U ] has non-empt y in terior. If X is a ompat spae and Y Hausdor, then a on tin uous surjetion f : X → Y is sk eletal if, and only if In t f [ U ] 6 = ∅ , for ev ery non-empt y and op en U ⊆ X . One an nd equiv alen t notions almost-op en or semi-op en in the literature, see [ 1℄ and [8℄. F ollo wing J. Mio duszewski and L. Rudolf [11 ℄ w e all su h maps sk eletal, ompare [14 ℄ p. 413. In a fat, one an use the next prop osition as a denition for sk eletal funtions. Prop osition 6. L et f : X → Y b e a skeletal funtion. If an op en set V ⊆ Y is dense, then the pr eimage f − 1 ( V ) ⊆ X is dense, to o. Pr o of. Supp ose that a non-empt y op en set W ⊆ X is disjoin t with f − 1 ( V ) . Then the image cl f [ W ] has non-empt y in terior and cl f [ W ] ∩ V = ∅ , a on tradition.  There are top ologial spaes with no sk eletal map on to a dense in itself metrizable spae. F or example, the remainder of the e h-Stone ompatiation β N . Also, if I is a ompat segmen t of onneted Souslin line and X is metrizable, then ea h sk eletal map f : I → X is onstan t. Indeed, let Q b e a oun table and dense subset of f [ I ] ⊆ X . 5 Supp ose a sk eletal map f : I → X is non onstan t. Then the preim- age f − 1 ( Q ) is no where dense in I as oun table union of no where dense subset of a Souslin line. So, for ea h op en set V ⊆ I \ f − 1 ( Q ) there holds In t f [ V ] = ∅ , a on tradition. Regular Baire spae X with a at- egory measure µ , for a denition of this spae see [12 , pp. 86 - 91℄, giv es an another example of a spae with no sk eletal map on to a dense in itself, separable and metrizable spae. In [3℄ A. Bªaszzyk and S. Shelah are onsidered separable extremally disonneted spaes with no sk eletal map on to a dense in itself, separable and metrizable spae. They form ulated the result in terms of Bo olean algebra: Ther e is a nowher e dense ultr alter on ω if, and only if ther e is a  omplete, atom- less, σ - enter e d Bo ole an algebr a whih  ontains no r e gular, atomless,  ountable sub algebr a . A family P of op en subsets of a spae X is alled a skeletal family , whenev er for ev ery non-empt y op en set V ⊆ X there exists W ∈ P su h that U ⊆ W and ∅ 6 = U ∈ P implies U ∩ V 6 = ∅ . The follo wing prop osition explains onnetion b et w een sk eletal maps and sk eletal families. Prop osition 7. L et f : X → Y b e a  ontinuous funtion and let B b e a π -b ase for Y . The family { f − 1 ( V ) : V ∈ B } is skeletal if, and only if f is a skeletal map. Pr o of. Assume, that f is a sk eletal map. Fix a non-empt y op en set V ⊆ X . There exists W ∈ B su h that W 6 = ∅ and W ⊆ Int cl f [ V ] . Also, for an y U ∈ B su h that ∅ 6 = U ⊆ W there holds f − 1 ( U ) ∩ V 6 = ∅ . Indeed, if f − 1 ( U ) ∩ V = ∅ , then U ∩ cl f [ V ] = ∅ , a on tradition. Th us the family { f − 1 ( V ) : V ∈ B } is sk eletal. Assume, that funtion f : X → Y is not sk eletal. Then there exists a non-empt y op en set U ⊆ X su h that In t cl f [ U ] = ∅ . Sine B is a π -base for Y , then for ea h W ∈ B there exists V ∈ B su h that V ⊆ W and V ∩ f [ U ] = ∅ . The family { f − 1 ( V ) : V ∈ B } is not sk eletal.  It is w ell kno w - ompare a ommen t follo wing the denition of om- pat op en-generated spaes in [15℄ - that all limit pro jetions are op en in an y in v erse system with op en b onding maps. And on v ersely , if all limit pro jetions of an in v erse system are op en, then so are all b onding maps. Similar fat holds for sk eletal maps. Prop osition 8. If { X σ , π σ  , Σ } is a inverse system suh that al l b onding maps π σ  ar e skeletal and al l pr oje tions π σ ar e onto, then any pr oje tion π σ is skeletal. 6 Pr o of. Fix σ ∈ Σ . Consider a non-empt y basi set π − 1 ζ ( V ) for the limit lim ← − { X σ , π σ  , Σ } . T ak e τ ∈ Σ su h that ζ ≤ τ and σ ≤ τ . W e get π − 1 ζ ( V ) = π − 1 τ (( π τ ζ ) − 1 ( V )) . Hene π τ [ π − 1 ζ ( V ) ] = π τ [ π − 1 τ (( π τ ζ ) − 1 ( V ) )] = ( π τ ζ ) − 1 ( V ) , the set π τ [ π − 1 ζ ( V )] is op en and non-empt y . W e ha v e π σ [ π − 1 ζ ( V )] = π τ σ [ π τ [ π − 1 ζ ( V )]] , sine π τ σ ◦ π τ = π σ . The b onding map π τ σ is sk eletal, hene the losure of π σ [ π − 1 ζ ( V ) ] has non-empt y in terior.  4. The open-open game Pla y ers are pla ying at a top ologial spae X in the op en-op en game. Pla y er I  ho oses a non-empt y op en subset A 0 ⊆ X at the b eginning. Then Pla y er I I  ho oses a non-empt y op en subsets B 0 ⊆ A 0 . Pla y er I  ho oses a non-empt y op en subset A n ⊆ X at the n -th inning, and then Pla y er I I  ho oses a non-empt y op en subset B n ⊆ A n . Pla y er I wins, whenev er the union B 0 ∪ B 1 ∪ . . . ⊆ X is dense. One an assume that Pla y er I I wins for other ases. The spae X is alled I- favor able whenev er Pla y er I an b e insured that he wins no matter ho w Pla y er I I pla ys. In other w ords, Pla y er I has a winning strategy . A strategy for Pla y er I ould b e dened as a funtion σ : [ {T n : n ≥ 0 } → T , where T is a family of non-empt y and op en subsets of X . Pla y er I has a winning strategy , whenev er he kno ws ho w to dene A 0 = σ ( ∅ ) and sueeding A n +1 = σ ( B 0 , B 1 , . . . , B n ) su h that for ea h game ( σ ( ∅ ) , B 0 , σ ( B 0 ) , B 1 , σ ( B 0 , B 1 ) , B 2 , . . . , B n , σ ( B 0 , B 1 , . . . , B n ) , B n +1 , . . . ) the union B 0 ∪ B 1 ∪ B 2 ∪ . . . ⊆ X is dense. F or more details ab out the op en-op en game see P . Daniels, K. Kunen and H. Zhou [4℄. Consider a oun table sequene σ 0 , σ 1 , . . . of strategies for Pla y er I. F or a family Q ⊆ T let P ( Q ) b e the minimal family su h that Q ⊆ P ( Q ) ⊆ T , and if { B 0 , B 1 , . . . , B n } ⊆ P ( Q ) , then σ k ( B 0 , B 1 , . . . , B n ) ∈ P ( Q ) , and σ k ( ∅ ) ∈ P ( Q ) , for all σ k . W e sa y that P ( Q ) is the losur e of Q under str ate gies σ k . In partiular, if σ is a winning strategy and the losure of Q under σ equals Q , then Q is losed under a winning strategy . 7 Lemma 9. If P is lose d under a winning str ate gy for Player I, then for any op en set V 6 = ∅ ther e is W ∈ P suh that whenever U ∈ P and U ⊆ W then U ∩ V 6 = ∅ . Pr o of. Let σ b e a winning strategy for Pla y er I. Consider an op en set V 6 = ∅ . Supp ose that for an y W ∈ P there is U W ∈ P su h that U W ⊆ W and U W ∩ V = ∅ . Then Pla y er I I wins an y game whenev er he alw a ys  ho oses sets U W ∈ P , only . In partiular, the game σ ( ∅ ) , U σ ( ∅ ) , σ ( U σ ( ∅ ) ) , U σ ( U σ ( ∅ ) ) , σ ( U σ ( ∅ ) , U σ ( U σ ( ∅ ) ) ) , U σ ( U σ ( ∅ ) ,U σ ( U σ ( ∅ ) ) ) , . . . w ould b e winning for him, sine all sets  hosen b y Pla y er I I: U σ ( ∅ ) , U σ ( U σ ( ∅ ) ) , U σ ( U σ ( ∅ ) ,U σ ( U σ ( ∅ ) ) ) , . . . ; are disjoin t with V , a on tradition.  Theorem 10. If a ring P of op en subsets of X is lose d under a winning str ate gy and P ⊆ P seq , then X/ P is a  ompletely r e gular sp a e and the Q P -map q : X → X/ P is skeletal. Pr o of. T ak e a nonempt y op en subset V ⊆ X . Sine P is losed under a winning strategy , there exists W ∈ P su h that if U ∈ P and U ⊆ W , then U ∩ V 6 = ∅ , b y Lemma 9. This follo ws q [ U ] ∩ q [ V ] 6 = ∅ , for an y basi set q [ U ] su h that U ⊆ W and U ∈ P . Therefore q [ W ] ⊆ cl q [ V ] , sine { q [ U ] : U ∈ P } is a base for the Q P -top ology . The Q P -map q : X → X/ P is on tin uous b y Lemma 1. By Theorem 5, the spae X/ P is ompletely regular.  Fix a π -base Q for a spae X . F ollo wing [4℄, ompare [10℄, an y family C ⊂ [ Q ] ω is alled a lub lter whenev er: The family C is losed under ω - hains with resp et to inlusion, i.e. if P 1 ⊆ P 2 ⊆ . . . is an ω - hain whi h onsists of elemen ts of C , then P 1 ∪ P 2 ∪ . . . ∈ C ; F or an y oun table subfamily A ⊆ Q , where Q is the π − base xed ab o v e, there exists P ∈ C su h that A ⊆ P ; and ( S ) . F or any non-empty op en set V and e ah P ∈ C ther e is W ∈ P suh that if U ∈ P and U ⊆ W , then U me ets V , i.e. U ∩ V 6 = ∅ . In fat, the ondition ( S ) giv es reasons to lo ok in to I-fa v orable spaes with resp et to sk eletal families. An y P losed under a winning strategy for Pla y er I fullls ( S ) , b y Lemma 9. There holds, see [4℄ Theorem 1.6, ompare [10℄ Lemmas 3 and 4: A top olo gi al sp a e has a lub lter if, and only if it is I-favor able . In the next part w e mo dify a little the denition of lub lters. W e in tro due T -lubs, i.e. lub lters with some additional prop erties. 8 Supp ose a ompletely regular spae X is I-fa v orable. Let T b e the family of all ozero subsets of X . F or ea h W ∈ T x sequenes { U W n : n ∈ ω } and { V W n : n ∈ ω } witnessing W ∈ T seq . First, for ea h k  ho ose σ ∗ k ( ∅ ) ∈ T . Next, put σ ∗ 2 n ( W ) = U W n and σ ∗ 2 n +1 ( W ) = V W n , and σ ∗ k ( S ) = σ ∗ k ( ∅ ) for other S ∈ S {T n : n ≥ 0 } . Then, a family P ⊆ T is losed under strategies σ ∗ k , whenev er P ⊆ P seq . Also, P is losed under nite unions, whenev er it is losed under the strategy whi h assigns the union A 0 ∪ A 1 ∪ . . . ∪ A n to ea h sequene ( A 0 , A 1 , . . . , A n ) . And also, P is losed under nite in tersetions, whenev er it is losed under the strategy whi h assigns the in tersetion A 0 ∩ A 1 ∩ . . . ∩ A n to ea h ( A 0 , A 1 , . . . , A n ) . Consider a olletion C = {P ( Q ) : Q ∈ [ T ] ω } . Assume that ea h P ∈ C is oun table and losed under a winning strategy for Pla y er I and all strategies σ ∗ k , and losed under nite in tersetions and nite unions. Then, the family C is alled T - lub . By the denitions, an y T - lub C is losed under ω - hains with resp et to the inlusion. Ea h P ∈ C is a oun table ring of sets and P ⊆ P seq and it is losed under a winning strategy for Pla y er I. By Theorem 10, the Q P -map q : X → X/ P is sk eletal and on to a metrizable separable spae, for ev ery P ∈ C . Th us, w e are ready to build an in v erse system with sk eletal b onding maps on to metrizable separable spaes. An y T -lub C is direted b y the inlusion. F or ea h P ∈ C it is assigned the spae X/ P and the sk eletal funtion q P : X → X/ P . If P , R ∈ C and P ⊆ R , then put q R P ([ x ] R ) = [ x ] P . Th us, w e ha v e dened the in v erse system { X/ R , q R P , C } . Spaes X/ R are metrizable and separable, b onding maps q R P are sk eletal and the direted set C is σ -omplete. Theorem 11. L et X b e a I-favor able  ompletely r e gular sp a e. If C is a T -lub, then the limit Y = lim ← − { X/ R , q R P , C }  ontains a dense subsp a e whih is home omorphi to X . Pr o of. F or an y P ∈ C , put f ( x ) P = q P ( x ) . W e ha v e dened the fun- tion f : X → Y su h that f ( x ) = { f ( x ) P } . If R , P ∈ C and P ⊆ R , then q R P ( f ( x ) R ) = f ( x ) P . Th us f ( x ) is a thread, i.e. f ( x ) ∈ Y . The funtion f is on tin uous. Indeed, let π P b e the pro jetion of Y to X/ P . By [5℄ Prop osition 2.5.5, the family { π − 1 P ( q P [ U ]) : U ∈ P ∈ C } is a base for Y . Also, f − 1 ( π − 1 P ( q P [ U ])) = q − 1 P ( q P [ U ]) = U holds for an y U ∈ P ∈ C . 9 V erify that f is injetion. Let x, y ∈ X and x 6 = y . T ak e P ∈ C su h that x ∈ U and y ∈ V for some disjoin t sets U, V ∈ P . Sets q P [ U ] and q P [ V ] are disjoin t, hene π − 1 P ( q P [ U ]) and π − 1 P ( q P [ V ]) are disjoin t neigh b ourho o ds of f ( x ) and f ( y ) , resp etiv ely . There holds f [ U ] = f [ X ] ∩ π − 1 P ( q P [ U ]) , whenev er U ∈ P ∈ C . Indeed, f [ U ] ⊆ π − 1 P ( q P [ U ]) implies f [ U ] ⊆ f [ X ] ∩ π − 1 P ( q P [ U ]) . Supp ose, there exists y ∈ π − 1 P ( q P [ U ]) ∩ f [ X ] su h that y 6∈ f [ U ]) . T ak e x ∈ X su h that f ( x ) = y and x 6∈ U . W e get π P ( f ( x )) = q P ( x ) 6∈ q P [ U ] , but this follo ws f ( x ) 6∈ π − 1 P ( q P [ U ]) , a on tradition. Th us, f is op en, sine T = S C is a base for X . But f [ X ] ⊆ Y is dense, sine the family { π − 1 P ( q P [ U ]) : U ∈ P ∈ C } is a base for Y .  5. Reonstr ution of I-f a v orable sp a es No w, w e are ready to pro v e the announe analog of Sh hepin's op enly generated spaes. Theorem 12. If X is a I-favor able  omp at sp a e, then X = lim ← − { X σ , π σ  , Σ } , wher e { X σ , π σ  , Σ } is a σ - omplete inverse system, al l sp a es X σ ar e  omp at and metrizable, and al l b onding maps π σ  ar e skeletal and onto. Pr o of. Let C b e a T -lub. Put { X σ , π σ  , Σ } = { X/ R , q R P , C } . Ea h spae X σ = X/ R has oun table base, b y the denition of T -lub. Also, ea h Q R -map q R : X → X/ R is on tin uous, b y Lemma 1. Hene, an y spae X σ is ompat and metrizable, b y Lemma 4. Ea h Q R -map q R : X → X σ is sk eletal, b y Theorem 10 . Th us, all b onding maps π σ  are sk eletal, to o. The spae X is homeomorphi to a dense subspae of lim ← − { X σ , π σ  , Σ } , b y Theorem 11. W e get X = lim ← − { X σ , π σ  , Σ } , sine X is ompat. The in v erse system { X σ , π σ  , Σ } is σ -omplete. Indeed, supp ose that P 0 ⊆ P 1 ⊆ . . . and all P n ∈ C . Let P = S {P n : n ∈ ω } ∈ C . Put ( h ([ x ] P )) P n = q P P n ([ x ] P ) = [ x ] P n . Sine maps q P P n are on tin uous, w e ha v e dened a on tin uous funtion h : X/ P → lim ← − { X/ P n , q P n +1 P n } . Whenev er { [ x n ] P n } is a thread in the in v erse system { X/ P n , q P n +1 P n } , then there exists x ∈ T { [ x n ] P n : n ∈ ω } , sine sets [ x n ] P n onsists of a en tered family of nonempt y losed sets 10 in a ompat spae X . Th us h − 1 ( { [ x n ] P n } ) = [ x ] P ∈ X/ P , hene h is a bijetion.  T o obtain the on v erse of Theorem 12 one should onsider an in v erse system of ompat metrizable spaes with all b onding maps sk eletal. Su h assumptions are unneessary . So, w e assume that spaes X σ ha v e oun table π -bases, only . Theorem 13. L et { X σ , π σ  , Σ } b e a σ - omplete inverse system suh that al l b onding maps π σ  ar e skeletal and al l pr oje tions π σ ar e onto. If al l sp a es X σ have  ountable π -b ase, then the limit lim ← − { X σ , π σ  , Σ } is I-favor able. Pr o of. Let ≤ denotes the relation whi h direts Σ . Desrib e the fol- lo wing strategy for a mat h pla ying at the limit X = lim ← − { X σ , π σ  , Σ } . Assume that Pla y ers pla y with basi sets of the form π − 1 σ ( V ) , where V is non-empt y and op en in X σ and σ ∈ Σ . Pla y er I  ho oses an op en non-empt y set A 0 ⊆ X at the b eginning. Let B 0 = { B 0 } b e a resp ond of Pla y er I I. T ak e σ 0 ∈ Σ su h that B 0 = π − 1 σ 0 ( V 0 0 ) ⊆ A 0 . Fix a oun table π -base { V 0 0 , V 0 1 , . . . } for X σ 0 . Assume, that w e ha v e just settled indexes σ 0 ≤ σ 1 ≤ . . . ≤ σ n and π -bases { V k 0 , V k 1 , . . . } for X σ k , where 0 6 k 6 n . A dditionally assume, that for an y V k m there exists V k +1 j su h that π − 1 σ k +1 ( V k +1 j ) = π − 1 σ k ( V k m ) . No w, Pla y er I pla ys ea h set from A n +1 = { π − 1 σ k ( V k m ) : k 6 n and m 6 n } one after the other. Let B n +1 denote the family of all resp onds of Pla y er I I, for innings from A n +1 . Cho ose σ n +1 ≥ σ n and a oun table π -base { V n +1 0 , V n +1 1 , . . . } for X σ n +1 whi h on tains the family { ( π σ n +1 σ k ) − 1 ( V k m ) : k 6 n and m ∈ ω } and su h that for an y V ∈ B n +1 there exists V n +1 j su h that π − 1 σ n +1 ( V k +1 j ) = V . Let σ = sup { σ n : n ∈ ω } ∈ Σ . An y set π σ n [ S { S B n : n ∈ ω } ] is dense in X σ n , sine it in tersets an y π -basi set V n j ⊆ X σ n . The in v erse system is σ -omplete, hene the set π σ [ S { S B n : n ∈ ω } ] is dense in X σ . The pro jetion π σ is sk eletal b y Prop osition 8. So, the set S { S B n : n ∈ ω } is dense in X b y Prop osition 6.  A on tin uous and op en map is sk eletal, hene ev ery ompat op enly generated spae is I-fa v orable. 11 Corollary 14. A ny  omp at op enly gener ate d sp a e is I-favor able.  The on v erse is not true. F or instane, the e h-Stone ompati- ation β N of p ositiv e in tegers with the disrete top ology is I-fa v orable and extremally disonneted. But β N is not op enly generated, sine a ompat extremally disonneted and op enly generated spae has to b e disrete, see Theorem 11 in [ 13 ℄. A  kno wledgemen t The authors wish to thank to referees for their areful reading of a rst v ersion of this pap er and for ommen ts that ha v e b een v ery useful to impro v e the nal form of the pro ofs of some results. Referenes [1℄ A. Arhangelskii, On op en and almost-op en mappings of top olo gi al sp a es , (Russian) Dokl. Ak ad. Nauk SSSR 147 (1962), 999 - 1002. [2℄ B. Balar, T. Je h and J. Zapletal, Semi-Cohen Bo ole an algebr as , Ann. of Pure and Appl. Logi 87 (1997), no. 3, 187 - 208. [3℄ A. Bªaszzyk and S. Shelah, R e gular sub algebr as of  omplete Bo ole an algebr as , J. 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Rudolf H-lose d and extr emal ly dis onne te d Hausdor sp a es , Dissertationes Math. 66 (1969). [12℄ J. Oxtob y , Me asur e and Cate gory , Graduate T exts in Mathematis, v ol. 2, Springer, New Y ork, (1971). [13℄ E.V. Sh hepin, T op olo gy of limit sp a es with un ountable inverse sp e tr a , (Rus- sian) Usp ehi Mat. Nauk 31 (1976), no. 5 (191), 191 - 226. [14℄ E.V. Sh hepin, On κ -metrizable sp a es , (Russian) Izv. Ak ad. Nauk SSSR Ser. Mat. 43 1979), no. 2, 442 - 478. [15℄ E.V. Sh hepin, F untors and un ountable p owers of  omp ata , (Russian) Us- p ekhi Mat. Nauk 36 (1981), no. 3(219), 3 - 62. 12 Andrzej Kuharski, Institute of Ma thema tis, University of Silesia, ul. Bank o w a 14, 40-007 Ka to wie E-mail addr ess : akuharux2.mat h.u s. ed u.p l Szymon Plewik, Institute of Ma thema tis, University of Silesia, ul. Bank ow a 14, 40-007 Ka to wie E-mail addr ess : plewikux2.math .us .e du .pl 13