Skeletal maps and I-favorable spaces

SKELET AL MAPS AND I-F A V ORABLE SP A CES ANDRZEJ KUCHARSKI AND SZYMON PLE WIK Abstract. W e show that th e clas s of all compact Hausdorff and I -fav orable spaces is adeq uate for the class of skeletal maps. 1. Introduction In [6] E. V. Shc hepin introduced so called adequate pairs. Supp o se X is a class of compact Hausdorff space s and Φ is a class of contin uous maps. The pair ( X , Φ) fulfills the cond ition of closur e , if the in v erse limit of a con tin uous sequence { X α , p β α , Σ } b el ong to X and all pro- jections π β ∈ Φ , whenev er all spaces X α ∈ X of and all b ounding maps p β α ∈ Φ . Th e pair ( X , Φ) fulfills the condition of de c omp osability , whenev er ev ery non-metrizable space X ∈ X is the in ve rse limit of a con tinuous sequence { X α ; p β α ; α < β < w( X ) } , whe re w( X α ) < w ( X ) and X α ∈ X a nd p β α ∈ Φ . Finally , ( X , Φ) is an ade quate p air , whenev er it fulfills c losure and decomposability . The aim of this note is to sho w that the class of compact Hausdorff and I -f av orable spaces is adequate for the c lass of s k eletal maps. A directed set Σ is said to b e σ - c o mplete if an y coun table chain of its elemen ts has least upper b ound in Σ . An in v erse system { X σ , π σ  , Σ } is said to be a σ - c omplete , whenev er Σ is σ -complete and for ev ery c hain { σ n : n ∈ ω } ⊆ Σ , suc h th at σ = sup { σ n : n ∈ ω } ∈ Σ , there holds X σ = lim ← − { X σ n , π σ n +1 σ n } . W e consider in v erse systems where b ounding maps are surjections, only . Details ab out in vers e systems one can find in [4], pages 135 - 14 4. A con tin uous surjection is called skeletal , whene v er f o r a n y non- empt y op en sets U ⊆ X the closure of f [ U ] has non- empty in terior. If X is a compact s pace and Y Hausdorff, then a contin uous surjection 2000 M athematics Su bje ct C lassific ation. Primary: 54B1 0, 91A4 4 ; Secondary: 54C10, 91 A05. Key wor ds and phr ases. Adequate pair , In verse system; I-favorable space; Skele- tal map. 1 f : X → Y is sk eletal if , and only if Int f [ U ] 6 = ∅ , for ev ery non- empty and op en U ⊆ X . It is w ell kno w - compare a commen t follow ing the definition of com- pact op e n-generated spaces in [7] - that each in v erse system with op en b ounding maps has op en limit pro jections . And con v ersely , if a ll limit pro jections of an in v erse system are op e n, then so are all b o unding maps. The following fact is stated in [2, Lemma 3]. Its pro of i s giv en in [5, Prop osition 8]: If { X σ , π σ  , Σ } is an inverse system such that al l b ounding maps π σ  ar e skele tal and al l pr oje ctions π σ ar e onto, then any pr oje ction π σ is skeletal . 2. On I -f a vorable sp a ces Let X b e a top ologigal space equipp ed with a to p ology T . The space X is called I- favor able , whenev er there exists a function σ : [ {T n : n ≥ 0 } → T suc h that for eac h seque nce B 0 , B 1 , . . . c onsisting of non-empt y eleme n ts of T with B 0 ⊆ σ ( ∅ ) and B n +1 ⊆ σ (( B 0 , B 1 , . . . , B n )) , for eac h n ∈ ω , the union B 0 ∪ B 1 ∪ B 2 ∪ . . . is dense in X . The function σ is called a winning str ate g y . In fact, one can take a π -base (or a base) instead of a top ology in the definition o f a winning strategy . The definition of I - fa v ora ble spaces w as in tro duced b y P . Daniels, K. Kunen and H. Z hou [3]. The next the lemma one can conclude from [1, Theorem 4.1]. Lemma 1. A skele tal im age of I -favor able sp ac e is a I -favor able sp ac e. Pr o of. Let f : X → Y b e a sk eletal map. Supp ose a f unction σ X : S {T n X : n ≥ 0 } → T X witnesse s that X is I - fa v ora ble. Put σ Y ( ∅ ) = Int cl f [ σ X ( ∅ )] . If V 0 ⊆ σ Y ( ∅ ) , then put B 0 = f − 1 ( V 0 ) ∩ σ X ( ∅ ) . Supp o se that non- empt y sets V 0 ⊆ σ Y ( ∅ ) a nd V n ⊆ σ Y (( V 0 , V 1 , . . . , V n − 1 )) are c ho ose n and sets B 0 , B 1 , . . . , B n − 1 are defined, a lso. Put B n = f − 1 ( V n ) ∩ σ X ( B 0 , B 1 , . . . , B n − 1 ) and σ Y (( V 0 , V 1 , . . . , V n )) = Int cl f [ σ X ( B 0 , B 1 , . . . , B n )] . The function σ Y witnesse s that Y is I -fav orable, si nce V 0 ∪ V 1 ∪ . . . con ta ins a dense set f [ B 0 ∪ B 1 ∪ . . . ] .  2 3. Decomposability W e shall pro v e that the class of compact Hausdorff and I -fa vorable spaces and the class of sk eletal maps fulfill decomp osabilit y . Recall notions and facts whic h are stated in [5]. If P is family o f subsets of X , then y ∈ [ x ] P denotes that, for e ac h V ∈ P , x ∈ V if, and only if y ∈ V ; and X / P is the family of a ll classes [ x ] P ; and X / P is equipped with the coarser top ology con taining all sets { [ x ] P : x ∈ V } , where V ∈ P . S upp ose X is a top o lo gical space and P consists of op en subsets o f X . If P is closed under finite in tersection, then the map q ( x ) = [ x ] P is contin uous, see [5, Lemma 1]. Let P seq b e the family of all sets W whic h satisfy the followi ng condition: There exist sequenc es of sets { U n : n ∈ ω } ⊆ P and { V n : n ∈ ω } ⊆ P suc h that U k ⊆ ( X \ V k ) ⊆ U k +1 , for any k ∈ ω , a nd S { U n : n ∈ ω } = W . If a ring P of op en subsets of X is closed under a winning strategy and P ⊆ P seq , then X/ P is a c ompletely regular sp ace and the map q : X → X/ P is sk eletal, see [5, Theorem 10]. In fact, we shall impro v e and generalize the follo wing: If X is a I - f a v ora ble compact space, t hen X = lim ← − { X σ , π σ  , Σ } , where { X σ , π σ  , Σ } is a σ - complete in v erse system, all spaces X σ are compact and metrizable, and all b o nding maps π σ  are sk eletal a nd onto, see [5, Theorem 12 ]. Let { X α : α ∈ Σ } b e a family of Hausdorff top ological spaces, where (Σ , < ) is an up ward directed set. Supp ose that there are giv en con tin u- ous functions p β α : X β → X α suc h that p γ α = p γ β ◦ p β α whenev er α < β < γ . Th us S = { X α ; p β α ; Σ } is the inv erse system with con tinuous b ounding maps. Lemma 2. L et X b e a Hausdorff top olo gic al sp ac e and S = { X α ; p β α ; Σ } an inverse system with c ontinuous b ound i n g m aps. I f ther e exist maps π β : X → X β such that e ach π β = p α β ◦ π α is onto X β and for e ach two differ ent p oin ts x , y ∈ X ther e ex i s ts α ∈ Σ with π α ( x ) 6 = π α ( y ) , then ther e exists a o ne-to-one c ontinuous map f : X → lim ← − S onto a d e nse subsp ac e of lim ← − S . Pr o of. F or an y x ∈ X , put f ( x ) = { π α ( x ) } . The function f is a required one, compare [4] or [5, Theorem 11].  A dditionally , o ne concludes that X has to be homeomorphic with lim ← − S , assuming that X is compact in the ab ov e lemma. No w, w e apply Lemma 2 to an in v erse seq uence with a directed se t Σ whic h consists of infinite o r dinal nu m b ers less than the w eigh t w( X ) of a space X . 3 Theorem 3. Any a c om p act non-me trizab le and I -favor able sp ac e X is hom e omorphic with the inverse limit of a c ontinuous se quenc e { X α ; p β α ; ω ≤ α < β < w( X ) } , wher e e ach X α is a c omp act Hausdorff and I -fav o r able sp ac e, with w ( X α ) < w ( X ) , and such that any b ounding map p β α is sk e l e tal. Pr o of. F or eac h cozero set W ⊆ X fix a contin uous function f W : X → [0 , 1] suc h that W = f − 1 W ((0 , 1]) . Put σ 2 n ( W ) = f − 1 W (( 1 n , 1]) and σ 2 n +1 ( W ) = f − 1 W ([0 , 1 n )) . Assume that σ = σ 0 . Fix a base { V α : α < w( X ) } consisting of cozero sets of X . If ω ≤ β < w ( X ) , t hen P β ⊇ { V α : α < β } is the least family consisting o f cozero sets and closed under finite unions and under finite inte rsections and closed under all functions σ n . Thus |P β | = | β | and P γ ⊆ P β , whenev er ω ≤ γ ≤ β . Also, w e get P β = S {P γ : γ < β } for a limit ordinal β . Put X β = X / P β and q β ( x ) = [ x ] P β , thus maps q P β P γ : X β → X γ are sk eletal. By Lemma 2, the in v erse limit lim ← − { X β ; q P β P γ ; ω ≤ γ < β < w( X ) } is homeomorphic to X a nd eac h inv erse limit lim ← − { X β ; q P β P γ ; ω ≤ γ < β < α } is homeomorphic to X α . All spaces X α are sk eletal images of X , hence they are I -fa v or a ble by Lem ma 1. W e get w( X β ) < w ( X ) , since the family {{ [ x ] P β : x ∈ V } : V ∈ P β } is a base for X β , compare [5 , Lemma 1].  4. Closur e W e shall pro v e that the class of compact Hausdorff and I -fa vorable spaces and the class of sk eletal maps fulfill closure. In fact, w e shall impro ve [5 , Theorem 13], not assuming that spaces X σ ha v e countable π -bases. Theorem 4. If S = { X α , π α  , Σ } is a σ -c omplete inverse s ystem which c onsists of c omp act Hausdorff and I -fa vor able sp ac es and skeletal b ounding maps π α  , then the inverse li m it lim ← − S is a c omp act Hausdorff and I -favor able sp ac e. Pr o of. Let B b e a base for lim ← − S whic h consists of all sets π − 1 τ ( V ) , where τ ∈ Σ and eac h V is an op en subset of X τ . F or eac h α ∈ Σ , let σ α b e a winning strategy in the op en-o p en game pla y ed on X α . Fix τ 0 ∈ Σ and 4 infinite and pairwise disjoin t sets A n suc h that S { A n : n ∈ ω } = ω and n ∈ A k implies k ≤ n . Put σ ω ( ∅ ) = π − 1 τ 0 ( σ τ 0 ( ∅ )) ⊆ lim ← − S . Supp ose that n ∈ A k and all σ ω (( B 0 , B 1 , . . . , B n − 1 )) has been already defined suc h that π − 1 τ m ( V m ) = B m ⊆ σ ω (( B 0 , B 1 , . . . , B m − 1 )) for 0 ≤ m ≤ n . Th us indexes τ 0 < τ 1 < . . . < τ n are fixed. If n is t he least elemen t of A k , then τ k ≤ τ n . Put σ ω (( B 0 , B 1 , . . . , B n )) = π − 1 τ k ( σ τ k ( ∅ )) . If { i 0 , i 1 , . . . , i j } = A k ∩ { 0 , 1 , . . . , n } and τ i 0 < τ i 1 < . . . < τ i j ≤ τ n , then let D i 0 = Int π τ k ( B i 0 ) , D i 1 = In t π τ k ( B i 1 ) , . . . , D i j = Int π τ k ( B i j ) ⊆ X τ k and σ ω (( B 0 , B 1 , . . . , B n )) = π − 1 τ k ( σ τ k (( D i 0 , D i 1 , . . . , D i j ))) ⊆ lim ← − S . F or other cases, put σ ω (( B 0 , B 1 , . . . , B n )) ∈ B arbitrarily . The strategy σ ω : S {B n : n ≥ 0 } → B is just defined. V erify that σ ω is a winning strategy . Let τ 0 < τ 1 < . . . and B 0 , B 1 , . . . b e sequences suc h that π − 1 τ 0 ( V 0 ) = B 0 ⊆ σ ω ( ∅ ) and π − 1 τ n +1 ( V n +1 ) = B n +1 ⊆ σ ω (( B 0 , B 1 , . . . , B n )) , where all τ k ∈ Σ and each B k ∈ B . If τ ∈ Σ is the least upp er b ound of { τ k : k ∈ ω } , then alwa ys π − 1 τ ( π τ ( B k )) = π − 1 τ ( π τ ( π − 1 τ k ( V k ))) = π − 1 τ (( π τ τ k ) − 1 ( V k ))) = B k . T ak e an arbitrary base set ( π τ τ k ) − 1 ( W ) ⊆ X τ , where W is an op en subset in X τ k . Suc h sets consis t of a bas e for X τ , since the in vers e system is σ -complete and [4, 2.5.5. Prop osition]. If σ τ k is a w inning strategy on X τ k , then there exists j ∈ A k suc h that W ∩ Int π τ k ( B j ) 6 = ∅ . Hence ( π τ τ k ) − 1 ( W ) ∩ π τ ( B j ) 6 = ∅ , Indeed, supp ose that ( π τ τ k ) − 1 ( W ) ∩ π τ ( B j ) = ∅ . Then ∅ = π τ τ k [( π τ τ k ) − 1 ( W ) ∩ π τ ( B j )] = W ∩ π τ τ k [ π τ ( B j )] ⊇ W ∩ In t π τ k ( B j ) , a con tradiction. Thus , the union π τ ( B 0 ) ∪ π τ ( B 1 ) ∪ π τ ( B 2 ) ∪ . . . is dense in X τ . But π τ is a sk eletal map, hence π − 1 τ ( π τ ( B 0 ) ∪ π τ ( B 1 ) ∪ π τ ( B 2 ) ∪ . . . ) = B 0 ∪ B 1 ∪ B 2 ∪ . . . has to b e dense in lim ← − S .  5 The class of compact Hausdorff and I - fa v or a ble spaces and the class of sk eletal maps fulfills closure, since ev ery con tin uous inv erse sequence is a σ - complete in v erse system, to o. An equiv alen t v ersion of Theorem 4 for con tin uous in vers e sequence, (using Bo olean algebras notions) one can find in [1, p. 197]. References [1] B. Balca r, T. Jec h a nd J. Zapletal, Semi-Co hen Bo ole an algebr as , Ann. o f Pure and Appl. Lo gic 87 (199 7), no. 3, 187 - 20 8. [2] A. Błaszc zyk, Souslin numb er and inverse limits , T op ology and mea sure I I I, Pro c. Conf., Vitte/Hiddensee 19 8 0, Part 1, 21 – 2 6 (1 9 82). [3] P . Daniels, K. Kunen a nd H. Zhou, On the op en-op en game , F und. Math. 145 (1994), no. 3 , 20 5 - 22 0. [4] R. Engelking, Gener al top olo gy , Polish Scientific Publishers, W ar szaw a (1977) [5] A. Kucharski and Sz. Plewik, Inverse systems and I -favor able s p ac es , T o p ology Appl. 1 56 (2008), no. 1, 110 – 116. [6] E.V. Shchepin, T op olo gy of limit sp ac es with unc ount able inverse sp e ct r a , (Rus- sian) Uspe hi Mat. Nauk 31 (1976), no. 5 (19 1), 19 1 - 226. [7] E.V. Shchepin, F unctors and unc oun table p owers of c omp acta , (Russian) Us- pek hi Mat. Nauk 36 (198 1), no. 3(219), 3 - 62 . Andrzej Kucharsk i, Institute of Ma thema tics, University of Silesia, ul. Bankow a 14, 40-007 Ka tow ice E-mail addr ess : akuc har@u x2.ma th.us.edu.pl Szymon Plewik, Institute of Ma thema tics, University of Silesia, ul. Bankow a 14, 40-007 Ka towice E-mail addr ess : plew ik@ma th.us .edu.pl 6