Covering dimension and finite-to-one maps
Hurewicz' characterized the dimension of separable metrizable spaces by means of finite-to-one maps. We investigate whether this characterization also holds in the class of compact F-spaces of weight c. Our main result is that, assuming the Continuum…
Authors: Klaas Pieter Hart, Jan van Mill
CO VERING DIMENSION AND FINITE-TO-ONE MAPS KLAAS PIETER HAR T AND JAN V AN MILL T o Ken Kunen on t he o c ca sion of his retir e ment fr om te aching Abstract. Hurewicz’ c haracterized the dimension of separable m etrizable spaces b y means of finite-to-one maps. W e inv estigate whether this char- acterization also h olds in the c lass of compac t F -spaces of w eight c . Our main r esult is that, assuming the Contin uum Hyp othesis, an n -dimensional compact F -space of weigh t c is the cont inuo us image of a zero-dimensional compact Hausdorff space by an at most 2 n -to-1 map. Introduction The starting p oint for this note is a theore m of Hurewicz from [5], which char- acterizes the dimensio n of separ able metriza ble spaces in terms of maps. Theorem. L et X b e a sep ar able and metrizable sp ac e and let n b e a natur al nu mb er. Then dim X 6 n iff ther e ar e a zer o-dimensional sep ar able metrizable sp ac e Y and a c ontinuous and close d surje ction f : Y → X such that f − 1 ( x ) 6 n + 1 for al l x . Our aim is to generaliz e this result to a wider class of spa ces. One half of Hurewicz’s theorem is a s pec ia l case of the theorem on dimens ion raising ma ps. This sp ecial case c an b e gener alized to the class of no rmal space s (the hin t to Pr oblem 1.7.F (c) in [3] provides a pr o of ): Theorem. L et f : Y → X b e a close d c ontinuous surje ct ion b etwe en normal sp ac es and n a natu r al numb er. If Y is str ongly zer o-dimensional and if f is such that f − 1 ( x ) 6 n + 1 for al l x then Ind X 6 n . Hurewicz’ pro o f of the other half was based o n the interpla y b etw een the large inductive dimensio n and the cov ering dimensio n, using finite co llections of closed sets of order n + 1 to constr uc t the pr eimage. Also Kura towski’s q uantit ative pr o of in [7] used covering dimensio n to show that in the case where X has no iso lated po int s the set of surjections with at mo st n + 1 - po int fib ers is residual in the space of a ll sur jections from the Cantor set onto X . This all suggests that we should lo ok for cla sses o f c o mpact Hausdo rff spaces where the covering dimension and the larg e inductiv e dimension c o incide. W e shall show that the co mpact F - s paces of weight c for m such a class, as suming the Contin uum Hyp othesis. Under that assumption these spaces are one step up from compact metrizable spaces: the weight is ℵ 1 and the F -space prop erty lets one do countably many t hings at a time, which is quite helpful in inductions a nd r ecursions of leng th ω 1 . In Section 1 w e give a v ar ia tion of Hurewicz’ pro of of the second half his theo rem; we do this with an eye to the pr o of of our main result a nd to make Hurewicz’ argument b etter known. Next w e establish, in Section 2, that the three ma jor Date : W ednesda y 30-09-2009 at 23:48:57 (cest). 2000 Mathematics Subj e ct Classific ation. Prim ary: 54F45. Secondary: 54C10, 54G05. Key wor ds and phr ases. cov eri ng dimension, inductive dimension, finite-to-one m aps, F- space. 1 2 KLAAS PIETER HAR T AND JAN V AN MILL dimension functions co incide o n the c lass o f compa c t F -spa ces of weigh t c . In Sections 3 and 4 we s how that every n -dimensional compa ct F -spa ce is an at most 2 n -to-1 co nt inuous image of a zer o-dimensional compact space. This leav es o p en an o bvious ques tion: Question ( CH ) . Is e very n -dimensional co mpact F -space o f weigh t c a c o ntin uous image of some zero- dimensional co mpact spa ce by a map whose fib ers hav e s iz e at most n + 1? Some prel iminaries. W e follow Engelking’s bo o k, [3], reg arding dimension theor y except tha t by the or der of a family T at a po int x we simply mean the cardina lit y of { T ∈ T : x ∈ T } . T hus a (nor mal) space has cov ering dimension at most n iff every finite op en cover has an op e n refinement of or der at most n + 1 . An F -sp ac e is o ne in which s e parated F σ -sets hav e disjoint closures. Whenever X is a σ -compact, lo cally compa ct spa ce that is not compac t then the remainder β X \ X in its ˇ Cech-Stone compactification is an F -space. Gillman a nd Jer ison’s bo ok, [4], is still a go o d source of basic information on F -spaces. A s e t is r e gular op en if it is the interior of its clos ure. The family R O( X ) of regular op en sets in the space X forms a Bo o le an a lgebra under the following op erations: U ∨ V = in t cl( U ∪ V ), U ∧ V = U ∩ V and U ′ = X \ cl U . In the pro of of o ur main r esult we obtain our zer o-dimensional preima ge as the Stone space of a suba lgebra of RO( X ); we refer to K o ppe lber g’s b o ok, [6], for a co mprehensive treatment o f this c onstruction. 1. Making a z er o-dimensional preimag e As anno unced we preent in this section a v aria tion of Hurewicz’ co nstruction of a zer o-dimensional preimage of an n -dimensiona l compact metrizable space by a map who se fib ers have car dinality at most n + 1. The key notion is that of a t iling of a space, which w e define to b e a finite pair wise disjoint family of regular o p e n sets whose c lo sures form a cover of the space. Given a tiling T and a p oint x we put T x = { T ∈ T : x ∈ cl T } . In the pro o f we will co nstruct ever finer tilings of the spac e ; the fo llowing lemma will help us keep the cardinalities of the fa milies T x under c o nt rol. Lemma 1.1. L et { B i : i < k } b e a family of r e gular op en sets in a sp ac e X and let T b e a r e gu lar op en s et t hat is c over e d by this family. F or e ach i put C i = T ∩ B i \ S j 0 ther e is a k such that diam T < ε for a ll T ∈ T k . If T = h T k : k < ω i and S = h S k : k < ω i in Y are such that T k = S k then f ( T ) , f ( S ) ∈ cl T k and hence d f ( T ) , f ( S ) < ε . The map f is o nto: if x ∈ X then it is easy to find T ∈ Y s uch that x ∈ T k for all k ; then x = f ( T ). The ma p f is at most n + 1- to-one. Indeed, let x ∈ X and for ea ch k let T k,x = { T ∈ T k : x ∈ cl T } . Then |T k,x | 6 |T k +1 ,x | 6 n + 1 for all k ; this means that there is a k 0 such that |T k,x | = |T k 0 ,x | for k > k 0 . And this implies that T 7→ T k 0 is a bijective map fro m f − 1 ( x ) to T k 0 ,x and thus: f − 1 ( x ) 6 n + 1 . It remains to constr uc t the T k . W e set T 0 = { X } . W e assume we hav e found T k as a subset of the Bo olean algebra generated by { B i : i < l } fo r some l and that the assumptions of Lemma 1.2 are satisfied: for every x ∈ X there are at least |T k,x | − 1 indices i < l such tha t x ∈ F r B i . Because for e a ch iso lated p oint x the set { x } o ccurs as B i infinitely o ften we know that for every m the family { B i : i > m } is a base for X . W e can therefor e find pairwise disjo int finite subsets F T of ( l , ω ) for T ∈ T k such that cl T ⊆ S i ∈ F T B i and diam B i < 2 − k for all i . W e can use these, as in Lemma 1.2 to define tilings o f each T ∈ T k : for i ∈ F T put C T ,i = T ∩ B i \ S j ∈ F T ,j |T k,x | − 1; second, by ass umption on our base we have n > { i : x ∈ F r B i } . This implies |T k,x | − 1 6 n . As mentioned in the introductio n, in [7 ] K uratowski gav e a qua nt itative version of this half of Hurewicz’ theo rem: if X is compact, metr izable, n -dimensio nal and 4 KLAAS PIETER HAR T AND JAN V AN MILL without iso lated p oints then the set of maps all of whose fibers ha ve size n + 1 or les s is residual in the s pace of all sur jectio ns fro m the Cantor set to X . The cov er ing dimension is invok ed to s how that, given a natural num b er k , the set o f maps with a fib er of size at least n + 2 in which the p oints ar e at lea s t distance 2 − k apart is nowhere dens e (it is also readily seen to b e close d). 2. Equal ity of dimensions It is well k nown that the three fundamental dimension functions take on the same v alues for all se pa rable metrizable spaces. W e prov e that this also holds in the class of compact F -spa ces of weigh t c , provided the Contin uum Hyp othesis holds. In the pr o of we use Hemmingsen’s characterizatio n of covering dimensio n ([3 , Theorem 1.6 .9]): dim X 6 n iff every n +2 -element op en cov er has an op en s hrinking with empt y intersection. Theorem 2.1 ( CH ) . L et X b e a c omp act F -sp ac e of weight c . Th en dim X = ind X = Ind X . Pr o of. The inequa lities dim X 6 ind X 6 Ind X ar e well known. W e show Ind X 6 dim X b y showing that dim X 6 n implies that betw een any tw o disjoint closed sets F and G one ca n find a partition L that satisfies dim L 6 n − 1 . This is known to b e true in case n = 0, so we assume n > 1 fr om now o n. Fix a base B for X that consis ts of cozero s e ts, has ca rdinality ℵ 1 (b y the CH ) and is clo sed under countable unions and finite in tersections. Let hB α : α < ω 1 i enumerate the family of all n + 1-element subfamilies o f B with cofinal r ep e titions. W e wr ite B α = { B α,i : i 6 n } . W e construct, by recurs ion, tw o sequences h U α : α < ω 1 i and h V α : α < ω 1 i in B such tha t (1) F ⊆ U 0 and G ⊆ V 0 ; (2) cl U α ⊆ U β and cl V α ⊆ V β whenever α < β ; (3) cl U α ∩ cl V α = ∅ for all α ; (4) if U α ∪ V α ∪ S B α = X then there is a s ubfamily B ′ α = { B ′ α,i : i 6 n } of B that refines B α and is such that U α +1 ∪ V α +1 ∪ S B ′ α = X and T B ′ α ⊆ U α +1 ∪ V α +1 . Then L = X \ S α ( U α ∪ V α ) is a par tition b etw e e n F and G and dim L 6 n − 1. That L is a partition betw een F and G follows from (1), (2 ) and (3). T o see that dim L 6 n − 1 let O b e a n n + 1-element basic o pe n cov er of L . There is an α such that O = { L ∩ B α,i : i 6 n } and s uch that X \ ( U α ∪ V α ) ⊆ S B α . By constr uctio n O ′ = { L ∩ B ′ α,i : i 6 n } is a refinement of O such tha t T O ′ = ∅ . It remains to p erfor m the c o nstruction. W e obta in U 0 and V 0 using compactness and the fact that B is closed under finite unions. If α is a limit w e let U α = S β <α U β and V α = S β <α V β . Then U α , V α ∈ B by the assumption that B is closed under countable unions and cl U α ∩ cl V α = ∅ becaus e X is an F -space. T o deal with the succ e ssor case we first take elements C and D of B such that cl U α ⊆ C , cl V α ⊆ D and cl C ∩ c l D = ∅ . If U α ∪ V α ∪ S B α = X we put E i = B α,i \ (cl U α ∪ cl V α ) for i 6 n . Then we apply the inequa lity dim X 6 n to find a shr inking { B ′ α,i : i 6 n } ∪ { O } of { E i : i 6 n } ∪ { C ∪ D } such that cl O ∩ T i cl B ′ α,i = ∅ . Let O 1 = O ∩ C and O 2 = O ∩ D ; als o let K = T i cl B ′ α,i . Note that cl U α ⊆ O 1 and cl V α ⊆ O 2 and that the family { K , cl O 1 , cl O 2 } is pairwise disjoint. W e choose U α +1 and V α +1 in B with disjoint closur es such that K ∪ cl O 1 ⊆ U α +1 and cl O 1 ⊆ U α +1 . Then the conclusion of (4) is satisfied. If U α ∪ V α ∪ S B α 6 = X we let U α +1 = C and V α +1 = D . DIMENSION AND MAPS 5 R emark 2 .2 . This pr o of is similar to the one given in [3 ] of the analog ous result for compact metrizable spaces . That pro of used a metric to g uide the countably many steps tow ar d a partition of cov er ing dimension at most n − 1. Of cours e in an infinite compac t F - space there is no metric av ailable; in our pro of the role of the metric is taken ov er (in the background) b y the unique unifor mit y that generates the topo logy o f the compact F -spac e. R emark 2.3 . The seco nd-named author has cons tr ucted an example o f a compa ct F -space of w eight c + with non-coinciding dimensions, [8]. 3. Special bases In Section 1 we used the fact that a metrizable co mpact space X with dim X 6 n has a base { B i : i < ω } with the prop erty that T i ∈ F F r B i = ∅ whenever | F | = n + 1. This is a sp e cial case of a str o nger s tructural statement: every metriza ble compact space has a base { B i : i < ω } with the prop erty that dim T i ∈ F F r B i 6 dim F r B i 0 − | F | + 1, where i 0 = min F . Our goal is to prov e a similar sta temen t for compact F -spaces o f weigh t c , as- suming the Contin uum Hypothesis . In genera l, if X is a compact space of weight ℵ 1 we shall a ssume it is embedded in the Tyc honoff cub e [0 , 1] ω 1 and for α < ω 1 we write X α = { x ↾ α : x ∈ X } ; this is the pro jection of X o nto the fir st α co ordinates . W e denote this pro jection map by p α , w e r eserve π α for the pr o jection onto the α th co ordinate. Lemma 3 .1. Ther e is a close d and unb ounde d subset C of ω 1 such that dim X α = dim X for α ∈ C . Pr o of. The cub e [0 , 1] ω 1 has a nice s ubbase, which consists of the str ips π − 1 α [0 , q ) and π − 1 α ( q , 1] , where α < ω 1 and q ∈ Q ∩ (0 , 1). W e close this subbase under finite unions a nd in tersections to obtain a base B for the cub e. First we as sume dim X = n < ∞ . In this case if B ′ is a finite subfamily of B that cov er s X then ther e is another finite subfamily B ′′ of B that also covers X , refines B ′ and is such that { B ∈ B ′′ : x ∈ B } 6 n + 1 for all x ∈ X . Observe that each finite subfamily C of B is supp orted by a finite subse t F C of ω 1 (the co o rdinates of the strips used to make its elements). Next note that, given α < ω 1 , there are only co un tably many finite subfamilies of B whose supp or t lies below α . Thus we obta in a function f : ω 1 → ω 1 , defined b y f ( α ) is the first countable ordina l β such that whenever B ′ is a finite subfamily of B tha t covers X and whose supp ort lies b elow α then it has a refinement of orde r at most n + 1 whose supp ort lies below β . The set C = { δ < ω 1 : ( ∀ α )( α < δ → f ( α ) < δ ) } is c losed and unbounded and it sho uld b e clea r that dim X δ 6 n whenever δ ∈ C . T o get equality w e note that there is a lso a finite cov er C o f X by mem b ers of B for which every op en r efinement has order n + 1. Upon deleting an initial segment of C w e ca n as s ume that C is suppo rted below min C ; then C witnesses that dim X δ > n for all δ ∈ C . In case X is infinite- dimens io nal we hav e for each n a finite cov er C n such that every op en refinement has order at least n . F or any α ab ov e the supp orts of these cov er s we hav e dim X α = ∞ . The following pro po sition is instrumental in the construction of the type of ba se alluded to a bove. It a ls o provides ano ther pro of of Theo rem 2.1. In it w e use the notion of a P -set: a subset of a space is a P -set if whenever it is dis joint from an F σ - subset it is also disjoint the clo s ure of thet F σ -set; in ter ms of neigh b ouirho o ds: the int ersection of countably many neighbourho o ds of the set is ag a in a neighbour ho o d 6 KLAAS PIETER HAR T AND JAN V AN MILL of the set. The member s of o ur bas e will have nowhere dense clo sed P -sets for bo undaries. Prop ositio n 3. 2 ( CH ) . L et X b e a c omp act F -sp ac e of weight c . L et F and G b e disjoint close d subsets of X and let Q b e a family of n o mor e than ℵ 1 many nowher e dense close d P -sets in X . Ther e ar e disjoi nt r e gu lar op en sets U and V such that (1) F ⊆ U and G ⊆ V , (2) P = X \ ( U ∪ V ) is a nowher e dense P -s et , (3) if dim X < ∞ t hen dim P 6 dim X − 1 , (4) if Q ∈ Q then P ∩ Q is nowher e dense in Q and if dim Q < ∞ then dim( P ∩ Q ) 6 dim Q − 1 . Pr o of. W e may as well assume that dim X = n < ∞ and that Q has cardinality ℵ 1 . The pro of is easily mo dified in case either o f these is not the case. Cho ose a clo sed and un bo unded set C such that dim X δ = dim X for δ ∈ C and assume without loss o f gener ality (and by compactness) that p δ [ A ] ∩ p δ [ B ] = ∅ , where δ = min C . Enumerate Q a s { Q α : α < ω 1 } and ch o ose for ea ch α a closed and unbounded subset C α of C such that dim p δ [ Q α ] = dim Q α whenever δ ∈ C α . Because the int ersection of countably many closed and unbounded s ets is again closed and un- bo unded w e may a s well as sume that C β ⊆ C α whenever α < β . In case δ ∈ C α we can choose a zero- dimens io nal F σ -set Z α,δ in X δ such that Z α,δ is dens e in X δ , the intersection Z α,δ ∩ p δ [ Q α ] is dense in p δ [ Q α ], a nd such that (3.1) dim( X δ \ Z α,δ ) 6 n − 1 and dim( p δ [ Q α ] \ Z α,δ ) 6 dim Q α − 1 W e s ta rt our recurs ive constructio n of P . Along the wa y we construct a sequence h δ α : α < ω 1 i of or dina ls. Let δ 0 = min C 0 and cho ose a partition L in X δ 0 betw een p δ 0 [ F ] and p δ 0 [ G ] that is disjo in t from Z 0 ,δ 0 . Thus we obtain automatically that • dim L 6 n − 1, • dim L ∩ p δ 0 [ Q 0 ] 6 dim Q 0 − 1, and • L is nowhere dense in X δ 0 and L ∩ p δ 0 [ Q 0 ] is nowhere dense in p δ 0 [ Q 0 ]. W rite X δ 0 \ L = U ∪ V , where U and V are op en and disjoint sets ar ound p δ 0 [ F ] and p δ 0 [ G ] resp ectively . Le t V 0 = X δ \ cl U and U 0 = X δ \ cl V 0 ; then • U 0 and V 0 are regular op en, • P 0 = X δ \ ( U 0 ∪ V 0 ) is a subse t of L and a pa rtition betw een p δ 0 [ F ] and p δ 0 [ G ] with dim P 0 6 dim L 6 n − 1 , and • dim P 0 ∩ p δ 0 [ Q 0 ] 6 dim L ∩ p δ 0 [ Q 0 ] 6 dim p δ 0 [ Q 0 ] − 1 = dim Q 0 − 1. Observe that cl U 0 = U 0 ∪ P 0 and cl V 0 = V 0 ∪ P 0 . T o find δ 1 observe first that p − 1 δ 0 [ U 0 ] and p − 1 δ 0 [ V 0 ] are disjoint op en F σ -sets of X and hence have disjoint clo sures a s X is a n F -spa ce. As with F and G we can find an or dina l η s uch that p η cl p − 1 δ 0 [ U 0 ] and p η cl p − 1 δ 0 [ V 0 ] are disjo in t. P ick δ 1 ∈ C 1 ab ov e η . In X δ 1 we can find a partition L b etw een p δ 1 cl p − 1 δ 0 [ U 0 ] and p δ 1 cl p − 1 δ 0 [ V 0 ] that is dis joint from Z 0 ,δ 1 ∪ Z 1 ,δ 1 — this is p o ssible b ecause Z 0 ,δ 1 ∪ Z 1 ,δ 1 is zero- dimensional b y the countable clo sed s um theor em. W e now obtain, by (3.1): • dim L 6 n − 1, • dim L ∩ p δ 1 [ Q 0 ] 6 dim Q 0 − 1, and • dim L ∩ p δ 1 [ Q 1 ] 6 dim Q 1 − 1. Because of the density conditions on Z 0 ,δ 1 and Z 1 ,δ 1 we know that L is nowhere dense in X δ 1 , that L ∩ p δ 1 [ Q 0 ] is nowhere dense in p δ 1 [ Q 0 ] and that L ∩ p δ 1 [ Q 1 ] is nowhere dens e in p δ 1 [ Q 1 ]. DIMENSION AND MAPS 7 As above w e find disjoin t regula r op en sets U 1 and V 1 around p δ 1 cl p − 1 δ 0 [ U 0 ] and p δ 1 cl p − 1 δ 0 [ V 0 ] resp ectively such tha t P 1 = X δ 1 \ ( U 1 ∪ V 1 ) ⊆ L . Note als o that p δ 1 δ 0 [ P 1 ] ⊆ P 0 . A t stage α we consider the disjo int op en F σ -sets A = S β <α p − 1 δ β [ U β ] and B = S β <α p − 1 δ β [ V β ]. There is a δ α ∈ C α ab ov e { δ β : β < α } such that p δ α [cl A ] and p δ α [cl B ] are disjoin t. The union Z = S β 6 α Z β ,δ α is zero -dimensional b y the co unt able closed sum theorem so we ca n find a partition L in X δ α betw een p δ α [cl A ] and p δ α [cl B ] that is disjoint from Z . As befo re w e find • dim L 6 n − 1, • dim L ∩ p δ α [ Q β ] 6 dim Q β − 1 for β 6 α , a nd • L and L ∩ p δ α [ Q β ] ar e nowhere dense in X δ α and p δ α [ Q β ] re spe ctively . As befor e we find disjoint reg ula r open se ts U α and V α around p δ α [cl A ] and p δ α [cl B ] resp ectively such that P α = X δ α \ ( U α ∪ V α ) is a subset of L . A t the end let U = S α p − 1 δ α [ U α ] and V = S α p − 1 δ α [ V α ]. then U and V are disjoint op en sets a r ound F and G resp ectively , so that P = X \ ( U ∪ V ) is a pa rtition betw een F and G . T o s e e that P is a P -set obser ve that cl p − 1 δ β [ U β ] ⊆ p − 1 δ α [ U α ] whenever β < α ; by co mpactness this implies tha t cl K ⊆ U whenever K is an F σ -subset contained in U . The same applies for V , s o tha t P ∩ cl K = ∅ , whenever K is an F σ -set that is disjo in t from P . T o see that dim P 6 n − 1 o bs erve that p δ α [ P ] ⊆ P α for all α . Any finite basic op en cov e r of P is supp orted b elow δ α for some α ; b ecause dim P α 6 n − 1 this cov er has an op en refinement of order at mo st n that is also suppo rted be low δ α . T o see that P is nowhere dense let B b e any ba sic op en set in [0 , 1] ω 1 that meets X and cho ose α such that B is supp orted b elow δ α . As P α is nowhere dense there is a basic op en se t B ′ ⊆ B , also supp orted b elow δ α , that meets X δ α but is disjoint fro m P α . Reinterpreted in X this mea ns that B ′ ⊆ B , that B ′ meets X and that B ′ ∩ P = ∅ . T o s ee that dim P ∩ Q α < dim Q α and P ∩ Q α is nowhere dense in Q α apply the pr e vious tw o parag raphs inside the space Q α , b oth times taking suitable δ s inside C α . In case Q is countable one needs only o ne clo sed a nd un bo unded set: the in ter- section of the co un tably many ass o ciated to X and the mem ber s o f Q . In case dim X = ∞ one choos es the dense zer o-dimensional subsets Z α,δ as ab ov e, this to make a ll intersections P ∩ Q α nowhere dense, but o ne only worries ab out the v alue o f dim( p δ [ Q α ] \ Z α,δ ) in ca se dim Q α < ∞ . In the following theorem we adopt the conv ent ion that ∞ − n = ∞ whenever n is a natural num b er — in this wa y the statement will b e v alid b oth for finite- and infinite-dimensio nal spaces. Theorem 3.3 ( CH ) . L et X b e a c omp act F -sp ac e of weight c . Then X ha s a b ase B = { B α : α < ω 1 } such that dim F r B α 6 dim X − 1 for al l α and dim T α ∈ F F r B α 6 dim F r B min F − | F | + 1 whenever F is a finite subset of ω 1 . Pr o of. Let C b e a base for X of cardinality ℵ 1 and let h C α , D α i : α < ω 1 enum erate the set of pair s h C, D i ∈ C 2 that satisfy cl C ⊆ D . Apply Pr op osition 3 .2 rep ea tedly to find, for ea ch α , disjoint regula r op en sets U α and V α around cl C α and X \ D α resp ectively such that P α = X \ ( U α ∪ V α ) is a nowhere dense P -set that satisfies • P α = F r U α , 8 KLAAS PIETER HAR T AND JAN V AN MILL • dim P α 6 dim X − 1, • for ev ery finite subset F of α one has dim( P α ∩ T β ∈ F P β ) 6 dim T β ∈ F P β − 1. Then { U α : α < ω 1 } is the bas e that we seek. A sp ecial ca se of this theorem is the o ne that we shall use in the next section. Theorem 3.4 ( CH ) . L et X b e a c omp act F - s p ac e of weight c and of fin ite dimen- sion n . Then X has a b ase B = { B α : α < ω 1 } , c onsisting of r e gu lar op en sets, such t hat dim T α ∈ F F r B α = ∅ whenever F is an n + 1 -element su bset of ω 1 . Pr o of. Let { B α : α < ω 1 } b e a ba se a s in Theorem 3.3. Then dim F r B α 6 n − 1 for all α , so if | F | = n + 1 then dim T α ∈ F F r B α 6 n − 1 − ( n + 1 ) + 1 = − 1, which means that T α ∈ F F r B α = ∅ . 4. Finite-to-one maps The purp ose o f this section is to s how that, a ssuming the Contin uum Hyp othesis, every finite-dimensional compact F -space of weigh t c is a finite-to- o ne co ntin uous image of a compact zero -dimensional space of weigh t c . Theorem 4.1 ( CH ) . L et X b e a c omp act F -sp ac e of weight c of finite dimension n . Then X is the at most 2 n -to- 1 c ontinu ous image of a c omp act zer o-dimensional sp ac e of weight c . Pr o of. Let B = { B α : α < ω 1 } b e a base for X as in Theor em 3 .4. Let B be the Bo olean s ubalgebra of RO( X ) generated by this ba s e a nd let Y be the Stone space of B . If y ∈ Y then T { cl C : C ∈ y } consists of exactly o ne p oint, which we denote by f ( y ). Let x ∈ X and put F = { α : x ∈ F r B α } . If f ( y ) = x then y determines a function p y : F → 2 by p y ( α ) = 1 iff B α ∈ y ; in addition if α / ∈ F then x ∈ B α or x / ∈ cl B α . It follows that if f ( y ) = f ( z ) = x then B α ∈ y iff B α ∈ z fo r α / ∈ F , so if y 6 = z then p y 6 = p z . This implies that f − 1 ( x ) 6 2 | F | 6 2 n . Corollary 4. 2 ( CH ) . If X is a one-dimensional c omp act F -sp ac e of weight c then X is an at most 2 - to- 1 c ont inu ous image of a c omp act zer o-dimensional sp ac e of weight c . Thu s, for co mpa ct one - dimensional F -s pa ces we hav e a direct generaliza tion of Hurewicz’ theorem, a s 1 + 1 = 2. One can give a pro of o f Theo r em 4 .1 alo ng the lines of the pro o f in Section 1. W e take a base a s in Theorem 3.4 but enumerate it in such a way that every singleton op en set is counted cofina lly o ften. Again one constructs tilings T α of order n + 1 but one can o nly ens ure that T α +1 refines T α for every α . The r eason b ecomes a pparent a t stag e ω : the common refinement of the tilings T m will b e infinite and not us a ble a s a factor in a compact pro duct. What one can do is star t a fresh ω -s equence of tilings at each limit ordinal λ . The tilings T λ + m will b e constructed from the family { B α : α > δ } for some δ (dep ending on λ ). The ze r o-dimensiona l space Y will consist of the po int s h T α : α < ω 1 i ∈ Q α T α with the following pro pe r ties: • T α +1 ⊆ T α for all α ; • { T α : α < ω 1 } ha s the finite intersection pro pe r ty . F or each x ther e will b e a t most n limite or dinals λ such that x is on the b oundary of a tile in one of the T λ + k (and hence in T λ + l for l > k ). Let h λ i : i < p i enumerate these limit ordinals a nd for each i let m i be the maximum of { |T λ i + k,x | : k ∈ ω } . The fib er of x under the o bvious map from Y onto X ha s car dina lit y Q i
P i
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