We establish a characterization of the extraordinary dimension of perfect maps between metrizable spaces

Chigogidze, A. "Alex"

NC DOCKS at The University of North Carolina at Greensboro

English

The University of North Carolina at Greensboro

arXiv:math/0308236v1  [math.GN]  25 Aug 2003EXTRAORDINARY DIMENSION OF MAPSA. CHIGOGIDZE AND V. VALOVAbstract. We establish a characterization of the extraordinary dimensionof perfect maps between metrizable spaces.1. IntroductionThe paper deals with extensional dimension of maps, specially, with the ex-traordinary dimension introduced recently by Ščepin [10] and studied by thefirst author in [1]. If L is a CW -complex and X a metrizable space, we writee-dimX ≤ L provided L is an absolute extensor for X (in such a case we saythat the extensional dimension of X is ≤ L, see [3], [4]). The extraordinarydimension of X generated by a complex L, notation dimLX, is the smallestinteger n such that e-dimX ≤ ΣnL, where ΣnL is the n-th iterated suspensionof L (by Σ0L we always denote the complex L itself). If L is the 0-dimensionalsphere S0, then dimL coincides with the covering dimension dim. We also writedimL f ≤ n, where f : X → Y is a given map, provided dimL f−1(y) ≤ n forevery y ∈ Y . Next is our main result.Theorem 1.1. Let f : X → Y be a σ-perfect map of metrizable spaces, let L bea CW -complex and n ≥ 1. Consider the following properties:(1) dimL f ≤ n;(2) There exists an Fσ subset A of X such that dimLA ≤ n − 1 and therestriction map f |(X\A) is of dimension dimL f |(X\A) = 0;(3) There exists a dense and Gδ subset G of C(X, In) with the source limi-tation topology such that dimL(f × g) = 0 for every g ∈ G;(3′) There exists a map g : X → In is such that dimL(f × g) = 0.Then (3) ⇒ (3′) ⇒ (1) and (3′) ⇒ (2). Moreover, (1) ⇒ (3) provided Y is aC-space and L is countable.Here, f : X → Y is σ-perfect if X is the union of countably many closed setsXi such that f(Xi) ⊂ Y are closed and the restriction maps f |Xi are perfect.Theorem 1.1 is inspired by the following result of M. Levin and W. Lewis [7,Theorem 1.8]: If X and Y are metrizable compacta then (3) ⇒ (3′) ⇒ (1) and1991 Mathematics Subject Classification. Primary: 54F45; Secondary: 55M10, 54C65.Key words and phrases. extensional dimension, extraordinary dimension, C-space.The second named author was partially supported by his NSERC grant.12 A. Chigogidze and V. Valov(3) ⇒ (2′) ⇒ (1), where (2′) is obtained from our condition (2) by replacingdimL f |(X\A) ≤ 0 with dim f |(X\A) ≤ 0. Moreover, the implication (1) ⇒ (3)was also established in [7] for a finite-dimensional compactum Y and a countableCW -complex L.Therefore, we have the following characterization of extraordinary dimensionof perfect maps between metrizable spaces:Corollary 1.2. Let f : X → Y be a perfect surjection between metrizable spaceswith Y being a C-space. If L is a countable CW -complex, then the followingconditions are equivalent:(1) dimL f ≤ n;(2) There exists a dense and Gδ subset G of C(X, In) with the source limi-tation topology such that dimL(f × g) ≤ 0 for every g ∈ G;(3) There exists a map g : X → In is such that dimL(f × g) ≤ 0.If, in addition, X is compact, then each of the above three conditions is equiva-lent to the following one:(4) There exists an Fσ set A ⊂ X such that dimLA ≤ n− 1 and the restric-tion map f |(X\A) is of dimension dim f |(X\A) ≤ 0.The equivalence of the first three conditions follow from Theorem 1.1. Moreprecisely, by Theorem 1.1 we have the following implications: (2) ⇒ (3) ⇒(1) ⇒ (2). When X is compact, the result of Levin-Lewis which was mentionedabove yields that (2) ⇒ (4) ⇒ (1). Therefore, combining the last two chains ofimplications, we can obtain the compact version of Corollary 1.2.Corollary 1.2 is a parametric version of [1, Theorem 4.9]. For the cover-ing dimension dim such a characterization was obtained by Pasynkov [9] andToruńczyk [11] in the realm of finite-dimensional compact metric spaces and ex-tended in [12] to perfect maps between metrizable C-spaces. Since the class ofC-spaces contains the class of finite-dimensional ones as a proper subclass (see[5]), the compact version of Corollary 1.2 is more general than the Levin-Lewisresult [7, Theorem 1.8]. It is interesting to know whether all the conditions (1)-(4) in Corollary 1.2 remain equivalent without the compactness requirement onX and Y .The source limitation topology on C(X,M), where (M, d) is a metric space,can be described as follows: a subset U ⊂ C(X,M) is open if for every g ∈ Uthere exists a continuous function α : X → (0,∞) such that B(g, α) ⊂ U . Here,B(g, α) denotes the set {h ∈ C(X,M) : d(g(x), h(x)) ≤ α(x) for each x ∈ X}.The source limitation topology doesn’t depend on the metric d if X is paracom-pact and C(X,M) with this topology has the Baire property provided (M, d) isa complete metric space. Moreover, if X is compact, then the source limitationtopology coincides with the uniform convergence topology generated by d.Extraordinary dimension of maps 3All function spaces in this paper, if not explicitely stated otherwise, areequipped with the source limitation topology.2. Some preliminary resultsThroughout this section K is a closed and convex subset of a given Banachspace E and f : X → Y a perfect map with X and Y paracompact spaces.Suppose that for every y ∈ Y we are given a property P(y) of maps h : f−1(y) →K and let P = {P(y) : y ∈ Y }. By CP(X|H,K) we denote the set of allbounded maps g : X → K such that g|f−1(y) has the property P(y) for everyy ∈ H , where H ⊂ Y . We also consider the set-valued map ψP : Y → 2C∗(X,K),defined by the formula ψP(y) = C∗(X,K)\CP(X|{y}, K), where C∗(X,K) isthe space of bounded maps from X into K.Lemma 2.1. Suppose that P = {P(y)}y∈Y is a family of properties satisfyingthe following conditions:(a) CP(X|H,K) is open in C∗(X,K) with respect to the source limitationtopology for every closed H ⊂ Y ;(b) g ∈ CP(X|{y}, K) implies g ∈ CP(X|U,K) for some neighborhood U ofy in Y .Then the map ψP has a closed graph provided C∗(X,K) is equipped with theuniform convergence topology.Proof. The proof of this lemma follows the arguments from the proof of [12,Lemma 2.6]. Recall that a closed subset F of the metrizable apace M is said to be a Zm-setin M , if the set C(Im,M\F ) is dense in C(Im,M) with respect to the uniformconvergence topology, where Im is the m-dimensional cube. If F is a Zm-set inM for every m ∈ N, we say that F is a Z-set in M .Lemma 2.2. Suppose y ∈ Y and P(y) satisfy the following condition:• For every m ∈ N the set of all maps h ∈ C(Im × f−1(y), K) with eachh|({z} × f−1(y)), z ∈ Im, having the property P(y) (as a map fromf−1(y) into K) is dense in C(Im×f−1(y), K) with respect to the uniformconvergence topology.Then, for every α : X → (0,∞) and g ∈ C∗(X,K), ψP(y) ∩ B(g, α) is a Z-setin B(g, α) provided B(g, α) is considered as subset of C∗(X,K) equipped withthe uniform convergence topology and ψP(y) ⊂ C∗(X,K) is closed.Proof. See the proof of [12, Lemma 2.8] Proposition 2.3. Let Y be a C-space and P = {P(y)}y∈Y such that:(a) the map ψP has a closed graph;4 A. Chigogidze and V. Valov(b) ψP(y)∩B(g, α) is a Z-set in B(g, α) for any continuous function α : X →(0,∞), y ∈ Y and g ∈ C∗(X,K), where B(g, α) is considered as a sub-space of C∗(X,K) with the uniform convergence topology.Then the set {g ∈ C∗(X,K) : g ∈ CP(X|{y}, K) for every y ∈ Y } is dense inC∗(X,K) with respect to the source limitation topology.Proof. Let G = {g ∈ C∗(X,K) : g ∈ CP(X|{y}, K) for every y ∈ Y }. It suf-fices to show that, for fixed g0 ∈ C∗(X,K) and a positive continuous func-tion α : X → (0,∞), there exists g ∈ B(g0, α) ∩ G. We equip C∗(X,K) withthe uniform convergence topology and consider the constant (and hence, lowersemi-continuous) convex-valued map φ : Y → 2C∗(X,K), φ(y) = B(g0, α1), whereα1(x) = min{α(x), 1}. Because of the conditions (a) and (b), we can applythe selection theorem [6, Theorem 1.1] to obtain a continuous map h : Y →C∗(X,K) such that h(y) ∈ φ(y)\ψP(y) for every y ∈ Y . Observe that h is amap from Y into B(g0, α1) such that h(y) ∈ CP(X|{y}, K) for every y ∈ Y .Then g(x) = h(f(x))(x), x ∈ X, defines a bounded map g ∈ B(g0, α) such thatg|f−1(y) = h(y)|f−1(y), y ∈ Y . Therefore, g ∈ CP(X|{y}, K) for all y ∈ Y ,i.e., g ∈ B(g0, α) ∩G. 3. Proof of Theorem 1.1(1) ⇒ (3) Suppose that L is countable and Y is a C-space. Let Xi be closedsubsets of X such that each fi = f |Xi : Xi → Yi = f(Xi) is a perfect map andYi is closed in Y . Then all Yi’s are C-spaces, and since the restriction mapsπi : C(X, In) → C(Xi, In), πi(g) = g|Xi, are open, the proof of this implicationis reduced to the case when f is a perfect map. Consequently, we may assumethat f is perfect.By [13, Theorem 1.1] (see also [8]), there exists a map q from X into theHilbert cube Q such that f × q : X → Y × Q is an embedding. Let {Wi}i∈Nbe a countable finitely-additive base for Q. For every i we choose a sequence ofmappings hij : Wi → L, representing all the homotopy classes of mappings fromWi to L (this is possible because L is a countable CW -complex). Following thenotations from Section 2, for fixed i, j and y ∈ Y we say that a map g ∈ C(X, In)has the property Pij(y) providedthe map hij ◦ q : q−1(Wi) → L can be continuously extended to a map over theset q−1(Wi) ∪(f−1(y) ∩ g−1(t))for every t ∈ g(f−1(y)).Let Pij = {Pij(y) : y ∈ Y } and for every H ⊂ Y we denote CPij (X|H, In)by Cij(X|H, In). Hence, Cij(X|H, In) consists of all g ∈ C(X, In) having theproperty Cij(y) for every y ∈ H . Let ψij : Y → 2C(X,In) be the set-valued mapψij(y) = C(X, In)\Cij(X|{y}, In).Extraordinary dimension of maps 5Lemma 3.1. Let g ∈ Cij(X|{y}, In). Then, there exist a neighborhood Uy of yin Y and a neighbourhood Vt ⊂ In of each t ∈ g(f−1(y)) such that hij ◦ q can beextended to a map from q−1(Wi) ∪(f−1(Uy) ∩ g−1(Vt))into L.Proof. Since g ∈ Cij(X|{y}, In), hij ◦q can be extended to a map from q−1(Wi)∪(f−1(y) ∩ g−1(t))into L for every t ∈ g(f−1(y)). Because L is an absoluteneighborhood extensor for X, there exists and open set Gt ⊂ X containingf−1(y) ∩ g−1(t) and a map ht : q−1(Wi) ∪ Gt → L extending hij ◦ q. Usingthat f × g is a closed map, we can find a neighborhood U ty × Vt of (y, t) inY × In such that St = (f × g)−1(U ty × Vt) ⊂ Gt. Next, choose finitely manypoints t(k), k = 1, 2, ., m, with f−1(y) ⊂⋃k=mk=1 St(k) and a neighbothood Uyof y in Y such that Uy ⊂⋂k=mk=1 Ut(k)y and f−1(Uy) ⊂⋃k=mk=1 St(k) (this canbe done since f is perfect). If t ∈ g(f−1(y)), then t ∈ Vt(k) for some k andf−1(Uy) ∩ g−1(Vt(k)) ⊂ St(k). Since , St(k) ⊂ Gt(k), the map ht(k) is an extensionof hij ◦ q over the set q−1(Wi) ∪(f−1(Uy) ∩ g−1(Vt(k)))Lemma 3.2. The set Cij(X|H, In) is open in C(X, In) for any i, j and closedH ⊂ Y .Proof. We follow the proof of [12, Lemma 2.5]. For a fixed g0 ∈ Cij(X|H, In) weare going to find a function α : X → (0,∞) such that B(g0, α) ⊂ Cij(X|H, In).By Lemma 3.1, for every z = (y, t) ∈ (f × g0)((f−1(H)) there exists a neigh-borhood Uz in Y × In such that(1) hij ◦ q can be extended to a map from q−1(Wi) ∪ (f × g0)−1(Uz) into L.Obviously, K = (f × g0)((f−1(H)) is closed in Y × In, so there exists openG ⊂ Y × In with K ⊂ G ⊂ G ⊂ U =⋃{Uz : z ∈ K}. Then ν = {Uz : z ∈K} ∪ {(Y × In)\G} is an open cover of Y × In. Let γ be an open locally finitecover of Y × In such that the family(2) {St(W, γ) : W ∈ γ} refines ν and St(W, γ) ⊂ G provided W ∩K 6= ∅.Consider the metric ρ = d + d1 on Y × In, where d is a metric on Y and d1the usual metric on In, and define the function α : X → (0,∞) by α(x) =2−1 sup{ρ((f × g0)(x), (Y × In)\W): W ∈ γ}. Let show that B(g0, α) ⊂Cij(X|H, In). Take g ∈ B(g0, α), y ∈ H and t ∈ g(f−1(y)). Then, (y, t) =(f × g)(x) for some x ∈ f−1(y). Since g is α-close to g0, there exists W ∈ γsuch that W ∩ K 6= ∅ and W contains both (f × g)(x) and (f × g0)(x). Itfollows from (2) that (f × g)−1(W ) ⊂ (f × g0)−1(Uz) for some z ∈ K. Inparticular, f−1(y) ∩ g−1(t) ⊂ (f × g0)−1(Uz). Consequently, by (1), hij ◦ qis extendable to a map from q−1(Wi) ∪(f−1(y) ∩ g−1(t))into L. Therefore,B(g0, α) ⊂ Cij(X|{y}, In) for every y ∈ H which completes the proof. 6 A. Chigogidze and V. ValovBecause of Lemma 3.1 and Lemma 3.2, we can apply Lemma 2.1 to obtainthe following corollary.Corollary 3.3. For any i and j the map ψij has a closed graph.Lemma 3.4. Let g ∈ C(X, In), α : X → (0,∞) and y ∈ Y . Then, for any i, j,ψij(y)∩B(g, α) is a Z-set in B(g, α) provided B(g, α) is considered as a subsetof C(X, In) with the uniform convergence topology.Proof. It follows from [7, Theorem 1.8, (1) ⇒ (3)] that if m ∈ N, then all mapsg : Im × f−1(y) → In such that e-dim(({z} × f−1(y)) ∩ g−1(t))≤ L for everyz ∈ Im and t ∈ In, form a dense subset G of C(Im × f−1(y)) with the uniformconvergence topology. It is clear that, for every g ∈ G and z ∈ Im, the restrictiong|({z} × f−1(y)), considered as a map on f−1(y), has the following property:hij ◦ q can be extended to a map from q−1(Wi) ∪ (f−1(y) ∩ g−1(t)) into L forany t ∈ In. Hence, we can apply Lemma 2.2 to conclude that ψij(y) ∩ B(g, α)is a Z-set in B(g, α). Now, we can finish the proof of this implication. Because of Corollary 3.3and Lemma 3.4, we can apply Proposition 2.3 to obtain that the set Cij =Cij(X|Y, In) is dense in C(X, In) for every i, j. Since, by Lemma 3.2, all Cijare also open, their intersection G is dense and Gδ in C(X, In). Let show thatdimL(f × g) ≤ 0 for every g ∈ G, i.e., e-dim(f × g) ≤ L. We fix y ∈ Y andt ∈ In and consider the fiber (f × g)−1(y, t) = f−1(y) ∩ g−1(t). Take a closedset A ⊂ f−1(y) ∩ g−1(t) and a map h : A → L. Because the map qy = q|f−1(y)is a homeomorphism, h′= h ◦ q−1y : q(A) → L is well defined. Next, extendh′to a map from a neighborhood W of q(A) (in Q) into L and find Wi withq(A) ⊂Wi ⊂Wi ⊂W . Therefore, there exists a map h′′: Wi → L extending h′.Then h′′is homotopy equivalent to some hij , so are h′′◦q and hij ◦q (consideredas maps from q−1(Wi) into L). Since hij ◦ q can be extended to a map fromq−1(Wi)∪(f−1(y)∩g−1(t))into L, by the Homotopy Extension Theorem, thereexists a map h̄ : q−1(Wi) ∪(f−1(y) ∩ g−1(t))→ L extending h′′◦ q. Obviously,h̄|(f−1(y) ∩ g−1(t))extends h. Hence, e-dim(f−1(y) ∩ g−1(t))≤ L.(3) ⇒ (3′) ⇒ (1) The implication (3) ⇒ (3′) is trivial. It is easily seen that inthe proof of (3′) ⇒ (1) we can assume f is perfect. Let g : X → In be suchthat dimL(f × g) ≤ 0 and y ∈ Y . Then the restriction g|f−1(y) : f−1(y) → Inis a pefect map with all of its fibers having extensional dimension e-dim ≤ L.Hence, by [2, Corollary], e-dimf−1(y) ≤ ΣnL, i.e, dimL f ≤ n.(3′) ⇒ (2) Because of the countable sum theorem, we can suppose that f isperfect. We fix a map g : X → In such that dimL(f × g) ≤ 0. According to [12,Lemma 4.1], there exists an Fσ subset B ⊂ Y × In such that dimB ≤ n−1 andExtraordinary dimension of maps 7dim({y} × In)\B ≤ 0 for every y ∈ Y . Then, applying again [2, Corollary], weconclude that the set A = (f × g)−1(B) is as required.References[1] A. Chigogidze, Extraordinary dimension theories generated by complexes, TopologyAppl., to appear.[2] A. Chigogidze and V. Valov, Extension dimension and C-spaces, Bull. London. Math.Soc. 34:6 (2002), 708–716.[3] A. Dranishnikov, The Eilenberg-Borsuk theorem for mappings into an arbitrary complex,Russian Acad. Sci. Sb. 81 (1995), 467–475.[4] A. Dranishnikov and J. Dydak, Extension dimension and extension types, Proc. SteklovInst. Math. 212 (1996), 55–88.[5] R. Engelking, Theory of dimensions: Finite and Infinite (Heldermann Verlag, Lemgo,1995).[6] V. Gutev and V. Valov, Continuous selections and C-spaces, Proc. Amer. Math. Soc.130:1 (2001), 233–242.[7] M. Levin and W. Lewis, Some mapping theorems for extensional dimension, Israel J.Math. 133 (2003), 61–76.[8] B. Pasynkov, On geometry of continuous maps of countable functional weight, Fundam.Prikl. Matematika 4:1 (1998), 155–164 (in Russian).[9] , On geometry of continuous maps of finite-dimensional compact metric spaces,Trudy Steklov Mat. Inst. 212 (1996), 147–172 (in Russian).[10] E. V. Ščepin, Arithmetic of dimension theory, Russian Math. Surveys 53:5 (1998), 975–1069.[11] H. Toruńczyk, Finite-to-one restrictions of continuous functions, Fund. Math. 125:3(1985), 237–249.[12] M. Tuncali and V. Valov, On dimensionally restricted maps, Fund. Math. 175:1 (2002),35–52.[13] , , On finite-dimensional maps, Tsukuba J. Math., to appear.Department of Mathematical Sciences, University of North Carolina atGreensboro, P.O. Box 26170, Greensboro, NC 27402-6170, U.S.A.E-mail address : chigogidze@uncg.eduDepartment of Mathematics, Nipissing University, 100 College Drive, P.O.Box 5200, North Bay, ON, P1B 8L7, CanadaE-mail address : veskov@nipissingu.ca

Extraordinary Dimension of Maps

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