MAPS WITH DIMENSIONALLY RESTRICTED FIBERS VESKO VALOV 1 1 0 Abstract. Weprovethatiff: X →Y isaclosedsurjectivemap 2 between metric spaces such that every fiber f−1(y) belongs to a n classofspaceS,thenthereexistsanFσ-setA⊂X suchthatA∈S a and dimf−1(y)\A = 0 for all y ∈ Y. Here, S can be one of the J following classes: (i) {M :e−dimM ≤K} for some CW-complex 5 K; (ii) C-spaces; (iii) weakly infinite-dimensional spaces. We also establish that if S = {M : dimM ≤ n}, then dimf△g ≤ 0 for ] N almost all g ∈C(X,In+1). G . h t a m 1. Introduction [ All spaces in the paper are assumed to be paracompact and all maps 2 continuous. By C(X,M) we denote all maps from X into M. Un- v less stated otherwise, all function spaces are endowed with the source 5 5 limitation topology provided M is a metric space. 1 The paper is inspired by the results ofPasynkov [11], Torunczyk [15], 0 . Sternfeld [14] and Levin [8]. Pasynkov announced in [11] and proved 1 in [12] that if f: X → Y is a surjective map with dimf ≤ n, where X 0 1 and Y are finite-dimensional metric compacta, then dimf△g ≤ 0 for 1 almost all maps g ∈ C(X,In) (see [10] for a non-compact version of this : v result). Torunczyk [15] established (in a more general setting) that if i X f, X and Y are as in Pasynkov’s theorem, then for each 0 ≤ k ≤ n−1 r there exists a σ-compact subset A ⊂ X such that dimA ≤ k and a k k dimf|(X\A ) ≤ n−k −1. k NextresultsinthisdirectionwereestablishedbySternfeldandLevin. Sternfeld[14]provedthatifinthecitedaboveresultsY isnot-necessarily finite-dimensional, then dimf△g ≤ 1 for almost all g ∈ C(X,In) and there exists a σ-compact subset A ⊂ X such that dimA ≤ n−1 and dimf|(X\A) ≤ 1. Levin [8] improved Sternfeld’s results by showing that dimf△g ≤ 0 for almost all g ∈ C(X,In+1), and has shown that 2000 Mathematics Subject Classification. Primary 54F45; Secondary 54E40. Key words and phrases. extensional dimension, C-spaces, 0-dimensional maps, metric compacta, weakly infinite-dimensional spaces. The author was partially supported by NSERC Grant 261914-08. 1 2 V.VALOV thisisequivalent totheexistence ofann-dimensional σ-compactsubset A ⊂ X with dimf|(X\A) ≤ 0. The above results of Pasynkov and Torunczyk were generalized in [17] for closed maps between metric space X and Y with Y being a C-space (recall that each finite-dimensional paracompact is a C-space [6]). But the question whether the results of Pasynkov and Torunczyk remain valid without the finite-dimensionality assumption on Y is still open. In this paper we provide non-compact analogues of Levin’s results for closed maps between metric spaces. We say that a topological property of metrizable spaces is an S- property if the following conditions are satisfied: (i) S is hereditary with respect to closed subsets; (ii) if X is metrizable and {H }∞ is a i i=1 ∞ sequence of closed S-subsets of X, then Si=1Hi ∈ S; (iii) a metrizable space X ∈ S provided there exists a closed surjective map f: X → Y such that Y is a 0-dimensional metrizable space and f−1(y) ∈ S for all y ∈ Y; (iv) any discrete union of S-spaces is an S-space. Any map whose fibers have a given S-property is called an S-map. Here are some examples of S-properties (we identify S with the class of spaces having the property S): • S = {X : dimX ≤ n} for some n ≥ 0; • S = {X : dim X ≤ n}, where G is an Abelian group and dim G G is the cohomological dimension; • more generally, S = {X : e−dimX ≤ K}, where K is a CW- complex and e−dim is the extension dimension, see [4], [5]; • S = {X : X is weakly infinite-dimensional}; • S = {X : X is a C-space}. To show that the property e−dim ≤ K satisfies condition (iii), we apply [3, Corollary 2.5]. For weakly infinite-dimensional spaces and C-spaces this follows from [7]. Theorem 1.1. Let f: X → Y be a closed surjective S-map with X and Y being metrizable spaces. Then there exists an F -subset A ⊂ X σ such that A ∈ S and dimf−1(y)\A = 0 for all y ∈ Y. Moreover, if f is a perfect map, the conclusion remains true provided S is a property satisfying conditions (i)−(iii). Theorem 1.1 was established by Levin [9, Theorem 1.2] in the case X and Y are metric compacta and S is the property e−dim ≤ K for a given CW-complex K. Levin’s proof of this theorem remains valid for arbitrary S-property, but it doesn’t work for non-compact spaces. We say that a map f: X → Y has a countable functional weight (notation W(f) ≤ ℵ ), see [10]) if there exists a map g: X → Iℵ0 0 MAPS WITH DIMENSIONALLY RESTRICTED FIBERS 3 such that f△g: X → Y × Iℵ0 is an embedding. For example [12, Proposition 9.1], W(f) ≤ ℵ for any closed map f: X → Y such that 0 X is a metrizable space and every fiber f−1(y), y ∈ Y, is separable. Theorem 1.2. Let X and Y be paracompact spaces and f: X → Y a closedsurjective map with dimf ≤ n and W(f) ≤ ℵ . Then C(X,In+1) 0 equipped with the uniform convergence topology contains a dense subset of maps g such that dimf△g ≤ 0. It was mentioned above that this corollary was established by Levin [8, Theorem 1.6] for metric compacta X and Y. Levin’s arguments don’t work for non-compact spaces. We are using the Pasynkov’s tech- nique from [10] to reduced the proof of Theorem 1.2 to the case of X and Y being metric compacta. Our last results concern the function spaces C(X,In) and C(X,Iℵ0) equipped with the source limitation topology. Recall that this topol- ogy on C(X,M) with M being a metrizable space can be described as follows: the neighborhood base at a given map h ∈ C(X,M) con- sists of the sets B (h,ǫ) = {g ∈ C(X,M) : ρ(g,h) < ǫ}, where ρ is ρ a fixed compatible metric on M and ǫ : X → (0,1] runs over con- tinuous positive functions on X. The symbol ρ(h,g) < ǫ means that ρ(h(x),g(x)) < ǫ(x) for all x ∈ X. It is well known that for paracom- pact spaces X this topology doesn’t depend on the metric ρ and it has the Baire property provided M is completely metrizable. Theorem 1.3. Let f: X → Y be a perfect surjection between para- compact spaces and W(f) ≤ ℵ . 0 (i) The maps g ∈ C(X,Iℵ0) such that f△g embeds X into Y × Iℵ0 form a dense G -set in C(X,Iℵ0) with respect to the source δ limitation topology; (ii) If there exists a map g ∈ C(X,In) with dimf△g ≤ 0, then all maps having this property form a dense G -set in C(X,In) with δ respect to the source limitation topology. Corollary 1.4. Let f: X → Y be a perfect surjection with dimf ≤ n and W(f) ≤ ℵ , where X and Y are paracompact spaces. Then 0 all maps g ∈ C(X,In+1) with dimf△g ≤ 0 form a dense G -set in δ C(X,In+1) with respect to the source limitation topology. Corollary 1.4 follows directly from Theorem 1.2 and Theorem 1.3(ii). Corollary 1.5 below follows from Corollary 1.4 and [2, Corollary 1.1], see Section 3. Corollary 1.5. Let X, Y be paracompact spaces and f: X → Y a perfect surjection with dimf ≤ n and W(f) ≤ ℵ . Then for ev- 0 ery matrizable ANR-space M the maps g ∈ C(X,In+1 × M) such 4 V.VALOV that dimg(f−1(y)) ≤ n + 1 for all y ∈ Y form a dense G -set E δ in C(X,In+1 ×M) with respect to the source limitation topology. Finally, let us formulate the following question concerning property S(anaffirmative answer of this question yields that (strong) countable- dimensionality is an S-property): Question 1.6. Suppose f: X → Y is a perfect surjection between metrizable spaces such that dimY = 0 and each fiber f−1(y), y ∈ Y, is (strongly) countable-dimensional. Is it true that X is (strongly) countable-dimensional? 2. S-properties and maps into finite-dimensional cubes This section contains the proofs of Theorem 1.1 and Theorem 1.2. Proof of Theorem 1.1. We follow the proof of [18, Proposition 4.1]. Let us show first that the proof is reduced to the case f is a perfect map. Indeed, according to Vainstein’s lemma, the boundary Frf−1(y) of every fiber f−1(y) is compact. Defining F(y) to be Frf−1(y) if Frf−1(y) 6= ∅, andanarbitrarypoint fromf−1(y)otherwise, weobtain a set X0 = S{F(y) : y ∈ Y} such that X0 ⊂ X is closed and the re- strictionf|X isa perfect map. Moreover, eachf−1(y)\X isopeninX 0 0 and has the property S (as an F -subset of the S-space f−1(y)). Hence, σ X\X being the union of the discrete family {f−1(y)\X : y ∈ Y} of 0 0 S-set is an S-set. At the same time X\X is open in X. Consequently, 0 X\X is the union of countably many closed sets X ⊂ X, i = 1,2,... 0 i Obviously, eachX , i ≥ 1, also hastheproperty S. Therefore, itsuffices i to prove Theorem 1.1 for the S-map f|X : X → Y. 0 0 So, we may suppose that f is perfect. According to [10], there exists a map g: X → Iℵ0 such that g embeds every fiber f−1(y), y ∈ Y. Let g = △∞ g and h = f△g : X → Y × I, i ≥ 1. Moreover, we i=1 i i i choose countably many closed intervals I such that every open subset j of I contains some I . By [17, Lemma 4.1], for every j there exists a j 0-dimensional F -set C ⊂ Y × I such that C ∩ ({y} × I ) 6= ∅ for σ j j j j every y ∈ Y. Now, consider the sets A = h−1(C ) for all i,j ≥ 1 and ij i j let A be their union. Since f is an S-map, so is the map h for any i. i Hence, A has property S for all i,j. This implies that A has also the ij same property. It remains to show that dimf−1(y)\A ≤ 0 for every y ∈ Y. Let dimf−1(y )\A > 0 for some y . Since g|f−1(y ) is an embedding, there 0 0 0 existsanintegerisuchthatdimg (f−1(y )\A) > 0. Theng (f−1(y )\A) i 0 i 0 has a nonempty interior in I. So, g (f−1(y )\A) contains some I . i 0 j Choose t ∈ I with c = (y ,t ) ∈ C . Then there exists x ∈ 0 j 0 0 0 j 0 MAPS WITH DIMENSIONALLY RESTRICTED FIBERS 5 f−1(y )\A such that g (x ) = t . On the other hand, x ∈ h−1(c ) ⊂ 0 i 0 0 0 i 0 A ⊂ A, a contradiction. (cid:3) ij Proof of Theorem 1.2. We first prove next proposition which is a small modification of [10, Theorem 8.1]. For any map f: X → Y we consider the set C(X,Y × In+1,f) consisting of all maps g: X → Y ×In+1 such that f = π ◦g, where π : Y ×In+1 → Y is the projection n n onto Y. We also consider the other projection ̟ : Y ×In+1 → In+1. It n iseasilyseenthattheformulag → ̟ ◦g providesone-to-onecorrespon- n dence between C(X,Y ×In+1,f) and C(X,In+1). So, we may assume that C(X,Y × In+1,f) is a metric space isometric with C(X,In+1), where C(X,In+1) is equipped with the supremum metric. Proposition 2.1. Let f: X → Y be an n-dimensional surjective map between compact spaces with n > 0 and λ: X → Z a map into a metric compactum Z. Then the maps g ∈ C(X,Y × In+1,f) satisfying the condition below form a dense subset of C(X,Y ×In+1,f): there exists a compact space H and maps ϕ: X → H, h: H → Y × In+1 and µ: H → Z such that λ = µ◦ϕ, g = h◦ϕ, W(h) ≤ ℵ and dimh = 0. 0 Proof. We fix a map g ∈ C(X,Y × In+1,f) and ǫ > 0. Let g = 0 1 ̟ ◦g . Then λ△g ∈ C(X,Z×In+1). Consider also theconstant maps n 0 1 f′: Z ×In+1 → Pt and η: Y → Pt, where Pt is the one-point space. ′ So, we have η◦f = f ◦(λ△g ). According to Pasynkov’s factorization 1 theorem [13, Theorem 13], there exist metrizable compacta K, T and maps f∗: K → T, ξ : X → K, ξ : K → Z×In+1 and η∗: Y → T such 1 2 that: • η∗ ◦f = f∗ ◦ξ ; 1 • ξ ◦ξ = λ△g ; 2 1 1 • dimf∗ ≤ dimf ≤ n. If p: Z ×In+1 → Z and q: Z ×In+1 → In+1 denote the corresponding projections, we have p◦ξ ◦ξ = λ and q ◦ξ ◦ξ = g . 2 1 2 1 1 Since dimf∗ ≤ n, by Levin’s result [8, Theorem 1.6], there exists a map φ: K → In+1 such that φ is ǫ-close to q ◦ξ and dimf∗△φ ≤ 0. 2 Then the map φ◦ξ is ǫ-close to g , so g = f△(φ◦ξ ) is ǫ-close to g . 1 1 1 0 Denote ϕ = f△ξ , H = ϕ(X) and h = (id ×φ)|H. If ̟ : H → K is 1 Y H the restriction of the projection Y ×K → K on H, we have λ = p◦ξ ◦ξ = p◦ξ ◦̟ ◦ϕ, so λ = µ◦ϕ, where µ = p◦ξ ◦̟ . 2 1 2 H 2 H Moreover, g = f△(φ◦ξ ) = (id ×φ)◦(f△ξ ) = h◦ϕ. Since K is a 1 Y 1 metrizable compactum, W(φ) ≤ ℵ . Hence, W(h) ≤ ℵ . 0 0 6 V.VALOV To show that dimh ≤ 0, it suffices to prove that dimh ≤ dimf∗△φ. To this end, we show that any fiber h−1((y,v)), where (y,v) ∈ Y ×In+1, is homeomorphic to a subset of the fiber (f∗△φ)−1((η∗(y),v)). Indeed, let π be the restriction of the projection Y × K → Y on the set Y H. Since η∗ ◦ f = f∗ ◦ ξ , H is a subset of the pullback of Y and 1 K with respect to the maps η∗ and f∗. Therefore, ̟ embeds every H fiber π−1(y) into (f∗)−1(y), y ∈ Y. Let a = (y ,k ) ∈ H ⊂ Y × K, Y i i i i = 1,2, such that h(a ) = h(a ). Then (y ,φ(k )) = (y ,φ(k )), 1 2 1 1 2 2 so y = y = y and φ(k ) = φ(k ) = v. This implies ̟ (a ) = 1 2 1 2 H i k ∈ (f∗)−1(cid:0)η∗(π (a ))(cid:1) = (f∗)−1(η∗(y)), i = 1,2. Hence, ̟ embeds i Y i H the fiber h−1((y,v)) into the fiber (f∗△φ)−1((η∗(y),v)). Consequently, dimh ≤ dimf∗△φ = 0. (cid:3) We can prove now Theorem 1.2. It suffices to show every map from C(X,Y ×In+1,f) can be approximated by maps g ∈ C(X,Y ×In+1,f) with dimg ≤ 0. We fix g ∈ C(X,Y × In+1,f) and ǫ > 0. Since 0 W(f) ≤ ℵ , there exists a map λ: X → Iℵ0 such that f△λ is an 0 embedding. Let βf: βX → βY be the Cech-Stone extension of the map f. Then dimβf ≤ n, see [13, Theorem 15]. Consider also the maps βλ: βX → Iℵ0 and g¯ = βf△βg , where g = ̟ ◦g . According 0 1 1 n 0 to Proposition 2.1, there exists a map g¯ ∈ C(βX,βY ×In+1,βf) which is ǫ-close to g¯ and satisfies the following conditions: there exists a 0 compact space H and maps ϕ: βX → H, h: H → βY × In+1 and µ: H → Iℵ0 such that βλ = µ◦ϕ, g¯ = h◦ϕ, W(h) ≤ ℵ and dimh = 0. 0 We have the following equalities βf△βλ = (π ◦g¯)△(µ◦ϕ) = (π ◦h◦ϕ)△(µ◦ϕ) = (cid:0)(π ◦h)△µ(cid:1)◦ϕ, n n n where π denotes the projection βY ×In+1 → βY. This implies that ϕ n embeds X into H because f△λ embeds X into Y ×Iℵ0. Let g be the restriction of g¯ over X. Identifying X with ϕ(X), we obtain that h is an extension of g. Hence, dimg ≤ dimh = 0. Observe also that g is ǫ-close to g , which completes the proof. (cid:3) 0 3. Proof of Theorem 1.3 and Corollary 1.5 Proof ofTheorem1.3(ii). Wefirstprovecondition(ii). SinceW(f) ≤ ℵ , there exists a map λ: X → Iℵ0 such that f△λ embeds X into Y × 0 Iℵ0. Choose a sequence {γ } of open covers of Iℵ0 with mesh(γ ) ≤ k k≥1 k 1/k, and let ω = λ−1(γ ) for all k. We denote by C (X,In,f) k k (ωk,0) the set of all maps g ∈ C(X,In) with the following property: every z ∈ (f△g)(X) has a neighborhood V in Y ×In such that (f△g)−1(V ) z z can be represented as the union of a disjoint open in X family re- fining the cover ω . According to [17, Lemma 2.5], each of the sets k MAPS WITH DIMENSIONALLY RESTRICTED FIBERS 7 C (X,In,f), k ≥ 1, is open in C(X,In) with respect to the source (ωk,0) limitation topology. It follows from the definition of the covers ω k that Tk≥1C(ωk,0)(X,In,f) consists of maps g with dimf△g ≤ 0. Since C(X,In) with the source limitation topology has the Baire property, it remains to show that any C (X,In,f) is dense in C(X,In). (ωk,0) To this end, we fix a cover ω , a map g ∈ C(X,In) and a function m 0 ǫ: X → (0,1]. We are going to find h ∈ C (X,In,f) such that (ωm,0) ρ(g (x),h(x)) < ǫ(x) for all x ∈ X, where ρ is the Euclidean metric 0 on In. Then, by [1, Lemma 8.1], there exists an open cover U of X satisfying the following condition: if α: X → K is a U-map into a paracompact space K (i.e., α−1(ω) refines U for some open cover ω of K), then there exists a map q: G → In, where G is an open neighbor- hood of α(X) in K, such that g and q ◦α are ǫ/2-close with respect 0 to the metric ρ. Let U be an open cover of X refining both U and ω 1 m such that inf{ǫ(x) : x ∈ U} > 0 for all U ∈ U . 1 Since dimf△g ≤ 0 for some g ∈ C(X,In), according to [1, Theorem 6] there exists an opencover V of Y such that for any V-map β: Y → L into a simplicial complex L we can find a U -map α: X → K into a 1 simplicialcomplexK andaperfectPL-mapp: K → Lwithβ◦f = p◦α and dimp ≤ n. We can assume that V is locally finite. Take L to be the nerve of the cover V and β: Y → L the corresponding natural map. Then there exist a simplicial complex K and maps p and α satisfying the above conditions. Hence, the following diagram is commutative. α X −→ K f  p   y y β Y −→ L Since K is paracompact, the choice of the cover U guarantees the existence of a map ϕ: G → In, where G ⊂ K is an open neighborhood of α(X), such that g and h = ϕ ◦ α are ǫ/2-close with respect to 0 0 ρ. Replacing the triangulation of K by a suitable subdivision, we may additionally assume that no simplex of K meets both α(X) and K\G. So, the union N of all simplexes σ ∈ K with σ ∩ α(X) 6= ∅ is a subcomplex of K and N ⊂ G. Moreover, since N is closed in K, p = p|N: N → L is a perfect map. Therefore, we have the following N commutative diagram: 8 V.VALOV h X 0 - In f QαQsN ϕ(cid:26)(cid:26)> ? Y pN Q ? βQs L Using that α is a U -map and inf{ǫ(x) : x ∈ U} > 0 for all U ∈ 1 U , we can construct a continuous function ǫ : N → (0,1] and an 1 1 open cover γ of N such that ǫ ◦ α ≤ ǫ and α−1(γ) refines U . Since 1 1 dimp ≤ dimp ≤ n and L, being a simplicial complex, is a C-space, N we can apply [17, Theorem 2.2] to find a map ϕ ∈ C (N,In,p ) 1 (γ,0) N which is ǫ /2-close to ϕ. Let h = ϕ ◦α. Then h and h are ǫ/2-close 1 1 0 because ǫ ◦α ≤ ǫ. On the other hand, h is ǫ/2-close to g . Hence, g 1 0 0 0 and h are ǫ-close. It remains to show that h ∈ C (X,In,f). To this end, fix a (ωm,0) point z = (f(x),h(x)) ∈ (f△h)(X) ⊂ Y ×In and let y = f(x). Then w = (p △ϕ )(α(x)) = (β(y),h(x)). Since ϕ ∈ C (N,In,p ), there N 1 1 (γ,0) N exists a neighborhood V of w in L×In such that W = (p △ϕ )−1(V ) w N 1 w is a union of a disjoint open family in N refining γ. We can assume that V = V × V , where V and V are neighborhoods of w β(y) h(x) β(y) h(x) β(y) and h(x) in Y and In, respectively. Consequently, (f△h)−1(Γ) = α−1(W), where Γ = β−1(cid:0)V (cid:1)×V . Finally, observe that α−1(W) β(y) h(x) is a disjoint union of an open in X family refining ω . Therefore, m h ∈ C (X,In,f). (cid:3) (ωm,0) Proof of Theorem 1.3(i). Let λ and ω be as in the proof of Theorem k 1.3(i). Denote by C (X,Iℵ0,f) the set of all g ∈ C(X,Iℵ0) such that ωk f△g is an ω -map. It can be shown that every C (X,Iℵ0,f) is open k ωk in C(X,Iℵ0) with the source limitation topology (see [16, Proposition 3.1]). Moreover, Tk≥1Cωk(X,Iℵ0,f) consists of maps g with f△g em- bedding X into Y ×Iℵ0. So, we need to show that each C (X,Iℵ0,f) ωk is dense in C(X,Iℵ0) equipped with the source limitation topology. Toprovethisfactwefollowthenotationsandtheargumentsfromthe proofofTheorem1.3(ii)(thatC (X,In,f)aredenseinC(X,In))by (ωk,0) considering Iℵ0 instead of In. We fix a cover ω , a map g ∈ C(X,Iℵ0) m 0 and a function ǫ ∈ C(X,(0,1]). Since W(f) ≤ ℵ , we can apply 0 Theorem 6 from [1] to find an open cover V of Y such that for any V-map β: Y → L into a simplicial complex L there exists a U -map 1 α: X → K into a simplicial complex K and a perfect PL-map p: K → L with β ◦f = p◦α. Proceeding as before, we find a map h = ϕ ◦α 1 which is ǫ-close to g , where ϕ ∈ C (N,Iℵ0,p ). It is easily seen that 0 1 γ N MAPS WITH DIMENSIONALLY RESTRICTED FIBERS 9 ϕ ∈ C (N,Iℵ0,p ) implies h ∈ C (X,Iℵ0,f). So, C (X,Iℵ0,f) is 1 γ N ωm ωm dense in C(X,Iℵ0). (cid:3) Proof of Corollary 1.5. It follows from [2, Proposition 2.1] that the set E is G in C(X,In+1 × M). So, we need to show it is dense in δ C(X,In+1 × M). To this end, we fix g0 = (g0,g0) ∈ C(X,In+1 × M) 1 2 with g0 ∈ C(X,In+1) and g0 ∈ C(X,M). Since, by Corollary 1.4, the 1 2 set G = {g ∈ C(X,In+1) : dimf△g ≤ 0} 1 1 1 is dense in C(X,In+1), we may approximate g0 by a map h ∈ G . 1 1 1 Then, according to [2, Corollary 1.1], the maps g ∈ C(X,M) with 2 dimg (cid:0)(f△h )−1(z)(cid:1) = 0 for all z ∈ Y ×In+1 form a dense subset G 2 1 2 ofC(X,M). So, wecanapproximateg0 byamaph ∈ G . Letusshow 2 2 2 that the map h = (h ,h ) ∈ C(X,In+1) × M belongs to E. We de- 1 2 fine the map π : (f△h)(X) → (f△h )(X), π (cid:0)f(x),h (x),h (x)(cid:1) = h 1 h 1 2 (cid:0)f(x),h (x)(cid:1), x ∈ X. Because f is perfect, so is π . Moreover, 1 h (π )−1(cid:0)f(x),h (x)(cid:1) = h (cid:0)f−1(f(x)) ∩ h−1(h (x))(cid:1), x ∈ X. So, ev- h 1 2 1 1 ery fiber of π is 0-dimensional. We also observe that π (cid:0)h(f−1(y))(cid:1) = h h (f△h )(f−1(y)) and the restriction π |h(f−1(y)) is a perfect surjection 1 h between the compact spaces h(f−1(y))and(f△h )(f−1(y))for any y ∈ 1 Y. Since (f△h )(f−1(y)) ⊂ {y}×In+1, dim(f△h )(f−1(y)) ≤ n+ 1, 1 1 y ∈ Y. Consequently, applying the Hurewicz’s dimension-lowering the- orem [6] for the map π |h(f−1(y)), we have dimh(f−1(y)) ≤ n + 1. h Therefore, h ∈ E, which completes the proof. (cid:3) References 1. T. Banakh and V. Valov, General Position Properties in Fiberwise Geometric Topology, arXiv:math.GT/10012522. 2. T.BanakhandV. Valov, Spaces with fibered approximation property in dimen- sion n, Central European J. 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