APPROXIMATION BY LIGHT MAPS AND PARAMETRIC LELEK MAPS 8 TARAS BANAKH AND VESKO VALOV 0 0 2 Abstract. The class of metrizable spaces M with the follow- n ing approximation property is introduced and investigated: M ∈ a AP(n,0) if for every ε > 0 and a map g: In → M there exists J a 0-dimensional map g′: In → M which is ε-homotopic to g. It 0 2 is shown that this class has very nice properties. For example, if Mi ∈ AP(ni,0), i = 1,2, then M1×M2 ∈AP(n1+n2,0). More- ] over, M ∈ AP(n,0) if and only if each point of M has a local T base of neighborhoods U with U ∈ AP(n,0). Using the proper- G ties of AP(n,0)-spaces, we generalize some results of Levin and . h Kato-Matsuhashi concerning the existence of residual sets of n- t dimensional Lelek maps. a m [ 1 1. Introduction v 7 All spaces in the paper are assumed to be metrizable and all maps 0 continuous. By C(X,M) we denote all maps from X into M. Un- 1 3 less stated otherwise, all function spaces are endowed with the source 1. limitation topology. 0 One of the important properties of the n-dimensional cube In or the 8 Euclidean space Rn, widely exploited in dimension theory, is that any 0 : mapfromann-dimensionalcompactumX intoIn (resp.,Rn)canbeap- v i proximated by maps with 0-dimensional fibers. The aim of this article X is to introduce and investigate the class of spaces having that property. r a More precisely, we say that a space M has the (n,0)-approximation property (br., M ∈ AP(n,0)) if for every ε > 0 and a map g: In → M there exists a 0-dimensional map g′: In → M which is ε-homotopic to g. Here, g′ is ε-homotopic to g means that there is an ε-small homotopy h: In × I → M connecting g and g′. It is easily seen that this definition doesn’t depend on the metric generating the topology of M. If M is LCn, then M ∈ AP(n,0)) if and only if for every ε > 0 and g ∈ C(In,M) there exists a 0-dimensional map g′ which is ε-close 1991 Mathematics Subject Classification. Primary 54F45; Secondary 55M10. Key words and phrases. dimension, n-dimensional maps, n-dimensional Lelek maps, dendrites, Cantor n-manifolds, general position properties. The second author was partially supported by NSERC Grant 261914-03. 1 2 to g. We show that the class of AP(n,0)-spaces is quite large and it has very nice properties. For example, if M ∈ AP(n ,0), i = 1,2, i i then M × M ∈ AP(n + n ,0). We also prove that M ∈ AP(n,0) 1 2 1 2 if and only if each point of M has a local base of neighborhoods U with U ∈ AP(n,0). Moreover, every path-connected compactum M ∈ AP(n,0) is a (Vn)-continuum in the sense of P. Alexandroff [1]. In particular, according to [10], any such M is a Cantor n-manifold, as well as a strong Cantor n-manifold in the since of Hadˇziivanov [9]. All complete LC0-spaces without isolated points are AP(1,0), as well as, every manifold modeled on the n-dimensional Menger cube or the n-dimensional Nöbeling space has the AP(n,0)-property. The class of AP(n,0)-spaces is very natural for obtaining results about approximation by dimensionally restricted maps. We present such a result about n-dimensional Lelek maps. Recall that a map g: X → M between compact spaces is said to be an n-dimensional Lelek map [17] if the union of all non-trivial continua contained in the fibers of g is of dimension ≤ n. For convenience, if n ≤ 0, by an n- dimensional Lelek map we simply mean a 0-dimensional map. Every n-dimensional Lelek mapg between compacta is at most n-dimensional because the components of each fiber of g is at most n-dimensional (we say that g is n-dimensional if all fibers of g are at most n-dimensional). Lelek [15] constructed such a map from In+1 onto a dendrite. Levin [17] improved this result by showing that the space C(X,In) of all maps from X into In contains a dense G -subset consisting of (m−n)- δ dimensional Lelek maps for any compactum X with dimX = m ≤ n. Recently Kato and Matsuhashi [12] established a version of the Levin result with In replaced by more general class of spaces. This class consists of complete metric ANR-spaces having a piecewise embedding dimension ≥ n. In the present paper we generalize the result of Kato-Matsuhashi in twodirections. First, weestablishaparametricversionoftheirtheorem and second, we show that this parametric version holds for AP(n,0)- spaces (see Section 6 where it is shown that the class of AP(n,0)- spaces contains properly the class of ANR’s with piecewise embedding dimensions ≥ n). Here is our result about parametric Lelek maps (a more general version is established in Section 3; moreover we also present a version of Theorem 1.1 when M has a property weaker than AP(n,0), see Theorem 4.5). Theorem 1.1. Let f: X → Y be a perfect map with dim (f) ≤ m, △ where X and Y are metric spaces. If M ∈ AP(n,0) is completely Approximation by light maps 3 metrizable, then there exists a G -set H ⊂ C(X,M) such that ev- δ ery simplicially factorizable map in C(X,M) is homotopically approximated by maps from H and for every g ∈ H and y ∈ Y the restriction g|f−1(y) is an (m−n)-dimensional Lelek map. Let us explain the notions in Theorem 1.1. A map g ∈ C(X,M) is homotopically approximated by maps from H means that for every function ε ∈ C(X,(0,1]) there exists g′ ∈ H which is ε-homotopic to g. Here, the maps g and g′ are said to be ε-homotopic, if there is a homotopy h: X × I → M connecting g and g′ such that each set h({x}×I) has a diameter < ε(x), x ∈ X. The function space C(X,M) appearing in this theorem is endowed withthesourcelimitationtopologywhoseneighborhoodbaseatagiven function f ∈ C(X,M) consists of the sets B (f,ε) = {g ∈ C(X,M) : ̺(g,f) < ε}, ̺ where ̺ is a fixed compatible metric on M and ε : X → (0,1] runs over continuous positive functions on X. The symbol ̺(f,g) < ε means that ̺(f(x),g(x)) < ε(x) for all x ∈ X. Since X is metrizable, the source limitation topology doesn’t depend on the metric ρ [14] and it has the Baire property provided M is completely metrizable [20]. We say that a map g: X → M is simplicially factorizable [2] if there exists asimplicial complex Landtwo mapsg : X → Landg : L → M 1 2 suchthatg = g ◦g . Ineachofthefollowingcasesthesetofsimplicially 2 1 factorizable maps is dense in C(X,M) (see [2, Proposition 4]): (i) M is an ANR; (ii) dimX ≤ k and M is LCk−1; (iii) X is a C-space and M is locally contractible. The dimension dim (f) was defined in [2]: dim (f) of a map f : △ △ X → Y is equal to the smallest cardinal number τ for which there is a map g : X → Iτ such that the diagonal product f∆g : X → Y ×Iτ is a 0-dimensional map. For any perfect map f : X → Y between metric spaces we have: (i) dim(f) ≤ dim (f); (ii) dim (f) = dim(f) if Y is △ △ a C-space, see [21] and [27]; (iii) dim (f) ≤ dim(f)+ 1 if the spaces △ X,Y are compact, see [16]. Since every metric space admitting a perfect finite-dimensional map onto a C-space isalso a C-space [11], Theorem 1.1 implies the following (here, H is homotopically dense in C(X,M) if every g ∈ C(X,M) is homotopically approximated by maps from H): 4 Corollary 1.2. Let f : X → Y be a perfect m-dimensional map between metric spaces with Y being a C-space. If M ∈ A(n,0) is completely metrizable, then in each of the following cases there exists a homotopically dense G -subset H ⊂ C(X,M) consisting of maps g such δ that g|f−1(y) is an (m−n)-dimensional Lelek map for every y ∈ Y: • M is locally contractible; • X is finite dimensional and M ∈ LCk−1 for k = dimX. The paper is organized as follows. In Section 2 some preliminary results about AE(n,0)-spaces are presented. In Section 3 we establish a generalized version of Theorem 1.1. Section 4 is devoted to almost AP(n,0)-spaces. The final two sections contain some interesting prop- ertiesof AP(n,0)-spaces. InSection5 weprove thatAP(n,0)-property has a local nature, as well as, M ×M ∈ AP(n +n ,0) provided each 1 2 1 2 M ∈ AP(n ,0) is completely metrizable. It is also established in this i i section that every path connected compactum is a (Vn)-continuum. Section 6 is devoted to the interplay of AP(n,0)-property and the gen- {n,k} eral position properties m − DD -properties. In particular, it is shown that every point of a locally path-connected AP(n,0)-space X is a homological Z -point in X. Another result from this section n−1 states that every completely metrizable space possessing the disjoint (n−1)-disks property is an AP(n,0)-space. 2. Preliminary results about AP(n,0)-spaces In this section we established some preliminary results on AP(n,0)- spaces which are going to be used in next sections. Suppose M is a metric space and ε > 0. We write d (M) < ε if n M can be covered by an open family γ such that diamU < ε for all U ∈ γ and ord(γ) ≤ n + 1 (the last inequality means that at most n+1 elements of γ can have a common point). It is easily seen that if F ⊂ M is closed and d (F) < ε, then F is covered by an open in n M family γ of mesh < ε and ord ≤ n + 1. We also agree to denote by cov(M) the family of all open covers of M. Let us begin with the following technical lemma. Lemma 2.1. Let Z = A ∪ B be a compactum, where A and B are closed subsets of Z. Suppose C ⊂ Z is closed such that dimC ∩A ≤ 0 and d (C ∩B) < ε. Then d (C) < ε. 0 0 Proof. Since d (C ∩ B) < ε, there exists a disjoint open family γ = 0 {W ,..,W } in B of mesh < ε. We extend every W to an open set 1 k i W˜ in Z such that W˜ ∩ B = W and γ˜ = {W˜ : i = 1,..,k} is a i i i i disjoint family of mesh < ε (this can be done because B is closed Approximation by light maps 5 in Z). Observe that C = C\ i=kW˜ is a closed subset of C ∩ A 1 i=1 i disjoint from B. Since dimC ∩A ≤ 0, there exists a clopen set C in 2 S C ∩ A disjoint from B and containing C . Obviously, C is clopen in 1 2 C. Moreover, dimC ≤ 0. Hence, there exists a cover ω = {V : j = 2 j 1,..,m} ∈ cov(C ) consisting of clopen subsets of C with diamV < ε 2 2 j for every j. Then ω = ω ∪{W˜ ∩C\C } is a disjoint open cover of C 1 i 2 and mesh(ω ) < ε. So, d (C) < ε. (cid:3) 1 0 The proof of next lemma is extracted from [2, Theorem 4]. Lemma 2.2. Let G ⊂ C(X,M), where (M,̺) is a complete metric space. Suppose (U(i) is a sequence of open subsets of C(X,M) such i≥1 that • for any g ∈ G, i ≥ 1 and any function η ∈ C(X,(0,1]) there exists g ∈ B (g,η)∩U(i)∩G which is η-homotopic to g. i ̺ Then, for any g ∈ G and ε: X → (0,1] there exists g′ ∈ ∞ U(i) and i=1 an ε-homotopy connecting g and g′. Moreover, g′|A = g |A for some 0 T g ∈ C(X,M) and A ⊂ X provided g |A = g |A for all i. 0 i 0 Proof. For fixed g ∈ G and ε ∈ C(X,(0,1]) let g = g and ε = ε/3. 0 0 We shall construct by induction a sequence (g : X → M) ⊂ G, a i i≥1 sequence (ε ) of positive functions, and a sequence (H : X×[0,1] → i i≥1 i M) of ε -homotopies satisfying the conditions: i≥1 i−1 • H (x,0) = g (x) and H (x,1) = g (x) for every x ∈ X; i+1 i i+1 i+1 • g ∈ B (g ,ε )∩U(i+1)∩G; i+1 ̺ i i • ε ≤ ε /2; i+1 i • B (g ,3ε ) ⊂ U(i+1). ̺ i+1 i+1 Assumethat,forsomei,wehavealreadyconstructedmapsg ,...,g , 1 i positive numbers ε ,...,ε , and homotopies H ,...,H satisfying the 1 i 1 i above conditions. According to the hypotheses, there exists a map g ∈ B (g ,ε ) ∩ U(i + 1) ∩ G such that g is ε -homotopic to g . i+1 ̺ i i i+1 i i Let H : X × [0,1] → M be an ε -homotopy connecting the maps i+1 i g and g . Since the set U(i + 1) is open in C(X,M), there is a i i+1 positive function ε ≤ ε /2 such that B (g ,3ε ) ⊂ U(i+1). This i+1 i ̺ i+1 i+1 completes the inductive step. It follows from the construction that the function sequence (g ) i i≥1 converges uniformly to some continuous function g′ : X → M. Ob- viously, ̺(g′,g ) ≤ ∞ ε ≤ 2ε for every i. Hence, according to i j=i j i the choice of the sequences (ε ) and (g ), g′ ∈ U(i) for every i ≥ 1. i i So, g′ ∈ ∞ U(i).PMoreover, the ε -homotopies H compose an i=1 i−1 i T 6 ε-homotopy H : X ×[0,1] → M 1 1 1 H x,2i(t−1+ ) if t ∈ [1− ,1− ], i ≥ 1; H(x,t) = i 2i−1 2i−1 2i  g′((cid:0)x) (cid:1) if t = 1.  connecting g = g and g′. 0 If follows from our construction that g′|A = g |A if g |A = g |A for 0 i 0 all i ≥ 1. (cid:3) Proposition 2.3. Let M be a completely metrizable space with the AP(n,0)-property and X be an n-dimensional compactum. Then E = {g ∈ C(X,M) : dimg ≤ 0} is a G -subset of C(X,M) such that every simplicially factorizable map δ is homotopically approximated by maps from E. Proof. We fix a metric on X and a complete metric ̺ on M. For every ε > 0 let C(X,M;ε) be the set all maps g ∈ C(X,M) with d (g−1(a)) < ε for every a ∈ M. It is easily seen that C(X,M;ε) is 0 open in C(X,M) and E = ∞ C(X,M;1/i). So, E is a G -subset of i=1 δ C(X,M). T Claim. For every simplicially factorizable map h ∈ C(X,M) and positive numbers η and δ there exists a simplicially factorizable map h′ ∈ C(X,M;η) which is δ-homotopic to h. To prove the claim, fix a simplicially factorizable map h ∈ C(X,M) and positive numbers δ, η. Then there exists a simplicial complex L and maps q : X → L, q : L → M with h = q ◦ q . We can assume 1 2 2 1 that L is finite and q (L) ⊂ ∪γ, where γ = {B (y,δ/4) : y ∈ g(X)}. 2 ̺ Because L is an ANR and dimX ≤ n, there is a polyhedron K with dimK ≤ n and two maps f: X → K, α : K → L such that f is an η-map and q and α ◦ f are q−1(γ)-homotopic. So, h and q ◦ α ◦ f 1 2 2 are δ/2-homotopic. Moreover, there is a cover ω ∈ cov(K) such that diamf−1(W) < η for every W ∈ ω. Let θ be a Lebesgue number of ω. It remains to find a map h∗: K → M which is δ/2-homotopic to q ◦ α and h∗ ∈ C(K,M;θ) (then h′ = h∗ ◦ f would be a simplicially 2 factorizable map δ-homotopic to h and h′ ∈ C(X,M;η)). To find such a map h∗: K → M, let {σ ,..,σ } be an enumeration 1 m of the simplexes of K and K = j=i σ . We are going to construct i j=1 j by induction maps h : K → M, i = 0,..,m, satisfying the following i S conditions: • h = q ◦α; 0 2 • h |K belongs to C(K ,M;θ), 1 ≤ i ≤ m; i i i • h and h are (δ/2m)-homotopic, i = 0,..,m−1. i i+1 Approximation by light maps 7 Let V = {B (y,δ/4m) : y ∈ M} ∈ cov(M) and assume that h ̺ i has already been constructed. Since h |K ∈ C(K ,M;θ), every fiber i i i h−1(y)∩K of h |K , y ∈ h (K ), is covered by a finite openand disjoint i i i i i 1 family Ω(y) in K with mesh(Ω(y)) < θ. Using that h |K is a perfect i i i map, we find a cover V ∈ cov(M) such that V is a star-refinement of V i i and if St(y,V )∩h (K ) 6= ∅ for some y ∈ M, then there is z ∈ h (K ) i i i i i with h−1(St(y,V ))∩K ⊂ Ω(z). Since M has the AP(n,0)-property, i i i there exists a 0-dimensional map p : σ → M which is V -homotopic i i+1 i to h |σ . By the Homotopy Extension Theorem, p can be extended i i+1 i toamaph : K → M beingV -homotopictoh . Thenh is(δ/2m)- i+1 i i i+1 homotopic to h . To show that h |K belongs to C(K ,M;θ), we i i+1 i+1 i+1 observe that h−1 (y)∩K = h−1 (y)∩K ∪ h−1 (y)∩σ , y ∈ M. i+1 i+1 i+1 i i+1 i+1 According to our construction, we have h h−1 (y)∩ K ⊂ St(y,V ). (cid:0) i(cid:1) i+(cid:0)1 i (cid:1) i Hence, h−1 (y)∩K iscontainedinΩ(z)forsomez ∈ h (K ). Therefore, i+1 i (cid:0) i (cid:1)i d h−1 (y) ∩ K < θ. Since h−1 (y) ∩ σ is 0-dimensional, Lemma 0 i+1 i i+1 i+1 2.1 implies that d (h−1 (y)∩ K ) < θ. Obviously the map h∗ = h (cid:0) (cid:1) 0 i+1 i+1 m is δ/2-homotopic to q ◦ α and h∗ ∈ C(K,M;θ). This completes the 2 proof of the claim. To finish the proof of the proposition, we apply Lemma 2.2 with G being the set of all simplicially factorizable maps from C(X,M) and U(i) = C(X,M;1/i), i ≥ 1 (we can apply Lemma 2.2 because of the claim). Hence, for every simplicially factorizable map g ∈ C(X,M) and a positive number ε there exists a map g′ ∈ ∞ U(i) which is i=1 ε-homotopic to g. Finally, we observe that ∞ U(i) consists of 0- i=1 T (cid:3) dimensional maps. T Proposition 2.4. Let M be a completely metrizable space with the AP(n,0)-property, X a compactum and Z an F -subset of X with σ dimZ ≤ n − 1. Then there exists a G -subset H ⊂ C(X,M) with δ the following properties: • Z is contained in the union of trivial components of the fibers of g for all g ∈ H; • for every simplicially factorizable map g ∈ C(X,M) and every ε > 0 there exists g′ ∈ H which is ε-homotopic to g. Proof. We represent Z as the union of an increasing sequence (Z ) i i≥1 with all Z being closed in Z. For every ε > 0 let i H(Z ,ε) = {g ∈ C(X,M) : F(g,ε)∩Z = ∅}, i i where F(g,ε) = ∪{C : C is a component of a fiber of g with diamC ≥ ε}. 8 It is easily seen that each H(Z ,ε) is open in C(X,M). So, the set i H = ∞ H(Z ,1/i) is G and Z is contained in the union of trivial i=1 i δ components of the fibers of g for every g ∈ H. T To prove the second item of our proposition, we first consider the particular case when X is a polyhedron and Z is a (compact) subpolyhedron of X. Claim 1. Suppose that, in addition to hypotheses of Proposition 2.4, X is a polyhedron and Z is a subpolyhedron of X. Then, for every g ∈ C(X,M) and δ > 0 there exists g ∈ H = ∞ H(Z,1/i) which is 0 i=1 δ-homotopic to g . 0 T The proof of Claim 1 is a slight modification of the proof of [12, Theorem 2.2]. Let g ∈ C(X,M) and δ > 0. We take an open 0 neighborhood W of Z in X, a retraction r: W → Z and a function α: X → I such that α−1(0) = Z, α−1(1) = X\W. Since X is a compact ANR, we can choose W so small that g |W is δ/2-homotopic 0 to (g ◦ r)|W. Next, denote by π: Z × I → Z the projection and 0 consider the map ϕ: W → Z × I, ϕ(x) = (r(x),α(x)). Obviously, (g ◦ r)|W = (g ◦ π ◦ ϕ)|W. So, (g ◦ π ◦ ϕ)|W is δ/2-homotopic to 0 0 0 g |W. Because Z is a polyhedron, so is Z × I. Hence, every map 0 from C(Z×I,M) is simplicially factorizable. Moreover, dimZ×I ≤ n and M ∈ AP(n,0). Therefore, by Proposition 2.3, there exists a 0- dimensional map h: Z × I → M which is δ/2-homotopic to the map g ◦π. Then (h◦ϕ)|W is δ/2-homotopicto(g ◦π◦ϕ)|W. Consequently, 0 0 (h◦ϕ)|W is δ-homotopic to g |W. By the Homotopy Extension Theo- 0 rem, there exits a map g ∈ C(X,M) such that g is δ-homotopic to g 0 and g|U = (h◦ϕ)|U, where U is an open neighborhood of Z in X with U ⊂ W. To finish the proof of Claim 1, it remains to show that g ∈ H. Let C be a subcontinuum of g−1(y) for some y ∈ M and let Z ∩ C 6= ∅. We are going to prove that C ⊂ Z. Otherwise, there would be a subcontinuum C′ ⊂ C ∩U such that C′ ∩Z 6= ∅ and C′\Z 6= ∅. Then g(C′) = h(ϕ(C′)) = y and, according to the definition of α, ϕ(C′) is a non-degenerate continuum in h−1(y). Since h is 0-dimensional, this is a contradiction. Hence, C ⊂ Z. Using again that dim(h) ≤ 0, we conclude that ϕ(C) is a point. On the other hand ϕ(C) = C ×{0} ⊂ Z ×I. Therefore, C should be a trivial continuum. Now, consider the general case of Proposition 2.4. Claim 2. Let g ∈ C(X,M) be a simplicially factorizable map and δ, 0 η positive numbers. Then for any i there exists a simplicially factorizable map g ∈ H(Z ,η) which is δ-homotopic to g . i 0 Approximation by light maps 9 Since g is simplicially factorizable, there exist a finite simplicial 0 complex L and maps q : X → L, q : L → M with g = q ◦ q . Let 1 2 0 2 1 V = {B (y,δ/4) : y ∈ M} ∈ cov(M) and W = q−1(V) ∈ cov(L). Next, ̺ 2 choose a finite cover U ∈ cov(X) of X with mesh(U) < η such that: • at most n elements of the family γ = {U ∈ U : U ∩Z 6= ∅} can i have a common point; • there exists a map h: N(U) → L such that h ◦ f is W- U homotopic to q , where N(U) is the nerve of U and f : X → 1 U N(U) is the natural map. Then q ◦ h ◦ f is δ/2-homotopic to g and the subpolyhedron K 2 U 0 of N(U) generated by the family γ is of dimension ≤ n − 1. So, according to Claim 1, there exists a map g : N(U) → M such that 1 g is δ/2-homotopic to q ◦h and K is contained in the union of trivial 1 2 components of the fibers of g . Since allfibers of f are of diameter < η 1 U and f (Z ) ⊂ K, the map g = g ◦f belongs to H(Z ,η). Moreover, U i 1 U i g is obviously simplicially factorizable and δ-homotopic to g . This 0 completes the proof of Claim 2. Finally, the proof of Proposition 2.4 follows from Lemma 2.2 (with G being the set of all simplicially factorizable maps from C(X,M) and U(i) = H(Z ,1/i) for every i ≥ 1) and Claim 2. (cid:3) i 3. Parametric Lelek maps In this section we are going to prove Theorem 1.1. Everywhere in this section we suppose that (M,̺) is a given complete metric space and µ = {W : ν ∈ Λ}, µ = {G : ν ∈ Λ} are locally finite open covers ν 1 ν of M such that (*) G ⊂ W and W ∈ AP(n ,0) with 0 ≤ n for every ν ∈ Λ. ν ν ν ν ν Obviously, Theorem 1.1 follows directly from next theorem. Theorem 3.1. Let f: X → Y be a perfect map between metrizable spaces with dim△(f) ≤ m. Suppose (M,̺) is a complete metric space and µ, µ two locally finite open covers of M satisfying condition (∗). 1 Then there is a G -set H ⊂ C(X,M) such that any simplicially fac- δ torizable map in C(X,M) can be homotopically approximated by maps from H and every g ∈ H has the following property: for any y ∈ Y and ν ∈ Λ the restriction g| f−1(y)∩g−1(G ) is an (m−n )-dimensional ν ν Lelek map from f−1(y)∩g−1(G ) into G . (cid:0) ν ν(cid:1) The proof of Theorem 3.1 consists of few propositions. We can assume that n ≤ m for all ν ∈ Λ. For every y ∈ Y, ε > 0, ν ∈ Λ and ν g ∈ C(X,M) we denote by F (g,ε,y) the union of all continua C of ν diameter ≥ ε such that C ⊂ f−1(y)∩g−1(z ) for some z ∈ G . Note C C ν 10 that each F (g,ε,y) is compact as a closed subset of f−1(y). Then, ν for a fixed η > 0, let H(y,ε,η) be the set of all g ∈ C(X,M) such that d F (g,ε,y) < η for every ν ∈ Λ. The last inequality means m−nν ν that F (g,ε,y) can be covered by an open family γ in X such that ν (cid:0) (cid:1) mesh(γ) < η and no more m − n + 1 elements of γ have a com- ν mon point. If V ⊂ Y, then H(V,ε,η) denotes the intersection of all H(y,ε,η), y ∈ V. Lemma 3.2. For every y ∈ Y and every g ∈ H(y,ε,η) there exists a neighborhood V of y in Y and δ > 0 such that if y′ ∈ V and y y y ̺(h(x),g(x)) < δ for all x ∈ f−1(y′), then h ∈ H(y′,ε,η). y Proof. Since µ is locally finite and f−1(y) is compact, for every y ∈ Y 1 and g ∈ C(X,M) there exists a neighborhood O (y) of g(f−1(y)) such g that the family Λ (y) = {ν ∈ Λ : O (y)∩G 6= ∅} is finite. g g ν Assume the lemma is not true for some y ∈ Y and g ∈ H(y,ε,η). Then there exists a local base of neighborhoods V of y, points y ∈ i i V and functions g ∈ C(X,M) such that g |f−1(y ) is 1/i-close to i i i i g|f−1(y ) but g 6∈ H(y ,ε,η). It is easily seen that for some k and i i i all i ≥ k we have g (f−1(y )) ⊂ O (y) . Consequently, for every i ≥ k i i g thereexistsν(i) ∈ Λ (y)suchthatd F (g ,ε,y ) ≥ η (because g m−nν(i) ν(i) i i g 6∈ H(y ,ε,η)). So, each F (g ,ε,y ) doesn’t have any open cover i i ν(i) i i (cid:0) (cid:1) of mesh < η and order ≤ m − n + 1. Since the family Λ (y) is ν(i) g finite, there exists ν(0) ∈ Λ (y) with ν(i) = ν(0) for infinitely many g i. With out loss of generality, we may suppose that ν(i) = ν(0) for all i ≥ k. This implies that g (f−1(y ))∩G 6= ∅, i ≥ k, where G = G . i i ν(0) Consequently, g(f−1(y)) ∩ G 6= ∅. Since g ∈ H(y,ε,η), F (g,ε,y) ν(0) can be covered by an open family γ in X of order ≤ m − n + 1 ν(0) and mesh(γ) < η. Let U = ∪γ. To obtain a contradiction, it suffice to show that F (g ,ε,y ) ⊂ U for some i ≥ k. Indeed, otherwise for ν(i) i i every i ≥ k there would exist points x ∈ F (g ,ε,y )\U, z ∈ G, and i ν(i) i i i a component C of f−1(y ) ∩ g−1(z ) containing x with diamC ≥ ε. i i i i i i Using that P = f−1({y }∞ ∪ {y}) is a compactum, we can suppose i i=k that {x }∞ converges to a point x ∈ f−1(y), {z }∞ converges to a i i=k 0 i i=k point z ∈ G and {C }∞ (considered as a sequence in the space of all 0 i i=k closed subsets of P equipped with the Vietoris topology) converges to a closed set C ⊂ f−1(y)∩g−1(z )). It is easily seen that C is connected 0 and diamC ≥ ε. Hence, C ⊂ F (g,ε,y) ⊂ U. So, x ∈ C ⊂ U for ν(0) i i some i, a contradiction. (cid:3) Now, we are in a position to show that the sets H(Y,ε,η) are open in C(X,M).

Approximation by light maps and parametric Lelek maps

Similar Approximation by light maps and parametric Lelek maps