OPEN MAPS HAVING THE BULA PROPERTY 8 VALENTIN GUTEV AND VESKO VALOV 0 0 2 Abstract. Everyopencontinuousmapf fromaspaceX ontoaparacompact n C-spaceY admitstwodisjointclosedsetsF0,F1 ⊂X,withf(F0)=Y =f(F1), a provided all fibers of f are infinite and C∗-embedded in X. Applications are J demonstrated for the existence of “disjoint” usco multiselections of set-valued 2 l.s.c. mappings defined on paracompact C-spaces, and for special type of fac- 1 torizations of open continuous maps from metrizable spaces onto paracompact C-spaces. This settles several open questions raised in [13]. ] N G . h 1. Introduction t a m Allspaces inthispaperareassumed tobeatleast completely regular. Following Kato and Levin [15], a continuous surjective map f: X −→ Y is said to have the [ Bula property if there exist two disjoint closed subsets F and F of X such that 1 0 1 v f(F0) = Y = f(F1). Inthesequel, suchapair(F0,F1)willbecalledaBula pair for 4 f. Bula [2] proved that every open continuous map f from a compact Hausdorff 0 space onto a finite-dimensional metrizable space has this property provided all 9 1 fibers of f are dense in themselves. This result was generalized in [9] to the case . 1 Y is countable-dimensional. Recently, Levin and Rogers [16] obtained a further 0 generalization with Y being a C-space. The question whether the Levin-Rogers 8 result remains true for open maps between metrizable spaces was raised in [13, 0 : Problem 1514] (if Y is not a C-space, this is not true, see [6] and [16]). Here, we v i provide a positive answer to this question: X r Theorem 1.1. Let X be a space, Y be a paracompact C-space, and let f: X −→ Y a be an open continuous surjection such that all fibers of f are infinite and C∗- embedded in X. Then, f has the Bula property. The C-space property was originally defined by W. Haver [11] for compact metric spaces. Later on, Addis and Gresham [1] reformulated Haver’s defini- tion for arbitrary spaces: A space X has property C (or X is a C-space) if for Date: February 2, 2008. 2000 Mathematics Subject Classification. Primary 54F45, 54F35, 54C60, 54C65; Secondary 55M10, 54C35, 54B20. Key words and phrases. Dimension, C-space, set-valued mapping, selection. Research of the first author is supported in part by the NRF of South Africa. The second author was partially supported by NSERC Grant 261914-03. 1 2 VALENTIN GUTEV AND VESKOVALOV every sequence {W : n = 1,2,...} of open covers of X there exists a sequence n {V : n = 1,2,...}ofpairwisedisjoint openfamiliesinX suchthateachV refines n n W and {V : n = 1,2,...} is a cover of X. It is well-known that every finite- n n S dimensionalparacompactspace, aswellaseverycountable-dimensional metrizable space, is a C-space [1], but there exists a compact metric C-space which is not countable-dimensional [24]. Let us also remark that a C-space X is paracompact if and only if it is countably paracompact and normal. Finally, let us recall that a subset A ⊂ X is C∗-embedded in X if every bounded real-valued continuous function on A is continuously extendable to the whole of X. Theorem1.1hasseveralinterestingapplications. InSection4,weapplythisthe- orem to the graph of l.s.c. set-valued mappings defined on paracompact C-spaces and with point-images being closed and infinite subsets of completely metrizable spaces. Thus, we get that any such l.s.c. mapping has a pair of “disjoint” usco multiselections (see, Corollaries 4.3 and 4.4), which provides the complete affir- mative solution to [13, Problem 1515] and sheds some light on [13, Problem 1516]. In this regard, let us stress the attention that, in Theorem 1.1, no restrictions on X are called a priori. In Section 5, we use Theorem 1.1 to demonstrate that every open continuous map f from a metric space (X,d) onto a paracompact C-space Y admits a special type of factorization (Y ×[0,1] throughout), provided all fibers of f are dense in themselves and complete with respect to d, see Theorem 5.1. This result is a common generalization of [9, Theorem 1.1] and [16, Theorem 1.2] (see, Corollary 5.10), and provides the complete affirmative solution to [13, Problem 1512]. Finally, a word should be said also for the proof of Theorem 1.1 itself. Briefly, a preparation for this is done in the next section. It is based on the existence of a continuous function g : X −→ [0,1] such that g is not constant on each fiber f−1(y), y ∈ Y, of f (see, Theorem 2.1). Having already established this, the proof of Theorem 1.1 will be accomplished in Section 3 relying on a “parametric” version of an idea in the proof of [16, Theorem 1.3]. 2. Bula property and fiber-constant maps Suppose that (F ,F ) is a Bula pair for a map f : X −→ Y, where X is a 0 1 normal space. Then, there exists a continuous function g : X −→ [0,1] such that g↾f−1(y) is not constant for every y ∈ Y. Indeed, take g : X −→ [0,1] to be such that F ⊂ g−1(i), i = 0,1. In this section, we demonstrate that the map f i in Theorem 1.1 has this property as well. Namely, the following theorem will be proved. OPEN MAPS HAVING THE BULA PROPERTY 3 Theorem 2.1. Let X be a space, Y be a paracompact C-space, and let f: X −→ Y be an open continuous surjection such that all fibers of f are infinite and C∗- embedded in X. Then, there exists a continuous function g : X −→ [0,1] such that g↾f−1(y) is not constant for every y ∈ Y. To prepare for the proof of Theorem 2.1, let us recall some terminology. For spaces Y and Z, we will use Φ : Y Z to denote that Φ is a set-valued mapping, i.e. a map from Y into the nonempty subsets of Z. A mapping Φ : Y Z is lower semi-continuous, or l.s.c., if the set Φ−1(U) = {y ∈ Y : Φ(y)∩U 6= ∅} is open in Y for every open U ⊂ Z. A mapping Φ : Y Z has an open (closed) graph if its graph Graph(Φ) = (y,z) ∈ Y ×Z : z ∈ Φ(y) (cid:8) (cid:9) is open (respectively, closed) in Y × Z. A map g : Y −→ Z is a selection for Φ : Y Z if g(y) ∈ Φ(y) for every y ∈ Y. Finally, let us recall that a space Z is Cm for some m ≥ 0 if every continuous image of the k-dimensional sphere Sk (k ≤ m) in Z is contractible in Z. In what follows, I = [0,1]and C(Z,I) denotes the set of all continuous functions from Z to I. Also, C(Z) = C(Z,R) is the set of all continuous functions on Z, and C∗(Z) — that of all bounded members of C(Z). As usual, C∗(Z) is equipped with the sup-metric d defined by ∗ d(g,h) = sup |g(z)−h(z)| : z ∈ Z , g,h ∈ C (Z). (cid:8) (cid:9) It should be mentioned that C∗(Z) is a Banach space, and C(Z,I) is a closed convex subset of C∗(Z). In the sequel, for g ∈ C(Z,I) and ε > 0, we will use Bd(g) to denote the open ε-ball Bd(g) = {h ∈ C(Z,I) : d(g,h) < ε}. ε ε The next statement is well-known and easy to prove. Lemma 2.2. Let X be a space, and let A ⊂ X be a C∗-embedded subset of X. Then, the restriction map π : C(X,I) −→ C(A,I) is an open continuous A surjection. For a subset B of a space Z, let Θ (B,I) be the set of all members of C(Z,I) Z which areconstant onB. If B = Z, then we will denote this set merely by Θ(Z,I). Note that Θ(Z,I) is, in fact, homeomorphic to I. Proposition 2.3. Let X be a space, and let A ⊂ X be an infinite C∗-embedded subset of X. Then, the set C(X,I)\Θ (A,I) is Cm for every m ≥ 0. X Proof. Consider the restriction map π : C(X,I) −→ C(A,I), and take a contin- A uous map g : Sn −→ C(X,I) \ Θ (A,I) for some n ≥ 0. Then, by Lemma 2.2, X the composition π ◦g : Sn −→ C(A,I)\Θ(A,I) is also continuous. Observe that A 4 VALENTIN GUTEV AND VESKOVALOV C(A,I) is an infinite-dimensional closed convex subset of C∗(A) because A is in- finite. From another hand, Θ(A,I) is one-dimensional being homeomorphic to I. Then, by [20, Lemma 2.1], C(A,I) \ Θ(A,I) is Cm for all m ≥ 0. Hence, there exists acontinuous extension ℓ : Bn+1 −→ C(A,I)\Θ(A,I)ofπ ◦g over the(n+1)- A dimensional ball Bn+1. Consider the set-valued mapping Φ : Bn+1 C(X,I) de- fined by Φ(t) = {g(t)} if t ∈ Sn and Φ(t) = π−1(ℓ(t)) otherwise. Since g is a A selection for π−1 ◦ ℓ↾Sn and, by Lemma 2.2, the restriction map is π is open, A A the mapping Φ is l.s.c. (see, [18, Examples 1.1∗ and 1.3∗]). Also, Φ is closed and convex-valued in C(X,I), hence in the Banach space C∗(X) as well. Then, by the Michael’s selection theorem [18, Theorem 3.2′′], Φ has a continuous selection h : Bn+1 −→ C(X,I) which is, infact, a continuous extension ofg over Bn+1. More- over, π (h(t)) = ℓ(t) ∈/ Θ(A,I) for all t ∈ Bn+1, which completes the proof. (cid:3) A A function ξ : X −→ R is lower (upper) semi-continuous if the set {x ∈ x : ξ(x) > r} (respectively, {x ∈ X : ξ(x) < r}) is open in X for every r ∈ R. Suppose that f : X −→ Y is a surjective map. Then, to any g : X −→ I we will associate the functions inf[g,f],sup[g,f] : Y −→ I defined for y ∈ Y by inf[g,f](y) = inf g(x) : x ∈ f−1(y) , (cid:8) (cid:9) and, respectively, sup[g,f](y) = sup g(x) : x ∈ f−1(y) . (cid:8) (cid:9) Finally, we will also associate the function var[g,f] : X −→ I defined by var[g,f](y) = sup[g,f](y)−inf[g,f](y), y ∈ Y. Observe that g : X −→ I is not constant on any fiber f−1(y), y ∈ Y, if and only if var[g,f] is positive-valued. The following property is well-known, [12] (see, also, [7, 1.7.16]). Proposition 2.4 ([12]). Let X and Y be spaces, f : X −→ Y be an open surjective map, and let g ∈ C(X,I). Then, sup[g,f] is lower semi-continuous, while inf[g,f] is upper semi-continuous. In particular, var[g,f] is lower semi-continuous. We finalize the preparation for the proof of Theorem 2.1 with the following proposition. Proposition 2.5. Let X and Y be spaces, and let f : X −→ Y be an open surjective map. Then the set-valued mapping Θ : Y C(X,I) defined by Θ(y) = Θ (f−1(y),I), y ∈ Y, has a closed graph. X Proof. Take a pointy ∈ Y andg ∈/ Θ(y). Then, var[g,f](y) > 2δ for some positive number δ > 0. By Proposition 2.4, there exists a neighbourhood V of y such that var[g,f](z) > 2δ for every z ∈ V. Then, V ×Bd(g) is an open set in Y ×C(X,I) δ OPEN MAPS HAVING THE BULA PROPERTY 5 such that V ×Bd(g) ∩Graph(Θ) = ∅. Indeed, take z ∈ V and h ∈ Bd(g). Since (cid:0) δ (cid:1) δ var[g,f](z) > 2δ, there are points x,t ∈ f−1(z) such that |g(x)−g(t)| > 2δ. Since h ∈ Bd(g), we have |h(x)−g(x)| < δ and |h(t)−g(t)| < δ. Hence, h(x) 6= h(t), δ which implies that var[h,f](z) > 0. Consequently, h ∈/ Θ(z). (cid:3) Proof of Theorem 2.1. Consider the set-valued mapping Φ : Y C(X,I) defined by Φ(y) = C(X,I) \ Θ(y), y ∈ Y, where Θ is as in Proposition 2.5. Then, by Proposition2.5, Φ hasanopengraph, while, by Proposition2.3, each Φ(y), y ∈ Y, is Cm for all m ≥ 0. Since Y is a paracompact C-space, by the Uspenskij’s selec- tion theorem [26, Theorem 1.3], Φ has a continuous selection ϕ : Y −→ C(X,I). Define a map g : X −→ I by g(x) = [ϕ(f(x))](x), x ∈ X. Since f and ϕ are contin- uous, so is g (see, the proof of [10, Theorem 6.1]). Since g↾f−1(y) = ϕ(y)↾f−1(y) and ϕ(y) ∈/ Θ(y) for every y ∈ Y, g is as required. (cid:3) 3. Proof of Theorem 1.1 Suppose that X, Y and f : X −→ Y are as in Theorem 1.1. By Theorem 2.1, there exists a function g ∈ C(X,I) such that inf[g,f](y) < sup[g,f](y) for every y ∈ Y. Since inf[g,f] is upper semi-continuous and sup[g,f] is lower semi- continuous (by Proposition 2.4), and Y is paracompact, by a result of [3] (see, also, [5, 14]), there are continuous functions γ ,γ : Y −→ I such that 0 1 inf[g,f](y) < γ (y) < γ (y) < sup[g,f](y), y ∈ Y. 0 1 Let α = γ ◦f : X −→ I, i = 0,1. Then, i i (3.1) inf[g,f](f(x)) < α (x) < α (x) < sup[g,f](f(x)) for every x ∈ X. 0 1 Next, define a continuous function ℓ : X ×I −→ R by letting t−α (x) ℓ(x,t) = 0 , (x,t) ∈ X ×I. α (x)−α (x) 1 0 Observe that ℓ(x,α (x)) = 0 and ℓ(x,α (x)) = 1 for every x ∈ X. Hence, 0 1 (3.2) ℓ {x}×[α (x),α (x)] = [0,1], for every x ∈ X, 0 1 (cid:0) (cid:1) because ℓ is linear for every fixed x ∈ X. Finally, define a continuous function h : X −→ R by h(x) = ℓ(x,g(x)), x ∈ X. According to (3.1) and (3.2), we now have that, for every y ∈ Y, h−1 (−∞,0] ∩f−1(y) 6= ∅ 6= h−1 [1,+∞) ∩f−1(y). (cid:0) (cid:1) (cid:0) (cid:1) Then, F = h−1 (−∞,0] and F = h−1 [1,+∞) are as required. The proof of 0 1 (cid:0) (cid:1) (cid:0) (cid:1) Theorem 1.1 completes. 6 VALENTIN GUTEV AND VESKOVALOV 4. Bula pairs and multiselections A set-valued mapping ϕ : Y Z is called a multiselection for Φ : Y Z if ϕ(y) ⊂ Φ(y) for every y ∈ Y. In this section, we present several applications of Theorem 1.1 about multiselections of l.s.c. mappings based on the following consequence of it. Corollary 4.1. Let Y be a paracompact C-space, Z be a normal space, and let Φ : Y Z be an l.s.c. mapping such that each Φ(y), y ∈ Y, is infinite and closed in Z. Then, there exists a closed-graph mapping θ : Y Z such that Φ(y)∩θ(y) 6= ∅ 6= Φ(y)\θ(y) for every y ∈ Y. Proof. LetX = Graph(Φ)bethegraphofΦ,andletf : X −→ Y betheprojection. Then, f is an open continuous map (because Φ is l.s.c.) such that all fibers of f are infinite. Let us observe that each f−1(y), y ∈ Y, is C∗-embedded in X. Indeed, take a point y ∈ Y, and a continuous function g : f−1(y) −→ I. Since f−1(y) = {y} × Φ(y), we may consider the continuous function g : Φ(y) −→ I 0 defined by g (z) = g(y,z), z ∈ Φ(y). Since Z is normal, there exists a continuous 0 extension h : Z −→ I of g . Finally, define h : X −→ I by h(t,z) = h (z) for every 0 0 0 t ∈ Y and z ∈ Φ(t). Then, h is a continuous extension of g. Thus, by Theorem 1.1, there are disjoint closed subsets F ,F ⊂ X such that f(F ) = Y = f(F ). 0 1 0 1 Finally, take a closed set F ⊂ Y ×Z, with F ∩X = F , and define θ : Y Z by 0 Graph(θ) = F. This θ is as required. (cid:3) To prepare for our applications, we need also the following observation about l.s.c. multiselections of l.s.c. mappings. Proposition 4.2. Let Y be a paracompact space, Z be a space, Φ : Y Z be an l.s.c. closed-valued mapping, and let Ψ : Y Z be an open-graph mapping, with Φ(y)∩ Ψ(y) 6= ∅ for every y ∈ Y. Then, there exists a closed-valued l.s.c. mapping ϕ : Y Z such that ϕ(y) ⊂ Φ(y)∩Ψ(y) for every y ∈ Y. Proof. Whenever y ∈ Y, there are open sets V ⊂ Y and W ⊂ Z such that y y y ∈ V ⊂ Φ−1(W )andV ×W ⊂ Graph(Ψ). Indeed, takeapointz ∈ Φ(y)∩Ψ(y). y y y y Since Ψ has an open graph, there are open sets O ⊂ Y and W ⊂ Z such that y y y ∈ O , z ∈ W and O × W ⊂ Graph(Ψ). Then, V = O ∩ Φ−1(W ) is as y y y y y y y required. Now, for every y ∈ Y, define a closed-valued mapping ϕ : V Z by y y letting that ϕ (t) = Φ(t)∩W , t ∈ V . According to [18, Propositions 2.3 and y y y 2.4], each ϕ , y ∈ Y, is l.s.c. Next, using that Y is paracompact, take a locally- y finite open cover U of Y refining {V : y ∈ Y} and a map p : U −→ Y such that y U ⊂ V , U ∈ U . Finally, define a mapping ϕ : Y Z by letting that p(U) ϕ(y) = ϕ (y) : U ∈ U and y ∈ U , y ∈ Y. [(cid:8) p(U) (cid:9) This ϕ is as required. (cid:3) OPEN MAPS HAVING THE BULA PROPERTY 7 In what follows, a mapping ψ : Y Z is upper semi-continuous, or u.s.c., if the set Φ#(U) = {y ∈ Y : Φ(y) ⊂ U} is open in Y for every open U ⊂ Z. Motivated by [19], we say that a pair (ϕ,ψ) of set-valued mapping ϕ,ψ : Y Z is a Michael pair for Φ : Y Z if ϕ is compact- valued and l.s.c., ψ is compact-valued and u.s.c., and ϕ(y) ⊂ ψ(y) ⊂ Φ(y) for every y ∈ Y. The following consequence provides the complete affirmative solution to [13, Problem 1515]. Corollary 4.3. Let (Z,ρ) be metric space, Y be a paracompact C-space, and let Φ : Y Z be an l.s.c. mapping such that each Φ(y), y ∈ Y, is infinite and ρ- complete. Then Φ has Michael pair (ϕ,ψ) : Y Z such that Φ(y)\ψ(y) 6= ∅ for every y ∈ Y. Proof. By Corollary 4.1, there is a closed-graph mapping θ : Y Z such that Φ(y)∩θ(y) 6= ∅ 6= Φ(y)\θ(y) for every y ∈ Y. Consider the set-valued mapping Ψ : Y Z defined by Graph(Ψ) = (Y × Z) \ Graph(θ). On one hand, by the properties of θ, we have that Φ(y)∩Ψ(y) 6= ∅ for every y ∈ Y. On another hand, Φ is closed-valued having ρ-complete values. Hence, by Proposition 4.2, there exists a closed-valued l.s.c. mapping Φ : Y Z such that Φ (y) ⊂ Φ(y)∩Ψ(y) 0 0 for every y ∈ Y. Then, Φ has also ρ-complete values and, by a result of [19], it 0 has a Michael pair (ϕ,ψ). This (ϕ,ψ) is as required. (cid:3) We conclude this section with the following further application of Theorem 1.1 that sheds some light on [13, Problem 1516]. Corollary 4.4. Let (Z,ρ) be metric space, Y be a paracompact C-space, and let Φ : Y Z be an l.s.c. mapping such that each Φ(y), y ∈ Y, is infinite and ρ- complete. Then Φ has Michael pairs (ϕ ,ψ ) : Y Z, i = 0,1, such that ψ (y)∩ i i 0 ψ (y) = ∅ for every y ∈ Y. 1 Proof. According to Corollary 4.3, Φ has a Michael pair (ϕ ,ψ ) : Y Z such 0 0 that Φ(y) \ ψ (y) 6= ∅ for every y ∈ Y. Note that ψ has a closed-graph being 0 0 u.s.c. Then, just like in the proof of Corollary 4.3, there exists a Michael pair (ϕ ,ψ ) : Y Z for Φ such that ψ (y) ⊂ Φ(y) \ ψ (y), y ∈ Y. These (ϕ ,ψ ), 1 1 1 0 i i i = 0,1, are as required. (cid:3) 5. Open maps looking like projections Throughout this section, by a dimension of a space Z we mean the covering dimension dim(Z) of Z. In particular, Z is 0-dimensional if dim(Z) = 0. We say that a continuous map f : X −→ Y has dimension ≤ k if all fibers of f have dimension ≤ k. A continuous map f : X −→ Y is light if it is 0-dimensional, 8 VALENTIN GUTEV AND VESKOVALOV i.e. if f has 0-dimensional fibers. Also, for convenience, we shall say that a map f : X −→ Y is compact if each fiber f−1(y), y ∈ Y, is a compact subset of X. Suppose that f : X −→ Y is a surjective map. A subset F ⊂ X will be called a section for f if f(F) = Y. In particular, we shall say that a section F for f is open (closed) if F is an open (respective, a closed) subset of X. In this section, we demonstrate the following factorization theorem which is a partial generalization of [9, Theorem 1.1], also it provides the complete affirmative solution to [13, Problem 1512]. Theorem 5.1. Let (X,d) be a metric space, Y be a paracompact C-space, and let f : X −→ Y be an open continuous surjection such that each fiber of f is dense in itself and d-complete. Also, let U ⊂ X be an open section for f. Then, there exists a continuous surjective map g : X −→ Y ×I, a closed section H ⊂ X for f, with H ⊂ U, and a copy C ⊂ I of the Cantor set such that (a) f = P ◦ g, where P : Y × I −→ Y is the projection, i.e. the following Y Y diagram is commutative. g X - Y ×I @ @ f P @ Y @@R ? Y (b) g(H) = Y × I and each g−1(y,c) ∩ H, (y,c) ∈ Y × C, is compact and 0-dimensional. In particular, H = H ∩g−1(Y ×C) is a closed section for f such that f ↾H is C C a compact light map. To prepare for the proof of Theorem 5.1, we introduce some terminology. For a metric space (X,d), a nonempty subset A ⊂ X and ε > 0, as in Section 2, we let Bd(A) = {x ∈ X : d(x,A) < ε}. ε Also, we will use diam (A) to denote the diameter of A with respect to d. d Following [9], to every nonempty subset F ⊂ X we associate the number δ(F,X) = inf 1,ε : ε > 0 and (cid:8) F ⊂ Bd(S) for some nonempty finite S ⊂ F . ε (cid:9) In what follows, we let Ω(f) to be the set of all open sections for f. Also, for every U ∈ Ω(f), we introduce the d-mesh of U with respect to f by letting mesh (U,f) = sup δ(f−1(y)∩U,X) : y ∈ Y . d (cid:8) (cid:9) OPEN MAPS HAVING THE BULA PROPERTY 9 Proposition 5.2. Let (X,d) be a metric space, Y be a paracompact C-space, and let f : X −→ Y be an open continuous surjection such that each fiber of f is dense in itself and d-complete. Then, for every U ∈ Ω(f) there are disjoint open sections U ,U ∈ Ω(f) such that U ⊂ U, i = 0,1. 0 1 i Proof. Consider U endowed with the compatible metric 1 1 ρ(x,y) = d(x,y)+(cid:12) − (cid:12), x,y ∈ U. (cid:12)d(x,X \U) d(y,X \U)(cid:12) (cid:12) (cid:12) Next, define an l.s.c. mapp(cid:12)ing Φ : Y X by Φ(y) =(cid:12)f−1(y)∩U, y ∈ Y. Then, each Φ(y), y ∈ Y, is infinite and ρ-complete in U because each f−1(y), y ∈ Y, is dense in itself and d-complete. Hence, by Corollary 4.4, Φ has compact-valued u.s.c. multiselections ψ ,ψ : Y U such that ψ (y)∩ψ (y) = ∅ for every y ∈ Y. 0 1 0 1 In fact, ψ and ψ are compact-valued and u.s.c. as mappings from Y into the 0 1 subsets of X. Hence, each F = {ψ (y) : y ∈ Y}, i = 0,1, is a closed subset of i i X, with F ⊂ U and f(F ) = Y. SSince F ∩F = ∅, we can take disjoint open sets i i 0 1 U ,U ⊂ X such that F ⊂ U ⊂ U ⊂ U, i = 0,1. This completes the proof. (cid:3) 0 1 i i i In our next considerations, to every nonempty subset F of a metric space (X,d) we associate (the possibly infinite) number td (F) = sup diam (C) : C ⊂ F is connected . d d (cid:8) (cid:9) Next, for a surjective map f : X −→ Y and a section U ∈ Ω(f), we let td (U,f) = sup td (f−1(y)∩U) : y ∈ Y . d d (cid:8) (cid:9) In the proof of our next lemma and in the sequel, ω denotes the first infinite ordinal. Lemma 5.3. Let (X,d) be a metric space, Y be a paracompact C-space, and let f : X −→ Y be an open continuous surjection. Then, for every ε > 0, every G ∈ Ω(f) contains an U ∈ Ω(f), with mesh (U,f) ≤ ε and td (U,f) ≤ ε. d d Proof. Let ε > 0 and G ∈ Ω(f). Whenever y ∈ Y and n < ω, take an open subset Wn ⊂ G such that y ∈ f(Wn) and diam (Wn) < ε · 2−(n+1). Since f y y d y is open, each family W = f(Wn) : y ∈ Y , n < ω, is an open cover of Y. n (cid:8) y (cid:9) Since Y is a paracompact C-space, there now exists a sequence {V : n < ω} of n pairwise disjoint open families of Y such that each V , n < ω, refines W and n n V = {V : n < ω} is a locally-finite cover of Y. For convenience, for every n n < ωS, define a map p : V −→ Y by V ⊂ f Wn , V ∈ V , and next set n n (cid:0) pn(V)(cid:1) n U = f−1(V)∩Wn . We are going to show that pn(V) pn(V) U = U : V ∈ V and n < ω [(cid:8) pn(V) n (cid:9) is as required. Since V is a cover of Y, U is a section for f, and clearly it is open. Take a point y ∈ Y, and set V = {V ∈ V : y ∈ V}. Then, V is finite y y 10 VALENTIN GUTEV AND VESKOVALOV and V ∩V ≤ 1 for every n < ω (recall that each family V , n < ω, is pairwise y n n (cid:12) (cid:12) disjo(cid:12)int). Hen(cid:12)ce, we can numerate the elements of V as V : k ∈ K(y) so that y k V ∈ V , k ∈ K(y), where K(y) = {n < ω : V ∩V 6= ∅}.(cid:8)Next, set U =(cid:9)U , k k y n k pk(Vk) k ∈ K(y). Since (5.1) diam(U ) < ε·2−(k+1) for every k ∈ K(y), k f−1(y)∩U ⊂ Bd(S) for every finite subset S ⊂ f−1(y)∩U, with S ∩U 6= ∅ for ε k all k ∈ K(y). Thus, δ(f−1(y)∩U,X) ≤ ε which completes the verification that mesh (U,f) ≤ ε. d To show that td (U,f) ≤ ε, take a nonempty connected subset C ⊂ f−1(y)∩U, d and points x,z ∈ C. Since C is connected and C ⊂ {U : k ∈ K(y)}, there is a k S sequence k ,...,k of distinct elements of K(y) such that x ∈ U , z ∈ U and 1 m k1 km U ∩U 6= ∅ if and only if |i−j| ≤ 1, see [7, 6.3.1]. Therefore, by (5.1), ki kj m d(x,z) ≤ diam (U ) ≤ diam (U ) X d ki X d k i=1 k∈K(y) ∞ < ε·2−(k+1) < ε· 2−(k+1) = ε. X X k∈K(y) k=0 Consequently, diam (C) ≤ ε, which completes the proof. (cid:3) d Recallthatapartiallyorderedset (T,(cid:22))iscalledatree iftheset{s ∈ T : s ≺ t} is well-ordered for every point t ∈ T. Here, as usual, “s ≺ t” means that s (cid:22) t and s 6= t. A chain η in a tree (T,(cid:22)) is a subset η ⊂ T which is linearly ordered by (cid:22). A maximal chain η in T is called a branch in T. Through this paper, we will use B(T) to denote the set of all branches in T. Following Nyikos [21], for every t ∈ T, we let (5.2) U(t) = {β ∈ B(T) : t ∈ β}, and next we set U (T) = {U(t) : t ∈ T}. It is well-known that U (T) is a base for a non-Archimedean topology on B(T), see [21, Theorem 2.10]. In fact, one can easily see that s ≺ t if and only if U(t) ⊂ U(s), while s and t is incomparable if and only if U(s)∩U(t) = ∅. In the sequel, we will refer to B(T) as a branch space if it is endowed with this topology. For a tree (T,(cid:22)), let T(0) be the set of all minimal elements of T. Given an ordinal α, if T(β) is defined for every β < α, then we let T ↾α = {T(β) : β ∈ α}, [ and we will use T(α) to denote the minimal elements of T \ (T ↾α). The set T(α) is called the αth-level of T. The height of T is the least ordinal α such that T ↾α = T. In particular, we will say that T is an α-tree if its height is α. Finally,