Open maps having the Bula property

OPEN MAPS HA VING THE BULA PR OPER TY V ALENTIN GUTEV AND VESKO V ALOV Abstract. Ev ery open c ontin uous map f from a s pace X onto a para compact C -space Y admits t w o disjoint closed sets F 0 , F 1 ⊂ X , with f ( F 0 ) = Y = f ( F 1 ), provided all fiber s of f are infinite and C ∗ -embedded in X . Applications are demonstrated for the exis tence of “ disjoint” usco m ultiselections of s et-v alued l.s.c. mappings defined o n paraco mpact C -s paces, and for spec ial type o f fac- torizations of op en contin uous maps from metriza ble s paces o nto paracompact C -spaces. This settles several op en que stions ra ised in [13]. 1. Intr oduction All spaces in this pap er are assume d to b e a t least completely regular. F ollowing Kato and Levin [15], a contin uous surjectiv e map f : X − → Y is said to hav e the Bula pr op erty if there exis t t w o disjoin t c losed subsets F 0 and F 1 of X suc h that f ( F 0 ) = Y = f ( F 1 ). In the sequel, suc h a pair ( F 0 , F 1 ) will b e called a Bula p air for f . Bula [2] pro v ed that ev ery op en con tinuous map f fr om a compact Hausdorff space on to a finite-dimens ional metrizable space has this prop ert y pro vided all fib ers of f are dense in themselv es. This result w as generalized in [9] to the case Y is coun table-dimensional. Recen tly , L evin and Rogers [16] o bta ined a fur t her generalization with Y b eing a C -space. T he que stion whether t he Levin-Rogers result remains true for op en maps b et wee n metrizable space s w as raised in [13, Problem 15 1 4] (if Y is not a C -space, this is not true, see [6] and [16]). Here, w e pro vide a p ositive answ er to this question: Theorem 1.1. L et X b e a sp ac e, Y b e a p ar ac omp act C -sp ac e, a nd let f : X − → Y b e an op en c ontinuous surje ction such that al l fib ers of f ar e infinite and C ∗ - emb e dde d in X . Then , f has the Bula pr op erty. The C -space pro p ert y was originally defined b y W. Ha v er [11] for compact metric spaces. Later o n, Addis and Gresham [1 ] reformulated Ha ve r’s defini- tion for arbitrary spaces: A space X has prop ert y C (or X is a C -sp ac e ) if f o r Date : Octob er 31, 2021. 2000 Mathematics Subje ct Classific ation. Primary 54 F45, 54F35 , 54C60, 54C65 ; Secondar y 55M10, 54 C35, 54B 20. Key wor ds and phr ases. Dim ension, C -space, set-v alued mapping, selectio n. Research o f the fir st a uthor is supp orted in part b y the NRF of South Africa. The second author was partia lly supp orted by NSERC Grant 2 6191 4-03 . 1 2 V A LENTIN GU TEV AND VESKO V ALOV ev ery sequence { W n : n = 1 , 2 , . . . } of op en co v ers o f X there exists a s equence { V n : n = 1 , 2 , . . . } of pairwise disjoin t op en families in X suc h that eac h V n refines W n and S { V n : n = 1 , 2 , . . . } is a co ver of X . It is well-kno wn that every finite- dimensional paracompact space, a s w ell as ev ery countable-dimen sional metrizable space, is a C - space [1], but there exists a compact metric C -space whic h is not coun table-dimensional [24]. Let us also remark that a C - space X is par a compact if and only if it is coun ta bly paracompact and normal. Finally , let us recall that a subset A ⊂ X is C ∗ -emb e dde d in X if eve ry b ounded real-v alued con tinuous function on A is con tinuously extendable to the whole o f X . Theorem 1.1 has sev eral interesting a pplications. In Section 4, w e apply this the- orem to the gra ph of l.s.c. set-v alued mappings defined on par a compact C -spaces and with p oint-images b eing closed and infinite subsets of completely metrizable spaces. Th us, we get that any s uc h l.s.c. mapping has a pair of “disjoin t” usco m ultiselections (see, Corollar ies 4 .3 and 4.4), whic h provide s the complete affir- mativ e solution to [13, Problem 1515] and sheds some lig ht on [13, Problem 1516]. In this regard, let us stress the atten tion that, in Theorem 1.1, no restrictions on X are called a priori. In Section 5, w e use Theorem 1.1 to demonstrate that eve ry op en contin uous ma p f from a metric space ( X , d ) on to a para compact C -space Y admits a sp ecial t yp e of fa ctorization ( Y × [0 , 1] throughout) , provided all fib ers of f are dense in themselv es and complete with resp ect 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), a nd provides the complete a ffirmativ e solution to [1 3, Problem 1512]. Finally , a w ord should b e said also for the pro of of Theorem 1 .1 itself. Briefly , a preparation for this is do ne in the next section. It is based on the existence of a contin uous function g : X − → [0 , 1] suc h tha t g is not constan t on eac h fib er f − 1 ( y ), y ∈ Y , of f (se e, Theorem 2.1). Ha ving already established this, the pro of o f Theorem 1.1 will b e accomplishe d in Section 3 relying on a “parametric” v ersion of an idea in the pro o f of [16, Theorem 1 .3]. 2. Bula p roper ty and fiber-const ant maps Supp ose that ( F 0 , F 1 ) is a Bula pair for a map f : X − → Y , where X is a normal space. Then, there exists a contin uous function g : X − → [0 , 1] such that g ↾ f − 1 ( y ) is not constan t for ev ery y ∈ Y . Indeed, take g : X − → [0 , 1] to b e suc h that F i ⊂ g − 1 ( i ), i = 0 , 1. In this section, w e demonstrate that the map f in Theorem 1.1 has this prop ert y as w ell. Namely , the follo wing theorem will b e pro ved. OPEN MAPS HA VING THE BULA PROPER TY 3 Theorem 2.1. L et X b e a sp ac e, Y b e a p ar ac omp act C -sp ac e, and le t f : X − → Y b e an op en c ontinuous surje ction such that al l fib ers of f ar e infinite and C ∗ - emb e dde d in X . Then, ther e ex i s ts a c ontinuous function g : X − → [0 , 1] s uch that g ↾ f − 1 ( y ) is not c onstant for every y ∈ Y . T o prepare fo r the pro of of Theorem 2.1, let us recall some terminology . F or spaces Y and Z , w e will use Φ : Y Z to denote that Φ is a set-v alued mapping, i.e. a map from Y into the nonempt y subsets of Z . A mapping Φ : Y Z is lo w er semi-c ontinuous , or l.s.c., if the set Φ − 1 ( U ) = { y ∈ Y : Φ( y ) ∩ U 6 = ∅ } is op en in Y for ev ery op en U ⊂ Z . A mapping Φ : Y Z has an op en ( close d ) graph if it s graph Graph(Φ) =  ( y , z ) ∈ Y × Z : z ∈ Φ( y )  is op en (resp ectiv ely , closed) in Y × Z . A map g : Y − → Z is a sele ction for Φ : Y Z if g ( y ) ∈ Φ( y ) for ev ery y ∈ Y . Finally , let us recall that a space Z is C m for some m ≥ 0 if ev ery contin uous image of the k -dimensional sphere S k ( k ≤ m ) in Z is contractible in Z . In what follow s, I = [0 , 1 ] and C ( Z, I ) denotes the set of all contin uous functions from Z t o I . Also, C ( Z ) = C ( Z, R ) is t he set of all con tinuous functions o n Z , and C ∗ ( Z ) — that of all b ounded mem b ers of C ( Z ). As usual, C ∗ ( Z ) is equipp ed with the sup-m e tric d defined b y d ( g , h ) = sup  | g ( z ) − h ( z ) | : z ∈ Z  , g , h ∈ C ∗ ( Z ) . It should b e men tioned that C ∗ ( Z ) is a Banac h space, and C ( Z , I ) is a closed con ve x subset of C ∗ ( Z ). In the sequel, for g ∈ C ( Z , I ) and ε > 0, w e will use B d ε ( g ) to denote the op en ε -ball B d ε ( g ) = { h ∈ C ( Z, I ) : d ( g , h ) < ε } . The next statemen t is w ell-known and easy to pro ve. Lemma 2.2. L et X b e a sp ac e, and let A ⊂ X b e a C ∗ -emb e dde d subset of X . Then, the r estriction map π A : C ( X, I ) − → C ( A, I ) is an op en c ontinuous surje ction. F or a subset B of a space Z , let Θ Z ( B , I ) b e the set of all members of C ( Z , I ) whic h are constan t on B . If B = Z , then we will denote this set merely b y Θ( Z , I ). Note that Θ( Z , I ) is, in fact, homeomorphic to I . Prop osition 2.3. L et X b e a sp ac e, and let A ⊂ X b e an infinite C ∗ -emb e dde d subset of X . Then, the set C ( X , I ) \ Θ X ( A, I ) is C m for every m ≥ 0 . Pr o of. Consider the restriction map π A : C ( X , I ) − → C ( A, I ), and tak e a contin- uous map g : S n − → C ( X , I ) \ Θ X ( A, I ) for some n ≥ 0 . Then, b y Lemma 2.2, the comp osition π A ◦ g : S n − → C ( A, I ) \ Θ( A, I ) is a lso contin uous. Observ e that 4 V A LENTIN GU TEV AND VESKO V ALOV C ( A, I ) is an infinite-dimensional closed con vex subset of C ∗ ( A ) b ecause A is in- finite. F rom ano ther hand, Θ( A, I ) is one-dimensional b eing ho meomorphic to I . Then, b y [20, Lemma 2.1], C ( A, I ) \ Θ( A, I ) is C m for all m ≥ 0. Hence, there exists a con tin uo us extension ℓ : B n +1 − → C ( A, I ) \ Θ( A, I ) of π A ◦ g ov er the ( n + 1)- dimensional ball B n +1 . Consider the set-v alued mapping Φ : B n +1 C ( X , I ) de- fined by Φ( t ) = { g ( t ) } if t ∈ S n and Φ( t ) = π − 1 A ( ℓ ( t )) otherwise. Since g is a selection for π − 1 A ◦ ℓ ↾ S n and, by Lemma 2 .2 , t he restriction map is π A is op en, the mapping Φ is l.s.c. (see, [18, Examples 1.1 ∗ and 1.3 ∗ ]). Also, Φ is closed and con ve x-v alued in C ( X, I ), hence in the Banac h space C ∗ ( X ) as we ll. Then, by the Mic hael’s selection theorem [18, Theorem 3.2 ′′ ], Φ has a con tin uous selection h : B n +1 − → C ( X, I ) which is, in fact, a contin uous extension of g ov er B n +1 . More- o ver, π A ( h ( t )) = ℓ ( t ) / ∈ Θ( A, I ) for all t ∈ B n +1 , whic h completes the pro of.  A function ξ : X − → R is lower ( upp er ) semi-c on tinuous if the set { x ∈ x : ξ ( x ) > r } (resp ectiv ely , { x ∈ X : ξ ( x ) < r } ) is op en in X for ev ery r ∈ R . Supp ose tha t f : X − → Y is a surjectiv e map. Then, to any g : X − → I w e will asso ciate the functions inf [ g , f ] , sup[ g , f ] : Y − → I defined for y ∈ Y b y inf [ g , f ]( y ) = inf  g ( x ) : x ∈ f − 1 ( y )  , and, resp ectiv ely , sup[ g , f ]( y ) = sup  g ( x ) : x ∈ f − 1 ( y )  . Finally , w e will also asso ciate the function v ar[ g , f ] : X − → I defined b y v ar[ g , f ]( y ) = sup[ g , f ]( y ) − inf [ g , f ]( y ) , y ∈ Y . Observ e that g : X − → I is not constan t on an y fib er f − 1 ( y ), y ∈ Y , if and only if v ar[ g , f ] is p ositive-v alued. The following prop ert y is well-kno wn, [12] (see, also, [7, 1.7.16]). Prop osition 2.4 ([12]) . L et X and Y b e sp ac es, f : X − → Y b e a n op en surje ctive map, a nd let g ∈ C ( X, I ) . Then, sup[ g , f ] is lower se m i-c ontinuous, while inf [ g , f ] is upp er semi-c ontinuous. In p articular, v ar[ g , f ] i s lower semi-c ontinuous. W e finalize the preparation for t he pro of of Theorem 2.1 with the fo llo wing prop osition. Prop osition 2.5. L et X and Y b e sp ac es, and let f : X − → Y b e an op en surje ctive map. Then the set-value d mapping Θ : Y C ( X, I ) define d by Θ( y ) = Θ X ( f − 1 ( y ) , I ) , y ∈ Y , has a close d gr aph . Pr o of. T ak e a p oint y ∈ Y and g / ∈ Θ( y ). Then, v ar[ g , f ]( y ) > 2 δ for some p ositiv e n umber δ > 0. By Prop osition 2 .4, there exists a neighbourho o d V of y suc h that v ar[ g , f ]( z ) > 2 δ for ev ery z ∈ V . Then, V × B d δ ( g ) is an op en set in Y × C ( X , I ) OPEN MAPS HA VING THE BULA PROPER TY 5 suc h that  V × B d δ ( g )  ∩ Graph(Θ) = ∅ . Indeed, tak e z ∈ V and h ∈ B d δ ( g ). Since v ar[ g , f ]( z ) > 2 δ , there are p oints x, t ∈ f − 1 ( z ) suc h that | g ( x ) − g ( t ) | > 2 δ . Since h ∈ B d δ ( g ), w e hav e | h ( x ) − g ( x ) | < δ and | h ( t ) − g ( t ) | < δ . Hence, h ( x ) 6 = h ( t ), whic h implies that v ar[ h, f ]( z ) > 0. Consequen tly , h / ∈ Θ( z ).  Pr o of of The or em 2.1. Consider the set-v alued mapping Φ : Y C ( X, I ) defined b y Φ( y ) = C ( X, I ) \ Θ( y ), y ∈ Y , where Θ is as in Prop osition 2.5. The n, b y Prop osition 2.5, Φ has an op en graph, while, b y Prop o sition 2.3, each Φ( y ), y ∈ Y , is C m for all m ≥ 0. Since Y is a pa racompact C - space, b y the Usp enskij’s selec- tion theorem [26, Theorem 1.3], Φ has a con tinuous selection ϕ : Y − → C ( X, I ). Define a map g : X − → I b y g ( x ) = [ ϕ ( f ( x ))]( x ), x ∈ X . Since f and ϕ are contin- uous, so is g (see, the pro of of [10, Theorem 6 .1]). Since g ↾ f − 1 ( y ) = ϕ ( y ) ↾ f − 1 ( y ) and ϕ ( y ) / ∈ Θ( y ) for ev ery y ∈ Y , g is as required.  3. Proo f of Theorem 1.1 Supp ose 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 ) suc h that inf [ g , f ]( y ) < sup[ g , f ]( y ) for ev ery y ∈ Y . Since inf [ g , f ] is upp er semi-con tinuous and sup[ g , f ] is low er semi- con tinuous (by Prop osition 2.4), and Y is para compact, by a result of [3] (see, also, [5, 14 ]), there are con tin uous functions γ 0 , γ 1 : Y − → I suc h t ha t inf [ g , f ]( y ) < γ 0 ( y ) < γ 1 ( y ) < sup[ g , f ]( y ) , y ∈ Y . Let α i = γ i ◦ f : X − → I , i = 0 , 1 . Then, (3.1) inf [ g , f ]( f ( x )) < α 0 ( x ) < α 1 ( x ) < sup[ g , f ]( f ( x )) for eve ry x ∈ X . Next, define a con tinuous function ℓ : X × I − → R b y letting ℓ ( x, t ) = t − α 0 ( x ) α 1 ( x ) − α 0 ( x ) , ( x, t ) ∈ X × I . Observ e that ℓ ( x, α 0 ( x )) = 0 and ℓ ( x, α 1 ( x )) = 1 for ev ery x ∈ X . Hence, (3.2) ℓ  { x } × [ α 0 ( x ) , α 1 ( x )]  = [0 , 1] , for ev ery x ∈ X , b ecause ℓ is linear for ev ery fixed x ∈ X . Fina lly , define a con tinuous function h : X − → R b y h ( x ) = ℓ ( x, g ( x )), x ∈ X . According to (3.1) a nd (3 .2), we now ha ve that, for eve ry y ∈ Y , h − 1  ( −∞ , 0]  ∩ f − 1 ( y ) 6 = ∅ 6 = h − 1  [1 , + ∞ )  ∩ f − 1 ( y ) . Then, F 0 = h − 1  ( −∞ , 0]  and F 1 = h − 1  [1 , + ∞ )  are as r equired. The pro of o f Theorem 1.1 completes. 6 V A LENTIN GU TEV AND VESKO V ALOV 4. B ula p airs and mul t iselections A set-v alued mapping ϕ : Y Z is called a multisele ction for Φ : Y Z if ϕ ( y ) ⊂ Φ( y ) fo r ev ery y ∈ Y . In this section, w e presen t sev eral applications of Theorem 1 .1 ab out m ultiselections of l.s.c. mappings based on the follo wing consequenc e of it. Corollary 4.1. L et Y b e a p ar ac omp ac t C -sp a c e, Z b e a norm a l s p ac e, and let Φ : Y Z b e an l.s.c . mapping such that e a ch Φ( y ) , y ∈ Y , is infinite and close d in Z . Then, ther e exists a clo s e d- gr aph mapping θ : Y Z such that Φ( y ) ∩ θ ( y ) 6 = ∅ 6 = Φ( y ) \ θ ( y ) for every y ∈ Y . Pr o of. Let X = Graph(Φ) b e the gr aph of Φ, and let f : X − → Y b e the pro jection. Then, f is a n op en contin uous ma p (b ecause Φ is l.s.c.) suc h that all fib ers of f are infinite. Let us observ e tha t eac h f − 1 ( y ), y ∈ Y , is C ∗ -em b edded in X . Indeed, tak e a p oin t y ∈ Y , and a con tinuous function g : f − 1 ( y ) − → I . Since f − 1 ( y ) = { y } × Φ( y ), w e ma y consider the contin uous function g 0 : Φ( y ) − → I defined b y g 0 ( z ) = g ( y , z ), z ∈ Φ( y ). Since Z is normal, there exis ts a contin uous extension h 0 : Z − → I of g 0 . Fina lly , define h : X − → I by h ( t, z ) = h 0 ( z ) for ev ery t ∈ Y and z ∈ Φ( t ). Then, h is a con tinuous extension of g . Th us, by Theorem 1.1, there a r e disjoin t closed subsets F 0 , F 1 ⊂ X suc h that f ( F 0 ) = Y = f ( F 1 ). Finally , take a closed set F ⊂ Y × Z , with F ∩ X = F 0 , and define θ : Y Z by Graph( θ ) = F . This θ is as required.  T o prepare for our applications, w e need also the following observ ation ab o ut l.s.c. multis elections of l.s.c. mappings. Prop osition 4.2. L et Y b e a p ar ac omp act sp ac e, Z b e a sp ac e, Φ : Y Z b e an l.s.c. close d-value d mapping, and let Ψ : Y Z b e an op en-gr aph mappin g, with Φ( y ) ∩ Ψ( y ) 6 = ∅ for every y ∈ Y . Then, ther e exists a close d-value d l.s . c . mapping ϕ : Y Z such that ϕ ( y ) ⊂ Φ( y ) ∩ Ψ( y ) for every y ∈ Y . Pr o of. Whenev er y ∈ Y , there are op en sets V y ⊂ Y and W y ⊂ Z suc h that y ∈ V y ⊂ Φ − 1 ( W y ) and V y × W y ⊂ Gra ph(Ψ). Indeed, t a k e a p oint z ∈ Φ( y ) ∩ Ψ( y ). Since Ψ has a n op en g r a ph, t here are op en sets O y ⊂ Y and W y ⊂ Z suc h that y ∈ O y , z ∈ W y and O y × W y ⊂ Graph(Ψ). Then, V y = O y ∩ Φ − 1 ( W y ) is as required. No w, for eve ry y ∈ Y , define a closed-v alued mapping ϕ y : V y Z b y letting that ϕ y ( t ) = Φ( t ) ∩ W y , t ∈ V y . According to [18, Prop ositions 2.3 and 2.4], each ϕ y , y ∈ Y , is l.s.c. Nex t, using that Y is paracompact, ta k e a lo cally- finite op en co v er U of Y refining { V y : y ∈ Y } and a map p : U − → Y suc h that U ⊂ V p ( U ) , U ∈ U . Finally , define a mapping ϕ : Y Z b y letting tha t ϕ ( y ) = [  ϕ p ( U ) ( y ) : U ∈ U and y ∈ U  , y ∈ Y . This ϕ is as required.  OPEN MAPS HA VING THE BULA PROPER TY 7 In what follow s, a mapping ψ : Y Z is upp er semi-c ontinuous , or u.s.c., if the set Φ # ( U ) = { y ∈ Y : Φ( y ) ⊂ U } is op en in Y for ev ery op en U ⊂ Z . Motiv ated by [19], w e sa y that a pair ( ϕ, ψ ) of set-v alued mapping ϕ, ψ : Y Z is a Michael p air for Φ : Y Z if ϕ is compact- v alued a nd l.s.c., ψ is compact-v alued and u.s.c., and ϕ ( y ) ⊂ ψ ( y ) ⊂ Φ( y ) for ev ery y ∈ Y . The fo llo wing consequence provide s the complete affirmativ e solution to [13 , Problem 15 1 5]. Corollary 4.3. L et ( Z , ρ ) b e metric sp ac e, Y b e a p ar ac omp act C -sp ac e, and let Φ : Y Z b e an l.s.c. mapping such that e ach Φ( y ) , y ∈ Y , is infinite and ρ - c omplete. Then Φ has Michael p ai r ( ϕ, ψ ) : Y Z s uch that Φ( y ) \ ψ ( y ) 6 = ∅ for every y ∈ Y . Pr o of. By Corolla r y 4.1, there is a closed-graph mapping θ : Y Z suc h that Φ( y ) ∩ θ ( y ) 6 = ∅ 6 = Φ( y ) \ θ ( y ) for ev ery y ∈ Y . Consider the set-v alued mapping Ψ : Y Z defined b y G r a ph(Ψ) = ( Y × Z ) \ Gra ph( θ ). On one ha nd, by the prop erties of θ , w e hav e that Φ( y ) ∩ Ψ( y ) 6 = ∅ for ev ery y ∈ Y . On anot her hand, Φ is closed-v alued ha ving ρ -complete v alues. Hence, b y Prop osition 4.2, there exists a closed-v alued l.s.c. mapping Φ 0 : Y Z suc h that Φ 0 ( y ) ⊂ Φ( y ) ∩ Ψ( y ) for ev ery y ∈ Y . Then, Φ 0 has also ρ -complete v alues and, by a result of [1 9], it has a Mic hael pa ir ( ϕ, ψ ). This ( ϕ, ψ ) is as required.  W e conclude this section with the follo wing further application of Theorem 1.1 that sheds some ligh t on [13, Problem 1516]. Corollary 4.4. L et ( Z , ρ ) b e metric sp ac e, Y b e a p ar ac omp act C -sp ac e, and let Φ : Y Z b e an l.s.c. mapping such that e ach Φ( y ) , y ∈ Y , is infinite and ρ - c omplete. Then Φ ha s Michael p airs ( ϕ i , ψ i ) : Y Z , i = 0 , 1 , s uch that ψ 0 ( y ) ∩ ψ 1 ( y ) = ∅ for every y ∈ Y . Pr o of. According t o Coro llary 4.3, Φ has a Mic hael pair ( ϕ 0 , ψ 0 ) : Y Z suc h that Φ( y ) \ ψ 0 ( y ) 6 = ∅ for ev ery y ∈ Y . Note that ψ 0 has a closed-gra ph b eing u.s.c. Then, just lik e in t he pro of of Corollary 4.3, there exists a Mic ha el pair ( ϕ 1 , ψ 1 ) : Y Z for Φ suc h that ψ 1 ( y ) ⊂ Φ( y ) \ ψ 0 ( y ), y ∈ Y . These ( ϕ i , ψ i ), i = 0 , 1, are as required.  5. Op en maps looking like projections Throughout this section, b y a dimens i on o f a space Z we mean the co v ering dimension dim( Z ) of Z . In part icular, Z is 0 - d imensional if dim ( Z ) = 0. W e sa y that a con tinu ous map f : X − → Y has dimension ≤ k if all fib ers of f ha ve dimension ≤ k . A con tin uo us map f : X − → Y is light if it is 0-dimensional, 8 V A LENTIN GU TEV AND VESKO V ALOV i.e. if f has 0- dimensional fib ers. Also, for conv enience, w e shall sa y tha t a map f : X − → Y is c omp act if each fib er f − 1 ( y ), y ∈ Y , is a compact subset of X . Supp ose that f : X − → Y is a surjectiv e map. A subset F ⊂ X will b e called a se ction for f if f ( F ) = Y . In particular, w e shall sa y that a section F for f is op en ( close d ) if F is an op en (respective , a closed) subset of X . In this section, w e demonstrate the fo llowing factorization theorem which is a partial generalization of [9, Theorem 1.1], a lso it provides the complete affirmative solution to [1 3 , Problem 1 512]. Theorem 5.1. L et ( X , d ) b e a me tric sp ac e, Y b e a p ar ac omp act C -sp ac e, and let f : X − → Y b e an op en c ontinuous surje ction such that e ach fib er of f is d ense in itself and d -c omplete. Also, let U ⊂ X b e an o p en se ction for f . Then, ther e exists a c ontinuous surje ctive m ap g : X − → Y × I , a clo s e d se ction H ⊂ X fo r f , with H ⊂ U , and a c opy C ⊂ I of the Cantor set such that (a) f = P Y ◦ g , wher e P Y : Y × I − → Y is the pr oje ction, i.e. the fol lowing diagr am is c ommutative. X Y × I ✲ g f ❅ ❅ ❅ ❅ ❅ ❘ Y ❄ P Y (b) g ( H ) = Y × I and e ach g − 1 ( y , c ) ∩ H , ( y , c ) ∈ Y × C , is c omp act and 0 -dimensiona l. In p articular, H C = H ∩ g − 1 ( Y × C ) is a close d se ction fo r f such that f ↾ H C is a c omp act light ma p. T o prepare for the pro of of Theorem 5.1, w e introduce some terminology . F o r a metric space ( X , d ), a no nempty subset A ⊂ X and ε > 0 , as in Section 2, w e let B d ε ( A ) = { x ∈ X : d ( x, A ) < ε } . Also, w e will use diam d ( A ) to denote the diameter of A with respect to d . F ollo wing [9], to eve ry nonempt y subset F ⊂ X we a sso ciate the n umber δ ( F , X ) = inf  1 , ε : ε > 0 and F ⊂ B d ε ( S ) for some no nempt y finite S ⊂ F  . In what follow s, w e let Ω( f ) to b e the set of all op en sections for f . Also, for ev ery U ∈ Ω( f ), we in tro duce the d -mesh of U with resp ect to f by letting mesh d ( U, f ) = sup  δ ( f − 1 ( y ) ∩ U, X ) : y ∈ Y  . OPEN MAPS HA VING THE BULA PROPER TY 9 Prop osition 5.2. L et ( X, d ) b e a metric sp ac e, Y b e a p ar ac omp act C -sp ac e, and let f : X − → Y b e an op en c ontinuous surje ction such that e ach fib er of f is d ense in itself and d -c omplete. Then, for every U ∈ Ω( f ) ther e ar e disjoint op en se ctions U 0 , U 1 ∈ Ω( f ) such that U i ⊂ U , i = 0 , 1 . Pr o of. Consider U endo w ed with the compatible metric ρ ( x, y ) = d ( x, y ) +     1 d ( x, X \ U ) − 1 d ( y , X \ U )     , x, y ∈ U. Next, define an l.s.c. mapping Φ : Y X b y Φ( y ) = f − 1 ( y ) ∩ U , y ∈ Y . Then, eac h Φ( y ), y ∈ Y , is infinite and ρ - complete in U b ecause each f − 1 ( y ), y ∈ Y , is dense in itself and d -complete. Hence, b y Corollary 4.4, Φ has compact-v alued u.s.c. multis elections ψ 0 , ψ 1 : Y U suc h that ψ 0 ( y ) ∩ ψ 1 ( y ) = ∅ for ev ery y ∈ Y . In fact, ψ 0 and ψ 1 are compact-v alued and u.s.c. as mappings fro m Y in to the subsets of X . Hence, each F i = S { ψ i ( y ) : y ∈ Y } , i = 0 , 1 , is a closed subset of X , with F i ⊂ U and f ( F i ) = Y . Since F 0 ∩ F 1 = ∅ , we can tak e disjoin t o p en sets U 0 , U 1 ⊂ X suc h that F i ⊂ U i ⊂ U i ⊂ U , i = 0 , 1. This completes the pro of.  In our next considerations, to ev ery nonempt y subset F of a metric space ( X , d ) w e asso ciate (the p o ssibly infinite) num b er td d ( F ) = sup  diam d ( C ) : C ⊂ F is connected  . Next, for a surjectiv e map f : X − → Y and a section U ∈ Ω( f ), w e let td d ( U, f ) = sup  td d ( f − 1 ( y ) ∩ U ) : y ∈ Y  . In the pro of of our next lemma and in the seque l, ω denotes the first infinite ordinal. Lemma 5.3. L et ( X , d ) b e a metric sp ac e, Y b e a p ar ac omp act C -sp ac e, and let f : X − → Y b e an op en c ontinuous surje ction. The n , fo r every ε > 0 , every G ∈ Ω( f ) c ontains an U ∈ Ω( f ) , with mesh d ( U, f ) ≤ ε and td d ( U, f ) ≤ ε . Pr o of. Let ε > 0 and G ∈ Ω( f ). Whenev er y ∈ Y and n < ω , tak e an op en subset W n y ⊂ G suc h that y ∈ f ( W n y ) a nd diam d ( W n y ) < ε · 2 − ( n +1) . Since f is op en, each family W n =  f ( W n y ) : y ∈ Y  , n < ω , is a n op en cov er o f Y . Since Y is a paracompact C -space, there now exists a sequence { V n : n < ω } of pairwise disjoin t o p en families of Y suc h that each V n , n < ω , refines W n and V = S { V n : n < ω } is a lo cally-finite co ve r of Y . F or conv enience, for ev ery n < ω , define a map p n : V n − → Y by V ⊂ f  W n p n ( V )  , V ∈ V n , and next set U p n ( V ) = f − 1 ( V ) ∩ W n p n ( V ) . W e are going to sho w that U = [  U p n ( V ) : V ∈ V n and n < ω  is as required. Since V is a cov er of Y , U is a section fo r f , and clearly it is op en. T ak e a p o in t y ∈ Y , a nd set V y = { V ∈ V : y ∈ V } . Then, V y is finite 10 V A LENTIN GU TEV AND VESKO V ALOV and   V y ∩ V n   ≤ 1 for ev ery n < ω (recall that eac h family V n , n < ω , is pairwise disjoin t). Hence, w e can nume rate the elemen ts of V y as  V k : k ∈ K ( y )  so that V k ∈ V k , k ∈ K ( y ), where K ( y ) = { n < ω : V y ∩ V n 6 = ∅ } . Next, set U k = U p k ( V k ) , k ∈ K ( y ). Since (5.1) diam( U k ) < ε · 2 − ( k + 1) for ev ery k ∈ K ( y ), f − 1 ( y ) ∩ U ⊂ B d ε ( S ) fo r ev ery finite subset S ⊂ f − 1 ( y ) ∩ U , with S ∩ U k 6 = ∅ for all k ∈ K ( y ). Th us, δ ( f − 1 ( y ) ∩ U, X ) ≤ ε whic h completes the v erification that mesh d ( U, f ) ≤ ε . T o sho w that td d ( U, f ) ≤ ε , take a nonempt y connected subset C ⊂ f − 1 ( y ) ∩ U , and p o in ts x, z ∈ C . Since C is connected and C ⊂ S { U k : k ∈ K ( y ) } , there is a sequence k 1 , . . . , k m of distinct elemen ts of K ( y ) such that x ∈ U k 1 , z ∈ U k m and U k i ∩ U k j 6 = ∅ if and only if | i − j | ≤ 1, see [7 , 6.3.1]. Therefore, by (5.1), d ( x, z ) ≤ m X i =1 diam d ( U k i ) ≤ X k ∈ K ( y ) diam d ( U k ) < X k ∈ K ( y ) ε · 2 − ( k + 1) < ε · ∞ X k =0 2 − ( k + 1) = ε . Consequen tly , diam d ( C ) ≤ ε , whic h completes the pro of.  Recall that a partially ordered set ( T ,  ) is called a tr e e if the set { s ∈ T : s ≺ t } is w ell-o r dered for ev ery p o int t ∈ T . Here, as usual, “ s ≺ t ” means that s  t and s 6 = t . A chain η in a tree ( T ,  ) is a subset η ⊂ T whic h is linearly o r dered b y  . A maximal c hain η in T is called a br anch in T . Through this pap er, w e will use B ( T ) t o denote the set of all branche s in T . F ollow ing Nyik os [21 ], for ev ery t ∈ T , w e let (5.2) U ( t ) = { β ∈ B ( T ) : t ∈ β } , and next w e set U ( T ) = { U ( t ) : t ∈ T } . It is w ell- kno wn tha t U ( T ) is a base for a non-Arch imedean top ology on B ( T ), see [21, Theorem 2.10]. In f act, 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, w e will refer to B ( T ) as a br anch sp ac e if it is endo w ed with this top ology . F or a tree ( T ,  ), let T (0) b e the set o f all minimal elemen ts of T . Given a n ordinal α , if T ( β ) is defined for ev ery β < α , then we let T ↾ α = [ { T ( β ) : β ∈ α } , and w e will use T ( α ) to denote the minimal elemen t s of T \ ( T ↾ α ). The set T ( α ) is called the α th -level of T . The he i g ht of T is the least ordinal α suc h that T ↾ α = T . In particular, we will sa y that T is an α -tr e e if its heigh t is α . Finally , OPEN MAPS HA VING THE BULA PROPER TY 11 w e can a lso define the height of an elemen t t ∈ T , denoted by h t( t ), whic h is the unique ordinal α suc h that t ∈ T ( α ). In what follows , we will b e mainly in terested in ω -trees, and the following realization of the Can tor set as a branc h space. Namely , let S b e a set which has at least 2 distinct p oints , S n b e the set of all maps t : n − → S (i.e., the n th -p ow er of S ), and let S <ω = [ { S n +1 : n < ω } . Whenev er t ∈ S <ω , let dom( t ) b e the doma i n of t . Consider the partial order  on S <ω defined for s, t ∈ S <ω b y s  t if and only if dom( s ) ⊂ dom( t ) and t ↾ dom( s ) = s. Then, ( S <ω ,  ) is a n ω -tree suc h that its branc h space B ( S <ω ) is the Baire space S ω . In part icular, t he branch space B (2 <ω ) is the Cantor set 2 ω . In the sequel, w e will refer to the tree (2 <ω ,  ) as the Cantor tr e e . By Prop o sition 5.2 and Lemma 5.3, using an induction on the lev els of the Can tor tree (2 <ω ,  ), w e get t he follo wing immediate consequence. Corollary 5.4. L et (2 <ω ,  ) b e the Cantor tr e e, and let ( X , d ) , Y , f : X − → Y and U ∈ Ω( f ) b e as in The or em 5.1. Then, ther e exists a map h : 2 <ω − → Ω( f ) such that, for every two distinct memb ers s, t ∈ 2 <ω , (a) h ( t ) ⊂ h ( s ) ⊂ U if s ≺ t , (b) h ( s ) ∩ h ( t ) = ∅ if s and t ar e inc omp ar able, (b) mesh d ( h ( t ) , f ) ≤ 2 − ht( t ) and td d ( h ( t ) , f ) ≤ 2 − ht( t ) . W e finalize the preparation for t he pro of of Theorem 5.1 with the fo llo wing sp ecial case of it. Lemma 5.5. L et ( X , d ) , Y , f : X − → Y and U ∈ Ω( f ) b e as in The or em 5 .1. Then, ther e exists a close d se ction H ⊂ X of f , w i th H ⊂ U , an d a surje ctive c omp act li g ht ma p ℓ : H − → Y × C such that f ↾ H = P Y ◦ ℓ , i.e. the fol lowing diagr am is c o m mutative. H Y × C ✲ ℓ f ↾ H ❅ ❅ ❅ ❅ ❅ ❘ Y ❄ P Y In p articular, f ↾ H is also a c omp act light map. Pr o of. Let h : 2 <ω − → Ω( f ) b e as in Corollary 5 .4. Whenev er n < ω , consider the n th -lev el of the Cantor tree 2 <ω , whic h is, in fact, 2 n +1 . Then, set H n = h (2 n +1 ), n < ω , and H = T { H n : n < ω } . By (a) of Corollary 5.4, H n +1 ⊂ H n ⊂ U for 12 V A LENTIN GU TEV AND VESKO V ALOV ev ery n < ω b ecause eac h lev el of 2 <ω is finite. Hence, H is a closed subse t of X , with H ⊂ U . Let us see that H is a section for f . Indeed, ta k e a p oin t y ∈ Y , a nd a branc h β ∈ B (2 <ω ). Then, eac h H t ( y ) = h ( t ) ∩ f − 1 ( y ), t ∈ β , is a nonempt y subset of f − 1 ( y ) (b ecause h ( t ) ∈ Ω( f )) suc h that H t ( y ) ⊂ H s ( y ) for s ≺ t (by (a) of Corollary 5 .4) and lim t ∈ β δ ( H t ( y ) , X ) = 0 (b y (c) of Coro llary 5.4). Hence, by [9, Lemma 3.2], H β ( y ) = T { H t ( y ) : t ∈ β } is a nonempt y compact subset of X . Clearly , H β ( y ) ⊂ H ∩ f − 1 ( y ) whic h completes the v erification that H is a section for f . In fact, this defines a compact-v alued mapping ϕ : Y × B (2 <ω ) H b y letting ϕ ( y , β ) = H β ( y ) = T { h ( t ) ∩ f − 1 ( y ) : t ∈ β } , ( y , β ) ∈ Y × B (2 <ω ). Since ϕ ( y , β ) ⊂ f − 1 ( y ), ( y , β ) ∈ Y × B (2 <ω ), the mapping ϕ is the in ve rse ℓ − 1 of a surjectiv e single-v alued map ℓ : H − → Y × B (2 <ω ). Also, ℓ ( x ) = ( y , β ) if a nd only if x ∈ ϕ ( y , β ) ⊂ f − 1 ( y ), hence f ↾ H = P Y ◦ ℓ . T o sho w that ℓ is contin uous and ligh t , tak e an o p en set V ⊂ Y , t ∈ 2 <ω , and let U ( t ) b e as in (5.2). Then, h ( t ) is an op en set in X suc h that, by (b) of Corollary 5.4, ℓ − 1 ( y , β ) = ϕ ( y , β ) ⊂ h ( t ) if and o nly if t ∈ β (i.e., β ∈ U ( t )) . Consequen tly , ℓ − 1 ( V × U ( t )) = f − 1 ( V ) ∩ h ( t ) ∩ H is op en in H . Finally , tak e a nonempt y connected subset C ⊂ ℓ − 1 ( y , β ) = ϕ ( y , β ) for a p oin t y ∈ Y and a branc h β ∈ B (2 <ω ). T hen, C ⊂ h ( t ) ∩ f − 1 ( y ) for ev ery t ∈ β and therefore, b y (c) o f Corollary 5.4, dia m d ( C ) = 0. Hence, C is a singleton, which implies that ℓ − 1 ( y , β ) is 0- dimensional b eing compact. T o sho w finally that f ↾ H is a compact light map, tak e a p oint y ∈ Y , a nd let us observ e that ℓ ↾  f − 1 ( y ) ∩ H  is p erfect. Indeed, tak e a branch β ∈ B (2 <ω ) and a neigh b ourho od W of ℓ − 1 ( y , β ) in X . Then, by [9, Lemma 3.2], there exists a t ∈ β , with H t ( y ) = h ( t ) ∩ f − 1 ( y ) ⊂ W . In this case, ℓ − 1 ( y , γ ) ⊂ W for ev ery γ ∈ U ( t ), where U ( t ) is as in (5 .2). Namely , γ ∈ U ( t ) implies that t ∈ γ and, therefore, ℓ − 1 ( y , γ ) ⊂ H t ( y ) ⊂ W . Th us, ℓ ↾  f − 1 ( y ) ∩ H  is p erfect, whic h implies that f − 1 ( y ) ∩ H = ℓ − 1 ( { y } × B (2 <ω )) is compact b ecause so is B (2 <ω ). Since B (2 <ω ) is zero-diemnsional and ℓ is a ligh t map, according to the classical Hurewicz theorem (see, [8]), this also implies that dim ( f − 1 ( y ) ∩ H ) = 0 whic h completes the pro of.  Pr o of of The or em 5.1. W e rep eat the arguments of [2, Theorem 1 ]. Briefly , let ( X , d ), Y , f : X − → Y and U ∈ Ω( f ) b e as in Theorem 5.1. By Lemma 5.5, there exists a closed section H ⊂ X of f , with H ⊂ U , and a con tin uous surjectiv e map ℓ : H − → Y × C suc h that f ↾ H is a compact ligh t map, and f ↾ H = P Y ◦ ℓ . T ak e a con t inuous surjectiv e map ρ : C − → I suc h that the set D = { t ∈ I : | ρ − 1 ( t ) | > 1 } is coun table. Also, let P C : Y × C − → C b e the pro jection. Then, using the Tietze- Urysohn theorem, extend ρ ◦ P C ◦ ℓ to a contin uous map u : X − → I . In this w ay , OPEN MAPS HA VING THE BULA PROPER TY 13 w e ha ve that u ( f − 1 ( y ) ∩ H ) = I for ev ery y ∈ Y . Then, w e can define our g : X − → Y × I by g ( x ) = ( f ( x ) , u ( x )), x ∈ X . As f o r the the second part of Theorem 5.1, t a k e a cop y C of the Cantor set in I \ D , whic h is p ossible b ecause D is countable. Then, by the prop erties o f ρ , w e ha v e that g − 1 ( Y × { c } ) ∩ H = ℓ − 1 ( Y × { c } ) for every c ∈ C . Hence, Lemma 5.5 completes the pro of.  W e finalize this pap er with sev eral applications of Theorem 5.1 . In what fo llows, for a space X , let F ( X ) b e the set of all nonempt y closed subsets of X . Recall that the Vietoris top olo gy τ V on F ( X ) is generated b y all collections of the form h V i = n S ∈ F ( X ) : S ⊂ [ V and S ∩ V 6 = ∅ , whenev er V ∈ V o , where V runs ov er the finite f a milies o f op en subsets of X . In the sequel, any subset D ⊂ F ( X ) will carry the relativ e Vietoris to p ology τ V as a subspace of ( F ( X ) , τ V ). In fact, we will b e mainly in terested in the subset F ( f ) = { H ∈ F ( X ) : f ( H ) = Y } , where f : X − → Y is a surjectiv e map. Corollary 5.6. L et ( X , d ) b e a metric sp ac e, Y b e a p ar ac omp act C - s p ac e, and let f : X − → Y b e an op en c ontinuous surje ction such that e ach fib er of f is d ense in itself and d -c omplete. Then, the set L ( f ) =  H ∈ F ( f ) : f ↾ H is a c omp act light map  is dense in F ( f ) with r esp e ct to the Vietoris top olo gy τ V . Pr o of. T ak e a closed section F ∈ F ( f ), and a finite family U of op en subsets of X , with F ∈ h U i . Then, U = S U is a n op en section for f , so, b y Theorem 5.1, it contains a closed section H ⊂ U suc h that f ↾ H is a compact light map. T ak e a finite set S ∈ h U i , and then set Z = H ∪ S . Clearly , Z ∈ L ( f ) ∩ h U i , whic h completes the pro of.  Prop osition 5.7. Whenever Y is a metrizable sp ac e, ther e exists a close d 0 - dimensional subset A ⊂ Y × C such that P Y ( A ) = Y . Pr o of. W e follow the idea of [25, Lemma 4.1]. Fix a 0-dimensional metrizable space M and a p erfect surjectiv e map h : M − → Y . By [22, Prop osition 9.1 ], there exists a con tin uous map g : M − → Q , where Q is the Hilbert cub e, suc h that h △ g : M − → Y × Q is a n em b edding. Next, take a Milyutin map p : C − → Q , i.e. a surjectiv e con tinuous map admitting an a v eraging op erator b et w een the function spaces C ( C ) a nd C ( Q ), see [23]. According to [4], there exists a compact-v alued lo wer semi-contin uous map ϕ : Q C suc h that ϕ ( z ) ⊂ p − 1 ( z ) fo r a ll z ∈ Q . Applying Mic hael’s 0-dimensional selection theorem [17], there exists a contin uous 14 V A LENTIN GU TEV AND VESKO V ALOV map ℓ : M − → C , with ℓ ( x ) ∈ ϕ ( g ( x )) for any x ∈ M . Then, h △ ℓ em b eds M as a closed subset A of Y × C . Ob viously , A is 0-dimensional and P Y ( A ) = Y .  Corollary 5.8. L et X b e a metrizable s p ac e, Y b e a m etrizable C -sp a c e, and let f : X − → Y b e an o p en p erfe ct surje ction such that e ach fib er of f is dens e in itself. Then, the set F 0 ( f ) =  H ∈ F ( f ) : dim( H ) = 0  is dense in F ( f ) with r esp e ct to the Vietoris top olo gy τ V . Pr o of. T ak e a closed section F ∈ F ( f ), and a finite family U of op en subsets of X , with F ∈ h U i . Then, U = S U is an op en section for f , so, by Theorem 5.1, there exists a closed section H f o r f , with H ⊂ U , a con tin uous surjectiv e map g : X − → Y × I , and a cop y C ⊂ I of the Can tor set such that f = P Y ◦ g , g ( H ) = Y × I , and f ↾  H ∩ g − 1 ( Y × C  is a ligh t map. By Prop o sition 5.7, Y × C contains a closed 0-dimensional set A , with P Y ( A ) = Y . Finally , tak e B = H ∩ g − 1 ( A ) whic h is a closed section for f b ecause P Y ( A ) = Y . Since f is p erfect, so is g . Hence, g ↾ B is a perfect ligh t map a nd, according to t he classical Hurewicz theorem, dim( B ) = 0. Then, Z = B ∪ S ∈ F 0 ( f ) ∩ h U i for some (eve ry) finite set S ∈ h U i .  T o prepare for o ur last consequence, let us also observ e the followin g prop ert y of 0-dimensional sections. Prop osition 5.9. L et X b e a c omp act metrizable sp ac e, and le t F 0 ( X ) b e the subset of al l 0 -dime nsional memb ers of F ( X ) . Then , F 0 ( X ) i s a G δ -subset o f F ( X ) . Pr o of. T ak e a metric d on X compatible with the top ology of X . Next, fo r ev ery H ∈ F 0 ( X ) and n ≥ 1, take a pairwise disjoin t finite family V n ( H ) of op en subsets of X suc h that diam( V ) < 1 / n , V ∈ V n ( H ), and H ∈ h V n ( H ) i . Then, eac h V n = S  V n ( H ) : H ∈ F 0 ( X )  , n ≥ 1, is τ V -neigh b ourho o d of F 0 ( X ), and clearly F 0 ( X ) = T  V n ( F ) : n = 1 , 2 , . . .  .  According to Corollary 5.8 and Prop osition 5.9, w e ha v e the follo wing immediate consequenc e, whic h is the Levin-Rogers [16, Theorem 1.2]. Corollary 5.10 ([16]) . 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Usp ens kij, A sele ction the or em for C - s p ac es , T op olo gy Appl. 85 (1998 ), 351–3 74. 16 V A LENTIN GU TEV AND VESKO V ALOV School of Ma thema tical Sciences, University of Kw aZ ulu-Na t al, W estville Campus, Priv a te Ba g X 54001, Durban 4000, South Africa E-mail addr ess : gutev @ukzn .ac.z a Dep ar tment of Computer Scie nce and Ma thema tics, Nipissing University, 10 0 College Drive, P.O. Box 5002, Nor th Ba y, ON, P1B 8L7, Canad a E-mail addr ess : vesko v@nip issin gu.ca