Selections, Extensions and Collectionwise Normality

SELECTIONS, EXTENSI ONS AND COLL ECTIONWISE NORMALITY V ALENTIN GUTEV AND NAR CISSE ROLAND LOUFOUMA MA KALA Abstract. W e demonstrate that the classical Michael selection theorem for l.s.c. ma ppings with a c o llectionwise nor ma l domain can b e reduced o nly to compact-v alued mappings mo dulo Do wker’s extension theor em for suc h spaces. The idea used to achieve this reduction is a lso applied to get a simple direc t pro of of that sele ction theor e m o f Michael’s. Some other p o ssible applications are demonstrated as well. 1. Intr oduction F or a top ological space E , let 2 E b e the family of all nonempt y subs ets of E ; F ( E ) — the subfamily of 2 E consisting of all closed mem bers of 2 E ; a nd let C ( E ) b e that one of all compact m em bers o f F ( E ) . A lso, let C ′ ( E ) = C ( E ) ∪ { E } . A set-v alued mapping ϕ : X → 2 E is lower semi-c ontinuous , or l.s.c., if the set ϕ − 1 ( U ) = { x ∈ X : ϕ ( x ) ∩ U 6 = ∅ } is op en in X for ev ery op en U ⊂ E . A map f : X → E is a sele ction f o r ϕ : X → 2 E if f ( x ) ∈ ϕ ( x ) for ev ery x ∈ X . Recall that a space X is c ol le ctionwise norm al if it is a T 1 -space and for ev ery discrete collection D of closed subs ets of X there exists an op en discrete family  U D : D ∈ D  suc h t hat D ⊂ U D for ev ery D ∈ D . Ev ery collection wise normal space is normal, but the con v erse is not neces sarily true [1], see, also, [7, 5 .1.23 Bing’s Example ]. It is w ell kno wn that a T 1 -space X is collection wise normal if and only if for ev ery closed s ubset A ⊂ X , ev ery con tinuous map from A to a Banac h space E can b e con t in uously extended to the whole of X , Do wk er [5]. Generalizing this result, Mic hael [1 6 ] stated the follo wing theorem. Theorem 1.1 ([16]) . F or a T 1 -sp ac e X , the fol lowing ar e e quivalent : (a) X is c ol le ctionwis e n o rmal. (b) If E is a Banach sp ac e an d ϕ : X → C ′ ( E ) is an l.s.c. c onvex-value d mapping, then ϕ ha s a c ontinuous sele ction. Date : August 18, 2009. 2000 Mathematics Subje ct Classific ation. P rimary 54C60 , 54C 6 5; Secondary 54C2 0, 54C55. Key wor ds a nd phr ases. Set-v a lued mapping, lower s e mi-contin uo us, selection, extens io n. The work of the first author is bas ed up o n res earch supp or ted b y the NRF of So uth Africa. 1 2 V ALENTIN GUTEV AND NARCISSE ROLAND LOUF OUMA MAK ALA Ho w ev er, the a rgumen ts in [16] for (a) ⇒ (b) of Theorem 1.1 w ere incom- plete and, in fact, w orking only fo r the case of C ( E )-v alued mappings. The first complete pro of of t his implication w as given by Choban and V alov [2 ] using a dif- feren t tec hnique. W e are now ready to state also the main purp ose of this pap er. Namely , in this pap er we pro ve the follo wing theorem whic h demonstrates t ha t the origina l Mic hael argumen ts in [16] hav e been actually adequate to the pro of of Theorem 1.1. Theorem 1.2. F or a Banach sp ac e E , the fol lowing ar e e quivalent : (a) If X is a c ol le ctionwise norm a l sp ac e and ϕ : X → C ( E ) is an l.s.c. c onvex- value d m a pping, then ϕ has a c on tinuous sele ction. (b) If X is a c ol l e ctionwise norm al sp ac e and ϕ : X → C ′ ( E ) is an l.s.c. c onvex-va lue d mapping, then ϕ has a c ontinuous sele ction. Let us emphasize that the pro of of Theorem 1.2 is based only on Do wk er’s extension theorem [5]. This pro of is presen ted in the next se ction and its main ingredien t is the fact that if ϕ : X → C ′ ( E ) and g : X → E , then ϕ ( x ) is compact for ev ery x ∈ X for whic h g ( x ) / ∈ ϕ ( x ). This is further applied in Section 3 to get with ease a direct pro of of a natural generalization of Theorem 1.1. Section 4 deals with controlled selections for set-v a lued mappings define d o n coun tably para com- pact or collectionw ise no r ma l spaces whic h are naturally interrelated to the idea of Theorem 1.2. 2. Proo f of Theorem 1.2 It only suffices to prov e t ha t (a) ⇒ ( b). T o this end, supp ose that (a) of Theorem 1.2 holds, E is a Banac h space, and X is a collection wise normal space. Here, and in t he se quel, w e will use d to denote the me tric o n E generated b y the norm of E . Recall that a map f : X → E is an ε -sele ction for ψ : X → 2 E if d ( f ( x ) , ψ ( x )) < ε for e v ery x ∈ X . The ke y elemen t in the pro of of this implication is the follo wing construction of appro ximate selections . Claim 2.1. L et ψ : X → 2 E b e an l.s.c. c onvex-value d m apping and g : X → E b e a c ontinuous map such that ψ ( x ) is c omp act w h enever x ∈ X a nd g ( x ) / ∈ ψ ( x ) . Then, for every ε > 0 , ψ has a c ontinuous ε - s e le ction. Pr o of. Let ε > 0 and A =  x ∈ X : d ( g ( x ) , ψ ( x )) ≥ ε  . Then, A ⊂ X is closed b ecause ψ is l.s.c. and g is contin uous. Since ψ ↾ A : A → C ( E ) and A is itself a collection wise normal space, by (a) o f Theorem 1.2, ψ ↾ A has a con tin uous selection h 0 : A → E . Since X is collectionw ise normal, b y Dowk er’s extension theorem [5], there exists a c on tin uous map h : X → E suc h that h ↾ A = h 0 . Consider t he set U =  x ∈ X : d ( h ( x ) , ψ ( x )) < ε  whic h con tains A and is op en b ecause ψ is l.s.c. and h is con tin uous. Finally , tak e a con tinuous function SELECTIONS, EXTENSIO NS AND COLLECTIONWISE NORMALITY 3 α : X → [0 , 1] suc h that A ⊂ α − 1 (0) and X \ U ⊂ α − 1 (1), and then define a con tin uous map f : X → E b y f ( x ) = α ( x ) · g ( x ) +  1 − α ( x )  · h ( x ) , x ∈ X . This f is as required. Indeed, ta k e a point x ∈ X . If x ∈ A , then α ( x ) = 0 and, therefore, f ( x ) = h ( x ) = h 0 ( x ) ∈ ψ ( x ). If x ∈ X \ U , then α ( x ) = 1 and w e no w ha ve that f ( x ) = g ( x ), so d ( f ( x ) , ψ ( x )) = d ( g ( x ) , ψ ( x )) < ε b ecause x / ∈ A . Supp ose finally that x ∈ U \ A . In this cas e, d ( h ( x ) , ψ ( x )) < ε and d ( g ( x ) , ψ ( x )) < ε . Since ψ ( x ) is conv ex and f ( x ) = α ( x ) · g ( x ) +  1 − α ( x )  · h ( x ), this implies that d ( f ( x ) , ψ ( x )) < ε . The pro of is completed.  Ha ving already es tablished Claim 2.1, we pro cee d to the proof of (a) ⇒ (b) whic h is based on standard argumen ts for constructing con tin uous selections, see [16]. In this pro of , and in wh at follow s, for a nonempt y subset S ⊂ E and ε > 0, w e will use B ε ( S ) = { y ∈ E : d ( y , S ) < ε } to denote the op en ε -neighb ourho o d of S . In particular, for a point y ∈ E , w e set B ε ( y ) = B ε ( { y } ). Let ϕ : X → C ′ ( E ) b e an l.s.c. conv ex-v alued mapping. If g : X → E is an y contin uous map, sa y a constant one, then ϕ ( x ) is compact for ev ery x ∈ X for whic h g ( x ) / ∈ ϕ ( x ). Hence , b y Claim 2.1 , ϕ has a con tin uous 2 − 1 -selection f 0 : X → E . Define ϕ 1 : X → F ( E ) b y ϕ 1 ( x ) = ϕ ( x ) ∩ B 2 − 1 ( f 0 ( x )) , x ∈ X . According to [16, Prop ositions 2.3 and 2.5], ϕ 1 is l.s.c., and clearly it is con vex - v alued. Finally , o bserv e that if f 0 ( x ) / ∈ ϕ 1 ( x ) for some x ∈ X , then f 0 ( x ) / ∈ ϕ ( x ) and, therefore, ϕ ( x ) is compact b ecause it is C ′ ( E )-v alued. Since ϕ 1 ( x ) is a closed subset of ϕ ( x ), it is also compact. Hence, b y Claim 2.1, ϕ 1 has a con tin uous 2 − 2 - selection f 1 . In particular, f 1 is a con tin uous 2 − 2 -selection for ϕ suc h that d ( f 1 ( x ) , f 0 ( x )) ≤ 2 − 1 < 2 0 , for ev ery x ∈ X . Th us, b y induction, w e get a sequence { f n : n < ω } of con tin uous maps suc h that, for ev ery n < ω a nd x ∈ X , d ( f n ( x ) , ϕ ( x )) < 2 − ( n +1) , (2.1) d ( f n +1 ( x ) , f n ( x )) < 2 − n . (2.2) By (2.2), { f n : n < ω } is a C auc h y sequence in the complete metric space ( E , d ), so it m ust conv erge to some con tin uous f : X → E . By (2.1), f ( x ) ∈ ϕ ( x ) for ev ery x ∈ X . Hence, (b) holds and the pro of of Theorem 1.2 is completed. 3. More on Sele ctions and Collectionwise Normality A space X is τ -c ol le ctionwise norm al , where τ is an infinite cardinal n um b e r, if it is a T 1 -space and for ev ery discrete collection D of closed subsets of X , with | D | ≤ τ , there exists a disc rete collection { U D : D ∈ D } of op en subs ets of X 4 V ALENTIN GUTEV AND NARCISSE ROLAND LOUF OUMA MAK ALA suc h that D ⊂ U D for eve ry D ∈ D . Clearly , a space X is collectionw ise normal if and only if it is τ - collection wise normal f or ev ery τ . Also, it is w ell know n tha t X is normal if and only if it is ω -collection wise normal. Ho w eve r, for ev ery τ there exists a τ -collection wise normal space w hic h is not τ + -collection wise normal [19], where τ + is the immediate succ essor of τ . The pro of of Theorem 1 .2 s uggests an easy direct pro of of the following natural generalization of Theorem 1.1 in [2]. Theorem 3.1 ([2]) . L et X b e a τ -c ol le ctionwise normal s p ac e, E b e a Banach sp ac e with a top o lo gic al weight w ( E ) ≤ τ , and let ϕ : X → C ′ ( E ) b e an l.s.c. c onvex-va lue d mapping. Th en, ϕ has a c on tinuous sele ction. Pr o of. It only suffices to prov e the statemen t of Claim 2.1 for this particular case. So, suppose that ψ : X → 2 E is l.s.c. a nd con v ex-v alued, a nd g : X → E is a con tin uous map such that ψ ( x ) is compact whenev er g ( x ) / ∈ ψ ( x ). Als o, le t ε > 0 and let V b e an op en and lo cally finite co v er of E suc h that diam d ( V ) < ε fo r ev ery V ∈ V . Since g is contin uous, U 1 = { g − 1 ( V ) ∩ ψ − 1 ( V ) : V ∈ V } is a lo cally finite family of op e n subse ts of X which refines { ψ − 1 ( V ) : V ∈ V } . T hen, A = X \ S U 1 is a close d subse t of X , while ψ ↾ A is compact-v alued. Indeed, if g ( x ) ∈ ψ ( x ), then x ∈ ψ − 1 ( V ) whenev er V ∈ V and g ( x ) ∈ V . T hat is, x ∈ A implies g ( x ) / ∈ ψ ( x ), so, in t his case, ψ ( x ) must b e compact. Th us, { ψ − 1 ( V ) : V ∈ V } is an o p en (in X ) and point-finite (in A ) cov er of A suc h that | V | ≤ τ b ecause V is locally- finite and w ( E ) ≤ τ . Sinc e X is τ -collectionw ise normal, b y [18, Lemma 1.6], { ψ − 1 ( V ) : V ∈ V } has a n o p en and lo cally finite (in X ) refinemen t U 2 whic h co v ers A . Then, U = U 1 ∪ U 2 is an op en and lo cally finite co ve r of X whic h refines { ψ − 1 ( V ) : V ∈ V } . F or eve ry U ∈ U tak e a fixed V U ∈ V suc h that U ⊂ ψ − 1 ( V U ) and a p oint e ( U ) ∈ V U pro vided V U 6 = ∅ . Next, tak e a partition of unit y { ξ U : U ∈ U } on X whic h is index sub o rdinated to the co v er U , see [15]. Finally , de fine a contin uous map f : X → E b y f ( x ) = P { ξ U ( x ) · e ( U ) : U ∈ U } , x ∈ X . This f is an ε -selection for ψ .  As far a s the role of the fa mily C ′ ( E ) is concerned, the argumen ts in the pro of of Theorems 1.2 a nd 3.1 w ere based only o n the prop erty that if ϕ : X → C ′ ( E ) and g : X → E is an ε - selection for ϕ for some ε > 0, then the set-v alued mapping ψ ( x ) = ϕ ( x ) ∩ B ε ( g ( x )) , x ∈ X , is suc h that ψ ( x ) is compact whenev er g ( x ) / ∈ ψ ( x ). That is, this resulting ψ is alw a ys as in Claim 2.1, and the inductiv e construction can b e carried on. Motiv ated b y this, we shall sa y that a mapping ψ : X → F ( E ) has a sele ction C ( E ) -deficiency if there exists a con tin uo us g : X → E suc h that ψ ( x ) ∈ C ( E ) for ev ery x ∈ X for whic h g ( x ) / ∈ ψ ( x ). Clearly , ev ery ϕ : X → C ′ ( E ) has this prop erty , for instance ta ke g : X → E to b e an y constan t map. How ev er, there SELECTIONS, EXTENSIO NS AND COLLECTIONWISE NORMALITY 5 are nat ural examples of mappings ϕ : X → F ( E ) whic h hav e a selection C ( E )- deficiency and a re not C ′ ( E )-v alued, see nex t section. Related to this, w e don’t kno w if suc h mappings ma y hav e con tin uous selections in the case of collectionwis e normal spaces. Question 1. Let X b e a c ollection wise normal space, E b e a Banac h space, and let ϕ : X → F ( E ) b e an l.s.c. con ve x-v alued mapping whic h has a selection C ( E )-deficienc y . Then, is it t r ue that ϕ has a contin uous selection? Another aspect of impro ving Theorem 1.1 is related to the range o f the s et- v alued mapping. In this r egard, Theorem 3.1 r emains v alid without any change in the argumen ts if the Banach space E is replaced b y a closed con v ex subset Y of E . On the other hand, if Y is a completely metrizable absolute retract for the metrizable spaces , then fo r ev ery collection wise normal space X and closed A ⊂ X , eve ry con tin uous map g : A → Y can be con tin uously extended to the whole of X , see, e.g., [19]. In particular, this is true for eve ry conv ex G δ -subset Y of a Banac h space E . Namely , Y is an absolute retract for me trizable spaces b eing con v ex (b y Dug undji’s extension t heorem [6]), and is also completely metrizable b eing a G δ -subset of a complete metric space. Motiv ated b y this and the r elatio ns b et w een extensions and selections demonstrated in the pro of o f Theorem 1.2, w e ha v e also the follo wing question. Question 2. Let E b e a Banach space, Y ⊂ E b e a conv ex G δ -subset of E , X b e a collectionwis e normal space, and let ϕ : X → C ′ ( Y ) b e an l.s.c. con vex -v alued mapping. Then, is it true that ϕ has a con tin uous selection? Question 2 is similar to Mic hael’s G δ -problem [1 7, Problem 3 96] if for a para- compact space X and a conv ex G δ -subset Y of a Banac h space, ev ery l.s.c. con v ex- v alued ϕ : X → F ( Y ) has a con tin uo us selection. In general, the a nsw er to this latter problem is in the negativ e due to a coun terexample constructed by Filipp o v [8, 9]. Ho w ev er, Mic hael’s G δ -problem w as r esolv ed in t he affirmativ e in a n um ber of partial cases . The solution in some of these case s remains v alid for Question 2 as w ell. F or instance, if Y is a coun table in tersection of op en con v ex sets, then t he closure con v ex-hull con v( K ) of ev ery compact subs et of Y will b e still a subset of Y , see [17]. In this case, b y a result of [2], ϕ will hav e a n l.s.c. con v ex-v alued selection ψ : X → C ( Y ) (i.e., ψ ( x ) ⊂ ϕ ( x ) for all x ∈ X ). Hence, ϕ will ha v e a contin uous selection because, by Theorem 1.1, s o does ψ . If the cov ering di- mension of X is bounded (i.e., dim( X ) < ∞ ), then the answ er to Question 2 is also “yes ”, this follo ws directly from a selec tion theorem in [10]. The answ er to Question 2 is also “ye s” if X is strongly coun table-dimensional (i.e., a countable union of closed finite-dimensional subse ts). In this case, there exists a me trizable (strongly) countable-dimens ional space Z , a contin uous map g : X → Z and an l.s.c. mapping ψ : Z → C ( Y ) suc h that ψ ( g ( x )) ⊂ ϕ ( x ) f or ev ery x ∈ X , see, fo r instance, the pro of of [18, Theorem 5.3]. Next, define a mapping Φ : Z → F ( Y ) 6 V ALENTIN GUTEV AND NARCISSE ROLAND LOUF OUMA MAK ALA b y Φ( z ) = c on v( ψ ( z )) Y , z ∈ Z , where the closu re is in Y . Acc ording to [16, Prop ositions 2 .3 and 2.6], Φ remains l.s.c., and, b y [11, Corollary 1.2 ], it admits a c on tin uous selec tion h : Z → Y . Then, f = h ◦ g is a contin uous selection for ϕ b ecause Φ( g ( x )) ⊂ ϕ ( x ) for all x ∈ X . 4. Controlled Selections and Count able P aracomp actness A function ξ : X → R is lower ( upp er ) sem i -c ontinuous if the set { x ∈ X : ξ ( x ) > r } (resp ectiv ely , { x ∈ X : ξ ( x ) < r } ) is op en in X fo r e v ery r ∈ R . If ( E , d ) is a metric space, ϕ : X → 2 E and η : X → (0 , + ∞ ), then w e shall sa y that g : X → E is an η -sele ction f o r ϕ if d ( g ( x ) , ϕ ( x )) < η ( x ) for ev ery x ∈ X . In this sec tion, w e first pro v e the follow ing c haracterization of coun t a bly para- compact normal spaces. Theorem 4.1. F or a T 1 -sp ac e X , the fol lowing ar e e quivalent : (a) X is c ountably p ar ac omp act and normal. (b) If E is a sep ar able Ba nach sp ac e, ϕ : X → F ( E ) is an l.s . c . c onvex-va l ue d mapping, η : X → (0 , + ∞ ) is lower sem i-c ontinuous and g : X → E is a c ontinuous η -sele ction for ϕ , then ϕ has a c o n tinuous sele ction f : X → E such that d ( f ( x ) , g ( x )) < η ( x ) for al l x ∈ X . (c) If ϕ : X → C ( R ) is an l.s.c. c onvex-value d m apping, ε > 0 a n d g : X → R is a c ontinuous ε -se le ction for ϕ , then ϕ has a c on tinuous se l e ction f : X → R s uch that d ( f ( x ) , g ( x )) < ε f o r al l x ∈ X . Pr o of. (a) ⇒ (b). Let X b e a coun tably paracompact normal space, and let E , ϕ , η and g be as in (b). Since ϕ is l.s.c. and g is con tinuous , ξ ( x ) = d ( g ( x ) , ϕ ( x )), x ∈ X , is an upp er semi-contin uous function suc h that ξ ( x ) < η ( x ) for all x ∈ X b ecause g is an η -selection for ϕ . Sinc e X is coun tably paracompact and normal, b y a result of [3, 4, 14] (see, also, [7, 5 .5 .20]) there exists a con tin uous function α : X → R suc h that ξ ( x ) < α ( x ) < η ( x ) for ev ery x ∈ X . Then, define an l.s.c. mapping ψ : X → F ( E ) by ψ ( x ) = ϕ ( x ) ∩ B α ( x ) ( g ( x )) , x ∈ X . Since ψ is con v ex-v alued, b y [16, Theorem 3.1 ′′ ], ψ has a contin uous selection f : X → E . In particular, d ( f ( x ) , g ( x )) ≤ α ( x ) < η ( x ) for all x ∈ X . Since (b) ⇒ (c) is obvious , we complete the pro of sho wing that (c) ⇒ (a). So, s upp ose that (c) holds. If A and B are disjoin t clos ed subsets of X , then ϕ ( x ) = { 0 } if x ∈ A , ϕ ( x ) = { 1 } if x ∈ B , a nd ϕ ( x ) = [0 , 1] otherwise , is an l.s.c. con v ex-v alued ma pping ϕ : X → C ( R ). If g ( x ) = 1 2 , x ∈ X , then g is a con tin uous 1-selection for ϕ , and, by (c), ϕ has a contin uous selection f : X → R . According to the definition of ϕ , w e get that A ⊂ f − 1 (0) and B ⊂ f − 1 (1), hence X is normal. T o s ho w that X is countably paracompact, let { F n : n < ω } be a decreasing SELECTIONS, EXTENSIO NS AND COLLECTIONWISE NORMALITY 7 sequence of closed subse ts o f X suc h that F 0 = X and T { F n : n < ω } = ∅ . Next, f or ev ery x ∈ X , let n ( x ) = max { n < ω : x ∈ F n } . Then, define a con v ex-v alued mapping ϕ : X → C ( R ) by ϕ ( x ) =  0 , 2 − n ( x )  , x ∈ X . Observ e that ϕ is l.s.c. b ec ause z ∈ X \ F n ( x )+1 implies that n ( z ) ≤ n ( x ) and, therefore, ϕ ( x ) =  0 , 2 − n ( x )  ⊂  0 , 2 − n ( z )  = ϕ ( z ). Finally , observ e tha t g ( x ) = 1, x ∈ X , is a con tinuous 1 -selection for ϕ . Hence, by (c), ϕ has a c on tin uous selection f : X → R s uc h that | g ( x ) − f ( x ) | = 1 − f ( x ) < 1 for ev ery x ∈ X , or, in other w ords, f ( x ) > 0 for all x ∈ X . F inally , define W n = f − 1 (( −∞ , 2 − n +1 )), n < ω . Th us, w e get a s equence { W n : n < ω } of op en op en subse ts of X suc h that F n ⊂ W n for ev ery n < ω . Indeed, x ∈ F n implies n ≤ n ( x ), so f ( x ) ∈ ϕ ( x ) =  0 , 2 − n ( x )  ⊂ [0 , 2 − n ] ⊂ [0 , 2 − n +1 ]. Since W n ⊂ f − 1 (( −∞ , 2 − n +1 ]), n < ω , and f ( x ) > 0 fo r ev ery x ∈ X , w e hav e that T  W n : n < ω  = ∅ . That is, X is coun tably paracompact, see [7, Theorem 5.2.1].  F or collection wise normal spaces w e ha v e a very similar result w hic h, in par- ticular, illustrat es the difference with coun tably paracompact ones (see, (c) of Theorem 4.1). Prop osition 4.2. L et E b e a Banach sp ac e , X b e a c ol le ctionwise normal sp ac e, ϕ : X → C ′ ( E ) b e an l.s.c. c onvex-value d mapp ing, and let g : X → E b e a c ontinuous ε -se l e ction for ϕ for some ε > 0 . Then, ϕ has a c ontinuous sele ction f : X → E such that d ( f ( x ) , g ( x )) ≤ ε for every x ∈ X . Pr o of. Define a mapping ψ : X → F ( E ) b y ψ ( x ) = B ε ( g ( x )), x ∈ X . Then, ψ is conv ex-v alued and d -proximal contin uous in the sense of [1 2]. Define ano ther mapping θ : X → F ( E ) b y θ ( x ) = ϕ ( x ) ∩ B ε ( g ( x )), x ∈ X . According to [16, Prop ositions 2 .3 and 2.5], θ is l.s.c. and clearly it is also con v ex-v alued. Finally , observ e that θ ( x ) ⊂ ψ ( x ) for ev ery x ∈ X , while θ ( x ) 6 = ψ ( x ) implies that θ ( x ) is compact. Then, by [13, Lemma 4.2 ], θ has a contin uous sele ction f : X → E . This f is as require d.  F ollowing the idea Prop osition 4.2 , one can extend Theorem 4.1 to the case of coun tably paracompact and τ -collection wise normal spaces. Theorem 4.3. F or a T 1 -sp ac e X and an in fi nite c ar dinal numb er τ , the fol lowi n g ar e e quivalent : (a) X is c ountably p ar ac omp act and τ -c ol le ctionwise normal. (b) If E is a Banach sp ac e with w ( E ) ≤ τ , ϕ : X → C ′ ( E ) is l.s.c. and c onvex- value d, η : X → (0 , + ∞ ) is lower sem i-c ontinuous, and g : X → Y is a c ontinuous η -sele ction for ϕ , then ϕ has a c o n tinuous sele ction f : X → E such that d ( f ( x ) , g ( x )) < η ( x ) for al l x ∈ X . Pr o of. (a) ⇒ (b). As in (a) of the proof of T heorem 4.1, t here exists a con- tin uous function α : X → (0 , + ∞ ) suc h that d ( g ( x ) , ϕ ( x )) < α ( x ) < η ( x ) for 8 V ALENTIN GUTEV AND NARCISSE ROLAND LOUF OUMA MAK ALA ev ery x ∈ X . Next, as in the pro of of Prop osition 4.2, define a ( d - pro ximal) con tin uous ψ ( x ) = B α ( x ) ( g ( x )) , x ∈ X , and another l.s.c. θ : X → F ( E ) b y θ ( x ) = ϕ ( x ) ∩ B α ( x ) ( g ( x )) , x ∈ X . As in the pro of of [13, Lemma 4.2], this θ has the Selection F actorization Property in the sense of [18]. Hence, by [18, Prop osi- tion 4.3], θ has a con t in uous selection f : X → E . According to the definition of θ , w e get that d ( f ( x ) , g ( x )) < η ( x ) for all x ∈ X . (b) ⇒ (a). This implication is based on standa r d a rgumen ts. In fact, X will b e coun tably paracompact and normal by Theorem 4.1. T o show tha t X is also τ -collection wise normal, let D b e a discrete family of closed subsets of X , with | D | ≤ τ , and let ℓ 1 ( D ) b e the Banach space of all functions y : D → R , with P  | y ( D ) | : D ∈ D  < ∞ , equipp ed with the no r m k y k = P  | y ( D ) | : D ∈ D  . Also, let ϑ ( D ) = 0, D ∈ D , b e the origin of ℓ 1 ( D ). F or ev ery D ∈ D , consider the function e D : D → R defined b y e D ( D ) = 1 and e D ( T ) = 0 for T ∈ D \ { D } . Then, e D ∈ ℓ 1 ( D ), D ∈ D , and k e D − ϑ k = 1 for ev ery D ∈ D . F inally , define an l.s.c. mapping ϕ : X → C ′ ( ℓ 1 ( D )) b y ϕ ( x ) = { e D } if x ∈ D f or some D ∈ D and ϕ ( x ) = ℓ 1 ( D ) o therwise. Then, ϕ is con v ex-v alued and g ( x ) = ϑ , x ∈ X , is a con tinuous 2 - selection for ϕ . Since w ( ℓ 1 ( D )) ≤ τ , b y (b), ϕ has a con tin uo us selection f : X → ℓ 1 ( D ). Then, U D = f − 1 ( B 1 ( e D )), D ∈ D , is a pairwise disjoin t family of op en subsets o f X suc h t hat D ⊂ U D , D ∈ D . Since X is normal, this implies that it is also τ -collection wise normal.  Exactly the same argumen ts as those in the pro of of Theorem 4.3 sh o w that if X is τ -collection wise norma l, E is a Ba nac h space with w ( E ) ≤ τ , ϕ : X → C ′ ( E ) is an l.s.c. conv ex-v alued mapping, η : X → (0 , + ∞ ) is con tin uous, and g : X → E is a contin uous η - selection, then ϕ has a con tinuous selection f : X → E suc h that d ( f ( x ) , g ( x )) ≤ η ( x ) for a ll x ∈ X . Motiv ated by t his and Prop o sition 4.2, w e ha ve the follo wing natural question. Question 3. Let X b e a collection wise normal space, E b e a Banac h space, ϕ : X → C ′ ( E ) b e an l.s.c. con v ex-v alued mapping, and let g : X → E b e a con tin- uous η -se lection for ϕ for some lo w er semi-con tin uous function η : X → (0 , + ∞ ). Then, do es ϕ ha v e a con tinuous selection f : X → E with d ( f ( x ) , g ( x )) ≤ η ( x ) for all x ∈ X ? Let us p oin t out that the answ er to Question 3 is “y es” if so is the answ er to Question 1. Indeed, if η : X → (0 , + ∞ ) is low er semi-con t in uous and g : X → E is con tin uous, then the mapping ψ ( x ) = B η ( x ) ( g ( x )), x ∈ X , will ha ve an op en graph. If g is also an η -selection for ϕ : X → C ′ ( E ), then θ ( x ) = ϕ ( x ) ∩ ψ ( x ), x ∈ X , will ha v e a se lection C ( E )- deficiency . Finally , if f : X → E is a con tin uous selection for θ , then d ( f ( x ) , g ( x )) ≤ η ( x ) for all x ∈ X . SELECTIONS, EXTENSIO NS AND COLLECTIONWISE NORMALITY 9 Reference s [1] R. H. Bing, Metrization of t op olo gic al sp ac es , Canad. J. Math. 3 (1951 ), 175– 1 86. [2] M. Choban and V. V alov, On a the or em of E. Michae l on sele ctions , C. R. Acad. Bulga re Sci. 28 (19 75), 871– 8 73, (in Russian). [3] J. Dieudonne, Une g´ en´ er alisation des esp ac es c omp acts , J . Math. Pures Appl. 23 (19 44), 65–76 . [4] C. H. Do wker, On c ountably p ar ac omp act sp ac es , Canad. J. of Math. 3 (1 951), 29 1–22 4 . [5] , On a the or em of Hanner , Ark. Mat. 2 (1 9 52), 307– 3 13. [6] J. Dugundji, An ext en sion of Tie tze’s the or em , Pacific J. Math. 1 (1951 ), 353–3 67. [7] R. Engelking, Gener al top olo gy, r evise d and c omplete d e dition , Heldermann V erlag, Berlin, 1989. [8] V. V. Filipp ov, On a question of E. A. M ichael , Commen t. Math. Univ. Carolinae 45 (2004), no. 4, 735– 737. [9] , O n a pr oblem of E. Micha el , Mat. Zametki 78 (200 5), no. 4 , 6 38–6 40, (in Russia n), Math. Notes 78 (34) (2005 ) 597–5 99. [10] V. Gutev, Continuous sele ctions and fi nite-dimensional sets in c ol le ctionwise normal sp ac es , C. R. Acad. Bulgare Sci. 39 (1986 ), no. 5, 9–1 2. [11] , Continu ous sele ctions, G δ -subsets of Banach sp ac es and usc o mappings , Co mment. Math. Univ. Car olin. 35 (199 4), no. 3, 5 3 3–53 8. [12] , We ak factorizations of c ont inuous set-value d m appings , T o p ology Appl. 102 (2000), 33–51 . [13] V. Gutev, H. Ohta, and K. Y amazaki, Sele ctions and sandwich-li ke pr op erties via semi- c ontinu ou s Banach-value d functions , J. Math. So c. Ja pan 55 (20 0 3), no. 2, 4 99–52 1. [14] M. Kat ˇ etov, On re al-value d functions in top olo gic al sp ac es , F und. Math. 38 (1951), 85–9 1. [15] E. Mic hael, A note on p ar ac omp act sp ac es , Pro c. Amer. Math. So c. 4 (19 53), 83 1–838 . [16] , Continuous sele ctions I , Ann. o f Math. 63 (19 56), 36 1–382 . [17] , Some pr oblems , Op en Problems in Topolo gy (J. v an Mill a nd G. M. Reed, eds.), North-Holland, Amsterdam, 1 990, pp. 2 73–2 7 8. [18] S. Nedev, Sele ction and factorization the or ems for set-value d mappings , Serdica 6 (1980), 291–3 17. [19] T. P rzymusi´ nski, Colle ctionwise normality and absolute r etr acts , F und. Math. 98 (1978), 61–73 . School of Ma thema tical S ciences, University of Kw aZul u-Na t al, Westville Campus, Priv a te Ba g X540 01, Durban 4000, South Africa E-mail addr ess : gutev @ukzn .ac.za School of Ma thema tical S ciences, University of Kw aZul u-Na t al, Westville Campus, Priv a te Ba g X540 01, Durban 4000, South Africa E-mail addr ess : rolan d@aim s.ac.z a