On Extension Of Functors

On Extension of F unctors L.Karchevsk a, T.Radul June 18, 2018 Departmen t of Me c hanics and Mathematics, Lviv National Univ ersit y , Univ ersytetsk a st.,1, 79000 Lviv, Ukraine. e-mail: tarasradul@y aho o.co.uk, crazymaths@ukr.net Key words and phrases: Chigogidze extension of functors, 1-preimage preserving prop- ert y 2000 Mathematics Sub ject Classifications. 18B30, 54B30, 57N20 Abstract A.Chigogidze d efined for eac h normal f u nctor on the category C omp an extension whic h is a normal functor on the cate gory T y ch . W e consider this extension for an y functor on the category C omp and inv estigate whic h prop erties it preserve s from the definition it preserves from the definition of normal fun cto r. W e in v estigate as well s ome top olog ical p rop ertie s of suc h extension. In tro duction. The general theory of functors acting in the category C omp of compact Hausdorff spaces (compacta) and contin uous mappings was fo unded by E.V. Shc hepin [15]. He distinguished some elemen tary prop erties of suc h functors and defined the notion of nor ma l functor that has become v ery fruitful. The class of normal functors includes man y classical constructions: the hyperspace exp , the f unctor o f probability measures P , t he p o w er functor and man y other functors (see [13],[9] for more details). But some imp ortan t functors do no t satisfy some of t he prop erties fro m the Shc hepin list. Omitting some prop erties w e obtain wider classes of functors suc h as we akly norma l functors and almost normal f unctors. The prop erties fro m the definition of normal functor could b e easily generalized for the func- tors on t he category T y ch of Tyc hono v spaces and contin uous maps. Let us remark that T y ch con ta ins C omp as a subcatego ry . A.Chigogidze defined for eac h normal functor on the category C omp an extension whic h is a normal functor on the catego ry T y ch [6]. This extension could b e considered for an y functor on the category C omp . But the situation is more complicated fo r wider classes of functors. F or example, the extension of the pro jectiv e p ow er functor (whic h is w eakly normal) do es not preserv e em b eddings, whic h mak es suc h extension useless (see for 1 example [13], p.67 ). How ev er, if we apply the Chigog idze extension to suc h we akly normal f unc- tors as the functor O of order- preserving functionals, the functor G of inclusion h yp erspaces , the sup erexte nsion, w e obtain functors on the category T y ch whic h preserv e em b eddings. The main aim of this pap er is to inv estigate whic h prop erties from the definition of normal functor are preserv ed b y Chigogidze extension, sp ecially w e concen trate our attention on t he preserving of em b eddings. The results dev oted to this problem are con t a ined in Section 2. W e define in this section t he 1 - preimages preserving prop erty whic h is crucial f or preserving of em b eddings. In Section 3 we consider whic h functors ha v e the 1-preimages preserving prop ert y . T.Banakh and R.Caut y obtained top ological classification of the Chigogidze extension of the functor of probability measures for separable metric spaces. W e generalize this result for con vex functors in Section 4 . 1. All spaces are assumed to b e T ychono v, all mappings are contin uous. All functors are assumed to b e co v arian t. In the presen t pap er w e will consider f unctor s acting in tw o categories: the category T y ch and it s subcategory C omp . Let us recall the definition of normal functor. A functor F : C omp → C omp is called monomorphic (epim orphic) if it preserv es em b eddings (surjections). F or a monomorphic functor F and an em b edding i : A → X we shall iden tify the space F ( A ) and the subspace F ( i )( F ( A )) ⊂ F ( X ). A monomo r phic functor F is said to b e pr e i mage-pr eserving if fo r eac h map f : X → Y and eac h close d subs et A ⊂ Y w e ha v e ( F ( f )) − 1 ( F ( A )) = F ( f − 1 ( A )). F or a monomorphic functor F the interse ction-pr eserving prop ert y is defined as follow s: F ( ∩{ X α | α ∈ A} ) = ∩{ F ( X α ) | α ∈ A} for ev ery family { X α | α ∈ A} of closed subsets of X . A functor F is called c on tinuous if it preserv es the limits o f in v erse system s S = { X α , p β α , A} o v er a directed set A . Let us also note that for a ny contin uous f unctor F : C omp → C omp the map F : C ( X , Y ) → C ( F X, F Y ) (the space C ( X , Y ) is considered with the compact-op en top ology) is con tin uous. Finally , a functor F is called weight-pr eserving if w ( X ) = w ( F ( X )) for every infinite X ∈ C omp . A f unctor F is called norm a l [15] if it is con tin uous, monomorphic, epimorphic, preserv es w eight,in tersections,preimages,sin gletons and the empty space. A f unc tor F is said to b e we ak ly normal ( a l m ost normal ) if it satisfies all the prop erties from the definition of a normal functor excepting p erhaps the preimage-preserving property (epimorphness)(see [13] for more details). Similarly , one can define the same prop erties for a functor F : T y ch → T y ch with t he only difference that the prop ert y of preserving surjec tions is replaced b y the prop ert y of sending k -co vering maps to surjections (recall that f : X → Y is a k - co v ering map if for an y compact set B ⊂ Y there exists a compact set A ⊂ X with f ( A ) = B ) (see [13], D ef.2.7.1). A.Chigogidze defined an extension construction of a functor in C omp onto T y ch the fol- lo wing wa y [6]. F or an y normal functor F : C omp → C omp a nd a n y X ∈ T y ch the 2 space F β ( X ) = { a ∈ F ( β X ) | there exists a compact set A ⊂ X with a ∈ F ( A ) } is considered with the top ology induced fro m F ( β X ), where β X is the Stone-Cec h compactification of the space X . Next, g iv en an y con tinuous mapping f : X → Y b et w een T yc honov sp aces, put F β ( f ) = F ( β f ) | F β ( X ) . Then F β forms a cov ariant functor in the catego ry T y ch . Chigogidze sho wed that in case F is normal, t he functor F β is also normal. 2. Let us mo dify the Chigogidze construction for an y functor F : C omp → C omp . F o r X ∈ T y ch w e put F β ( X ) = { a ∈ F ( β X ) | there exists a compact se t A ⊂ X with a ∈ F ( i A )( F ( A )) } where b y i A w e denote the natural embedding i A : A ֒ → X (w e do not assume tha t the map F ( i A ) is an embedding). Eviden tly F β preserv es empt y set and one-p oin t space iff F do es. No w w e consider the problem when F β preserv es embeddings. Extension of an y normal functor preserv es em b eddings, but, if w e drop the preimage preserving pro perty , the situation could b e different. Ho w ev er, the examples from the in t r oduction show that the preimage- preserving property is not neces sary . W e define some w eak er prop ert y whic h will giv e us a necessary and sufficien t condition. Definition 1. We say that a monom orphic functor F : C omp → C omp preserv es 1 - preimages , if for any f : X → Y , whe r e X , Y ∈ C omp , a n y close d A ⊂ Y such that f | f − 1 ( A ) is a home omorphis m , we have that ( F f ) − 1 ( F A ) = F ( f − 1 ( A )) . (L et us r emark it is e quivalent to the c o n dition that the m ap F f | ( F f ) − 1 ( F A ) is a hom e omorphism.) Let us note that this definition w as indep enden tly in tro duced b y T.Banakh and A.Kuch arski [3]. Prop osition 1. If F is a monomorphi c functor that pr eserve s 1 -pr eima ges in the class of op en mappings, then F pr eserve s 1-pr eimages. Pr o of. T ake an y mapping f : X → Y suc h that f | f − 1 ( A ) is a homeomorphism f or some closed subset A ⊂ Y . Let i 1 : X → X × Y b e an em b edding defined by the fo r m ula i 1 ( x ) = ( x, f ( x )). Denote Z = X × Y /ε , where the relation ε is giv en by ε = { pr − 1 Y ( a ) | a ∈ A } ( pr Y : X × Y is the resp ectiv e pro jection). Let q : X × Y → Z b e the quotien t mapping. The map h : Z → Y giv en b y the conditio ns h ( z ) = y f o r an y z = ( x, y ) ∈ Z \ q ( X × A ) a nd h ( z ) = a for any z = q ( pr − 1 Y ( a )), a ∈ A , is op en and satisfies the follow ing tw o conditions: pr Y = h ◦ q , h | h − 1 ( A ) is a homeomorphism. Apparen tly , the ma p i = q ◦ i 1 is an em b edding, moreo v er, h ◦ i = f . Since F preserv es 1-preimages in the class of op en mappings, w e ha ve ( F h ) − 1 ( F A ) = F ( h − 1 ( A )), whic h giv es us the equalit y ( F f ) − 1 ( F A ) = F ( f − 1 ( A )). Prop osition 2. If F is a monomorphic functor that pr eserv e s 1-pr eimages , then F β pr eserves emb e ddings. Pr o of. T ake any em b edding f : X → Y . Then the map F β ( f ) is closed as the restriction of a closed map onto a full preimage and, moreov er, injective , hence an em b edding. 3 F or any X ∈ T y ch and an y its compactification bX w e can define F b ( X ) = { a ∈ F ( bX ) | there is a compact subset A ⊂ X with a ∈ F ( A ) } ⊂ F ( bX ) and consider it with the resp ectiv e subspace top ology . Corollary 1. If F is a monom o rphic, 1 -pr eima ge-pr es e rving functor, then F β ( X ) ∼ = F b ( X ) for any T ychonov sp a c e X and its c omp actific ation bX . Prop osition 3. I f F is monom orphic, pr eserves 1-pr ei m ages and we i g ht, then F β pr eserves weight. Pr o of. The statemen t follows from the previous coro lla ry and the fact that for any X ∈ T y ch there exists its compactification bX whic h ha s the same w eigh t as X . As the following prop osition sho ws, the reve rse implication to that of Prop osition 2 also holds. Prop osition 4. L e t F b e a c ontinuous functor s uch that F β pr eserves emb e dding s. Then F pr eserves 1-pr eimages. Pr o of. Assume the con trary . Then there exist a map f : X → Y and a closed subset A ⊂ Y suc h that f | f − 1 ( A ) is a homeomorphism and F f − 1 ( F A ) 6 = F ( f − 1 ( A )). Henc e we can c ho ose ν ∈ F A and µ ∈ F X \ F ( f − 1 ( A )) suc h that F f ( µ ) = ν . W e will construct a space S ∈ T y ch and its compactification γ S suc h that the map F β ( id S ) : F β ( S ) → F β ( γ S ) = F ( γ S ) is not an em b edding, where id S : S → ( γ S ) is an iden tit y embedding. First put Z = X × α N , where the space of natural n umbers N is considered with the discrete top ology and α N = N ∪ { ξ } is the o ne-point compactification of N . Define a con tinuous function g : Z → Y by g ( x, n ) = f ( x ) f o r any x ∈ X, n ∈ α N . Let T = Z/ ε b e a quotient space, where ε is a n equiv alence relation defined b y its classes of equiv a lence {{ x }| x ∈ ( X \ A ) × N } ∪ { g − 1 ( y ) ∩ X × { ξ }| y ∈ Y \ A } ∪ {{ a } × α N | a ∈ A } . By q : Z → T w e denote the resp ec tiv e quotient mapping. Then the map h : T → Y defined by the equalit y g = h ◦ q is con tin uous. The set D = q ( X × { ξ } ) is compact as a con tin uous image of a compact set a nd moreo ve r h | D is one-to-one, hence a homeomorphism betw een D and Y . W e denote b y j : Y → T the in v erse em b edding. Also, for a n y n ∈ N the space S n = q ( X × { n } ) is homeomorphic to X and w e denote by j n : X → T the in v erse em b edding. Then w e hav e h ◦ j n = f . Finally note that T is a compactification of the space S = T \ q (( X \ A ) × { ξ } ). Put µ n = F ( j n )( µ ) for n ∈ N . The sequence j n con verges to j ◦ f in t he space C ( X , T ). Since F is con tin uous, the sequence F ( j n ) con v erg es to F ( j ◦ f ) in the space C ( F X , F T ). Hence the sequence µ n con verges to F ( j ◦ f )( µ ) = F ( j )( ν ) ∈ F ( q ( A × α N )). No w consider F β ( S ) as a subspace of F ( β S ). There exists a map s 1 : S → X suc h that s 1 ◦ j n = id X . Let s : β S → X be the extension o f s 1 . Then F s ( µ n ) = µ / ∈ F ( f − 1 ( A )). Then the sequence µ n do es not con ve rge to an y elemen t of F ( q ( A × α N )). The prop osition is pro v ed. 4 Prop ositions 2 and 4 yield the following Theorem 1. F or any c ontinuous monomorphic functor F the functor F β pr eserves emb e dd i n gs if and only if F pr ese rv e s 1-pr eimages. The pro of of the following prop osition is a routine che c king and w e o mit it. Prop osition 5. L e t F : C omp → C omp b e a functor. 1) if F pr es erves emb e ddings, 1-pr eimages and interse ctions then F β pr eserves i n terse ctions; 1) if F pr eserves emb e ddings an d pr eimages then F β pr eserves pr e images; 3) if F pr eserves surje c tions then F β sends k -c overi n g maps to surje ction s ; No w let us consider con tinuit y of the Chigog idze extension. The following ex ample sho ws that in the absence of the preimage-preserving prop ert y o f the functor F , it is difficult to sp eak of con tinuit y of F β , since ev en the extension of suc h known w eakly nor ma l functor as G do es not p osses s it. Example. Let us define the inclusion hy p erspace functor G . Recall that a closed subset A ∈ exp 2 X , where X ∈ C omp is called an inclusion hyperspace, if for ev ery A ∈ A and ev ery B ∈ exp X the inclusion A ⊂ B implies B ∈ A . Then GX is the space of all inclusion h yp erspaces with the induce d from exp 2 X top ology . F or an y map f : X → Y define Gf : GX → GY by Gf ( A ) = { B ∈ exp Y | f ( A ) ⊂ B f or some A ∈ A} . The functor G is weakly normal (see [1 3] for more details). In the next section we will see that the functor G preserv es 1-preimages. Let us sho w that the functor G β is not con tin uous. Consider the following in v erse sys- tem. F or a ny n ∈ N put X n = N × { 1 , ..., n } ( here the spaces N and { 1 , ..., n } are cons id- ered with the discrete topolo gy). D efine p m n : X m → X n , whe re m ≥ n , the following wa y: p m n ( x, k ) = ( x, min { k , n } ). W e obtained the inv erse system S = { X m , p m n , N } . Then the limit space X = lim S is homeomorphic to the space N × A ( here A = α N = N ∪ { ξ } is t he one-p oint compactification of N , i.e. a con v ergent sequence; also w e put ξ to b e greater than any natural n umber), and the limit pro jections p n : X → X n can b e give n by p n ( x, k ) = ( x, min { k , n } ), k ∈ A . The contin uit y of G β means tha t lim G β ( p n ) : G β (lim S ) → lim G β ( S ) is a homeomor- phism. Here bo t h G β (lim S ) and lim G β ( S ) can be though t as subspaces of G ( bX ), where b is a compactification of X with the prop erty bX = lim β S . No w w e will construct K ∈ lim G β ( S ) whic h do es not b elong to lim G β ( p n )( G β (lim S )). Consider t he space X imbedded into its compactification bX . F or a n y n ∈ A \{ ξ } put K n = { 1 , ..., n } × { n } . If w e w an t to obta in a closed family of sets, the set K ξ = N × { ξ } must b e added to the family e K = { K n } n ∈ N . No w put K = { B ⊂ bX | K n ⊂ B for some n ∈ A } . Then K ∈ lim G β ( S ). Ho w ev er, there is appar ently no elemen t C ∈ G β (lim S ) with lim G β ( p n )( C ) = K . Hence, lim G β ( p n ), b eing not surjectiv e, is not a homeomorphism. 5 3. W e start this section with definitions o f some functors we deal with in this pa per. Let X b e compactum. By C ( X ) we denote the Banac h space of all con tin uous functions φ : X → R with t he usual sup-norm. W e consider C ( X ) with natural order. Let ν : C ( X ) → R b e a functional (we do not suppose a priori that ν is linear or con tin uous). W e sa y that ν is 1) non-expanding if | ν ( ϕ ) − ν ( ψ ) | ≤ d ( ϕ, ψ ) for all ϕ , ψ ∈ C ( X ); 2) w eakly additiv e if for any function φ ∈ C ( X ) and a n y c ∈ R w e hav e ν ( φ + c X ) = ν ( φ ) + c (b y c X w e denote the constan t function); 3) preserv es order if for a n y ϕ, ψ ∈ C ( X ) suc h that ϕ ≤ ψ t he inequalit y ν ( ϕ ) ≤ ν ( ψ ) holds; 4) linear if for any α , β ∈ R and for a ny tw o functions ψ , φ ∈ C ( X ) w e ha v e ν ( α φ + β ψ ) = α ν ( φ ) + β ν ( ψ ). No w for an y space X denote V X = Q ϕ ∈ C ( X ) [min ϕ, max ϕ ]. F or an y mapping f : X → Y define the map V f as follo ws: V f ( ν )( ϕ ) = ν ( ϕ ◦ f ) for ev ery ν ∈ V X, ϕ ∈ C ( Y ). Then V is a co v aria n t functor in the categor y C omp [11]. Let us remark that the space V X could b e considered as the space of all functionals ν : C ( X ) → R with the only condition min ϕ ( X ) ≤ ν ( ϕ ) ≤ max ϕ ( X ) for ev ery ν ∈ V X , ϕ ∈ C ( Y ). By E X w e denote t he subset o f V X defined by the condition 1) (non-expanding func- tionals; see [5] for more details), b y E AX the subset defined b y the conditions 1) and 2 ). The conditions 2) a nd 3) define the subset O X (o rder-prese rving functionals, see [10]); finally , the conditions 3) a nd 4) define the well-kno wn subset P X (probabilit y measures, see for ex ample [ ? ]). F or a map f : X → Y the mapping F f , where F is one of P , O , E A , E , is defined a s the restriction of V f on F X . It is easy to c hec k t hat the constructions P , O , E A and E define subfunctors of V . It is kno wn that the functors O and E are w eakly normal (see [10 ] and [5]). Using the same ar gumen ts one can che c k that E A is weakly normal to o. The que stion arises naturally whic h of defined ab ov e functors hav e the property of preserving 1-preimages. It is easy t o c heck that we ha ve the inclusions P X ⊂ O X ⊂ E AX ⊂ E X ⊂ V X . W e will show t hat the functor E A satisfies this prop ert y and E do es no t. Since subfunctors inherit the 1-preimages preserving pro p erty , this is the complete answe r. Let us also remark that the resu lts of [11] and [12] sho w that man y other kno wn functor s could b e considered as subfunctors of E A , fo r example the sup erextens ion, the h yp erspace functor, the inclusion h yp erspace functor etc. This sho ws that the class of functors with the 1-preimages pres erving prop ert y is wide enough. W e start with a definition of a n AR -compactum. Recall that a compactum X is called an absolute retract (briefly X ∈ AR ) if for any em b edding i : X → Z of X in to compactum Z the image i ( X ) is a retract of Z . The next lemma will b e needed in the fo llo wing discussion. Lemma 1. L et F b e a monomorphic subfunctor o f V which p r e serves interse ctions and B b e a close d subset of a c o mp actum X . Then ν ∈ F B iff ν ( ϕ 1 ) = ν ( ϕ 2 ) for e ach ϕ 1 , ϕ 2 ∈ C ( X ) such that ϕ 1 | B = ϕ 2 | B . 6 Pr o of. Ne c e s s ity . The inclusion ν ∈ F B ⊂ F X means that there exists ν 0 ∈ F B with F ( i B )( ν 0 ) = ν , where i B : B → X is a natural em b edding. Hence, for a ny ϕ 1 , ϕ 2 ∈ C ( X ) suc h that ϕ 1 | B = ϕ 2 | B w e ha ve ν ( ϕ 1 ) = ν 0 ( ϕ 1 ◦ i B ) = ν 0 ( ϕ 2 ◦ i B ) = ν ( ϕ 2 ). Sufficiency . W e can find an em b edding j : B ֒ → Y , where Y ∈ AR . Define Z to b e the quotien t space o f the disjoin t union X ∪ Y obtained b y atta c hing X a nd Y b y B . Denote b y r : Z → Y the retraction mapping. No w tak e any ν ∈ F X ⊂ F Z with the prop ert y ν ( ϕ 1 ) = ν ( ϕ 2 ) for eac h ϕ 1 , ϕ 2 ∈ C ( X ) suc h that ϕ 1 | B = ϕ 2 | B . W e claim that F ( r )( ν ) = ν . Indeed, take an y ϕ ∈ C ( Z ). Then F ( r )( ν )( ϕ ) = ν ( ϕ ◦ r ) = ν ( ϕ ) since ϕ ◦ r | Y = ϕ | Y . Hence, ν ∈ F X ∩ F Y = F B . Prop osition 6. T he functor E A pr eserves 1-pr eimages. Pr o of. Let f : X → Y b e a con tin uous op en map b et w een compacta X and Y and B b e a closed subset of Y suc h that f | f − 1 ( B ) is a homeomorphism. Cho ose an y ν ∈ E A ( B ) ⊂ E A ( Y ). Using Lemma 1 w e can define µ 0 ∈ E A ( f − 1 ( B )) b y the condition µ 0 ( ϕ ) = ν ( ψ ) for eac h ϕ ∈ C ( X ) and ψ ∈ C ( Y ) suc h that ψ ◦ f | f − 1 ( B ) = ϕ | f − 1 ( B ) . It is enough to sho w that for eac h µ ∈ ( E A ( f )) − 1 ( ν ) w e hav e µ = µ 0 . Supp ose the con tr a ry . Then there exist ϕ ∈ C ( X ) and ψ ∈ C ( Y ) suc h that ψ ◦ f | f − 1 ( B ) = ϕ | f − 1 ( B ) and µ ( ϕ ) 6 = ν ( ψ ). W e can suppose that µ ( ϕ ) > ν ( ψ ). Define a function ψ ′ : Y → R by ψ ′ ( y ) = max ϕf − 1 ( y ) for any y ∈ Y . The function ψ ′ is con tin uous since f is op en. Put ξ = ( ψ ′ − D ) ◦ f , where D = sup { max ϕf − 1 ( y ) − min ϕf − 1 ( y ) | y ∈ Y } . Then d ( ξ , ϕ ) ≤ D but µ ( ϕ ) − µ ( ξ ) = µ ( ϕ ) − µ (( ψ ′ − D ) ◦ f ) = µ ( ϕ ) − ν ( ψ ′ ) + D = µ ( ϕ ) − ν ( ψ ) + D > D and w e obtain a contradiction. The pro of is similar fo r the case µ ( ϕ ) < ν ( ψ ). Hence, E A preserv es 1-preimages in the class of op en mappings, a nd, b y Proposition 1, w e are done. Prop osition 7. T he functor of nonexp anding func tion a ls E do es not pr eserve 1-pr eimages. Pr o of. Consider the mapping f : X → Y b et w een discre te spaces X = { x, y , s, t } and Y = { a, b, c } whic h is defined a s follows : f ( x ) = a , f ( y ) = b , f ( s ) = f ( t ) = c . Put A = { ϕ ∈ C ( X ) | ϕ ( s ) = ϕ ( t ) } . Define the f unc tional ν : A → R as f o llo ws: ν ( ϕ ) = min { ϕ ( x ) , ϕ ( y ) } if ϕ | { x,y } ≥ 0, ν ( ϕ ) = max { ϕ ( x ) , ϕ ( y ) } if ϕ | { x,y } ≤ 0, and ν ( ϕ ) = 0 otherwise. One can chec k that ν is nonexp anding. No w take the function ψ : X → R defined as follows ψ ( x ) = 1, ψ ( y ) = − 1, ψ ( s ) = 0, ψ ( t ) = 4. One can che c k that w e can extend ν to a nonexpanding functional o n A ∪ { ψ } b y defining its v alue o n ψ to b e − 1. This new functional can b e further extended to a nonexpanding functional on the whole C ( X ) [5]. D enote this extension b y e ν . Eviden tly , E f ( e ν ) ∈ E ( { a, b } ). On t he other hand, e ν / ∈ E ( { x, y } ). 4. W e consider in this section a monomorphic contin uous functor F whic h preserv es in- tersections, w eigh t, empt y set, p o in t and 1 -preimages. W e inv estigate top ology of t he space 7 F β Y where Y is a metrizable separable non-compact space. W e consider Y as a dense subset of metrizable compactum X . It fo llo ws from Corollary 1 that F β Y is homeomorphic t o F b Y ⊂ F X (where X is considered as a compactification bY of Y ) and in what f ollo ws w e identify F β Y with F b Y . Also, the pro perties w e imp ose on F imply that F β Y is a dense pro per subspace of F X . T.Banakh pro v ed in [1] that F β Y is F σ -subset of F X when Y is lo cally compact; F β Y is F σδ -subset when Y is G δ -subset. If Y is not a G δ -subset, then F β Y is not analytic. W e consider in the Hilb ert cu b e Q = [ − 1 , 1] ω the follo wing subse ts: Σ = { ( t i ) ∈ Q | sup i | t i | < 1 } ; σ = { ( t i ) ∈ Q | t i 6 = 0for finitely man y of i } and Σ ω ⊂ Q ω ∼ = Q . It is shown in [2] t ha t any a nalytic P β Y is homeomorphic t o one of the spaces σ , Σ or Σ ω . W e g ene ralize this result f or con v ex functors. By C onv we denote the categor y of con vex compacta (compact conv ex subsets o f lo cally con vex top ological linear spaces) and affine maps. Let U : C onv → C omp b e the forgetful functor. A functor F is called c onvex if there exists a f unctor F ′ : C omp → C onv suc h that F = U F ′ . It is easy to see that the functors V , E , E A , O and P a re con v ex. It is shown in [14] that for each conv ex functor F there exists a unique natur a l transformation l : P → F suc h that the map l X : P X → F X is a n affine embedding. Lemma 2. P β Y = ( l X ) − 1 ( F β Y ) . Pr o of. T ake any measure µ ∈ P ( X ) suc h that l X ( µ ) = µ ′ ∈ F β Y . By the definition of F β Y it means that µ ′ ∈ F B f or some compactum B ⊂ Y . W e will sh ow that µ ∈ P B ⊂ P β Y . Cho ose an absolute retract T w hic h contains B and define Z to b e t he quotient space of the disjoin t union X ∪ T obtained b y attaching X and T b y B . By r : Z → T denote the retraction. Since l is a natural tr a nsformation and r is an iden tit y on T ⊂ Z , w e hav e that F ( r ) ◦ l Z ( µ ) = µ ′ = l T ◦ P ( r )( µ ). Hence, µ = P ( r )( µ ) ∈ P ( T ) due to injectivit y o f l Z . Therefore, µ ∈ P X ∩ P T = P B . The lemma is pro v ed. W e need some notions from infinite-dimensional top ology . See [4] fo r mor e details. All spaces ar e assumed to b e metrizable and separable. A closed subset A of a compactum T is called Z - set if there exists a homotopy H : T × [0 ; 1] → T suc h that H | T ×{ 0 } = id T ×{ 0 } and H ( T × (0 , 1]) ∩ A = ∅ ; a subset B of T is called σ Z - se t if it is contained in coun table union of Z -sets of T . In what f ollo ws we will use t he follo wing f a cts . W e don’t know if F β Y is a σ Z -set in F X for any conv ex f unctor F . Th us, w e in tro duce some additio nal prop ert y . W e consider the compactum F X as a con v ex subset of a lo cally con vex linear space. Definition 2. A c onvex functor F : C omp → C omp is c al le d str ongly c onvex if for e ach c omp actum X , e ach close d subset A ⊂ X we have ( F X \ F A ) ∩ aff F A = ∅ . Prop osition 8. Ea ch c onv ex subfunctor F of the functor V is str ongl y c onvex. 8 Pr o of. By Lemma 1 an y elemen t from aff F A ta k es the same v alue at an y tw o functions from C ( X ) whic h coincide on A , whic h is not true f o r functionals from F X \ F A . Prop osition 9. L e t F b e a str ongl y c onvex functor. Then F β Y is a σ Z -set in F X . Pr o of. T ake an y y ∈ X \ Y . The n F β Y ⊂ F β ( X \{ y } ), and X \{ y } can b e represen ted as a coun ta ble union of its compact subsets A n with the property that A n ⊂ in t A n +1 , hence, F β ( X \{ y } ) = ∪ n ∈ N F ( A n ). Let us sho w tha t all F ( A n ) are Z - sets in F X . T ake an y ν ∈ F X \ F β ( X \ { y } ) and the set Z = { tν + (1 − t ) µ | t ∈ (0 , 1] , µ ∈ F β ( X \ { y } ) } . Sinc e F is strongly con v ex, w e hav e Z ∩ F β ( X \ { y } ) = ∅ . Since Z is con ve x and dense subset o f F X , there exists a homotop y H : F X × [0 , 1] → F X suc h that H ( F X × (0 , 1]) ⊂ Z (see, for example, Ex. 12, 13 to section 1.2 in [4]). No w, w e are going to obtain the complete top ological classification of the pair ( F X, F β Y ) where X is a metrizable compactum and Y its prop er dense G δ -subset. W e need some charac- terization theorems. Theorem A. [8] L et C b e an infinite-dim ensional den se σ Z c onvex subsp ac e of a a c onve x metrizable c o m p actum K , and additional ly let C b e a c ountable union of its fi n ite-dimensional c omp act subsp a c es. The n the p air ( K , C ) is home om orphic to ( Q, σ ) . Theorem B. [7 ] L et K b e a c onvex metrizable c om p actum, and let C ⊂ K b e its pr op er dense σ Z c onvex σ -c omp act subsp ac e that c ontains an infinite-dimensional c on v ex c om p actum. Then the p a ir ( K , C ) is home omorphic to the p air ( Q, Σ) . The following theorem follows from 5.3.6, 5.2.6 , 3.1.10 in [4]. Theorem C. L et K b e a c onvex c omp act subset lo c al ly c onve x line ar metric sp ac e, and let C ⊂ K b e its pr op er dense σ Z c onvex F σδ subsp ac e such that K \ C ) ∩ aff C = ∅ , and a d ditional ly ther e exists a c on tinuous em b e ddi n g h : Q → K such that h − 1 ( C ) = Σ ω . Then the p air ( K, C ) is home omorphic to the p a ir ( Q, Σ ω ) . Theorem 2. L et F b e a str ongly c onv e x functor, X is a m etrizable c omp actum an d Y is its pr op er dense G δ -subset. The p air ( F X , F β Y ) is home o morphic to 1. ( Q, σ ) , if Y is discr ete subsp ac e of X and F ( n ) is finite-di m ensional for e ac h n ∈ N ; 2. ( Q, Σ) , if Y is disc r e te subsp a c e of X and F ( n ) is infinite-dimensio nal for some n ∈ N or Y is lo c a l ly c omp act no n-discr ete subsp ac e of X ; 3. ( Q, Σ ω ) , i f Y is not lo c al ly c omp act. 9 Pr o of. It is easy to see that F β Y is a conv ex subset of F X . W e pro v e the first assertion. Since X is metrizable, Y is countable. W e can represen t Y = ∪ ∞ n =1 Y n where | Y n | = n . Then F β Y = ∪ ∞ n =1 F Y n . Since P Y n could b e considered as an n − 1- dime nsional subspace of F Y n , t he space F β Y is infinite-dimensional. Moreo ve r, F β Y is a σ Z -set b y Prop osition 9. Since eac h F Y n is a finite- dimensional compactum, w e can apply Theorem A. W e prov e the second assertion. In the case when Y is discrete, F Y n is infinite-dimens ional con vex compactum for some n . When Y is not discrete , it con tains an infinite compactum Y ′ and F Y ′ is infinite-dimensional con v ex compactum. W e a pply Proposition 9 and Theorem B. F or the third a ss ertion, note that the pair ( P X , P β Y ) is homeomorphic to ( Q, Σ ω ) [2]. Since F is strongly con v ex, w e hav e ( F X \ F β Y ) ∩ aff F β Y = ∅ . W e apply Lemma 2, Propo sition 9 and Theorem C. Corollary 2. Supp ose that F is a str ongly c onvex functor. Then for any sep ar abl e metrizable sp ac e X 1) X ∼ = N implies F β ( X ) ∼ = Q f in c ase F ( n ) is fi n ite-dimensional for any n ∈ N or F β ( X ) ∼ = Σ otherwise; 2) if X is lo c al ly c omp act n o n-discr ete and non-c omp act then F β ( X ) ∼ = Σ ; 3) if X is top olo gic al l y c omplete not lo c a l ly c o m p act then F β ( X ) ∼ = Σ ω . References [1] T. Banakh, D escriptive classes o f sets and top olo gic al func tors , Ukrain. Mat. 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