On functors preserving skeletal maps and skeletally generated compacta

ON FUNCTORS PRESER VING SKELET AL MAPS AND SKELET ALL Y GENERA TED COMP ACT A T ARA S BANAKH, ANDRZEJ KUCHARSKI, AND MAR T A MAR TYNENKO Abstra ct. A map f : X → Y b etw een top ologica l spaces is skeletal if the preimage f − 1 ( A ) of eac h no where dense subset A ⊂ Y i s no where dense in X . W e prov e that a normal functor F : Comp → Comp is skeletal (which means t hat F preserves skeletal epimorphisms) if and only if for any open su rjectiv e map f : X → Y b etw een metrizable zero-dimensional compacta with t wo-elemen t non-degeneracy set N f = { x ∈ X : | f − 1 ( f ( x )) | > 1 } the map F f : F X → F Y is skeleta l. This chara cterization implies that each op en n ormal functor is skel etal. The conv erse is not tru e even for normal functors of finite degree. The other main result of the p aper sa ys that eac h normal fun ctor F : Comp → Comp preserves the class of skeletally generated compacta. This contrasts with the known ˇ Sˇ cepin’s result saying that a normal fun ctor is op en if and only if it preserves the class of open ly generated compacta. 1. Introduction In this pap er we address the p r oblem of preserv ation of ske letal maps and sk eletally generated compacta by normal functors in the category C omp of co mpact Hausdorff spaces and their co n tin uou s maps. In the sequel all s paces are Hausdorff and all m aps are con tin u ous. A c omp actum is a compact Hausdorff sp ace. A map f : X → Y b etw een top ological sp aces is called skeletal if f or eac h n o where dense subs et A ⊂ Y th e pr eimage f − 1 ( A ) is no wh ere d ense in X , see [11]. It is easy to see that eac h op en map is sk eletal wh ile the con verse is not tru e. T o formulat e our main results, w e need to recall some defin itions from th e top ological theory of functors, s ee [14]. A fu nctor F : Comp → Comp in the category C omp is defined to b e • monomo rphic (resp. epimorphic ) if for eac h injectiv e (resp. surjectiv e) map f : X → Y b et w een compacta the map F f : F X → F Y is injectiv e (resp. sur jectiv e); • ( finitely ) op en if for any op en surjection f : X → Y b et ween (finite) compacta th e map F f : F X → F Y is op en; • ( finitely ) skeletal if for an y op en surjection f : X → Y b et w een (fi nite) compacta the map F f : F X → F Y is sk eletal; • weight-pr eserving if w ( F X ) ≤ w ( X ) f or eac h infinite compactum X . Here w ( X ) d enotes the weight (i.e. the smallest cardinalit y of a base of the top ology) of X . If a functor F : Comp → Comp is monomorph ic, then for eac h closed sub space X of a compact space Y the map F i : F X → F Y induced by the inclusion i : X → Y is a top ological em b eddin g, wh ic h allo ws us to identify the space F X with th e subsp ace F i ( F X ) of F Y . Next w e r ecall and define sev eral prop erties of functors related to the bicommutat ivit y . Let D b e a commutati v e square d iagram ˜ X p X   ˜ f / / ˜ Y p Y   X f / / Y consisting of con tin u ous maps b et w een compact spaces. 1991 Mathematics Subje ct Cl assific ation. 18B30, 54B30, 54C10, 54B35, 54D30. Key wor ds and phr ases. S kel etal map, functor, skel etally generated compact space. 1 2 T ARAS BANAKH , AN D RZEJ K UCHARSKI, AND MAR T A MAR TYNENKO The diagram D is called bic ommutative if ˜ f  p − 1 X ( x )  = p − 1 Y  f ( x )  for all x ∈ X , see [10, § 3.IV], [13, § 2.1]. W e sa y that a fu nctor F : Comp → Comp • is ( finitely ) bic ommutative if F preserv es the bicomm u tativit y of square diagrams D consisting of sur jectiv e maps f , ˜ f , p X , p Y and (fi n ite) compacta X, Y , ˜ X , ˜ Y ; • pr eserves ( finite ) pr eimages if F pr eserv es the b icomm utativit y of squ are diagrams D with injectiv e maps f , ˜ f (and fi nite space X ); • pr eserves ( finite ) 1-pr eimages if F preserv es the b icomm utativit y of square diagrams D with injectiv e maps f , ˜ f , bijectiv e map p X (and fi n ite space X ). It is clear that eac h bicomm u tativ e functor is finitely bicommutativ e. T he con v erse is true for nor- mal fun ctors with finite sup p orts, see Prop osition 2.10.1 of [14]. It is easy to see that a monomorp hic functor F : Comp → Comp p reserv es [finite] (1-)preimages if and only if for an y m ap f : X → Y b et w een compact spaces and a [fin ite] closed subs et Z ⊂ Y (such that f − 1 ( z ) is a singleton for ev ery z ∈ Z ) we get ( F f ) − 1 ( F Z ) = F  f − 1 ( Z )  . A functor F : C omp → Comp w ill b e called me c if F is monomorp h ic, epimorph ic, and con tinuous. A mec functor th at preserves finite 1-pr eimages will b e called a 1-me c fun ctor. A 1-mec fu nctor that preserv es weigh t of infinite compacta will b e called a 1-me c w f u nctor. T he class of 1-mecw functors includ es all normal f u nctors in the sense of ˇ S ˇ cepin [13], [14, § 2.3] (let u s r ecall that a functor F : Comp → C omp is we akly normal if it is monomorp hic, epimorp hic, contin uous and preserves in tersections, the empt y set, the singleton, and the weig h t of infinite compacta; F is normal if it is w eakly normal and preserv es preimages). Our pr im ary aim is to c h aracterize skelet al f unctors among 1-mec fu nctors. F or a top ological sp ace Z consider th e op en m ap 2 Z : Z ⊕ 2 → Z ⊕ 1 d efined by 2 Z : z 7→ ( z if z ∈ Z , 0 if z ∈ 2 . Here for a natur al num b er n by Z ⊕ n we denote the top ological sum of Z and the discrete space n = { 0 , . . . , n − 1 } . Theorem 1.1. A 1-me c (r esp. 1-me cw) functor F : C omp → C omp is skeletal if and only if for e ach zer o-dimensional c omp act (metrizable) sp ac e Z the map F 2 Z : F ( Z ⊕ 2) → F ( Z ⊕ 1) is skeletal. Since eac h op en map is sk eletal, the p receding theorem implies: Corollary 1.2. Each op en 1-me c functor F : Comp → C omp is skeletal. Examples 12.3–12.5 presente d in Section 12 sh o w that Corollary 1.2 cannot b e rev ers ed . Next, we discuss the in terpla y b etw een the skele talit y and the (finite) bicomm u tativit y of fu nctors. Theorem 1.3. A 1-me cw functor F : Comp → C omp is skeletal if it is finitely bic ommutative and finitely skeletal. This criterion should b e compared with the follo wing c h aracterizati on of op en fu nctors d ue to ˇ S ˇ cepin, see Prop ositio ns 3.18 and 3.19 of [13]. Theorem 1.4 ( ˇ S ˇ cepin) . A normal functor F : Comp → C omp is op en if and only if F is bic om- mutative and finitely op en. No w let us discus s the problem of preserv ation of sk eletally generated compacta by n ormal fun ctors. F ollo win g [15] we s a y that a compact Hausdorff sp ace X is skeletal ly gener ate d if X is homeomorphic to the limit of an in v erse contin uous ω -sp ectrum S = { X α , π β α , Σ } consisting of metrizable compacta and su rjectiv e sk eletal b onding pr o jectio ns π β α : X β → X α . According to [9] or [15 ], a compact Hausd orff space X is ske letally generated if and only if the first pla yer has a w inning strategy in the follo wing op en-op en game. Th e pla ye r I starts the game selecting a non-empt y op en set V 0 and the p la ye r I I resp ond s with a non-empt y op en set W 0 ⊂ V 0 . At the FUNCTORS PRESER VING SKELET AL MAPS 3 n -th innin g th e pla y er I c ho oses a non-empt y op en set V n and play er I I resp ond s with a non-empt y op en set W n ⊂ V n . At the end of the game the pla ye r I is declared the winner if the union S n ∈ ω W n is dens e in X . Otherwise the pla y er I I wins th e game. The class of ske letally generated compacta con tains all op enly generated compacta and all contin u- ous images of op en ly generated compacta, see [15]. In particular, eac h dy adic compactum is skele tally generated. Sk eletally generated compacta share some pr op erties of dy adic compacta. In particular, eac h skele tally generated compactum has coun table cellularit y , see [4] or [9]. It is kno wn that in general, n ormal fu nctors do not preserve op enly generated compacta. In fact, a n ormal fun ctor F : Comp → Comp p r eserv es the class of op enly generated spaces if and only if F is op en, s ee [13, § 4.1]. This con trasts with the f ollo wing theorem. Theorem 1.5. Each 1-me cw functor F : Comp → Comp pr eserves the class of skeletal ly gener ate d c omp acta. F or pr eimage preservin g mecw-functors F with F 1 6 = F 2 this theorem can b e improv ed as follo ws. Belo w we iden tify a natural n u m b er n w ith the discrete space n = { 0 , . . . , n − 1 } . Theorem 1.6. L et F : Comp → C omp b e a pr eimage pr eserving me cw-functor with F 1 6 = F 2 . A c omp act Hausdorff sp ac e X is skeletal ly gener ate d i f and only if the sp ac e F X is ske letal ly gener ate d. Remark 1.7. Among the prop erties comp osing the defi nition of 1-mec functor the least studied is the prop ert y of preserv ation of fin ite 1-preimages. It is clear that a functor F : Comp → Comp preserve s (fin ite) 1-preimages if it p reserv es (finite) preimages. On th e other h and, the f u nctor of sup erextension λ and the f u nctor of ord er-preserving functionals O preserve 1-preimages but fail to preserve finite pr eimages, see [7], [14, 2.3.2, 2.3.6]. A simple example of a mec functor th at do es not preserve finite 1-preimages is P r 3 , th e fu nctor of the third pro jectiv e p ow er, see [14, 2.5.3]. Another example of suc h a functor is E , the functor of non-expand ing functionals, see [7]. By Theorem 1 of [7], a con tinuous monomorphic functor F : Comp → Comp preserves 1-preimages if and only if its Chigogidze extension F β : T yc h → Tyc h preserv es em b eddings of Tyc honoff spaces. Theorems 1.1, 1.3, 1.5 and 1.6 will b e pro v ed in Sections 8 — 11 after some preliminary work done in S ections 2–6. Several examples of skelet al and non-ske letal f u nctors will b e giv en in Section 12. In that section w e also p ose s ome op en p roblems r elated to sk eletal fun ctors. 2. Skelet al map s an d skelet al s quares In this section we recall th e necessary inf ormation on sk eletal maps b et we en compact s p aces. First w e introdu ce the necessary definitions. Definition 2.1. A map f : X → Y b et wee n t wo top ological spaces is defined to b e • skeletal at a p oint x ∈ X if f or eac h neighborh o od U ⊂ X of x the closure cl Y ( f ( U )) of f ( U ) has non -emp t y in terior in Y ; • skeletal at a subset A ⊂ X if f is skele tal at eac h p oint x ∈ A ; • op en at a p oint x ∈ X if for eac h neigh b orho o d U ⊂ X of x the image f ( U ) is a n eigh b orho o d of f ( x ); • op en at a subset A ⊂ X if f is op en at eac h p oin t x ∈ A ; • densely op en if f is op en at some dense su b set A ⊂ X . It is easy to see that eac h densely op en map is skele tal. Th e con verse is tru e for skele tal maps b et w een metrizable compacta: Theorem 2.2. A map f : X → Y b etwe en c omp act metrizable sp ac es is skeletal if and only if it is op en at a dense G δ -subset of X . This theorem has b een pro ved in [1]. The metrizabilit y of X is essen tial in this theorem as sh o wn b y the pro jection pr : A → [0 , 1] of the Aleksandro v “t wo arro ws” space A on to the in terv al, see [5, 3.10.C]. Th is pr o jection is ske letal but is op en at no p oin t of A . 4 T ARAS BANAKH , AN D RZEJ K UCHARSKI, AND MAR T A MAR TYNENKO A c haracterization of skel etal maps b et w een non-metrizable compact s paces w as giv en in [1] in terms of morphism s of inv erse ω -sp ectra with s keleta l limit or b onding s quares. Definition 2.3. Let D denote a commutati v e diagram ˜ X ˜ f / / p X   ˜ Y p Y   X f / / Y consisting of maps b et w een compact s paces. The square D is defined to b e • op en at a p oint x ∈ X if for eac h neigh b orho o d U ⊂ X of x the p oin t f ( x ) has a n eigh b orhoo d V ⊂ Y su c h that V ⊂ f ( U ) and p − 1 Y ( V ) ⊂ ˜ f ( p − 1 X ( U )); • op en at a subset A ⊂ X if D is op en at eac h p oin t x ∈ A ; • densely op en if it is op en at s ome dense subset A ⊂ X ; • skeletal at a p oint x ∈ X if f or eac h neighborh o od U ⊂ X of x ther e is a non-empt y op en set V ⊂ Y su c h that V ⊂ f ( U ) and p − 1 Y ( V ) ⊂ ˜ f ( p − 1 X ( U )); • skeletal at a subset A ⊂ X if D is skelet al at eac h p oint x ∈ A ; • skeletal if D is sk eletal at X . Remark 2.4. Let A b e a subs et of th e space X in th e d iagram D . (1) If the diagram D is s k eletal (resp. op en) at A , th en the map f is s keleta l (resp. op en) at A ; (2) The d iagram D is sk eletal (resp. op en) at A if it is bicomm utativ e and the m ap f is s k eletal (resp. op en ) at A ; (3) If th e diagram D is op en at A , then it is bic ommutative at A in the sense that ˜ f ( p − 1 X ( x )) = p − 1 Y ( f ( x )) for all p oin ts x ∈ A . Remark 2.5. A map f : X → Y is sk eletal (resp. op en ) at a subset A ⊂ X if and only if the square X f / / id X   Y id Y   X f / / Y is skel etal (resp. op en) at the sub set A . It is easy to see that eac h densely op en square is s k eletal. The conv erse is tru e in the metrizable case. Th e follo wing lemma p r o ved in [1 ] is a “squ are” coun terpart of the charac terizatio n Th eorem 2.2. Lemma 2.6. If in the diagr am D the sp ac e X is metrizable and the map p Y is surje ctive, then the squar e D is skeletal if and only if D is op e n at a dense G δ -subset of X . In ord er to form ulate the sp ectral c h aracterizat ion of sk eletal maps, we need to recall some infor- mation ab out in v erse sp ectra from [5, § 2.5] and [3, Ch.1]. F or an in verse sp ectrum S = { X α , p β α , Σ } consisting of top ological spaces and con tinuous b onding maps, by lim S = { ( x α ) α ∈ Σ ∈ Y α ∈ Σ X α : ∀ α ≤ β p β α ( x β ) = x α } w e denote the limit of S and by p α : lim S → X α , p α : x 7→ x α , th e limit pro jections. An inv erse sp ectrum S = { X α , p β α , Σ } is called an ω -sp e ctrum if • eac h space X α , α ∈ Σ, has countable weig h t; • the ind ex set Σ is ω -c omplete in the sense that eac h countable su bset Σ ′ ⊂ Σ h as the s m allest upp er b oun d sup Σ ′ in Σ; • the sp ectrum S is ω -c ontinuous in the sense that for an y count able directed sub set Σ ′ ⊂ Σ with γ = sup Σ the limit map lim p γ α : X γ → lim { X α , p β α , Σ ′ } is a h omeomorphism. FUNCTORS PRESER VING SKELET AL MAPS 5 Let S X = { X α , p β α , Σ } and S Y = { Y α , π β α , Σ } b e t wo inv erse sp ectra in d exed by the same d irected partially ordered set Σ. A morphism { f α } α ∈ Σ : S X → S Y b et w een th ese sp ectra is a family of maps { f α : X α → Y α } α ∈ Σ suc h that f α ◦ p β α = π β α ◦ f β for any elemen ts α ≤ β in Σ. Eac h morph ism { f α } α ∈ Σ : S X → S Y b et w een inv erse sp ectra indu ces the limit map lim f α : lim S X → lim S Y , lim f α : ( x α ) α ∈ Σ 7→ ( f α ( x α )) α ∈ Σ b et w een the limit s p aces of these sp ectra. F ollo win g [1] we sa y th at a morphism { f α } α ∈ Σ : S X → S Y b et w een t w o inv erse sp ectra S X = { X α , p β α , Σ } and S Y = { Y α , π β α , Σ } : • is skeletal if eac h m ap f α : X α → Y α , α ∈ Σ, is sk eletal; • has skeletal limit squar es if for ev ery α ∈ Σ the commutativ e square lim S X lim f α / / p α   lim S Y π α   X α f α / / Y α is skel etal. W e sa y that tw o maps f : X → Y and f ′ : X ′ → Y ′ are home omorphic if th ere are homeomorphisms h X : X → X ′ and h Y : Y → Y ′ suc h that f ′ ◦ h X = h Y ◦ f . The follo win g s p ectral characte rizations of sk eletal maps w as prov ed in [1]. Theorem 2.7. F or a map f : X → Y b etwe en c omp act Hausdorff sp ac es the fol lowing c onditions ar e e quivalent: (1) f is skeletal. (2) f is home omorphic to the limit map lim f α : lim S X → S Y of a skeletal morphism { f α } : S X → S Y b etwe en two ω -sp e ctr a S X = { X α , p β α , Σ } and S Y = { Y α , π β α , Σ } with surje ctive limit pr oje ctions. (3) f is home omorphic to the limit map lim f α : lim S X → S Y of a morphism { f α } : S X → S Y with skeletal limit squar es b etwe e n two ω -sp e ctr a S X = { X α , p β α , Σ } and S Y = { Y α , π β α , Σ } with surje ctive limit pr oje ctions. 3. Some proper ties o f densel y op en square s In this section we assume that D is a comm u tativ e squ are ˜ X ˜ f / / p X   ˜ Y p Y   X f / / Y consisting of sur j ectiv e maps b et w een compact spaces. By D f = { y ∈ Y : | f − 1 ( y ) | = 1 } and D f = { x ∈ X : | f − 1 ( f ( x )) | = 1 } w e denote the lower and upp er de gener acy sets of the map f : X → Y , r esp ectiv ely . Lemma 3.1. The squ ar e D is op en at the upp e r de gener acy set D f ⊂ X of f . Pr o of. Giv en an op en n eigh b orho o d U ⊂ X of a p oint x ∈ D f , observ e that the set V = Y \ f ( X \ U ) is an op en neigh b orho o d of f ( x ) s u c h that f − 1 ( V ) ⊂ U . Ap plying to this inclusion the surj ectiv e map f , we get V ⊂ f ( U ). T o see that p − 1 Y ( V ) ⊂ ˜ f ( p − 1 X ( U )), fix an y p oin t ˜ y ∈ p − 1 Y ( V ) and using the surjectivit y of the map ˜ f , find a p oin t ˜ x ∈ ˜ X with ˜ f ( ˜ x ) = ˜ y . I t follo w s th at f ◦ p X ( ˜ x ) = p Y ◦ ˜ f ( ˜ x ) ∈ V and h ence p X ( ˜ x ) ∈ f − 1 ( V ) ⊂ U . Then ˜ x ∈ p − 1 X ( U ) and ˜ y = ˜ f ( ˜ x ) ∈ ˜ f ( p − 1 X ( U )).  6 T ARAS BANAKH , AN D RZEJ K UCHARSKI, AND MAR T A MAR TYNENKO Lemma 3.2. A ssume that the squar e D is op en at a p oint a ∈ X and the sp ac e X is first c ountable at a . Then ther e is a close d subset Z ⊂ X such that a ∈ D f | Z , f ( Z ) = Y and ˜ f ( p − 1 X ( Z )) = ˜ Y . Pr o of. Be ing first countable at a , the space X has a countable neigh b orho o d b ase ( W n ) n ∈ ω at a suc h that W n +1 ⊂ W n ⊂ W 0 = X for all n ∈ ω . Let U 0 = X and V 0 = Y . Using the fact that the s q u are D is op en at the p oin t a , by induction on n , we can construct a sequence ( U n ) ∞ n =1 of op en neighborh o o ds of a in X and a sequence ( V n ) ∞ n =1 of op en neigh b orho o ds of f ( a ) in Y su c h that • U n ⊂ W n ∩ U n − 1 ∩ f − 1 ( V n − 1 ), • V n ⊂ f ( U n ) ∩ V n − 1 and p − 1 Y ( V n ) ⊂ ˜ f ( p − 1 X ( U n )) for eve ry n ∈ N . W e claim that the set Z = { a } ∪ [ n ∈ ω U n \ f − 1 ( V n +1 ) is closed in X and h as the required p r op erties: a ∈ D f | Z , f ( Z ) = Y and ˜ f ( p − 1 X ( Z )) = ˜ Y . The definition of the set Z imp lies th at it is closed in X an d a ∈ D f | Z . T o show that ˜ f ( p − 1 X ( Z )) = ˜ Y , fix any p oint ˜ y ∈ ˜ Y . W e need to fi n d a p oint ˜ x ∈ p − 1 X ( Z ) suc h that ˜ f ( ˜ x ) = ˜ y . F or this we consider separately t wo cases. 1) The image y = p Y ( ˜ y ) of ˜ y coincides with f ( a ). I n this case for ev ery n ∈ ω we get ˜ y ∈ p − 1 Y ( V n ) ⊂ ˜ f ( p − 1 X ( U n )) and hen ce there is a p oint ˜ x n ∈ p − 1 X ( U n ) such that ˜ y = ˜ f ( ˜ x n ). By the compactness of ˜ X , the sequence ( ˜ x n ) n ∈ ω has an accumulatio n p oint ˜ x ∈ p − 1 X ( a ) ⊂ p − 1 X ( Z ). The con tinuit y of the m ap ˜ f guaran tees that ˜ f ( ˜ x ) = ˜ y . 2) The p oin t y = p Y ( ˜ y ) is not equal to f ( a ). Since V 0 = Y and T n ∈ ω f ( U n ) = T n ∈ ω V n = { f ( a ) } , there is a un ique num b er n ∈ ω suc h that y ∈ V n \ V n +1 . Th en ˜ y ∈ p − 1 Y ( V n ) ⊂ ˜ f ( p − 1 X ( U n )) and hence there is a p oin t ˜ x ∈ p − 1 X ( U n ) such that ˜ f ( ˜ x ) = ˜ y . Consider the image x = p X ( ˜ x ) ∈ U n and ob s erv e that f ( x ) = f ◦ p X ( ˜ x ) = p Y ◦ ˜ f ( ˜ x ) = p Y ( ˜ y ) = y / ∈ V n +1 . Consequently , x ∈ U n \ f − 1 ( V n +1 ) ⊂ Z and ˜ x ∈ p − 1 X ( x ) ⊂ p − 1 X ( Z ). Therefore ˜ f ( p − 1 X ( Z )) = ˜ Y . App lying to this equalit y the s u rjectiv e map p Y , we get f ( Z ) = f ◦ p X ( p − 1 X ( Z )) = p Y ◦ ˜ f ( p − 1 X ( Z )) = p Y ( ˜ Y ) = Y .  Lemma 3.3. If the squ ar e D is op en at a finite subset A ⊂ X , the sp ac e X is first c ountable at e ach p oint x ∈ A , and the r estriction f | A is inje ctive, then ther e is a close d subset Z ⊂ X suc h that f ( Z ) = Y , ˜ f ( p − 1 X ( Z )) = ˜ Y , and A ⊂ D f | Z . Pr o of. By Lemma 3.2, for eve ry p oin t a ∈ A there is a closed subset Z a ⊂ X suc h that a ∈ D f | Z a , f ( Z a ) = Y , and ˜ f ( p − 1 X ( Z a )) = ˜ Y . Since f | A is injectiv e, for eac h p oin t a ∈ A we can fi nd an op en neigh b orho o d W a ⊂ Y of f ( a ) in Y such th at the closur es W a , a ∈ A , are pairwise distinct. L et B = Y \ S a ∈ A W a and Z = f − 1 ( B ) ∪ [ a ∈ A f − 1 ( W a ) ∩ Z a . It is easy to c h ec k that the set Z is closed and has the r equired prop erties: A ⊂ D f | Z , f ( Z ) = Y and ˜ f ( p − 1 X ( Z )) = ˜ Y .  The pro of of the follo wing sim p le lemma is left to the reader. Lemma 3.4. A map f : X → Y is op en (r esp. skeletal) at a p oint x ∈ X pr ovide d for some subset Z ⊂ X that c ontains the p oint x the map f | Z : Z → Y is op en (r esp. skeletal) at the p oint x . The follo win g lemma is a “square” coun terpart of Lemma 3.4. FUNCTORS PRESER VING SKELET AL MAPS 7 Lemma 3.5. The squar e D is op en (r esp. skeletal) at a p oint x ∈ X pr ovide d for some subset Z ⊂ X that c ontains the p oint x , and its pr eimage ˜ Z = p − 1 X ( Z ) the squar e ˜ Z ˜ f | ˜ Z / / p X | ˜ Z   ˜ Y p Y   Z f | Z / / Y is op en (r esp. skeleta l) at the p oint x . 4. Preliminaries on func tors In this section we p r o ve s ome au x iliary results on fun ctors in the catego ry Comp of compacta. F rom now on we assum e that F : Comp → Comp is a monomorph ic epimorph ic cont in u ous functor. F or t w o compact Hausdorff sp aces X an d Y b y C ( X , Y ) w e denote the space of contin uous functions f : X → Y , endo wed w ith th e compact-op en top ology . A pr o of of the follo win g fact due to ˇ S ˇ cepin [13, § 3.2] can b e found in [14, 2.2.3]. Lemma 4.1. F or any c omp acta X , Y the map F : C ( X , Y ) → C ( F X, F Y ) , F : f 7→ F f , is c ontinuous. Next, w e discuss the notion of su pp ort. Let X b e a compact Hausdorff sp ace, and a ∈ F X . W e sa y that a p oin t a ∈ F X has finite supp ort if a ∈ F A for some fi nite sub space A ⊂ X . In this case w e defin e supp( a ) as the in tersection supp( a ) = ∩{ A : a ∈ F A, A ⊂ X is finite } . W e shall often use the follo wing fact pro ved in [2]: Lemma 4.2. L et a ∈ F X b e an element with finite supp ort. If supp( a ) 6 = ∅ , then a ∈ F (supp( a )) . If supp( a ) = ∅ , then a ∈ F A for any non-empty close d subsp ac e A ⊂ X . The set of all element s with finite su pp ort in F X will b e denoted by F ω ( X ). The follo wing lemma w as pr o ve d in [14, 2.2.1 ]. Lemma 4.3. The su b set F ω ( X ) is dense in F X . F or a top ological space Y by ˙ Y we shall denote the set of isolated p oin ts of Y . F or a sur jectiv e function f : X → Y let N f = { y ∈ Y : | f − 1 ( y ) | > 1 } = Y \ D f and N f = { x ∈ X : | f − 1 ( f ( x )) | > 1 } = X \ D f b e th e lower and upp er non-de gener acy sets of f , resp ectiv ely . Lemma 4.4. F or any skeletal map f : X → Y b etwe en c omp acta and any dense subset A ⊂ X , the set A 0 = { a ∈ F ω ( X ) : su pp( a ) ⊂ A, N f | supp( a ) ⊂ ˙ Y } is dense in F X . Pr o of. By Lemma 4.3, the set F ω ( X ) is dense in F X . So, it su ffi ces to c h ec k that A 0 is dense in F ω ( X ). Fix any elemen t a ∈ F ω ( X ) and a neigh b orho o d O a ⊂ F ω ( X ). W e need to find an elemen t b ∈ O a ∩ A 0 . If su pp( a ) = ∅ , then a ∈ A 0 ∩ O a b y the d efinition of A 0 . So we assum e that B = sup p( a ) is not empt y . By L emma 4.2, a ∈ F B . By Lemma 4.1, the map F : C ( B , X ) → C ( F B , F X ) , F : g 7→ F g is con tin u ous and so is th e map F a : C ( B , X ) → F X , F a : g 7→ F g ( a ) . 8 T ARAS BANAKH , AN D RZEJ K UCHARSKI, AND MAR T A MAR TYNENKO It follo ws from the conti n uit y of F a that the identit y in clus ion i B : B → X has a neighborh o o d O ( i B ) in the fun ction space C ( B , X ) such that F a ( g ) = F g ( a ) ∈ O a for any map g ∈ O ( i B ). W e claim that there is a m ap g ∈ O ( i B ) s u c h that N f ◦ g | B ⊂ ˙ Y . Since the compact-op en top ology on C ( B , X ) coincides with the top ology of p oint wise conv ergence, for eac h p oin t x ∈ B w e can find a neigh b orho o d O x ⊂ X suc h that a map g : B → X b elongs to the neighborh o o d O ( i B ) pro vid ed g ( x ) ∈ O x for all x ∈ B . Let C = B ∩ f − 1 ( ˙ Y ). W e claim that f or eac h p oin t x ∈ B \ C th e set f ( O x ) is infinite. Assu ming the conv erse, we can find a smaller neighb orh o o d U x of x such that f ( U x ) coincides with th e singleton { f ( x ) } which is op en in Y b ecause of the sk eletal pr op ert y of f . In this case f ( x ) ∈ ˙ Y and x ∈ C , whic h con tradicts the c hoice of x . Let B \ C = { x 1 , . . . , x n } b e an enumeratio n of the set B \ C . By finite induction for every i ≤ n c ho ose a p oin t x ′ i ∈ O x i ∩ A suc h that f ( x ′ i ) / ∈ f ( C ) ∪ { f ( x ′ j ) : j < i } . As f ( O x i ) is infinite and A is dense in X , the c hoice of x ′ i is alw a ys p ossible. Af ter completing the ind uctiv e construction, define a map g : B → X letting g ( x i ) = x ′ i for i ≤ n and g ( x ) ∈ O x ∩ A ∩ f − 1 ( f ( x )) for an y x ∈ C . By the construction, g ∈ O ( i B ) and the map f ◦ g | B \ C is injectiv e, which means that N f ◦ g | B ⊂ f ( C ) ⊂ ˙ Y . By the choice of the neighborho od O ( i B ), the elemen t b = F g ( a ) lies in the n eighb orh o o d O a . Since b ∈ F ( g ( B )), we get sup p( b ) ⊂ g ( B ) ⊂ A , witnessing that b ∈ A 0 .  Lemma 4.5. L et f : X → Y b e a skeletal map b e twe en c omp act Hausdorff sp ac es. If ˙ Y ⊂ D f , then for every dense subset A ⊂ X the set A 1 = { a ∈ F ω ( X ) : su pp( a ) ⊂ A, f | supp( a ) is 1-to-1 } is dense in F X . Pr o of. By Lemma 4.4, the set A 0 = { a ∈ F ω ( X ) : su pp( a ) ⊂ A, N f | supp( a ) ⊂ ˙ Y } is dense in F X . Ob serv e that for eac h a ∈ A 0 w e get N f | supp( a ) ⊂ f (sup p( a )) ∩ ˙ Y ⊂ f (su pp( a )) ∩ D f ⊂ D f | supp ( a ) , wh ic h implies N f | supp ( a ) = ∅ and a ∈ A 1 . No w we see that the d en sit y of the set A 0 implies the densit y of th e set A 1 ⊃ A 0 in F X .  5. 1-Mec functors and dens el y open squares In this section w e assume that F : Comp → Comp is a 1-mec functor an d stud y its action on densely op en squares. Let D b e a comm u tativ e square ˜ X ˜ f / / p X   ˜ Y p Y   X f / / Y consisting of surjectiv e maps b et we en compact Hausdorff sp aces. Applying the functor F to this square, we obtain the commutativ e square F D : F ˜ X F ˜ f / / F p X   F ˜ Y F p Y   F X F f / / F Y . Lemma 5.1. If the sp ac e X is first c ountable and the squar e D is op en at a non-empty subset A ⊂ X , then the squ ar e F D i s op en at the subset A 1 = { a ∈ F ω ( X ) : su pp( a ) ⊂ A, f | su pp( a ) is 1-to-1 } ⊂ F X . If ˙ Y ⊂ D f and the set A is dense in X , then the set A 1 is dense in F X and henc e the squar e F D is densely op e n. FUNCTORS PRESER VING SKELET AL MAPS 9 Pr o of. Fix an y p oin t b ∈ A 1 and consid er its sup p ort supp( b ). If it is not empt y , p ut B = supp( b ). If sup p( b ) is empt y , put B = { z } ⊂ A b e an y sin gleton in A . In b oth cases we ha v e that B ⊂ A , f | B is injectiv e, and b ∈ F B , see Lemma 4.2. Let C = f ( B ) and observe that f | B : B → C is a homeomorphism. By Lemma 3.3, there is a closed su bset Z ⊂ X su c h that B ⊂ D f | Z , f ( Z ) = Y and ˜ f ( p − 1 X ( Z )) = ˜ Y . Let ˜ Z = p − 1 X ( Z ), p Z = p X | ˜ Z , f Z = f | Z , ˜ f Z = ˜ f | ˜ Z and consider the comm u tativ e squ are D Z : ˜ Z ˜ f Z / / p Z   ˜ Y p Y   Z f Z / / Y that consists of surjectiv e maps. App lyin g to this squ are the ep imorphic fu n ctor F , we obtain the comm utativ e square F D Z : F ˜ Z F ˜ f Z / / F p Z   F ˜ Y F p Y   F Z F f Z / / F Y , also consisting of surjectiv e maps. T aking int o accoun t that B ⊂ D f Z and F pr eserves fi nite 1-preimages, w e conclude that F B ⊂ D F f Z . By Lemma 3.1, the square F D Z is op en at F B . Applying Lemma 3.5, w e conclude that th e square F D is op en at F B . In particular, F D is op en at the p oint b ∈ F B . If ˙ Y ⊂ D f , then by Lemma 4.5, th e set A 1 is dense in F X and h ence the squ are F D is densely op en.  6. 1-Mec funct ors and skelet al map s In this section w e study the action of 1-mec f unctors on some sp ecial types of ske letal m aps. As in the pr eceding section, F : C omp → Comp is a 1-mec fun ctor in the categ ory of compact Hausdorff spaces. Ou r principal result is the follo wing theorem. Theorem 6.1. F or any surje ctive skeletal map f : X → Y b etwe en c omp act Hausdorff sp ac es the map F f : F X → F Y is skeletal at the subse t A 1 = { a ∈ F ω ( X ) : f | su pp( a ) is 1-to-1 } . If ˙ Y ⊂ D f , then the set A 1 is dense in F X and henc e the map F f is skeletal. Pr o of. By Theorem 2.7 , the skel etal map f : X → Y can b e identified with th e limit map lim f α of a morphism ~ f = { f α } α ∈ Σ : S X → S Y b et w een some ω -sp ectra S X = { X α , p β α , Σ } and S Y = { Y α , π β α , Σ } with su rjectiv e limit pro jections such that for any α ∈ Σ th e limit square D α : X f / / p α   Y π α   X α f α / / Y α is skel etal. T o show that the map F f is ske letal at eac h p oin t a ∈ A 1 , fix an y op en neigh b orho o d U ⊂ F X of a . W e need to pro v e th at the image F f ( U ) has non-emp t y interior in F Y . The inclusion a ∈ A 1 implies that the restriction f | su p p( a ) is injectiv e. By the cont in u it y of the functor F , there is an in dex α ∈ Σ and an op en neighborh o o d U α ⊂ F X α of the p oin t a α = F p α ( a ) su c h that U ⊃ ( F p α ) − 1 ( U α ). Replacing α by a larger index, if n ecessary , w e can add itionally assume that the restriction p α | supp( a ) and π α ◦ f | su pp( a ) are injectiv e. Then 10 T ARAS BANAKH , AN D RZEJ K UCHARSKI, AND MAR T A MAR TYNENKO the map f α | p α (supp( a )) also is injectiv e. Since supp( a α ) ⊂ p α (supp( a )), th e restriction f α | supp( a α ) is injectiv e. By our assumption the limit square D α is sk eletal and by L emm a 2.6 it is op en at some dense subset A α ⊂ X α . Rep eating the argument from the pro of of Lemma 4.4, we can app ro ximate the elemen t a α b y an elemen t a ′ α ∈ U α suc h that s u pp( a ′ α ) ⊂ A α and th e m ap f α | supp( a ′ α ) is in jectiv e. By Lemma 5.1, the square F D α is op en at the p oint a ′ α . T hen f or the n eigh b orho o d U α of a ′ α there is a non-empty op en set V α ⊂ F Y α suc h th at V α ⊂ F f α ( U α ) and the op en su b set V = ( F π α ) − 1 ( V α ) of the space F Y lies in the image F f (( F p X ) − 1 ( U α )) ⊂ F f ( U ), whic h completes the p ro of of th e sk eletalit y of F f at a . If ˙ Y ⊂ D f , then the set A 1 is dense in F X b y Lemma 4.5 and hence the map F f : F X → F Y is sk eletal.  A map f : X → Y b et w een top ologica l spaces is called irr e ducible if f ( X ) = Y but f ( Z ) 6 = Y f or eac h closed subset Z ⊂ X . Corollary 6.2. F or e ach irr e duci b le map f : X → Y b etwe en c omp act Hausdorff sp ac es the map F f : F X → F Y is skeletal. Pr o of. This lemma follo ws from Th eorem 6.1 b ecause eac h closed irredu cible map f : X → Y is sk eletal and has ˙ Y ⊂ D f .  7. Preima ge pres er v ing functors and skel et al maps The follo w in g theorem imp lies th at for normal fu nctors F the sk eletalit y of a m ap f : X → Y b et w een compacta follo ws fr om the s keleta lit y of the map F f . Theorem 7.1. L et F : Comp → C omp b e a pr eimage pr eserving me c-functor such that F 1 6 = F 2 . A su rje c tiv e map f : X → Y b etwe en c omp act Hausdorff sp ac es is sk e letal if the map F f : F X → F Y is skeletal. Pr o of. Assume that the map F f is s keleta l. T o sh o w that f : X → Y is skele tal, fi x a no where dense su b set N ⊂ Y . W e need to sho w that its preimage f − 1 ( N ) is n o wh ere dense in X . Assume con versely that f − 1 ( N ) con tains s ome non-empty op en set U . T he set F ( X \ U ) is closed in F X and its complemen t U = F X \ F ( X \ U ) is op en in F X . Let us sh ow that the set U is not empt y . Fix an y p oin t u ∈ U and consider the closed subsp ace Z = ( X \ U ) ∪ { u } of X and the con tinuous map p : Z → 2 = { 0 , 1 } suc h that p − 1 (0) = X \ U and p − 1 (1) = { u } . By ou r h yp othesis, F 1 6 = F 2. So w e can find an elemen t b ′ ∈ F 2 \ F 1. Since the functor F is epimorphic, there is an elemen t a ′ ∈ F Z suc h that F p ( a ′ ) = b ′ . The elemen t a ′ do es n ot b elong to F ( X \ U ), whic h implies that the sets F Z \ F ( X \ U ) an d U = F X \ F ( X \ U ) are not empt y . Since the map F f : F X → F Y is ske letal, the image F f ( U ) of the non-empt y op en set U ⊂ F X h as non-empt y interior in F Y and h ence conta ins some non-empty op en subset V ⊂ F f ( U ) of the space F Y . Since the set N is no wh ere dense in Y , the subsp ace F ω ( X \ N ) = { a ∈ F ω ( X ) : su pp( a ) ⊂ X \ N } is dens e in F X . So, we can find an elemen t b ∈ F ω ( X ) ∩ V w ith fin ite su pp ort supp( b ) ⊂ Y \ N . Let A = sup p( b ) if supp( b ) 6 = ∅ and A = { y } ⊂ Y \ N b e any singleton in Y \ N if su p p( b ) = ∅ . By [2], b ∈ F ( A ) ⊂ F ( X \ N ). Since b ∈ V ⊂ F f ( U ), there is an elemen t a ∈ U = F X \ F ( X \ U ) with F f ( a ) = b . Ob serv e that f − 1 ( A ) ⊂ f − 1 ( Y \ N ) = X \ f − 1 ( N ) ⊂ X \ U . Since the functor F preserves preimages, we conclud e that a ∈ F ( f − 1 ( A )) ⊂ F ( X \ U ), whic h con tradicts the c hoice of a . This con tradiction sho ws that the set f − 1 ( N ) is no w here dens e in X and hence the map f is sk eletal.  8. Pr oof of Theore m 1.1 T o pro v e the “1-mec” part of Theorem 1.1, assume th at F : Comp → C omp is a 1-mec fun ctor suc h that f or eac h compact zero-dimensional space Z the map F 2 Z : F ( Z ⊕ 2) → F ( Z ⊕ 1) is sk eletal . Lemma 8.1. F or any surje ctive map f : A → B b etwe en finite discr ete sp ac es and any c omp act zer o-dimensional sp ac e Z the map F (id Z ⊕ f ) : F ( Z ⊕ A ) → F ( Z ⊕ B ) i s skeletal. FUNCTORS PRESER VING SKELET AL MAPS 11 Pr o of. Let n = | A | \ | B | and ( A i ) n i =0 b e an increasing sequ en ce of subsets of A suc h that | A 0 | = | B | , f ( A 0 ) = B , A n = A and | A i +1 \ A i | = 1 for ev ery i < n . F or every p ositiv e num b er i ≤ n c ho ose a surjectiv e map f i : A i → A i − 1 suc h that f ◦ f i = f | A i . Ob serv e that id Z ⊕ f = (id Z ⊕ f 1 ) ◦ · · · ◦ (id Z ⊕ f n ) and f or ev ery i ≤ n the map id Z ⊕ f i is homeomorphic to the map 2 Z i where Z i = Z ⊕ ( A i − 1 \ D f i ). By our assumption the map F ( 2 Z i ) is s k eletal and so is its homeomorphic cop y F (id Z ⊕ f i ). Since the comp osition of sk eletal maps b et w een compacta is skel etal, the map F (id Z ⊕ f ) = F (id Z ⊕ f 1 ) ◦ · · · ◦ F (id Z ⊕ f n ) is skel etal.  Lemma 8.2. F or any surje ctive map f : A → B b etwe en finite discr ete sp ac es and any c omp act sp ac e X the map F (id X ⊕ f ) : F ( X ⊕ A ) → F ( X ⊕ B ) is skeletal. Pr o of. By [5, 3.2.2, 3.1.C], the compact space X is the image of a compact zero-dimensional space Z under an irreducible map ξ : Z → X . Applying to the commutativ e diagram X ⊕ A id X ⊕ f / / X ⊕ B Z ⊕ A id Z ⊕ f / / ξ ⊕ id A O O Z ⊕ B ξ ⊕ id B O O the functor F , w e obtain the comm utativ e d iagram F ( X ⊕ A ) F (id X ⊕ f ) / / F ( X ⊕ B ) F ( Z ⊕ A ) F (id Z ⊕ f ) / / F ( ξ ⊕ id A ) O O F ( Z ⊕ B ) F ( ξ ⊕ id B ) O O in wh ich the map F ( ξ ⊕ id A ) is su rjectiv e, F (id Z ⊕ f ) is ske letal b y Lemma 8.1 and F ( ξ ⊕ id B ) is sk eletal by Corollary 6.2. Because of that the map F (id X ⊕ f ) is sk eletal.  The follo win g lemma yields the “1-mec” part of Th eorem 1.1. Lemma 8.3. F or any skeletal surje ction f : X → Y b etwe en c omp acta the map F f : F X → F Y i s skeletal. Pr o of. By Lemma 4.4, the set A 0 = { a ∈ F ω ( X ) : N f | supp( a ) ⊂ ˙ Y } is dense in F X . So, th e sk eletali t y of the map F f will follo w as so on as w e c heck its sk eletalit y at eac h p oint a ∈ A 0 . If f | supp( a ) is injectiv e, then F f is skel etal at a by Theorem 6.1. So, we assu me that f | supp( a ) is not in jectiv e. In this case the supp ort A = sup p( a ) is not empty and a ∈ F A by Lemma 4.2. By our assumption, N f | A ⊂ ˙ Y and h ence the complemen t Y \ N f | A is an op en-and-closed subset of Y . Con s ider the closed subspace ˜ X = f − 1 ( Y \ N f | A ) ∪ N f | A of X and the top ological sum ˜ Y = ( Y \ N f | A ) ⊕ N f | A . Next, consider the comm utativ e diagram X f / / Y ˜ X ˜ f / / i O O ˜ Y h O O where i : ˜ X → X is the em b edding, ˜ f is d efi ned by ˜ f | ˜ X \ N f | A = f | ˜ X \ N f | A and ˜ f | N f | A = id while h : ˜ Y → Y is defi ned by h | Y \ N f | A = id and h | N f | A = f | N f | A . 12 T ARAS BANAKH , AN D RZEJ K UCHARSKI, AND MAR T A MAR TYNENKO Applying th e functor F to this d iagram we get the comm utativ e diagram F X F f / / F Y F ˜ X F ˜ f / / F i O O F ˜ Y F h O O Since ˜ f is skele tal and the restriction ˜ f | A is injectiv e, th e m ap F ˜ f : F ˜ X → F ˜ Y is skelet al at a b y Theorem 6.1. By Lemma 8.2, the map F h is sk eletal. Consequ ently , the comp osition F h ◦ F ˜ f is sk eletal at a and then F f is skelet al at a by Lemma 3.4.  T o prov e the “1-mecw” part of Theorem 1.1, assum e that F : C omp → Comp is a 1-mecw functor suc h that for eac h zero-dimensional compact metrizable sp ace Z the map F 2 Z : F ( Z ⊕ 2) → F ( Z ⊕ 1) is skelet al. T he skelet alit y of the functor F w ill follo w f rom the “1-mec” part of Theorem 1.1 as so on as we chec k that for eac h zero-dimensional compact s pace Z the map F 2 Z : F ( Z ⊕ 2) → F ( Z ⊕ 1) is sk eletal. F or this we shall ap p ly the Characterization Theorem 2.7. By (the pro of of ) Prop ositio n 1.3.5 of [3 ], the zero-dimensional space Z is homeomorphic to the limit lim S Z of an ω -sp ectrum S Z = { Z α , p β α , Σ } with su r jectiv e limit p ro jections, consisting of zero- dimensional compact metrizable spaces Z α , α ∈ Σ. F or n ∈ { 1 , 2 } , consider the in v erse sp ectrum S Z ⊕ n = { Z α ⊕ n, p β α ⊕ id n , Σ } , where id n : n → n denotes the identit y map of the discrete space n = { 0 , . . . , n − 1 } . Next, consider the sk eletal morph ism { 2 Z α } α ∈ A : S Z ⊕ 2 → S Z ⊕ 1. Applying to this m orp hism the mecw functor F , w e obtain a morp hism { F 2 Z α } α ∈ Σ : F ( S Z ⊕ 2) → F ( S Z ⊕ 1). By our assumption, for ev ery α ∈ Σ the map F 2 Z α : F ( Z α ⊕ 2) → F ( Z α ⊕ 1) is sk eletal. Then Theorem 2.7 guaran tees that the limit map lim F 2 Z α : lim F ( S Z ⊕ 2) → lim F ( S Z ⊕ 1) of the sk eletal morphism { F 2 Z α } α ∈ Σ is skele tal. By the con tinuit y of the functor F , th is map is homeomorphic to the map F 2 Z : F ( Z ⊕ 2) → F ( Z ⊕ 1). 9. Pr oof of Theore m 1.3 W e need to prov e that a 1-mecw fu nctor F : Comp → Comp is sk eletal if it is finitely bicom- m u tativ e and fi nitely s keleta l. By Theorem 1.1, it suffices to c h ec k that for any zero-dimensional compact metrizable space Z the map F 2 Z : F ( Z ⊕ 2) → F ( Z ⊕ 1) is s k eletal. W rite th e space Z as the limit of an in verse sp ectrum S Z = { Z n , p m n , ω } consisting of finite spaces Z n , n ∈ ω , and surjectiv e b onding maps p m n : Z m → Z n , n ≤ m . Th en the map 2 Z : Z ⊕ 2 → Z ⊕ 1 can b e iden ti- fied with th e limit map of the morph ism { 2 Z n } n ∈ ω : S Z ⊕ 2 → S Z ⊕ 1 b et ween the inv erse sp ectra S Z ⊕ 2 = { Z n ⊕ 2 , p m n ⊕ id 2 , ω } and S Z ⊕ 1 = { Z n ⊕ 1 , p m n ⊕ id 1 , ω } . App lying to this morphism the con tinuous functor F , we obtain a m orp hism { F 2 Z n } n ∈ ω : F ( S Z ⊕ 2) → F ( S Z ⊕ 1) b etw een the in verse sp ectra F ( S Z ⊕ 2) = { F ( Z n ⊕ 2) , F ( p m n ⊕ id 2 ) , ω } and F ( S Z ⊕ 1) = { F ( Z n ⊕ 1) , F ( p m n ⊕ id 1 ) , ω } . The finite skelet alit y of the f unctor F implies that the morp hism { F 2 Z n } n ∈ ω consists of sk eletal maps F 2 Z n : F ( Z n ⊕ 2) → F ( Z n ⊕ 1) for all n ∈ ω . It is clear that for an y n ≤ m the b ond ing ↓ m n -square D m n Z m ⊕ 2 p m n ⊕ id 2   2 Z m / / Z m ⊕ 1 p m n ⊕ id 1   Z n ⊕ 2 2 Z n / / Z n ⊕ 1 is bicomm u tativ e and consists of finite spaces. S ince th e functor F is fi nitely bicommutat iv e, the b ondin g ↓ m n -square F D m n of th e morphism { F 2 Z n } n ∈ ω also is bicomm utative . FUNCTORS PRESER VING SKELET AL MAPS 13 By Prop osition 2.5 of [13], the bicommutativi t y of the b onding ↓ m n squares F D m n , n ≤ m , implies the bicomm utativit y of the limit ↓ n -square F D n F ( Z ⊕ 2) F ( p n ⊕ id 2 )   F ( 2 Z ) / / F ( Z ⊕ 1) F ( p n ⊕ id 1 )   F ( Z n ⊕ 2) F ( 2 Z n ) / / F ( Z n ⊕ 1) for ev ery n ∈ ω . This fact com bin ed with the skele talit y of the map F ( 2 Z n ) implies that the limit ↓ n -square F D n is sk eletal. No w Pr op osition 3. 1 of [1] guarantee s that the limit map F 2 Z : F ( Z ⊕ 2) → F ( Z ⊕ 1) of the m orp hism { F 2 Z n } n ∈ ω : F ( S Z ⊕ 2) → F ( S Z ⊕ 1) is ske letal. 10. Proof of The orem 1.5 Let F : Comp → Comp b e a 1-mecw f unctor. Given a sk eletally generated compact Hausd orff space X , w e need to pr ov e that the space F X is skele tally generated. Represen t X as the limit of a contin uous ω -sp ectrum S = { X α , π β α , Σ } w ith surj ective limit pro jec- tions p α : X → X α , α ∈ Σ. By [9], the space X , b eing skelet ally generated, has counta ble cellularit y . Consequent ly , the set ˙ X of isolate d p oin ts of X is at m ost countable. F or e ac h isola ted p oin t x ∈ ˙ X of X we can find an ind ex α x ∈ Σ suc h that { x } = π − 1 α x ( U x ) for some op en set U x ⊂ X α x , which must coincide with the singleton of π α x ( x ). Then for an y α ≥ sup { α x : x ∈ ˙ X } we get π α ( ˙ X ) ⊂ D π α ∩ ˙ X α . Replacing the index set Σ by its cofinal subset { α ∈ Σ : α ≥ sup { α x : x ∈ ˙ X }} , if necessary , we can assu me that π α ( ˙ X ) ⊂ D π α ∩ ˙ X α for all α ∈ Σ. Claim 10.1. The se t Σ ′ = { α ∈ Σ : π α ( ˙ X ) = ˙ X α } is close d and c ofinal in Σ . Pr o of. First w e prov e that Σ ′ is closed in Σ . Giv en a c h ain C ⊂ Σ ′ ha vin g the supremum sup C in Σ , w e need to sho w that sup C ∈ Σ ′ . If sup C ∈ C , then there is nothing to pro v e. So w e assume that γ = sup C / ∈ C . In this case by the cont in u it y of the sp ectrum S , th e space X γ is th e limit of the in verse sub sp ectrum S | C = { X α , π α β , C } . W e n eed to p ro ve that ˙ X γ ⊂ π γ ( ˙ X ). T ake any isolated p oint x ∈ ˙ X γ . By the definition of the top ology of the inv erse limit X γ = lim S | C , there is an in dex α ∈ C s uc h that { x } = ( π γ α ) − 1 ( y ) for some isolated p oin t y ∈ X α . Sin ce y ∈ ˙ X α = π α ( ˙ X ), there is a p oint z ∈ ˙ X with y = π α ( z ). No w consider the p oin t x ′ = π γ ( z ) ∈ X γ and ob s erv e that π γ α ( x ′ ) = π γ α ( π γ ( z )) = π γ ( z ) = y = π γ α ( x ) and y ∈ ˙ X α = π α ( ˙ X ) ⊂ D π α ⊂ D π γ α , w hic h implies x = x ′ ∈ π γ ( ˙ X ). Next, we prov e that Σ ′ is cofin al in Σ. Giv en any α 0 ∈ Σ we need to find α ∈ Σ ′ with α ≥ α 0 . F or any isolated p oin t x ∈ ˙ X α 0 \ π α 0 ( ˙ X ) th e p reimage π − 1 α 0 ( x ) is an op en subset of X contai ning no isolate d p oin ts of X . Since the metrizable compactum X α 0 has at most coun tably many isolated p oint s, and the in dex set Σ is ω -complete, there is an index α 1 ≥ α 0 suc h that for eac h isolated p oin t x ∈ ˙ X α 0 \ π α 0 ( ˙ X ) the p reimage ( π α 1 α 0 ) − 1 ( x ) is n ot a singleton, wh ich means that ˙ X α 0 \ π α 0 ( ˙ X ) ⊂ N π α 1 α 0 . Pro ceeding b y induction, w e can co nstruct an increasing c hain ( α n ) n ∈ ω in Σ su c h that ˙ X α n \ π α n ( ˙ X ) ⊂ N π α n +1 α n for all n ∈ ω . Since Σ is ω -complete, the c hain ( α n ) n ∈ ω has the smallest u pp er b ound α ω = s u p n ∈ ω α n in Σ. W e claim that α ω ∈ Σ ′ . Giv en any isolated p oint x ∈ ˙ X α ω , we need to prov e that x ∈ π α ω ( ˙ X ). The con tin u it y of the sp ectrum S guarantee s that th e space X α ω is the limits of the inv erse sequence { X α n , π α n +1 α n , ω } . By the definition of the top olog y of the in verse limit, there is a n u m b er n ∈ ω suc h that { x } = ( π α ω α n ) − 1 ( U n ) for some op en set U n ⊂ X α n whic h must coincide with the s ingleton of the isolated p oin t y = π α ω α n ( x ). W e claim that y ∈ π α n ( ˙ X ). In the other case the c h oice of the ind ex α n +1 guaran tees that the p r eimage ( π α n +1 α n ) − 1 ( y ) is not a singleton and then ( π α ω α n +1 ) − 1 (( π α n +1 α n ) − 1 ( y )) = ( π α ω α n ) − 1 ( y ) = { x } 14 T ARAS BANAKH , AN D RZEJ K UCHARSKI, AND MAR T A MAR TYNENKO is not a singleton, w hic h is a cont radiction. Th us y = π α n ( z ) for some isolated p oin t z ∈ ˙ X . T aking in to account th at y ∈ π α n ( ˙ X ) ⊂ D π α n ⊂ D π α ω α n and π α ω α n ( x ) = y = π α ω α n ( π α ω ( z )), w e can sh o w that x = π α ω ( z ) ∈ π α ω ( ˙ X ).  It follo ws from Claim 10.1 th at X is the limit of the ω -sp ectrum S = { X α , π α β , Σ ′ } consisting of metrizable compacta and surjectiv e skel etal b onding p ro jections and such that ˙ X α ⊂ D π α ⊂ D π β α for all α ≤ β in Σ ′ . By T heorem 6.1, the latter condition guarant ees that th e map F π β α : F X β → F X α is sk eletal. Sin ce the functor F is epimorp hic, contin uous and preserv es w eight, the space F X is sk eletally generated, b eing the limit of th e con tin u ous ω -sp ectrum { F X α , F π β α , Σ ′ } w ith surj ective sk eletal b onding pro jections F π β α : F X β → F X α . 11. Proof of The orem 1.6 Assume that F : C omp → Comp is a pr eimage pr eserving mecw-fun ctor with F 1 6 = F 2. W e n eed to pr o ve that a compact Hausdorff space X is sk eletall y generated if and only if so is the sp ace F X . If X is sk eletally generated, then b y Th eorem 1.5, so is the space F X . Now assume con versely that the space F X is sk eletally generated. W rite the sp ace X as the limit of an inv erse ω -sp ectrum S = { X α , p β α , Σ } with sur jectiv e b ond- ing p ro jectio ns. Applyin g to this sp ectrum the functor F , we get the in verse ω -sp ectrum F S = { F X α , F p β α , Σ } . By th e cont in u it y of the functor F , the limit space of the sp ectrum F S can b e iden tified with F X . The space F X , b eing sk eletally generated, is the limit of an in verse ω -sp ectrum with ske letal b onding p r o jectio n. By the S p ectral Theorem of ˇ S ˇ cepin [13], [3, 1.3.4], we can assume that the latter s p ectrum coincides with the su bsp ectrum S | Σ ′ = { F X α , F p β α , Σ ′ } for some ω -clo sed cofinal subset Σ ′ of the index set Σ. According to Theorem 7.1, the sk eletalit y o f the maps F p β α implies the sk eletalit y of the maps p β α for an y α ≤ β in Σ ′ . Consequently , the sp ace X is s k eletally generated, b eing homeomorphic to the limit sp ace of the inv erse sp ectrum S | Σ ′ = { X α , p β α , Σ ′ } with surjectiv e skel etal b ondin g p ro jections. 12. Some Examples and Open Problems In this section we shall present examples of skeleta l and non-sk eletal fun ctors. F or a n atural num b er n and a mec fu nctor F : Comp → Comp let F n b e the subfu nctor of F assigning to eac h compact space X the closed su bspace F n ( X ) = { a ∈ F X : ∃ ξ ∈ C ( n, X ) su c h that a ∈ F ξ ( F n ) } of F . First we observe that subfuctors F n of op en functors need not b e skele tal. Example 12.1. F or the op e n functor P : Comp → C omp of pr ob ability me asur es and every natur al numb er n ≥ 2 the subfunctor P n is not skeletal. This can b e sho wn applyin g Theorem 1.1. The non-skele tal functors P n are not finitary . W e recall that a functor F : C omp → Comp is finitary if for an y fin ite discrete s pace X the space F X is finite. A t yp ical example of a finitary fu nctor is the functor exp of h yp erspace, s ee [14, 2.1.1]. Th is functor is op en according to [14, 2.10.11]. Example 12.2. F or every n ≥ 3 the subfunctor exp n of the hyp ersp ac e functor is normal and finitary but not skeletal. By Corollary 1.2, eac h op en 1-mec functor is sk eletal. No w we present th ree examples sho win g that the reve rse implication do es not h old. By Prop osition 2.10.1 [14], a n ormal functor F : Comp → Comp with finite sup p orts is bicomm u- tativ e if and only if F is finitely bicomm u tative . In [14, p .85] A.T eleik o and M.Zaric hnyi constructed an example of a fi n itary normal functor F : Comp → Comp , whic h is finitely bicommutat iv e b ut not b icomm utativ e. Applying to this functor T heorems 1.3 and 1.4, we get: FUNCTORS PRESER VING SKELET AL MAPS 15 Example 12.3. Ther e is a finitary normal f unctor F : C omp → C omp which is finitely bi c ommu- tative and skeletal but is not bic ommutative and henc e not op en. By Prop osition 2.10.1 of [14], the functor f rom Example 12.3 h as infi n ite d egree. T h ere is also a finitary we akly normal functor of finite d egree, which is sk eletal but not op en. By λ : Comp → Comp we d enote the functor of su p erextension, see [14, 2.1.2]. It is kno w n that the fun ctor λ is op en, fi nitary , w eakly normal, p reserv es 1-pr eimages but fails to preserve preimages, see [7] and Pr op ositions 2.3.2, 2.10.13 of [14]. By [14, 2.10.19], for ev ery n ≥ 3 the sub functor λ 3 of λ is not op en. Usin g the c h aracterization Th eorem 1.1, one can easily chec k th at the fun ctor λ 3 is sk eletal. Thus w e obtain another: Example 12.4. The finitary we akly normal functor λ 3 is skeletal but not op en. The functor λ 3 is finitary and has finite degree but is not normal. Ou r final example is a normal functor of finite degree wh ic h is skele tal but not op en. Example 12.5. The fu nctor P 3 c ontains a normal subfu nc tor P ∆ , which is skeletal b ut not op en. Pr o of. In the stand ard 2-simplex ∆ 2 = { ( α, β , γ ) ∈ [0 , 1] 3 : α + β + γ = 1 } consider the closed sub sets ∆ 0 =  ( α, β , γ ) ∈ ∆ 2 : max { α, β , γ } = 1  , ∆ 1 =  ( α, β , γ ) ∈ ∆ 2 : min { α, β , γ } = 0 , max { α, β , γ } ≤ 11 12  , ∆ 2 =  ( α, β , γ ) ∈ ∆ 2 : min { α, β , γ } ≥ 1 12 , max { α, β , γ } ≥ 3 4  , and th eir union ∆ = ∆ 0 ∪ ∆ 1 ∪ ∆ 2 , w hic h lo oks as follo ws: r ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ r r ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ N N N No w consider the subfun ctor P ∆ ⊂ P of the functor of probabilit y measur es assigning to eac h compact space X the closed subspace P ∆ ( X ) = { αδ x + β δ y + γ δ z : ( α, β , γ ) ∈ ∆ , x, y , z ∈ X } ⊂ P ( X ) . Here δ x stands for the Dirac measures concen tr ated at a p oin t x . One can c h ec k th at P ∆ is a norm al functor of degree deg P ∆ = 3. In fact, P ∆ is a su bfun ctor of the functor P 3 ⊂ P . Theorem 2.10.2 1 [14] c h aracterizing op en normal fun ctors of fi nite degree implies that the fun ctor P ∆ is not op en. Applying th e c h aracterizat ion Theorem 1.1, one can c h ec k that the functor P ∆ is sk eletal.  Examples 12.1 — 12.5 suggest the follo win g op en Problem 12.6. A ssume that a finitary normal functor F : Comp → Comp of finite de gr e e is skeletal. Is F op en? Equivalently, is F finitely bic ommutative? Let us also ask some other questions ab out s keleta lit y of fu nctors. W e s hall sa y that a f unctor F : Comp → Comp is ( finitely ) squar e- skeletal if for eac h skele tal square D consisting of con tinuous s u rjectiv e maps b et ween (finite) compact spaces the squ are F D is sk eletal. Prop osition 12.7. L et F : C omp → Comp b e an epimorphic functor. 16 T ARAS BANAKH , AN D RZEJ K UCHARSKI, AND MAR T A MAR TYNENKO (1) If F is (finitely) squar e-skeletal, then F is (finitely) skeletal. (2) If F is (finitely) bic ommutative and (finitely) skeletal, then F is (finitely) squar e -skeletal. (3) If F is finitary, then F is finitely squar e-skeletal if and only if F is finitely bic ommutative only if F is skeletal. Pr o of. 1,2 . The fir st t wo statemen ts follo w f r om Remarks 2.5 and 2.4, r esp ectiv ely . 3. The third statemen t follo ws from T heorem 1.3 and an observ ation that a commutativ e square consisting of epimorph isms b etw een finite spaces is skel etal if and only if it is bicommutati v e.  Prop osition 12.7 suggests another tw o problems: Problem 12.8 . Is e ach (finitary) skeletal normal functor F : Comp → Comp (finitely) squar e- skeletal? Problem 12.9. Is a normal functor F : Comp → C omp skeletal if it is finitely sq u ar e-ske letal? (Theorem 1.3 implies th at the answe r is affirmativ e if the functor F is finitary). It is clear that eac h fu nctor that preserves (1-)preimages preserves fi n ite (1-)preimages. W e d o not kno w if the conv erse statemen t is true. Problem 12.10. Do es a me c - functor F : Comp → Comp pr eserve (1-)pr eimages i f F pr eserves finite (1-)pr eimages. 13. A c knowledgements The auth ors would lik e to thank V esk o V alo v for v aluable commen ts concerning sk eletally generated compacta. Referen ces [1] T. Banakh, A. K ucharski, M. Martynenko , A sp e ctr al char acterization of skeletal maps , preprint (arXiv:1108.4195 ) [2] T. Banakh, M. Mart ynenko, M. Zarichn yi, On monomorphic top olo gi c al f unctors with finite supp orts , preprint (arXiv:1004.04 57 ). [3] A. Chigogidze, Inverse Sp e ctr a , N orth-Holland Pub l. Co., Amsterdam, 1996. [4] P . Daniels, K. Kunen and H. Zhou, On the op en-op en game , F u nd. Math. 145 :3 ( 1994), 205–220 . [5] R . Engelking, Gener al T op olog y , H eldermann V erlag, Berlin, 1989. [6] V.V . F edorc huk, A. Chigogidze, Absolute R etr acts and Infinite-Dimensional Manif olds, “Nauk a”, Mosco w, 1992 (in Russian). [7] L. Karchevsk a, T. Radul, On extensions of functors , preprint (arXiv:1106.040 4 ). [8] A. Kucharski, Sz. Plewik, Game appr o ach to universal ly Kur atowski-Ulam sp ac es , T op ology Ap pl. 154 :2 (2007), 421–427 . [9] A. Kucharski, Sz. Plewik, Inverse systems and I -favor able sp ac es , T op ology Ap pl. 156 :1 (2008), 110–116. [10] K. Kuratows ki, T op olo gy, I , Academic Press, NY-Lond on; PWN, W arsa w, 1966. [11] J. Mi odu szewski, L. Rudolf, H-close d and extr emal ly disc onne cte d Hausdorff sp ac es , Dissert. Math. 66 (1969), 55pp. [12] L. Shapiro, The sp ac e of close d subsets of D ω 2 is not dyadic bi c omp actum , (Russian) Doklady Ak ad. Nauk S SSR. 228 :6 (1976), 1302–1305. [13] E.V. ˇ Sˇ cepin, F unctors and unc ountable p owers of c omp acta , Usp ekhi Mat. Nauk 36 (1981), 3–62 (in R ussian). [14] A. T eleiko, M. Z arichnyi, Cate goric al T op ol o gy of Comp act Hausdorff sp ac es , VNTL, Lviv, 1999. [15] V. V alo v, External char acterization of I-f avor able sp ac e , Mathematica Balk anica, 25 :1-2 (2011), 61–78. T. Banakh: F acul ty of Mechanics and Ma the ma tics, Iv an Franko Na tional University of L viv ( Ukraine) and Instytut Ma tema tyki, Jan Ko chanow ski Unive rsity, Kielce (Poland) E-mail addr ess : t.o.banak h@gmail.co m A.Kucharski: Institute of Ma thema tics, University of Silesia, ul. Banko w a 14, 40-007 Ka to wice (Poland) E-mail addr ess : akuchar@m ath.us.edu .pl M. Mar tynenko: F a cul ty o f Mecha nics and Ma the ma tics, Iv an Franko Na ti onal Unive rsity of L viv (Ukraine) E-mail addr ess : martamart ynenko@ukr .net