A spectral characterization of skeletal maps

A SPECTRAL CHARA CTERIZA TION OF SKELET AL MAPS T ARAS BANAKH, ANDRZEJ KUCHARSKI, AND MAR T A MAR TYNE NKO, Abstract. W e pro ve tha t a map betw een tw o realcompact spaces is skelet al if and only if it is ho meomorphic to the limi t map of a skeletal morphis m betw een ω -sp ectra wi th surj ectiv e li mit pro jections. In this pap er we present tw o characterizatio ns of skeletal ma ps be tw een re a lcompact top olog ical spaces. All maps considered in this pa pe r ar e co nt inuous and all spaces are T ychonoff. F or a subset A of a topo logical space X b y c l A we sha ll deno te the clo sure of A in X . A map f : X → Y is c alled skeletal if for ea ch nowhere dense subset A ⊂ Y the preimage f − 1 ( A ) is nowhere dense in X . This is equiv a lent t o s aying that for each non-empt y op en s e t U ⊂ X the closure f ( U ) has non-empty interior in Y , see [4]. The latter definition can b e lo calized as follows. A ma p f : X → Y betw een tw o top olo gical space s is c a lled • skeletal at a p oint x ∈ X if for eac h neig hborho o d U ⊂ X of x the closure cl Y ( f ( U )) of f ( U ) has non-empty interior in Y ; • skeletal at a subset A ⊂ X if f is skeletal a t ea ch p oint x ∈ A . It is clear that a map f : X → Y is skeletal if and only if f is skeletal at each p oint x ∈ X . 1. Chara cterizing sk elet al maps between metrizable Baire sp ace s It is clear that each o p e n map is skeletal. F or closed maps b etw een metr izable Ba ire spaces this implication can be par tly reversed. Let us recall that a top ologic a l spac e X is Bair e if fo r any sequence ( U n ) n ∈ ω of o p en dense subsets U n ⊂ X the in tersection T n ∈ ω U n is dense in X . W e s ha ll say that a ma p f : X → Y betw een top o lo gical spaces is • op en at a p oint x ∈ X if for ea ch neighbor ho o d U ⊂ X of x the ima ge f ( U ) is a neigh b orho o d of f ( x ); • op en at a subset A ⊂ X if f is open a t each p o int x ∈ A ; • densely op en if f is op en at some dense subset A ⊂ X . It is easy to see that eac h densely open map is skeletal. The con v erse is true for sk eletal maps betw een metrizable compacta , a nd more genera lly , for close d skeletal maps defined o n metrizable Baire s pa ces. Theorem 1 .1. F or a close d m ap f : X → Y define d on a metrizable Bair e sp ac e X the fol lowing c onditio ns ar e e quivalent: (1) f is skeletal; (2) f is skeletal at a dense subset of X ; (3) f is densely op en; (4) f is op en at a dense G δ -subset of X . Pr o of. The implications (4) ⇒ (3) ⇒ (2) ⇒ (1) a re trivial and hold without any conditions on f . T o prov e the implication (1) ⇒ (4), fix a metric d g enerating the topolo gy o f the metriza ble s pace X . F or every n ∈ N consider the family U n of all non-empty op e n subsets U ⊂ X s uch tha t diam( U ) < 1 / n and f ( U ) is op en in f . The skeletal pro p erty of f implies tha t the union S U n is dense in X . Since the spa ce X is Baire , the intersection A = T ∞ n =1 S U n is a dense G δ -set in X . It is clear that f is op en at the set A .  The following simple example shows that the metrizability of X is essential in Theo rem 1.1 and cannot b e weak ened to the first countabilit y . Example 1 .2. The pr oje ction pr : A → [0 , 1] fr om the A leksandr ov “two arr ows” sp ac e A = [0 , 1) × { 0 } ∪ (0 , 1] × { 1 } onto t he interval is skeletal. Y et it is op en at no p oint x ∈ A . 1991 Mathematics Subje ct Classific ation. 54B35, 54C10. Key wor ds and phr ases. Skeletal m ap, inv erse sp ectrum. 1 2 T ARAS BANAKH, ANDRZEJ KUCHARSKI, AND MAR T A MAR TYNENK O, 2. Skelet al and densel y open squares In this section the notions of skeletal and densely op en ma ps are genera lized to square diag rams. These generalized prop er ties will b e used in the sp ectral characterizatio n of skeletal maps given in the next section. Definition 2.1. L e t D be a comm utative diag ram ˜ X ˜ f / / p X   ˜ Y p Y   X f / / Y consisting of contin uous maps b etw een top ologica l s paces. The commutativ e square D is called • op en at a p oint x ∈ X if for each neig hborho o d U ⊂ X of x the p oint f ( x ) has a neighbor ho o d V ⊂ Y such 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 each p oint x ∈ A ; • densely op en if it is ope n at some dense subse t A ⊂ X ; • skeletal at a p oint x ∈ X if for each neighbo rho o d U ⊂ X of x there is a no n- empt y op en set V ⊂ Y such that V ⊂ cl f ( U ) and p − 1 Y ( V ) ⊂ cl ˜ f ( p − 1 X ( U )); • skeletal at a subset A ⊂ X if D is skeletal at each p oint x ∈ A ; • skeletal if D is skeletal at X . Remark 2 .2. If the squar e D is skeletal (at a p oint x ∈ X ), then the map f is skeletal (at the p oint x ). Remark 2 .3. A map f : X → Y is skeletal (r esp. op en ) at a subset A ⊂ X if and only if the squar e X f / / id X   Y id Y   X f / / Y is skeletal (r esp. op en) at the subset A . It is ea sy to see that each densely op en s quare is sk eletal. Under some conditions the con verse is also true. The following pr op osition is a “s quare” counterpart of the characterization Theo rem 2.4. Prop ositi on 2.4. L et D b e a c ommutative dia gr am ˜ X ˜ f / / p X   ˜ Y p Y   X f / / Y c onsisting of c ontinuou s m aps b etwe en top olo gic al sp ac es such t hat the map ˜ f : ˜ X → ˜ Y is close d, the pr oje ction p Y is surje ctive, and the sp ac e X is metrizable and Bair e. Then the fol lowing c onditions ar e e quivalent: (1) the squar e D is skeletal; (2) D is skeletal at a dense subset of X ; (3) D is densely op en; (4) D is op en at a dense G δ -subset of X . Pr o of. The implications (4) ⇒ (3) ⇒ (2) ⇒ (1) a re trivial and hold without any conditions on D . T o prove the implication (1) ⇒ (4), ass ume that the squa re D is skeletal. First let us pr ove tw o auxiliary claims. Claim 2.5. F or e ach non-empty op en s ubset U ⊂ X t her e is a non-empty op en set V ⊂ Y such that V ⊂ f ( U ) and p − 1 Y ( V ) ⊂ ˜ f ( p − 1 X ( U )) . A SPECTRAL CHARA CTERIZA TION OF SKE LET AL MAPS 3 Pr o of. Using the r egularity o f the spa ce X , find a no n-empty op en subset W ⊂ X whos e c losure W lie s in the o p en set U . Since the square D is s keletal, for the set W ther e is a non- empty op en set V ⊂ Y such that p − 1 Y ( V ) ⊂ cl ˜ f ( p − 1 X ( W )). T aking into account that the ma p ˜ f is clo sed, we see that the set ˜ f ( p − 1 X ( W )) is clo sed in ˜ Y and hence p − 1 Y ( V ) ⊂ cl ˜ f ( p − 1 Y ( W )) ⊂ ˜ f ( p − 1 X ( W )) ⊂ ˜ f ( p − 1 X ( U )) . Applying to these inclusions the surjective map p Y , w e get the inclusion V = p Y ( p − 1 Y ( V )) ⊂ p Y ◦ ˜ f ( p − 1 X ( U )) = f ◦ p X ( p − 1 X ( U )) ⊂ f ( U ) .  Claim 2 .6. Each non-empty op en set U ⊂ X c ontains a non-empty op en set W ⊂ U such that f ( W ) is op en in Y and ˜ f ( p − 1 X ( W )) = p − 1 Y ( f ( W )) . Pr o of. By Claim 2.5, there is a non- empt y op en set V ⊂ Y such that V ⊂ f ( U ) and p − 1 Y ( V ) ⊂ ˜ f ( p − 1 X ( U )). Then the op en set W = U ∩ f − 1 ( V ) ha s the requir ed prope r ties. Indeed, its ima g e f ( W ) = V is o pe n in Y . Also the inclusion p − 1 Y ( V ) ⊆ ˜ f ( p − 1 X ( U )) implies ˜ f ( p − 1 X ( W )) = ˜ f ( p − 1 X ( U ∩ f − 1 ( V ))) = ˜ f ( p − 1 X ( U ) ∩ p − 1 X ( f − 1 ( V ))) = = ˜ f ( p − 1 X ( U ) ∩ ˜ f − 1 ( p − 1 Y ( V ))) = ˜ f ( p − 1 X ( U )) ∩ p − 1 Y ( V ) = p − 1 Y ( V ) = p − 1 Y ( f ( W )) .  Let W b e the family of all non-empty op en sets W ⊂ X such that f ( W ) is op en in Y and p − 1 Y ( f ( W )) = ˜ f ( p − 1 X ( W )). Fix any metric d generating the topo logy of the metrizable s pa ce X and for every n ∈ ω co ns ider the subfamily W n = { W ∈ W : diam( W ) < 2 − n } . By Claim 2.6, the union S W n is an op en dense subs e t of X . Since X is a Ba ire spa ce, the intersection A = T n ∈ ω S W n is a de ns e G δ -set in X . T o finish the pr o of, observe that the diagram D is op en at the dense G δ -set A .  3. Skelet al squares and inverse spectra In this section we detect mor phisms be tw een inv erse sp ectr a, inducing skeletal maps b etw een their limit spaces. At first w e need to recall some standard infor mation a b o ut inverse s pe ctra, see [3, § 2.5] and [2, Ch.1] for more details. F or an inverse s p e ctrum S = { X α , p β α , A } consis ting of to p o logical spaces a nd co ntin uo us b onding maps, b y lim S = { ( x α ) α ∈ A ∈ Y α ∈ A X α : ∀ α ≤ β p β α ( x β ) = x α } we denote the limit of S and by p α : lim S → X α , p α : x 7→ x α , the limit pr o jections. Let S X = { X α , p β α , A } a nd S Y = { Y α , π β α , A } b e t wo in verse sp ectra indexed by the same directed partially ordered set A . A morphism { f α } α ∈ A : S X → S Y betw een these sp ectr a is a family of maps { f α : X α → Y α } α ∈ A such that f α ◦ p β α = π β α ◦ f β for any ele ments α ≤ β in A . Each morphism { f α } α ∈ A : S X → S Y of inv erse spec tra induces a limit map lim f α : lim S X → lim S Y , lim f α : ( x α ) α ∈ A 7→ ( f α ( x α )) α ∈ A betw een the limits of these inv erse spec tra. F or indices α ≤ β in A the comm utative squa res lim S X p α   lim f α / / lim S Y π α   X α f α / / Y α and X β p β α   f β / / Y β π β α   X α f α / / Y α are called re s p ectively the limit ↓ α -squar e and the b onding ↓ β α -squar e of the morphism { f α } . W e s ha ll say that the morphism { f α } α ∈ A : S X → S Y has • is skeletal if each map f α : X α → Y α , α ∈ A , is skeletal; • has skeletal limit squar es if for every index α ∈ A the limit ↓ α -square is skeletal; 4 T ARAS BANAKH, ANDRZEJ KUCHARSKI, AND MAR T A MAR TYNENK O, • has skeletal b onding squar es if for every indices α ≤ β in A the b o nding ↓ β α -square is skeletal. Our a im is to find conditions o n a morphism { f α } : S X → S Y of s pe ctra implying the skeletality of the limit map f = lim f α : lim S X → lim S Y . Prop ositi on 3.1. F or a morphism { f α } α ∈ A : S X → S Y b etwe en inverse sp e ctr a S X = { X α , p β α , A } and S Y = { X β , π β α , A } with surje ctive limit pr oje ctions, t he limit map lim f α : lim S X → lim S Y is skeletal if the morphism { f α } has skeletal limit squar es. Pr o of. W e need to show that the limit map f = lim f α : X → Y is s keletal, where X = lim S X , Y = lim S Y . Given any non-empty op en set U ⊂ X , we need to find a non-empty o p en set V ⊂ Y suc h that V ⊂ cl f ( U ). By the definition o f the top olog y of the limit space X = lim S X , there is an index α ∈ A and a non-empty op en set U α ⊂ X α such that U ⊃ p − 1 α ( U α ). Since the limit ↓ α -square X f / / p α   Y π α   X α f α / / Y α is skeletal, for the op en s et U α ⊂ X α there ex ists a non-empty op en set V α ⊂ Y α such that the o pe n set V = π − 1 α ( V α ) lies in the closure of the set f ( p − 1 α ( U α )), which lie s in the clo sure of the set f ( U ).  It turns out that in some cases the skeletality of squa res is preserved by limits. A partially o rdered set A is ca lle d κ -dir e cte d for a ca r dinal num ber κ if ea ch subset K ⊂ A of ca rdinality | C | ≤ κ has an upper b ound in A . F or a to p o logical space X by π w ( X ) we denote the π -weight of X , that is, the smallest ca rdinality |B | of a π - base B for X . W e recall that a family B o f non- empt y ope n subs e ts o f X is called a π - b ase for X if each non-empty op en subset of X contains a set U ∈ B . Prop ositi on 3. 2. L et { f α } α ∈ A : S X → S Y b e a morphism b etwe en inverse sp e ctr a S X = { X α , p β α , A } and S Y = { X β , π β α , A } with surje ct ive limit pr oje ct ions. If for some α ∈ A and t he c ar dinal κ = πw ( Y α ) the index set A is κ -dir e cte d, then t he limit ↓ α -squar e is skeletal pr ovide d that for any β ≥ α in A the b onding ↓ β α -squar e is skeletal. Pr o of. Assuming that the limit ↓ α -square is no t skeletal, we can find a non-empt y op en set U α ⊂ X α such that for any no n-empty op en set V α ⊂ Y α we get π − 1 α ( V α ) 6⊂ cl f ( U ) where U = p − 1 α ( U α ) and f = lim f α is the limit map. Fix a π -base B for the space Y α having cardina lity |B | = π w ( Y α ) ≤ κ . F or every set V ∈ B the op en s et π − 1 α ( V ) \ cl f ( U ) is not empty and hence con tains a set of the form π − 1 α V ( W V ) for some index α V ≥ α in A and some non-empty op en set W V ⊂ Y α V . Since the index se t A is κ -directed, the se t { α V : V ∈ B } has an upper bo und β ∈ A . By our hypothesis, the bonding ↓ β α -square is skeletal. Then for the op en subset U β = ( p β α ) − 1 ( U α ) of X β we can find a non-empty op en set V ⊂ Y α such tha t ( π β α ) − 1 ( V ) ⊂ cl f β ( U β ). W e lose no genera lity assuming that V ∈ B . In this cas e the choice of the set W V guarantees that π − 1 α V ( W V ) ⊂ π − 1 α ( V ) \ f ( U ). Then the op en s ubset W β = ( π β α V ) − 1 ( W V ) = π β ( π − 1 α V ( W V )) of ( π β α ) − 1 ( V ) do es not int ersect the s et π β ◦ f ( U ) = f β ◦ p β ( U ) = f β ( U β ) and hence canno t lie in cl f β ( U β ). This contradiction shows that the limit ↓ α -square is s keletal.  Corollary 3. 3. L et { f α } α ∈ A : S X → S Y b e a morphism b etwe en inverse sp e ctr a S X = { X α , p β α , A } and S Y = { X β , π β α , A } with surje ct ive limit pr oje ct ions. If for the c ar dinal κ = sup { π w ( Y α ) : α ∈ A } the index set A is κ -dir e cte d, then the morphism { f α } α ∈ A has skeletal limit squar es pr ovide d it has skeletal b onding squar es. F or π τ -sp ectra, Pr o p osition 3.1 can b e pa rtly reversed. First let us int ro duce the nec e s sary definitions. Let τ be an infinite car dinal n umber. W e sha ll say that an inv erse spectr um S = { X α , p β α , A } is a π τ - sp e ctrum (resp. a τ -sp e ctrum ) if • each space X α , α ∈ A , has π -weigh t π w ( X α ) ≤ τ (resp. weigh t w ( X α ) ≤ τ ); • the index set A is τ -dir e cte d in the sense that each subset B ⊂ A of cardinality | B | ≤ τ ha s an upp er bo und in A ; • the index s et A is ω - c omplete in the sense that ea ch countable chain C ⊂ A has the smalles t upp er bo und sup C in A ; A SPECTRAL CHARA CTERIZA TION OF SKE LET AL MAPS 5 • the sp ectrum S is τ -c ontinuous in the sense that for any directed subset C ⊂ A with γ = sup C the limit map lim p γ α : X γ → lim { X α , p β α , C } is a homeo morphism. A subset C o f a directed p ose t A is called • c ofinal if for any α ∈ A ther e is an index β ∈ C with α ≤ β ; • τ -close d if for each directed subset D ⊂ C that has the lowest upp er b ound s up D in A we get s up D ∈ C ; • τ -st ationary if C has non-empty in tersection with a ny cofinal τ -clo sed subse t of A . Theorem 3 .4. L et { f α } α ∈ A : S X → S Y b e a morphism b etwe en two π τ -sp e ctr a S X = { X α , p β α , A } and S Y = { Y α , π β α , A } with surje ct ive limit pr oje ctions. If the limit map lim f α : lim S X → lim S Y is skeletal, then for some c ofinal τ -close d subset B ⊂ A t he morphism { f α } α ∈ B is skeletal and has skeletal b onding and limit squar es. Pr o of. T o simplify denotations, let X = lim S X , Y = lim S Y , and f = lim f α : X → Y . First w e show that the set B = { α ∈ A : the limit ↓ α -square is s keletal } is cofinal and τ -clo sed in A . F o r this we shall prov e a n auxiliar y statemen t: Claim 3 .5. F or every α ∈ A ther e is β ∈ A , β ≥ α , such that for any non-empty op en set U ⊂ X α ther e is a non-empty op en set V ⊂ Y β such that π − 1 β ( V ) ⊆ cl f ( p − 1 α ( U )) . Pr o of. In the s pace X α fix a π -base B of cardinality |B | = π w ( X α ) ≤ τ . F or every set U ∈ B the preimage p − 1 α ( U ) is a non-empty o p en set in X = lim X α . Then the skeletalit y of the limit map f : X → Y yields a n op en set V U ⊂ Y suc h that V U ⊂ cl f ( p − 1 α ( U )). By the definition of the topo logy of the limit s pace Y , for some index α U ∈ A , α U ≥ α , ther e is a non-empty op en set W U ⊂ Y α U such that π − 1 α U ( W U ) ⊂ V U . Since the index set A is τ -directed, the set { α U : U ∈ B } has an upper b ound β in A . I t is easy to see that the index β has the prop er ty sta ted in Cla im 3 .5.  Claim 3 .6. The set B is c ofinal in A . Pr o of. Fix any index α 0 ∈ A . Using Claim 3.5, by induction we can constr uct a non-decr easing s equence ( α n ) n ∈ ω in A such that for any non-empty op en set U ⊂ X α n , n ∈ ω , ther e is a non-empty op en set V ⊂ Y α n +1 with π − 1 α n +1 ( V ) ⊆ cl f ( p − 1 α n ( U )). Since the set A is ω -complete, the set { α n } n ∈ ω has the smallest upp er b ound β = s up { α n } n ∈ ω ∈ A . The pro of of Cla im 3 .6 will b e complete as s o on as we chec k that β ∈ B , which means that the limit ↓ β -square is skeletal. Given any non-empty op en set U β ⊂ X β we need to find a non-empty open set V β ⊂ Y β such that π − 1 β ( V β ) ⊂ cl f ( p − 1 β ( U β )). Since the sp ectr um S X is τ -contin uous, the s pa ce X β can b e identified with the limit of the inv erse s p ec tr um { X α n , p α m α n , ω } and hence for the op en set U β ⊂ X β there ar e an index n ∈ N and a no n-empty op en set U ⊂ X α n such that ( p β α n ) − 1 ( U ) ⊂ U β . By the constr uction of the sequence ( α k ), for the set U ⊂ X α n there is a no n- empt y op en se t V ⊂ Y α n +1 such that π − 1 α n +1 ( V ) ⊂ cl f ( p − 1 α n ( U )). Consider the o p e n s e t V β = ( π β α n +1 ) − 1 ( V ) ⊂ Y β . T ak ing into account that the limit pr o jections p β and π β are surjective, w e conclude that V β = π β ( π − 1 α n +1 ( V )) ⊂ π β (cl f ( p − 1 α n ( U )) ⊂ cl π β ◦ f ( p − 1 α n ( U )) = = cl f β ◦ p β ( p − 1 α n ( U )) ⊂ cl f β (( p β α n ) − 1 ( U )) ⊂ cl f β ( U β ) , witnessing that β ∈ B .  Claim 3 .7. The set B is τ - close d in A . Pr o of. Let C ⊂ B b e a dire cted subset of c a rdinality | C | ≤ τ having the low est upp er b ound γ = sup C in A . W e need to show tha t γ ∈ B , which means that the limit γ -square is skeletal. Fix a non-empty op en subset U γ ⊂ X γ . Since the sp ectrum S X is τ -contin uous, the space X γ can be identifi ed with the limit space of the in verse sp ectrum { X α , p β α , C } . Then the o p en s et U γ ⊂ X γ contains the preimage ( p γ α ) − 1 ( U α ) of some non-empty op en set U α ⊂ X α , α ∈ C . Since α ∈ C ⊂ B , the limit ↓ α -square is s keletal. Conseq uently , for the 6 T ARAS BANAKH, ANDRZEJ KUCHARSKI, AND MAR T A MAR TYNENK O, set U α there is a non-empty op en set V α ⊂ Y α such that π − 1 α ( V α ) ⊂ cl f ( p − 1 α ( U α )). Then for the o pe n subset V γ = ( π γ α ) − 1 ( V α ) in X γ we get π − 1 γ ( V γ ) = π − 1 α ( V α ) ⊂ cl f ( p − 1 α ( U α )) = cl f  p − 1 γ (( p γ α ) − 1 ( U α ))  ⊂ cl f ( p − 1 γ ( U γ )) , witnessing that the limit ↓ γ -square is s keletal.  Claim 3 .8. F or any indic es α ≤ β in B the b onding ↓ β α -squar e is skeletal. Pr o of. T o show that the b o nding ↓ β α -square is skeletal, fix any o p en non-empty subset U ⊆ X α . Since α ∈ B , the limit ↓ α -square is s keletal and hence there exists op en no n-empty subset V ⊆ Y α such that π − 1 α ( V ) ⊆ cl f ( p − 1 α ( U )). Since the limit pro jections p β and π β are surjective, w e get ( π β α ) − 1 ( V ) = π β ( π − 1 α ( V )) ⊆ π β (cl f ( p − 1 α ( U ))) ⊆ cl π β ◦ f ( p − 1 α ( U )) = cl f β ◦ p β ( p − 1 α ( U )) = cl f β (( p β α ) − 1 ( U )) .  The definition of the set B and Remark 2.2 imply our last claim, which completes the pr o of of Theorem 3.4. Claim 3 .9. F or every α ∈ B t he map f α : X α → Y α is skeletal and henc e the morphi sm { f α } α ∈ B is skeletal.  The following theor em partly reverses T heo rem 3.4. Theorem 3. 10. L et { f α } α ∈ A : S X → S Y b e a morphism b etwe en two π τ -sp e ctra S X = { X α , p β α , A } and S Y = { Y α , π β α , A } with su rje ctive limit pr oje ctions. If the limit map lim f α : lim S X → lim S Y is not skeletal, then the set B = { α ∈ A : f α is not skeletal } is ω -stationary in A . Pr o of. Assume that the limit map f = lim f α : X → Y betw een the limit spa c e s X = lim S X and Y = lim S Y is not skeletal. Then the spac e X contains a non-empty o p en s et U ⊂ V whose ima ge f ( U ) is no where dense in Y . W e lose no genera lity assuming tha t the set U is of the form U = p − 1 o ( U o ) for some index o ∈ A and some non-empty o p en set U o ⊂ X o . T o prove our theor em, we need to check that the set B meets each cofinal ω -closed subset C o f A . Claim 3.11 . F or any index α ∈ C , α ≥ o , ther e is an index β ∈ C , β ≥ α , such that for any non- empty op en set V α ⊂ Y α ther e is a non-empty op en set W β ⊂ Y β such that π − 1 β ( W β ) ⊂ π − 1 α ( V α ) \ f ( U ) . Pr o of. Fix a π -ba se B for the spac e Y α having ca rdinality |B | = π w ( Y α ) ≤ κ . Since the set f ( U ) is nowhere dense, for every s e t V ∈ B the op en subset π − 1 α ( V ) \ cl f ( U ) of Y is not empty and hence contains a set of the form π − 1 α V ( W V ) for some index α V ≥ α in A and some no n-empty o p en set W V ⊂ Y α V . Since the index set A is κ -directed and the set C is cofinal in A , the se t { α V : V ∈ B } has an upper b ound β ∈ C . It is easy to see that the index β has the required prop erty .  Using Claim 3.11, b y induction constr uc t a non-decr easing sequence ( α n ) n ∈ ω in C such that α 0 ≥ o and for any non- e mpt y op en set V ⊂ Y α n , n ∈ ω , there is a non-empty open set W ⊂ Y α n +1 such tha t π − 1 α n +1 ( W ) ⊂ π − 1 α n ( V ) \ f ( U ). Since the se t C is ω -closed in the ω -complete set A the chain { α n } n ∈ ω ⊂ C has an low est upp er b ound β ∈ A , which belo ngs to the ω -close d set C . Claim 3 .12. β ∈ B ∩ C . Pr o of. W e need to show tha t the ma p f β : X β → Y β is not s keletal. Assuming the o pp osite, for the non- empt y op en subset U β = ( p β 0 ) − 1 ( U o ) = p β ( U ) of X β , we can find a non- empt y op en set V β ⊂ Y β that lies in the closure cl f β ( U β ). Since the sp ectrum S Y is ω -contin uous, the spa ce Y β can b e identified with the limit space of the in verse sp ectrum { Y α n , π α m α n , ω } . Therefor e, we lose no g enerality assuming that the set V β is o f the form V β = ( π β α n ) − 1 ( V ) for some o p e n set V ⊂ Y α n , n ∈ ω . By the choice of α n , there is a no n-empty op en set A SPECTRAL CHARA CTERIZA TION OF SKE LET AL MAPS 7 W ⊂ Y α n +1 such that π − 1 α n +1 ( W ) ⊂ π − 1 α n ( V ) \ f ( U ). Applying to this inclusion the surjective map π β , we o bta in that the non- e mpt y op en subse t ( π β α n +1 ) − 1 ( W ) = π β ( π − 1 α n +1 ( W )) ⊂ π β ( π − 1 α n ( V ) \ f ( U )) = = π β ( π − 1 α n ( V )) \ π β ◦ f ( U ) = ( π β α n ) − 1 ( V ) \ f β ◦ p β ( U ) = V β \ f β ( U β ) of V do es not intersect the s et f β ( U β ) and hence cannot lie in its closure. This co nt radiction shows that the map f β is not skeletal a nd hence β ∈ B ∩ C .   4. A spectral characteriza tion of skelet al maps between realcomp act sp aces In this section we prov e Theo rem 4.1 whic h characterizes skeletal maps b etw een rea lcompact spaces a nd is the main r esult of this pap er. This character iz ation has been applied in the pap er [1] detecting functor s that preserve skeletal maps b etw een compact Hausdorff spaces. Let us reca ll that a Tyc honoff space X is called re alc omp act if each C -embedding f : X → Y into a T ychonoff space Y is a closed embedding. An embedding f : X → Y is called a C -emb e dding if each contin uo us function ϕ : f ( X ) → R has a con tinuous extension ¯ ϕ : Y → R . By Theorem 3.11.3 [3], a topo logical space is realcompact of and only if it is ho meo morphic to a closed subspac e o f some p ow er R κ of the real line, see [3, § 3.11]. By [3, 3.11.12 ], each Lindel¨ of space is r ealcompact. W e say that tw o ma ps f : X → Y and f ′ : X ′ → Y ′ are home omorph ic if there are homeomorphisms h X : X → X ′ and h Y : Y → Y ′ such tha t f ′ ◦ h X = h Y ◦ f . It is clear that a map f : X → Y is skeletal if and only if it is homeomorphic to a skeletal map f ′ : X ′ → Y ′ . Theorem 4. 1 . F or a map f : X → Y b etwe en T ychonoff sp ac es the fol lowing c onditions ar e e quivalent: (1) f is skeletal and the sp ac es X, Y ar e r e alc omp act. (2) f is home omorphic t o t he limit map lim f α : lim S X → lim 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 , A } and S Y = { Y α , π β α , A } with surje ctive limit pr oje ctions. (3) f is home omorphi c to the limit map lim f α : lim S X → lim S Y of a morphism { f α } : S X → S Y with skeletal limit squ ar es b etwe en two ω -sp e ctr a S X = { X α , p β α , A } and S Y = { Y α , π β α , A } with surje ctive limit pr oje ct ions. (4) f is home omorphi c to the limit map lim f α : lim S X → lim S Y of a morphism { f α } : S X → S Y with skeletal b onding squ ar es b etwe en two ω -s p e ctr a S X = { X α , p β α , A } and S Y = { Y α , π β α , A } with surje ctive limit pr oje ct ions. Pr o of. W e sha ll prov e the implications (1) ⇒ (4) ⇒ (3) ⇒ (2) ⇒ (1). (1) ⇒ (4) Assume that the spaces X , Y a re r ealcompact. Then Pro p o sition 1.3.4 and 1 .3.5 o f [2] imply that the map f is homeo mo rphic to the limit map lim f α : lim S X → lim S Y of a morphism { f α } α ∈ A betw een tw o ω -sp ectr a S X = { X α , p β α , A } and S Y = { Y α , π β α , A } with surjective limit pro jections. If the ma p f is skeletal, then Theor em 3.4 yields a cofina l ω -b ounded subse t B ⊂ A such that the morphism { f α } α ∈ B has skeletal bo nding squa res. Since the set B is cofina l in A , f is homeomorphic to the limit map lim f α induced by the morphism { f α } α ∈ B with skeletal b onding squares betw een the in verse ω -s p ec tr a { X α , p β α , B } and { Y α , π β α , B } . The implications (4) ⇒ (3) a nd (3) ⇒ (2) follow from Cor ollary 3.3 and Remark 2.2, resp ectively . The fina l implica tion (2) ⇒ (1) follows from Theor em 3.10 and Pr op osition 1.3.5 [2] saying tha t a Tyc honoff space is homeomor phic to the limit spa ce of an ω -sp ectrum (with sur jective limit pro jections) if and only if it is realcompa c t.  Let us observe that Theo rem 4.1 do es not hold for arbitrar y sp ectra. Just take any non- s keletal map f : X → Y be tw een zero- dimensional (metrizable) compacta and apply the following lemma. Lemma 4.2. Each c ontinuous map f : X → Y fr om a t op olo gic al sp ac e X to a r e alc omp act sp ac e Y o f c overing top olo gic al dimension dim( Y ) = 0 is home omorphic t o the limit map lim f α : lim S X → S Y of a skeletal morphism { f α } α ∈ A : S X → S Y b etwe en inverse sp e ctr a S X = { X α , p β α , A } and S Y = { Y α , π β α , A } . Pr o of. By Lemma 6.5.4 of [2], the zero -dimensional realco mpa ct space Y is homeomor phic to a clo s ed subspace of the p ow er N τ for some cardinal τ . Let A = [ τ ] <ω be the family of finite subsets of τ , par tially or dered by the inclusion relation. F or every α ∈ A let Y α be the pr o jection o f the spa ce Y ⊂ N τ onto the fac e N α 8 T ARAS BANAKH, ANDRZEJ KUCHARSKI, AND MAR T A MAR TYNENK O, and π α : Y → Y α be the c orres p o nding pro jection map. F or any finite sets α ⊂ β let π β α : Y β → Y α be the corres p o nding bo nding pro jection. Then the space Y ca n b e ide ntified with the limit lim S Y of the inverse sp ectrum S Y = { Y α , π β α , A } c o nsisting of discrete spa ces Y α , α ∈ A . The space X can b e iden tified with the limit of the tr ivial sp ectr um S X = { X α , p β α , A } consis ting o f spaces X α = X and identit y b onding maps π β α : X β → X α . Then the map f is homeo morphic to the limit map lim f α : lim S X → S Y of the skeletal morphism { f α } α ∈ A : S X → S Y consisting o f the maps f α = π α ◦ f : X α = X → Y α , α ∈ A . Here we remar k that each map f α : X α → Y α is s keletal b eca use the space Y α is dis c rete.  References [1] T. Banakh, A. Kucharski, M. Martynen ko, On functors pr eserving ske letal maps and skelet al ly genera te d co mp acta , prepri nt [2] A. Chigogidze, Inverse Sp ectr a , Elsevier, 1996. [3] R. Engelking, Genera l T op olo gy , Heldermann V er lag, Berli n, 1989. [4] J. M io duszewski, L. Rudolf, H-close d and ex t r emal ly disc onne cte d Hausdorff sp ac es , Dissert. M ath. 66 (1969), 55pp. T. Banakh: F acul ty of Mecha nics and Mat hema tics, Iv an Franko Na tional Un iversity of L viv (Ukraine) and, Instytut Ma tema tyki, Jan Kochanow ski University, Kielce (Poland) E-mail addr e ss : t.o.banakh@ gmail.com A. Kucharski: Institute of Ma them a tics, University of S ilesia, ul. Bankow a 14, 40-007 Kat owice (Poland) E-mail addr e ss : akuchar@mat h.us.edu.p l M. Mar tynenko: F acul ty of Mechanics and M a thema tics, Iv an Franko Na tional University of L viv (Ukraine) E-mail addr e ss : martamartyn enko@ukr.n et