On the weight and the density of the space of order-preserving functionals

On the w eigh t and t he densit y of the space of order-preserving function al s Sh. A. Ayup o v 1 , A. A. Zai to v 2 No v em b er 19, 2018 Abstract In the presen t pap er it is pr o v ed that the fu nctors O τ of τ -smo oth order preserving functionals and O R of Radon order preserving functionals preserv e the w eigh t of infinite T yc honoff spaces. Moreo v er, it is established that the density and the w eak densit y of infi nite T ychonoff spaces d o n ot in crea se u nder these functors. 1 Institute of Mathematics and information tec hnologies, Uzb ekistan Academ y of Science, F. Ho djaev str. 29, 100125, T ashk ent (Uzb ekistan), e-mail: sh ayup ov@mail.ru, e ayup ov@hotmail.c om, mathinst@uzsci.net 2 Institute of Mathematics and informat io n tec hnologies, Uzb ekis tan Academ y of Science, F. Hodj a ev str. 29, 100125, T ashke n t (Uzb ekis tan), e-ma il: adilb ek − zaitov@mail.ru AMS Sub ject Classifications (2000): 60J55,(18B99, 46E27, 46M15) . Key words: Order-preserving functional, functor, w eigh t, densit y , w eak densit y . 1 0. In tro duction Let X b e a compact( ≡ com pact Hausdor ff top olo g ical space) and let C ( X ) b e the Banac h algebr a of al l con tin uous real -v a l ued functions with the usual algebr aic op erations and wi th the sup-nor m. F or functio ns ϕ, ψ ∈ C ( X ) w e shal l wr ite ϕ ≤ ψ i f ϕ ( x ) ≤ ψ ( x ) for all x ∈ X . If c ∈ R then b y c X w e denot e the constan t functi on identically equal to c. Recall that a functi o nal µ : C ( X ) → R i s said [2 ] to b e: 1) or der-pr eserving if for an y pa ir ϕ, ψ ∈ C ( X ) of functio ns the in- equali ty ϕ ≤ ψ i mplies µ ( ϕ ) ≤ µ ( ψ ); 2) we akly a d ditive if µ ( ϕ + c X ) = µ ( ϕ ) + c µ (1 X ) for all ϕ ∈ C ( X ) and c ∈ R ; 3) norme d if µ (1 X ) = 1. F or a compact X denote b y O ( X ) t he set of all order -p r eserving w eakly addi tiv e and normed functio nals µ : C ( X ) → R . By W ( X ) w e denote the set o f all f unctionals sat isfying only t he condit ions 1 ) and 2) of the a bov e definit ion. Note that acco rding to prop ositio n 1 [7] eac h order-pr eserving w eakl y additive funct ional is contin uo u s. F urther order- preserving weakly additi ve functional s a r e call ed order -p r eserving func- tiona ls [2]. Let X b e a T yc honoff spa ce and l et C b ( X ) b e the algebra of all b ounded contin uo us r eal-v alued functio ns with the p oin t wise alg eb r aic op erati o ns. F or a function ϕ ∈ C b ( X ) put k ϕ k = sup {| ϕ ( x ) | : x ∈ X } . C b ( X ) with this nor m is a Banach algebra. F or a net { ϕ α } ⊂ C b ( X ) ϕ α ↓ 0 X means that for ev ery p oin t x ∈ X one has ϕ α ( x ) ≥ ϕ β ( x ) at β ≻ α a nd li m α ϕ α ( x ) = 0 X . In t his case w e say tha t { ϕ α } is a m onotone decreasing net p oint wi se con vergen t to zero. F or a Tyc ho no ff space X b y β X denote its S t one- ˇ Cec h com pact ex- tension. Given an y functi on ϕ ∈ C b ( X ) consider its contin uous extensio n e ϕ ∈ C ( β X ) . T his gives an isom orphism b et w een the spaces C b ( X ) and 2 C ( β X ) mo reo ver, k e ϕ k = k ϕ k , i . e. this iso morphism is a n isomet ry , and top olog ical prop ert ies o f the ab o ve spaces coincide. Therefor e one may co nsi der an y funct ion from C b ( X ) as an element of C ( β X ) . Hence definitio ns 0.1 and 0. 2 from [1 ] may b e given in th e fol lo w ing form. Definition 1. An or der-preserving functi onal µ ∈ W ( β X ) is said to b e τ -smo oth if µ ( ϕ α ) → 0 for ea ch m o notone net { ϕ α } ⊂ C ( β X ) decreasing to zer o on X . Definition 2. An order-pr eserving functional µ ∈ O ( β X ) is sa id to b e R adon order -preserving functio nal i f µ ( ϕ α ) → 0 for eac h b ounded net { ϕ α } ⊂ C ( β X ) wh i c h unif ormly con v erges to zer o o n compact subsets of X . F or a Tyc honoff space X b y W τ ( X ) and W R ( X ) denote the set s of all τ -smo oth and Radon or der-preserving funct ionals from W ( β X ) , re- sp ectiv ely . The sets W τ ( X ) and W R ( X ) a r e equipp ed with th e p oint- wise conv ergence top ol ogy . The base of n ei gh b orho o ds of a functio nal µ ∈ W τ ( X ) (r esp ecti v ely , o f µ ∈ W R ( X )) i n the p oin t wise conv erg en ce top olo g y consists of the sets h µ ; ϕ 1 , ... , ϕ k ; ε i = { ν ∈ W ( β X ) : | ν ( ϕ i ) − µ ( ϕ i ) | < ε } ∩ W τ ( X ) (resp ectively , h µ ; ϕ 1 , ... , ϕ k ; ε i = { ν ∈ W ( β X ) : | ν ( ϕ i ) − µ ( ϕ i ) | < ε }∩ W R ( X )) where ϕ i ∈ C ( β X ) , i = 1 , . .., k a nd ε > 0. Put O τ ( X ) = { µ ∈ W τ ( X ) : µ (1 X ) = 1 } , O R ( X ) = { µ ∈ W R ( X ) : µ (1 X ) = 1 } . The o per a tions O τ and O R are functors [1, Theorem 0.3 ] i n the ca te- gory T y ch o f Tyc honoff spaces and their contin uo us maps. Let A b e a clo sed subset of the compact X . An o rder-preserving func- tiona l µ ∈ O ( X ) is said to b e supp orte d on A if µ ∈ O ( A ) [ 2]. The set supp µ = ∩{ A : µ ∈ O ( A ) a nd A i s closed i n X } 3 is call ed the supp ort of the order -preserving function a l µ. F or a Tyc hono ff space X put O β ( X ) = { µ ∈ O ( β X ) : supp µ ⊂ X } . The op era t ion O β transl ating a Tyc honoff space X to O β ( X ) , is a functor [3] in the cat egory T y ch . Obviously the i nclusions O β ( X ) ⊂ O R ( X ) ⊂ O τ ( X ) ⊂ O ( β X ) (1) are v alid for an y Tyc ho noff space X , and the equal ities O β ( X ) = O R ( X ) = O τ ( X ) = O ( β X ) are true for ar bitrary com pact X . Let X a nd Y b e compa cts and let f : X → Y be a con t in uo us map. Then the map O ( f ) : O ( X ) → O ( Y ) defined b y the for m ula O ( f )( µ )( ϕ ) = µ ( ϕ ◦ f ) is contin uo u s, where ϕ ∈ C ( Y ) and µ ∈ O ( X ) . No w let X a nd Y b e Tyc ho noff spaces and let f : X → Y b e a con tinuous map. P ut O τ ( f ) = O ( β f ) | O τ ( X ) , O R ( f ) = O ( β f ) | O R ( X ) and O β ( f ) = O ( β f ) | O β ( X ) where β f : β X → β Y is the Sto ne- ˇ Cec h extensio n of f . Note that the ab ov e map s O τ ( f ) : O τ ( X ) → O τ ( Y ) , O R ( f ) : O R ( X ) → O R ( Y ) a nd O β ( f ) : O β ( X ) → O β ( Y ) a re defined corr ectly and they al so ar e contin uo us. In the pap er [ 3 ] it w as shown t h a t the functo r O β preserves the weigh t of infini te Tyc honoff spaces. There exi sts a n example [4 , example 3] , 4 which shows t hat under the functo r O ( β · ) the w eight of a T yc ho noff space may stri ctly incr ease, more precisely , th e exa mple sho ws tha t the functor t ranslates a T yc ho noff space X with coun table w eight t o a com - pact O ( β X ) wit h con tin uum w ei gh t. The questi on whether the funct o rs O τ and O R preserve t he weigh t w a s op en. In this pap er we ob t ain a p os- itive a nsw er t o t his questio n . Moreov er we pr o ve tha t under the functors O τ and O R the densit y and the w eak densi t y of infinite Tyc ho noff spa ces do not increase. It is esta blished t hat the w ea k densities of the fol lo wing spaces coinci de: O ω ( X ) of order -preserving functio nals wit h finit e supp orts, O β ( X ) of order -preserving funct ionals wi th com pact supp orts, O R ( X ) of Radon or der-preserving funct ionals, O τ ( X ) of τ -smo oth order-preser ving functio na ls and O ( β X ) of all order-pr eser ving function a ls. Note that the space W τ ( X ) equipp ed with the p oin t wise conv ergence top olo g y may b e consider ed a s a subspace of t he top ol ogical pro duct Π = Q { R ϕ : ϕ ∈ C ( β X ) } of rea l li nes R ϕ = R . Since Π is a T yc honoff space the spaces W τ ( X ) and O τ ( X ) with the p oin t wise con v ergence to po logy are also Tyc honoff spaces. 1. Main res ults Let X b e a top olo g ical space. Recall t hat a weight of X i s t he cardina l n um b er w ( X ) defined b y the for m ula w ( X ) = min {| B | : B is a base o f th e top ology on X } . In this sect i on w e sha ll pro ve t h a t t he functors O τ of τ -smo oth order- preserving funct i onals and O R of Radon order-pr eserving functi onals pre- serv e the w eig h t of i nfinite T ychonoff spaces. T o do thi s, we need som e construct ions. 5 Let Y b e a subspace of a T ychonoff space X . Put C = { ψ | Y : ψ ∈ C b ( X ) } . The following not ion is well-kno wn. A subspace Y ⊂ X is call ed C - emb e dde d in X if fo r eac h f unction ϕ ∈ C b ( Y ) there ex ists a funct ion e ϕ ∈ C b ( X ) suc h , that e ϕ | Y = ϕ . If Y is a C -embedded subspace o f t he given space X then cl ea rly C ≡ C b ( Y ) . F or a C -emb edded subspace Y of a Tyc hono ff space X , a funct ional µ ∈ W ( β X ) and a fun ct ion ϕ ∈ C b ( Y ) put r X Y ( µ )( ϕ ) = = inf { µ ( ψ ) : ψ ∈ C b ( X ) , ψ ≥ ( inf { ϕ ( y ) : y ∈ Y } ) X , ψ | Y = ϕ } . (2) Lemma 1. F or e ach µ ∈ W ( β X ) we have r X Y ( µ ) ∈ W ( β Y ) . In other wor ds r X Y ( µ ) is an or der-pr eservi n g we akl y additive func tional on C b ( Y ) . Pro of. W e ha ve the foll o wi ng equality r X Y ( µ )(1 Y ) = µ (1 X ) , (3) which directly fo llo ws from (2). Let ϕ ∈ C b ( Y ) . T h en ϕ + c Y ∈ C for a ll c ∈ R . W e hav e r X Y ( µ )( ϕ + c Y ) = = inf { µ ( ψ ) : ψ ∈ C b ( X ) , ψ ≥ ( inf { ϕ ( y ) + c : y ∈ Y } ) X , ψ | Y = ϕ + c Y } = = inf { µ ( ψ ) : ψ ∈ C b ( X ) , ψ ≥ (i nf { ϕ ( y ) : y ∈ Y } ) X + c X , ψ | Y = ϕ + c Y } = = inf { µ ( ψ − c X ) + c · µ (1 X ) : ψ ∈ C b ( X ) , ψ − c X ≥ ( inf { ϕ ( y ) : y ∈ Y } ) X , ( ψ − c X ) | Y = ϕ } = = inf { µ ( ψ − c X ) : ψ ∈ C b ( X ) , ψ − c X ≥ ( inf { ϕ ( y ) : y ∈ Y } ) X , ( ψ − c X ) | Y = ϕ } + c · µ (1 X ) = = r X Y ( µ )( ϕ ) + c · µ (1 X ) = = (by vir tue of (3)) = 6 = r X Y ( µ )( ϕ ) + c · r X Y ( µ )(1 Y ) , i. e. r X Y ( µ )( ϕ + c Y ) = r X Y ( µ )( ϕ ) + c · r X Y ( µ )(1 Y ) . No w let us show tha t r X Y ( µ ) is a n order -p r eserving fun ct ional. Let ϕ i ∈ C b ( Y ) , i = 1 , 2 , and ϕ 1 ≤ ϕ 2 . Then r X Y ( µ )( ϕ 1 ) = = inf { µ ( ψ ) : ψ ∈ C b ( X ) , ψ ≥ ( inf { ϕ 1 ( y ) : y ∈ Y } ) X , ψ | Y = ϕ 1 } ≤ ≤ inf { µ ( ψ ) : ψ ∈ C b ( X ) , ψ ≥ ( inf { ϕ 2 ( y ) : y ∈ Y } ) X , ψ | Y = ϕ 1 } ≤ ≤ inf { µ ( ψ ) : ψ ∈ C b ( X ) , ψ ≥ ( inf { ϕ 2 ( y ) : y ∈ Y } ) X , ψ | Y = ϕ 2 } = = r X Y ( µ )( ϕ 2 ) , i. e. r X Y ( µ )( ϕ 1 ) ≤ r X Y ( µ )( ϕ 2 ) . T h us the functi onal r X Y ( µ ) is order- preserving and weakly a dd i tiv e on C b ( Y ) . Lemma 1 i s prov ed. The o rder-preserving functio nal r X Y ( µ ) defined as ab o v e is sai d to b e a r estriction of t he gi v en or der-preserving functio nal µ ∈ W ( β X ) and the map r X Y : W ( β X ) → W ( β Y ) is cal led the r es triction op er ator. F rom Lemm a 1 and the equa l it y (3) w e ha v e the fol lo w ing Prop osition 1. L et Y b e a C -emb e d de d sub s p ac e of T ychonoff sp ac e X . Then r X Y ( µ ) ∈ O ( β Y ) if and o n ly if µ ∈ O ( β X ) . Let Y ⊂ X , µ ∈ W ( β Y ) and ϕ ∈ C b ( X ) . Put e Y X ( µ )( ϕ ) = µ ( ϕ | Y ) . (4) The fol l o wi ng sta temen t is obvious. Prop osition 2. F or every µ ∈ W ( β Y ) we hav e e Y X ( µ ) ∈ W ( β X ) and e Y X ( µ )(1 X ) = µ (1 Y ) . Henc e, e Y X ( µ ) ∈ O ( β Y ) if and only if µ ∈ O ( β X ) . The o r der-preserving functional e Y X ( µ ) is sai d to b e the exten s ion of the given order -preserving functio nal µ, and the m ap e Y X : W ( β X ) → W ( β Y ) is call ed the extension op er ator. Lemma 2. L e t Y b e a C -emb e dde d subsp ac e of a T ychon o ff sp ac e X . Then r X Y ◦ e Y X = id W ◦ β ( Y ) . 7 Pro of. If µ ∈ W ( β Y ) t hen b y vi r tue of Pr op o sition 2 e Y X ( µ ) ∈ W ( β X ) . F rom Pro po sition 1 it foll o ws that r X Y ( e Y X ( µ )) ∈ W ( β Y ) . According to the construct ion of t he restr iction o per ator the restricti o n r X Y ( e Y X ( µ )) of the o rder-preserving functi onal e Y X ( µ ) i s defined on C = { ψ | Y : ψ ∈ C b ( X ) } ≡ C b ( Y ) . But a ccording to (2) and ( 4) we hav e r X Y ( e Y X ( µ ))( ϕ ) = µ ( ϕ ) for each ϕ ∈ C b ( Y ) . Lemma 2 i s prov ed. Lemma 3. L e t Y b e a C -emb e dde d subsp ac e of a T ychon o ff sp ac e X . A n or der-pr eserving f unctional µ ∈ W ( β Y ) is τ -smo oth if an d onl y if e Y X ( µ ) ∈ W ( β X ) is a τ -smo oth or der-pr eserving function a l. Pro of. Let µ ∈ W τ ( Y ) b e an arbit rary o rder-preserving funct ional and let { ϕ α } ⊂ C b ( X ) b e a net suc h that ϕ α ↓ 0 X . Then ϕ α | Y ↓ 0 Y . Hence, e Y X ( µ )( ϕ α ) = µ ( ϕ α | Y ) → 0 . So , e Y X ( µ ) ∈ W τ ( X ) . Let us establish t he conv erse statement. Let µ ∈ W ( β Y ) b e an a r- bitra ry order -preserving functional suc h that e Y X ( µ ) ∈ W τ ( X ) . Then for eac h net { ψ α } ⊂ C b ( X ) mo notone decreasin g to zer o on X we hav e e Y X ( µ )( ψ α ) → 0 . Hence from (4) it follows that fo r eac h net { ψ α } ⊂ C b ( X ) satisfyi ng ψ α | Y ↓ 0 Y one has µ ( ψ α | Y ) → 0. No w ta k e an ar bitrary net { ϕ α } ⊂ C b ( Y ) such that ϕ α ↓ 0 Y . Si n ce Y is C -em b edded in X there ex i sts a net { ψ α } ⊂ C b ( X ) such tha t ψ α | Y = ϕ α for a ll α. Hence, ψ α | Y ↓ 0 Y and therefor e µ ( ϕ α ) = µ ( ψ α | Y ) → 0 . Thus µ ∈ W τ ( Y ) . Lemm a 3 is p r o ved. Note that for a co mpact X each order -preserving w eakly addi tiv e func- tiona l µ : C ( X ) → R has a (contin uous) o rder-preserving w eakly addit iv e extension µ ′ : B ( X ) → R wit h µ ′ (1 X ) = µ (1 X ) [5] . Here B ( X ) is the space of all b ounded functions equipp ed with the uni form co n vergence top olo g y . As we ha ve noted ab o v e for each T ychonoff space X the nor med spaces C b ( X ) a nd C ( β X ) a re i sometrical ly i somorphic. T herefore an y τ - smo oth or der-preserving functiona l µ : C b ( X ) ∼ = C ( β X ) → R may b e also extended to B ( β X ) as well. W e shall use the sam e nota tion for an order-pr eserving functi onal from W ( β X ) and for its ext ension on B ( β X ). Let Y b e a subspace of a Tyc ho n o ff space X . Co n si der t he fo llo wing 8 set O ∗ Y ( X ) = { µ ∈ O τ ( X ) : µ ( χ K ) = 0 for ev ery compact K ⊂ X suc h that K ∩ Y = ∅} , where χ K is the characterist ic funct ion of th e set K . The equal ities (2) and ( 4 ) im ply the fol lo w ing Prop osition 3. L et Y b e a C -em b e dde d s u bsp ac e of a T ychonoff s p ac e X . Then e Y X ◦ r X Y | O ∗ Y ( X ) = id O ∗ Y ( X ) . Lemmas 2 , 3 a nd Pr op osition 3 yi eld that for a C -emb ed ded subspace Y of a Tyc honoff space X the fol lo w ing eq u a lities ho ld e Y X ( O τ ( Y )) = O ∗ Y ( X ) , r X Y ( O ∗ Y ( X )) = O τ ( Y ) . These equali ties impl y the foll o wi ng Prop osition 4. F or any T ycho n off sp ac e X the maps e X β X : O τ ( X ) → O ∗ X ( β X ) and r β X X : O ∗ X ( β X ) → O τ ( X ) ar e mutua l ly inverse home omorphi s m s. The next stat emen t is the k ey result . Theorem 1. F or an arbitr a ry T ychonoff sp ac e X and for e very its c omp actifi c ation bX the sp ac es O ∗ X ( β X ) and O ∗ X ( bX ) ar e home omorphi c . Pro of. A t first recall that a contin uous map f : b 1 X → b 2 X b et w een compact ifications b 1 X and b 2 X of the given T yc ho noff space X is called natur al , i f f ( x ) = x for al l x ∈ X [ 5, P . 4 7 ]. Let X b e a Tyc honoff space, and supp ose that b X is its arbitr ary compact extensi o n. Let f : β X → bX b e a natur al m ap. Assum e that µ ∈ O ∗ X ( β X ) and O ( f )( µ ) = ν. Co n si der an ar bi trary compa ct set F ⊂ bX \ X . By virt u e o f Theor em 3.5.7 [6 , P . 22 0] the inclusion f − 1 ( F ) ⊂ β X \ X hol ds. P ut K = f − 1 ( F ) . Then f ( K ) = F and χ K = 9 χ F ◦ f . W e hav e ν ( χ F ) = O ( f ) ( µ )( χ F ) = µ ( χ F ◦ f ) = µ ( χ K ) = 0 . So, O ( f )( O ∗ X ( β X )) ⊂ O ∗ X ( bX ) . In ot her w ords the foll o wi n g restri ction map is corr ectly defined O ( f ) | O ∗ X ( β X ) : O ∗ X ( β X ) → O ∗ X ( bX ) . (5) The map O ( f ) | O ∗ X ( β X ) is con tinuous as the rest riction of t h e con tinuous map O ( f ) : O ( β X ) → O ( bX ) . Let µ ∈ O ( β X ) \ O ∗ X ( β X ) . T h en there exi st s a compa ct set K ⊂ β X \ X suc h that µ ( χ K ) 6 = 0 . Applyi ng theorem 3 . 5.7 [6, P . 220 ] we obtai n f ( K ) ⊂ bX \ X a nd O ( f )( µ )( χ f ( K ) ) = µ ( χ f ( K ) ◦ f ) = µ ( χ K ) 6 = 0 . Hence, O ( f )( µ ) ∈ O ( bX ) \ O ∗ X ( bX ) . So O ( f )( O ∗ X ( β X )) = O ∗ X ( bX ) si nce the map O ( f ) is surjecti ve, i. e. the m ap (5 ) is surjecti v e. Note th a t for ev ery com pact ext ension bX of a Tyc ho no ff space X the inclusio n O β ( X ) ⊂ O ( bX ) , is true. Th us, for every Tyc honoff space X and i ts co m pact ext en si on bX one has O β ( X ) ⊂ O ( β X ) ∩ O ( bX ) . (6) F rom this i t fo llo ws O β ( X ) ⊂ O ∗ X ( β X ) ∩ O ∗ X ( bX ) . Lemma 4 [2] and the incl usion (6 ) imply O ( f )( µ ) = µ (7) for each order-pr eserving functio nal µ ∈ O β ( X ) . No w w e need the densit y lemma for order-pr eser ving funct ionals. Re- call tha t the density of a to p ol ogical space X is the least car dinal n um b er 10 of the form | A | where A runs ov er ev erywhere dense subsets of the space X , and | A | denotes t he cardinal it y of the set A. The density o f a top o- logi cal space X is denoted by d ( X ) . F or a Tyc hono ff space X put O ω ( X ) = { µ ∈ O ( β X ) : supp µ ⊂ X and supp µ is finite set } . The fol l o wi ng sta temen t m a y b e consi dered as a v ersi on o f t he density lemma 1.4 fro m [8] for o r der-preserving f un ct ionals. Lemma 4. F or an infinite T ycho noff sp ac e X and for its subsp a c e Y the se t O ω ( Y ) is everywher e den s e in O ( β X ) if and on l y if Y is every- wher e dense in X . Pro of. If Y is not everywhere dense in X then ther e ex i sts a nonem pt y op en set U ⊂ X suc h tha t U ∩ Y = ∅ . T ak e x ∈ U. Consi der a basic neighbor ho o d h δ x ; ϕ ; ϕ ( x ) i , where δ x is the Dirac m easure, defined as δ x ( ψ ) = ψ ( x ) , ψ ∈ C b ( X ) , and ϕ ∈ C b ( X ) i s a functio n suc h that ϕ ( x ) > 0 and ϕ ( y ) = 0 fo r all y ∈ X \ U. Then it is cl ea r that h δ x ; ϕ ; ϕ ( x ) i ∩ O ω ( Y ) = ∅ . Let now Y b e an everywhere dense in X . Then we hav e O ω ( Y ) ⊂ O β ( Y ) ⊂ (since O β is monomo rphic [3]) ⊂ O β ( X ) , and hence, O ω ( Y ) ⊂ O ω ( X ) . Let µ ∈ O ω ( X ) b e a n a rbitrar y or der- preserving functio nal, and let h µ ; ϕ 1 , ... , ϕ k ; ε i b e a neighbo rho o d of µ. Supp ose that supp µ = { x 1 , ... , x n } . One can choo se a set { y 1 , ... , y s } ⊂ Y and a n order -preserving functional ν ∈ O ω ( Y ) suc h that the fo l lo w ing conditi ons hold: (i) supp ν = { y 1 , ... , y n } ; (ii) | ν ( ϕ i ) − µ ( ϕ i ) | < ε, i = 1 , ..., k . This impli es ν ∈ h µ ; ϕ 1 , ... , ϕ k ; ε i , i. e. t h e set O ω ( Y ) i s everywhere dense in O ω ( X ) . F rom the ab ov e in part i cular it foll o ws th a t the set O ω ( X ) is every- where dense in O ω ( β X ). On the other hand acco r ding to prop ositi o n 3 11 [2] O ω ( β X ) is everywhere dense in O ( β X ). T h er efore O ω ( X ) is every- where dense in O ( β X ) and thus , O ω ( Y ) is everywhere dense in O ( β X ) . Lemma 4 is prov ed. Accordin g t o Lemm a 4 the set O β ( X ) is everywhere dense in t he spaces O ( β X ) and O ( bX ) . Hence, O β ( X ) i s ev erywhere dense in the sets O ∗ X ( β X ) and O ∗ X ( bX ). No w let us show t h a t the m ap ( 5) is o ne-to-one. F or t his pur po se ta k e an ar bitrary order-pr eserving fu n ct ional ν ∈ O ∗ X ( bX ) . Supp ose that th er e exist or der-preserving functiona ls µ 1 , µ 2 ∈ O ∗ X ( β X ) suc h that µ 1 6 = µ 2 and O ( f )( µ 1 ) = O ( f )( µ 2 ) = ν. Let { µ i α } ⊂ O β ( X ) , i = 1 , 2 , b e tw o nets conv ergi ng to the functi onals µ 1 and µ 2 , resp ectively . Since the m ap O ( f ) | O ∗ X ( β X ) : O ∗ X ( β X ) → O ∗ X ( bX ) is co ntin uous, t he nets { O ( f ) ( µ i α ) } , i = 1 , 2 , con v erge to ν. On the other hand accor ding t o (7) o ne has O ( f )( µ i α ) = µ i α for i = 1 , 2 a nd for al l α. Hence, µ 1 = li m α µ 1 α = ν = lim α µ 2 α = µ 2 . W e obta in a con tradict ion which sho ws t hat our assumpt ion i s false. F rom the ab ov e, in part i cular, it foll o ws that the map ( O ( f ) | O ∗ X ( β X )) − 1 : O ∗ X ( bX ) → O ∗ X ( β X ) , in v erse t o (5 ) , is al so contin uo us. Thus, the map (5) i s a homeom orphism of the spaces O ∗ X ( β X ) and O ∗ X ( bX ) . Theor em 1 is prov ed. Since each T yc honoff space X has a compa ct extension bX such tha t w ( X ) = w ( bX ), Pr o po sition 4 and Theo rem 1 im ply the fol lo w ing Corollary 1. The functor O τ pr eserves t h e w eight of every infinit e T ychon off sp ac e X , i. e. w ( O τ ( X )) = w ( X ) . Accordin g to (1) we hav e Corollary 2. The functor O R pr eserves the weight of every infi nite T ychon off sp ac e X , i. e. w ( O R ( X )) = w ( X ) . Th us, w ( O β ( X )) = w ( O R ( X )) = w ( O τ ( X )) = w ( X ) 12 for ev ery infinit e Tyc hono ff space X . Note that from Lemma 4 o ne ca n also obtai n a strength en ed v er sion of theorem s 1.8 and 2.6 fr om [1] . Let X b e a n infini te Tyc ho noff space. By vi rtue of the inclusio ns (1) a nd O ω ( X ) ⊂ O β ( X ) i t fol l o ws tha t O β ( X ) is everywhere dense in the spa ces O R ( X ), O τ ( X ) a nd O ( β X ) . On the other hand a cco rding to results of [3] one has d ( O β ( X )) ≤ d ( X ) for every infini te T yc honoff space X . T hus, a streng thening of theo rems 1.8 and 2 .6 from [1 ] may b e to stated as fo l lo w s Corollary 3. The den s i t y of an infinite T ych o noff sp ac e do es not incr e ase under the f u nctors: O ( β · ) of a l l or de r-pr eserving functionals, O τ of τ -smo oth or der-pr eserving functionals, O R of R adon or der-pr e serving fun c tionals and O β of or der-pr eserving functionals with c omp act supp orts. Moreov er, for every infinite Tyc honoff space X we hav e d ( O ( β X ) ≤ d ( O τ ( X )) ≤ d ( O R ( X )) ≤ d ( O β ( X )) ≤ d ( X ) . Recall the fo llo wing notio n. Definition 3[3]. The w e ak density w d ( X ) of a top ol o gical spa ce X is the least cardina l n umb er τ such t h a t X has a π -base whic h i s the union of τ cen tered famil ies of op en set s i n X . W e need the fo llo wing prop erti es of t he weak densi t y [3] : (A) If Y i s ev erywhere dense in X t hen w d ( Y ) = w d ( X ); (B) If X is com pact then w d ( X ) = d ( X ) . The pr o per t y (A) of t he weak densit y , Lemma 4 and the inclusi ons (1) impl y: Prop osition 5. F or e ach T ychonoff sp ac e X one has w d ( O ω ( X )) = w d ( O β ( X )) = w d ( O R ( X )) = w d ( O τ ( X )) = w d ( O ( β X )) . (8) 13 Moreov er acco r ding to (A) and (B) one has w d ( X ) = w d ( β X ) = d ( β X ) and w d ( O ( β X ) ) = d ( O ( β X )) . Ther efore Lemma 4 and Pr op osi- tion 5 impl y Corollary 4. The den s i t y of an infinite T ych o noff sp ac e do es not incr e ase under the f u nctors O ( β · ) , O τ , O R and O β . In connect ion wit h Pr op o sition 5 the fol lo wing question ari ses Question. Are equali ties simil ar to (8 ) v a lid for the densi t y of a T yc h o noff space X ? The next result g i v es a p ositi v e a nsw er for a par ticular case. Prop osition 6. The sp ac e O R ( X ) is s e p ar able if and only if the s p ac e O β ( X ) is sep ar able. Pro of. Since O β ( X ) i s ev er ywhere dense i n O R ( X ) o ne has the inequa l- ity d ( O R ( X )) ≤ d ( O β ( X )) . Let us sho w that the opp osite i n eq uality is also true. Let { µ n } ⊂ O R ( X ) b e a cou ntable everywhere dense subset of o r der- preserving functio nals. F or every or d er -preserving funct ional µ n and ea ch p ositive in teger m there exists a compa ct set K n,m ⊂ X such that µ n ( ϕ ) < 1 m (9) where ϕ ∈ C b ( X ) is a n arbit rary functio n sat isfying t h e fol lo wing in- equali t ies 0 ≤ ϕ ≤ χ ( X \ K n,m ) . (10) Define an order -preserving funct i onal µ n,m on C b ( X ) by the for mula µ n,m = r X K n,m ( µ n ) . (11) Then µ n,m ∈ O ( K n,m ) ⊂ O β ( X ) . W e hav e | µ n ( ϕ ) − µ n,m ( ϕ ) | = ( according to (1 1)) = | µ n ( ϕ ) − r X K n,m ( µ n )( ϕ ) | = = | µ n ( ϕ ) − r X K n,m ( µ n )( ϕ | K n,m ) | = (acco rding to (10 ) ) = = | µ n ( ϕ ) | < ( according to (9 )) < 1 m 14 for all n, m. F rom this it fol lo w s that the sequence { µ n,m } ∞ m =1 p oin t wise con v erges t o µ n . Hence, M ≡ { µ n,m : m, n = 1 , 2 , ... } is ev erywhere dense in O R ( X ) . On the other hand, M ⊂ O β ( X ) ⊂ O R ( X ) . This means that M is ev erywhere dense in O β ( X ) , i. e. d ( O β ( X )) ≤ d ( O R ( X )) . Prop osit ion 6 i s prov ed. Ac kn o wledgmen ts. The au t h ors would lik e to ackn o w le dge the hospitality of the ”Institut f ¨ ur Angewandte Mathemati k ”, Universit¨ at Bonn ( Germany). This work i s sup p orte d in p art by t h e DF G 43 6 USB 113/10/0-1 pr oje ct (Germany) a n d t he F undamental R ese ar ch F o unda- tion of the U zb ekistan A c ad emy of Scienc es. 15 References [1] A. A. 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