Normally preordered spaces and utilities

Normally preordered spaces and utilities ∗ E. Minguzzi † Abstract In applications it is useful to know whether a top ological preordered space is normally preordered. It is prov ed that every k ω -space equipp ed with a closed preorder is a normally preordered space. F urthermore, it is prov ed that second countable regularly preordered spaces are p erfectly normally preordered and admit a countable utility represen tation. 1 In tro duction In applica tions such as dynamica l sys tems [1], general rela tivit y [2 7] or micro e- conomics [6] it is useful to know if a top ologica l pr eordered spac e , usually a top ological manifold, is nor ma lly pr e ordered. 1 The preor der aris es from the orbit dynamics of the dyna mical system; from the causal preorder of the space- time manifold; or from the preferences of the a gent in micro eco nomics. The condition of preor der nor mality ca n be regarded as just o ne first step in o rder to prov e that the space is quasi- uniformizable or even quasi-pseudometriza ble in such a wa y that it admits o rder co mpletions and order compactifications. The case of a preorder is often as important as the case of an order. Indee d, dynamical systems ar e esp ecially int eres ting in the presence of dy na mical cycles, and, analogo usly , spacetimes are par ticularly interesting in prese nc e of causality violations. Also the cas e of a preor der is the usua l one considered in micro eco- nomics as an agent may b e indifferent with resp ect to tw o pos sibilities in the space o f alterna tiv es (pros pect space ) (actually the agent can e ven b e unable to compare them, this p ossibility is called indecisiveness o r incompar abilit y [2]). It is well k nown that a topo logical space eq uipped with a closed or de r is Hausdorff. The remov al of the an tisymmetry condition for the order sugges ts to remov e the Haus do rff condition for the to p olo gy . Indeed, quite often in applications, the preorder is so tight ly linked with the top olog y that o ne has that tw o po in ts which are indistinguisha ble acco r ding to the preorder (i.e. x ≤ y and y ≤ x ) are also indistinguisha ble according to the topo logy , so that even ∗ This version differs from that published in Order as it cont ains t wo pr oofs of theorem 2.7. † Dipartiment o di Matematica Appli cata “G. Sansone”, Universit` a degli Studi di Firenze, Via S. Marta 3, I-50139 Firenze, Italy . E-mail: ettore.minguzzi@unifi.it 1 Domain theory [12] has applications to computer science and i s related in a natural wa y to ordered top ological s paces. In this field a topological space equipp ed wi th a closed or der is called p osp ac e and a normally ordered s pace is call ed monotone normal p osp ac e . 1 impo sing the T 0 prop erty could b e to o strong. This fact is not completely appreciated in the litera ture on topolo gical preo rdered spaces. Nach bin, in his foundational b o ok [29], uses at some crucial step the Hausdorff co ndition implied by the closed orde r assumption [29 , Theor . 4, Cha p. I]. In this w ork w e sha ll remov e altog ether the Hausdorff a ssumption on the topo logy and in fact ev en the T 0 assumption. The idea is that the s eparability conditions for the to polo gy should preferably come from their preorder versions and should not b e added to the a ssumptions. So fa r the only result which a llows us to infer that a top olo gical preo rdered space is normally preorder e d is Nach bin’s theorem [29, Theor. 4, Chap. I], which states that a compa ct space eq uipped with a closed o rder is nor mally ordered. There are other r esults of this t yp e [10, Theo r. 4.9] but they assume the totality of the order . Our main o b jective is to prove a r esult that holds at least for top olog ic al manifolds and in the preor dered ca se so as to b e used in the mentioned a pplications. Indeed, we shall pr ov e that the k ω -spaces equipp ed with a clo sed preorder a re normally preo rdered. Since top ologica l manifolds are second countable and lo cally compact and these pr ope r ties imply the k ω -space prop erty , the theorem will achieve our g o al. 1.1 T opological preliminaries Since in this work we do not assume Hausdor ffness, it is necessa ry to clar ify that in our termino logy a to p olo gical space is lo c al ly c omp act if e v ery p oint admits a c ompact neig h b orho o d. A top olog ic al space E is a k-sp ac e if O ⊂ E is op en if a nd only if, for every compact set K ⊂ E , O ∩ K is o pen in K . W e r e mark that we are using here the definition given in [3 2], thus we do not include Hausdorffness in the definition as done in [8, Co r. 3 .3 .19]. Using our definition of lo cal compactness it is not difficult to prove that every first countable or lo cally co mpa ct space is a k -spa ce (mo difying slightly the pro of in [32, Theor. 43.9]). A rela ted notion is that of k ω -sp ac e which ca n b e characterized through the following pr o per t y [11]: there is a co un table sequence K i of compact sets such that S ∞ i =1 K i = E and for every O ⊂ E , O is o pen if and only if O ∩ K i is op en in K i with the induced top ology (ag ain, her e E is not required to b e Hausdorff ). The sequence K i is ca lled admissible . By r eplacing K i → S i j =1 K j one chec ks that it is p ossible to a ssume K i ⊂ K i +1 ; a lso the replac e men t K i → K ∪ K i shows that in the a dmissible sequence K 1 can b e chosen to b e any compact set. W e hav e the chain of implications: co mpact ⇒ k ω -space ⇒ σ -compa ct ⇒ Lindel¨ of, and the fact that lo cal compac tness makes the last thre e prop erties coincide. W e sha ll b e interested o n the b ehavior o f the k ω -space c ondition under quo- tien t maps. Rema r k ably , the next pro of shows tha t the Hausdo r ff condition in Morita’s theorem [28, Lemmas 1-4] can b e dro pp ed. Theorem 1.1. Every σ -c omp act lo c al ly c omp act (lo c al ly c omp act) sp ac e is a k ω -sp ac e (r esp. k -sp ac e). The quotient of a k ω -sp ac e ( k -sp ac e) is a k ω -sp ac e 2 (r esp. k -sp ac e). Every k ω -sp ac e ( k -sp ac e) is the qu otient of a σ -c omp act (r esp. p ar ac omp act) lo c al ly c omp act sp ac e. Pr o of. The first statement has b een a lready mentioned. Let K α , α ∈ Ω, b e an admissible sequence (r esp. the family o f all the co mpact sets) in E . Let π : E → ˜ E b e a q uotien t map and let ˜ K α = π ( K α ); then the sets ˜ K α , α ∈ Ω, form a coun table family of co mpact sets (resp. a subfamily of the family of a ll the compact sets). Supp ose that C ⊂ ˜ E is s uch that C ∩ ˜ K α is closed in ˜ K α . W e hav e π − 1 ( C ∩ ˜ K α ) = π − 1 ( C ) ∩ π − 1 ( ˜ K α ); thus π − 1 ( C ) ∩ K α = π − 1 ( C ) ∩ π − 1 ( ˜ K α ) ∩ K α = π − 1 ( C ∩ ˜ K α ) ∩ K α . Let C α be a closed s et on ˜ E such that C α ∩ ˜ K α = C ∩ ˜ K α ; then π − 1 ( C ) ∩ K α = π − 1 ( C α ∩ ˜ K α ) ∩ K α = π − 1 ( C α ) ∩ π − 1 ( ˜ K α ) ∩ K α = π − 1 ( C α ) ∩ K α . As the set on the right-hand side is closed in K α for every α we get that π − 1 ( C ) is clo sed and hence C is closed by the definition of quotient top ology . (Note that ˜ K α is a n admissible sequence in the k ω -space case.) Let K α , α ∈ Ω, be a n admissible se quence (res p. the family o f a ll the compact sets) in E . Let ˜ K α = { ( x, α ) , x ∈ K α } , E ′ = ∪ α ˜ K α be the dis join t union [3 2] and let g : E ′ → E b e the map given by g (( x, α )) = x so that ϕ α := g | ˜ K α : ˜ K α → K α is a homeo morphism. L e t C ⊂ E be s uch that g − 1 ( C ) is closed then for each α , g − 1 ( C ) ∩ ˜ K α is closed in ˜ K α th us g ( g − 1 ( C ) ∩ ˜ K α ) = ϕ α ( g − 1 ( C ) ∩ ˜ K α ) = C ∩ K α is closed in K α bec ause ϕ α is a homeomor phis m. Thu s C is closed and hence g is a quotient map. It is trivial to chec k tha t E ′ is σ -compact (resp. pa racompact) a nd lo cally compact. 1.2 Order theoretical preliminaries F or a top ological pre ordered s pace ( E , T , ≤ ) o ur ter minology and no tation fol- low Nach bin [29]. Thus by pr e or der we mean a reflexive and tr ansitive relation. A preor der is an or der if it is a ntisymmetric. With i ( x ) = { y : x ≤ y } and d ( x ) = { y : y ≤ x } we denote the increasing a nd decre a sing hulls. A top olog ic a l preorder ed s pace is semiclose d pr e or der e d if i ( x ) and d ( x ) are closed for every x ∈ E , and it is close d pr e or der e d if the graph of the pr eorder G ( ≤ ) = { ( x, y ) : x ≤ y } is closed. A subset S ⊂ E , is ca lle d incr e asing if i ( S ) = S and de cr e asing if d ( S ) = S . I ( S ) denotes the s mallest clo s ed increasing set containing S , a nd D ( S ) denotes the smallest closed decr easing s e t co n taining S . It is under sto od that the set inclusion is reflexive, X ⊂ X . A to polo gical preorder ed space is a n ormal ly pr e or der e d sp ac e if it is semi- closed preorder ed and for every closed decreasing set A and clos e d increasing set B which are disjoint, A ∩ B = ∅ , it is p ossible to find a n op en decrea s ing set U and an op en increasing set V which s eparate them, namely A ⊂ U , B ⊂ V , and U ∩ V = ∅ . A r e gularly pr e or der e d sp ac e is a semiclosed preordered space such that, if x / ∈ B with B a closed increasing s e t, there is an op en decreas ing set U ∋ x and an o pen increa s ing se t V ⊃ B , such that U ∩ V = ∅ , and a nalogously the dual prop erty must ho ld for y / ∈ A with A a closed decreas ing set. 3 W e hav e the implicatio ns : no rmally preo rdered space ⇒ r egularly preo rdered space ⇒ clos e d preo rdered space ⇒ semiclosed preorde r ed space. A top olo gical preorder ed space is c onvex [29] if for every x ∈ E , and op en se t O ∋ x , there are an op en decreas ing se t U and an op en incr easing set V such that x ∈ U ∩ V ⊂ O . 2 Preorders on compact spaces and k ω -spaces W e are interested in establishing in which wa y compactness and countabilit y assumptions improve the preorder s e parability pro per ties of a top ological pre- ordered spac e . The example b elow shows that these c onditions do not pro mote semiclosed preor dered spaces to closed preo r dered spaces , no t even under con- vexit y , and thus that the closed preorder pro per t y is in fact muc h mor e inter- esting since, as we shall see, it allows us to reach b etter separ ability prop erties. Example 2.1 . Let E = [0 , 1] 2 with the usual pr o duct top ology T . Eviden tly ( E , T ) has very g o o d top ological prop erties: it is second countable, Hausdo rff, compact and even complete with resp ect to the Euclidea n metric. Let ( x, y ) b e co ordinates on E a nd let ≤ b e the order defined as follows i (( x, y )) = { ( x ′ , y ′ ) : x ′ = x and , if x > 0 , y ≤ y ′ ; if x = 0 and y ≤ 1 / 2 , y ≤ y ′ ≤ 1 / 2; if x = 0 and y > 1 / 2 , y = y ′ } . With this choice ≤ is completely determined, a nd ( E , T , ≤ ) can be chec ked to be a conv ex semiclo sed or dered space which is no t a clo sed o r dered space . W e need to s ta te the next t wo prop ositions that g eneralize to pr e orders t wo corres p onding prop ositions due to Nach bin [29, Pr op. 4,5, Chap. I]. Actually the pro o fs given by Nach bin for the case of an order work als o in this case without any mo dification. F or this rea son they ar e omitted. Prop osition 2. 2. L et E b e a close d pr e or der e d s p ac e. F or every c omp act K ⊂ E , we hav e d ( K ) = D ( K ) and i ( K ) = I ( K ) , that is, t he de cr e asing and incr e asing hu l ls ar e close d. Prop osition 2. 3. L et E b e a c omp act close d pr e or der e d sp ac e. L et F ⊂ V wher e F is incr e asing and V is op en, t hen t her e is an op en incr e asing set W such that F ⊂ W ⊂ V . An analo gous statement holds in the de cr e asing c ase. Every compa c t s pace equipp ed with a clo sed order is no rmally or dered [2 9, Theor. 4 , Chap. I]. W e s hall need a s light ly str onger statement. Theorem 2.4. Every c omp act sp ac e E e quipp e d with a close d pr e or der is a normal ly pr e or der e d sp ac e. Pr o of. If A ∩ B = ∅ with A closed decrea sing and B clos ed incre asing, for every x ∈ A and y ∈ B we ha ve d ( x ) ∩ i ( y ) = ∅ , thus there is (see [29, Pro p. 1, Chap. I]) a decrea sing neighborho o d U ( x, y ) of x and an increasing neig h b orho o d V ( y , x ) 4 of y (they are not necess a rily op en) such that U ( x, y ) ∩ V ( y , x ) = ∅ . Since A and B are c lo sed subse ts of a compact set they ar e compact and ∪ y ∈ B V ( y , x ) ⊃ B th us there are po in ts y i ∈ B , i = 1 , · · · , k , such tha t, de fined the increa sing neighborho o d of B , V ( x ) := ∪ i V ( y i , x ), and the decr easing neighbor hoo d of x , U ( x ) := ∩ i U ( x, y i ) we have U ( x ) ∩ V ( x ) = ∅ . The neighbo rho o ds U ( x ), x ∈ A are such that ∪ x ∈ A U ( x ) ⊃ A thus we can find x j , j = 1 , · · · , n such that defined the decreasing neighborho o d of A , U ′ := ∪ j U ( x j ), and the inc r easing neighborho o d of B , V ′ := ∩ j V ( x j ), we have U ′ ∩ V ′ = ∅ . Finally , by Pr op. 2.3 there are an op en decrea sing set U suc h that A ⊂ U ⊂ U ′ , and an op en increasing set V s uc h that V ′ ⊃ V ⊃ B , fro m which the thesis i.e. U ∩ V = ∅ . A subset S ⊂ E with the induced top olo gy T S and the induced preorder ≤ S is a to po lo gical preor dered space hence calle d subsp ac e . In g eneral it is no t true that every op en increasing (decreasing ) s et o n S is the intersection of a n op en increasing (resp. decreas ing) set on E with S . If this is the ca se S is called a pr e or der e d subsp ac e [3 1, 24, 17]. Prop osition 2.5. Every su bsp ac e of a (s emi)close d pr e or der e d sp ac e is a (semi)- close d pr e or der e d sp ac e. Pr o of. Let ( E , T , ≤ ) b e semiclosed preorder ed. Given x ∈ S , let i S ( x ) = { y ∈ S : x ≤ S y } = { y ∈ S : x ≤ y } . F ro m this express io n we hav e i S ( x ) = i ( x ) ∩ S , th us i S ( x ) is closed in the induced to polo gy . Analog ously , d S ( x ) is closed in the induced top ology , that is ( S, T S , ≤ S ) is s emiclosed preorder ed. Let ( E , T , ≤ ) be clos ed preo rdered, thus G ( ≤ ) is closed, a nd hence the graph of ≤ S , G ∩ ( S × S ) is closed in the induced (pro duct) top olo gy ( T × T ) S × S . The equality ( T × T ) S × S = ( T S × T S ) prov es that the gr aph o f ≤ S is clo sed. Prop osition 2.6. In a close d pr e or der e d sp ac e every c omp act su bsp ac e S is a pr e or der e d subsp ac e. Pr o of. If A ⊂ S is clo sed decreas ing then it is compact, thus d ( A ) is closed decreasing and such that A = d ( A ) ∩ S . The pr o o f in the increasing case is analogo us. Using the previous r esults it is p ossible to follow the strateg y of the pro of that a Hausdorff k ω -space is a normal space [11] in order to obtain the next theorem. Theorem 2.7 . L et ( E , T , ≤ ) b e a k ω -sp ac e endowe d with a close d pr e or der, 2 then ( E , T , ≤ ) is a n ormal ly pr e or der e d sp ac e. W e shall give a nother pro of in the next section. It must b e noted that Hausdorff loca lly compa ct spaces need no t b e normal thus in theor em 2.7 the k ω -space condition ca nnot be weak ened to the k - space condition (consider the discrete order). 2 W e shall call these spaces: cl osed preordered k ω -spaces. Ho we ver, this termi nology is differen t from that used in [16] where in their closed ordered k -spaces the term “ k -space” includes an additional condition on the upp er and low er top ologies. 5 Pr o of. 3 Let K n , K n ⊂ K n +1 , S n K n = E , b e an admissible sequence of compact sets (i.e. that app ear ing in the definition of k ω -space). A set O is ope n if a nd only if O ∩ K n is op en in K n . Let A, B b e r espe c tiv ely a closed decr easing and a close d increasing set such that A ∩ B = ∅ and denote A n = A ∩ K n , B n = B ∩ K n . B y pro p os itio n 2 .5 and theor em 2.4 every K n with the induced top olog y and pr e order is a normally preorder ed space. Let ˜ A 1 = A 1 , ˜ B 1 = B 1 and let U 1 , V 1 ⊂ K 1 be resp ectively op en decrea sing a nd op en increasing sets in K 1 (with resp ect to the induced top ology and preor der) such that D 1 ( U 1 ) ∩ I 1 ( V 1 ) = ∅ , ˜ A 1 ⊂ U 1 , ˜ B 1 ⊂ V 1 , where D 1 , I 1 are the closure op erators fo r the pre o rdered space K 1 . They exist, it suffices to apply the pre ordered nor mality o f the s pace three times. The set I 1 ( V 1 ) be ing a close d set in K 1 is, rega rded as a subset of E , the intersection betw een a closed set a nd a compact set thus it is compa ct. The set i ( I 1 ( V 1 )) is closed (Prop. 2 .2 ) and analo g ously , d ( D 1 ( U 1 )) is closed. No w, let us consider the clos e d increas ing set o n K 2 given by ˜ B 2 = [ i ( I 1 ( V 1 )) ∩ K 2 ] ∪ B 2 and the closed decreasing set given by ˜ A 2 = [ d ( D 1 ( U 1 )) ∩ K 2 ] ∪ A 2 (see figure 2.7). P S f r a g r e p l a c e m e n t s A B V 1 U 1 U 2 V 2 K 1 K 2 d ( D 1 ( U 1 )) i ( I 1 ( V 1 )) Figure 1: The idea o f the pro of of theor em 2.7. They a re disjoint b ecause i ( I 1 ( V 1 )) ∩ d ( D 1 ( U 1 )) = ∅ (otherwise I 1 ( V 1 ) ∩ D 1 ( U 1 ) 6 = ∅ , a contradiction), B 2 ∩ A 2 = ∅ , and B 2 ∩ d ( D 1 ( U 1 )) = ∅ as i ( B 2 ) ∩ D 1 ( U 1 ) ⊂ B 2 ∩ D 1 ( U 1 ) ⊂ V 1 ∩ D 1 ( U 1 ) = ∅ . Analo gously , i ( I 1 ( V 1 )) ∩ A 2 = ∅ . Th us, a rguing as b efore we can find U 2 , V 2 ⊂ K 2 resp ectively op en decr eas- ing a nd op en incr easing sets in K 2 (with r esp e c t to the induced topo logy and preorder ) such that D 2 ( U 2 ) ∩ I 2 ( V 2 ) = ∅ , ˜ A 2 ⊂ U 2 , ˜ B 2 ⊂ V 2 . Co ntin uing in this way we define at each step ˜ B j +1 = [ i ( I j ( V j )) ∩ K j +1 ] ∪ B j +1 , ˜ A j +1 = [ d ( D j ( U j )) ∩ K j +1 ] ∪ A j +1 . Arguing as b efore ˜ A j +1 , ˜ B j +1 are disjoint closed decreasing and closed increasing subsets of K j +1 and since the latter is a nor- mally preo rdered spac e there ar e U j +1 , V j +1 ⊂ K j +1 resp ectively o p en decrea s- 3 This first proof do es not appear in the version of the paper published in Order. 6 ing a nd op en incr easing sets in K j +1 such that D j +1 ( U j +1 ) ∩ I j +1 ( V j +1 ) = ∅ , ˜ A j +1 ⊂ U j +1 , ˜ B j +1 ⊂ V j +1 . Note that V j ⊂ ˜ B j +1 ⊂ V j +1 and a na logously , U j ⊂ U j +1 . L et V = S j V j and U = S j U j . The set V contains B b e cause B j ⊂ ˜ B j ⊂ V j th us B = S j B j ⊂ V . Ana lo gously , U co n tains A . The set V is op en b ecause V ∩ K s = S j ≥ 1 ( V j ∩ K s ) = S j ≥ s ( V j ∩ K s ), and the set V j ⊂ K j is op en in K j so that, since for j ≥ s , K s ⊂ K j , V j ∩ K s is op en in K s and so is the union V ∩ K s . The k ω -space prop erty implies that V is o pen. Analog ously , U is o pen. Finally , let us prov e that V is increa s ing. Let x ∈ V then ther e is some j ≥ 1 such that x ∈ V j ⊂ K j . Let y ∈ i ( x ), then we can find so me r ≥ j such tha t y ∈ K r . Since V j ⊂ V r , x ∈ V r , and since V r is incr easing on K r , y ∈ V r th us y ∈ V . Analogously , U is decrea sing which completes the pro of. Corollary 2.8 . Every lo c al ly c omp act σ -c omp act sp ac e e qu ipp e d with a close d pr e or der is a normal ly pr e or der e d sp ac e. W e can reg ard as a cor ollary the known r esult [11], Corollary 2.9. Every Hausdorff k ω -sp ac e is a normal sp ac e. Pr o of. Apply theorem 2.7 to the discrete order and use the fact that the Haus- dorff condition is equiv alent to the closure of the graph of the discrete or der, namely the diagona l ∆ = { ( x, y ) : x = y } . By a result due to Milnor [2 6, Lemma 2.1] Hausdo rff k ω -spaces are finitely pro ductive. The lo cally compa ct σ -compact spaces are finitely pro ductive even if they a re not Ha usdorff [5], thus closed preo rdered lo cally co mpact σ -compact spaces a re finitely pro ductive. The closed preordered lo ca lly co mpa ct σ -compact spaces pr ovide our main example b ecause, by using the no n-Hausdorff genera l- ization of Mo rita’s theorem, we get the following result. Corollary 2. 10. Every close d pr e or der e d k ω -sp ac e ( k -sp ac e) E is the quotient of a close d pr e or der e d σ - c omp act (r esp. p ar ac omp act) lo c al ly c omp act sp ac e E ′ , π : E ′ → E , in su ch a way that G ( ≤ ′ ) = ( π × π ) − 1 ( G ( ≤ )) . Pr o of. Ob vious fro m theorem 1.1 bec a use ≤ ′ so defined is a closed preorder. 3 A pr o of based on an extension theorem A function f : E → R is isotone , if x ≤ y ⇒ f ( x ) ≤ f ( y ). Nach bin proved tha t a c ont inuous isotone function f : S → [0 , 1] defined o n a compac t subset S of a normally o rdered spac e E can b e extended to a function F : E → [0 , 1] o n the whole s pa ce preserv ing contin uit y and the isotone pro per t y [29, Theo r. 6 , Cha p. I]. 4 Unfortunately , he uses the order c o ndition (and hence the implied Haus do rff 4 The fact that the theorem holds wi th the functions f , F , taking v al ues in [0 , 1] is eviden t from Nac hbin’s proof but is not stated in the ori ginal theorem. 7 condition) and we nee d therefore to gener alize the theo rem to the preo rdered case. Levin g ives a similar result for closed preo rders o n a compact set E [2 0, Lemma 2], [2 1] [2 2, Theor. 6.1], in the context of the mass tra nsfer problem, nevertheless we prefer to give a pro of closer in spirit to Nach bin’s top olog ical approach. Theorem 3.1. L et E b e a normal ly pr e or der e d sp ac e and let S b e a subsp ac e. L et f : S → [0 , 1 ] b e c ont inuous and isotone on S . In or der that t he function f b e exten dible to a c ontinuous isotone funct ion F : E → [0 , 1] , it is ne c essary and su fficient that ξ , ξ ′ ∈ [0 , 1] , ξ < ξ ′ ⇒ D ( f − 1 ([0 , ξ ])) ∩ I ( f − 1 ([ ξ ′ , 1])) = ∅ . (1) The pr o o f of this theorem is the same a s the pro of of [29, Theo r. 2]. Indeed, in the b o dy of that pr o of the image of the or iginal a nd extended functions is in [0,1 ] and, rather surprising ly , a close chec k o f the pr o of (also of the omitted contin uit y par t [1 5, Cha p. 4, Lemma 3 ]) s hows tha t it do es not dep end on the closure c ondition on S whic h is imp osed in the theorem statement. R emark 3.2 . Instead of rechecking Nac hbin’s pro o f [2 9, Theor. 2] one might just follow the arg ument b elow to show that f in theor em 3.1 ca n b e ex tended to S preser ving co n tinuit y , the iso tone prop erty a nd Eq. (1). Then one co uld apply Nach bin’s res ult [29, Theor . 2] and get theorem 3.1. Under the hypo theses o f The o rem 3 .1, let x ∈ S \ S . Let ( U α ) α be a neigh- bo rho o d basis of x . E very U α meets S . Th us ( U α ∩ S ) α is a filter basis on S . Let η = lim inf α f ( U α ∩ S ) a nd η ′ = lim sup α f ( U α ∩ S ). If η < η ′ we ca n find η < ξ < ξ ′ < η ′ . By hypothesis D ( f − 1 ([0 , ξ ])) ∩ I ( f − 1 ([ ξ ′ , 1])) = ∅ . But x belo ngs to this intersection as it b elongs to f − 1 ([0 , ξ ]) ∩ f − 1 ([ ξ ′ , 1]), which gives a contradiction. Th us η = η ′ and f can be extended contin uously to x . The extended function ˜ f : S → [0 , 1] is isotone. Indeed, the inequality ˜ f ( y ) < ˜ f ( x ) implies that there are ζ < ζ ′ such that ˜ f ( y ) < ζ < ζ ′ < ˜ f ( x ). By c o n tinuit y o f ˜ f , x ∈ f − 1 ([ ζ ′ , 1]) and y ∈ f − 1 ([0 , ζ ]) and using D ( f − 1 ([0 , ζ ])) ∩ I ( f − 1 ([ ζ ′ , 1])) = ∅ , we co nclude that x  y . Finally , let ξ , ξ ′ ∈ [0 , 1], ξ < ξ ′ and c ho ose ζ , ζ ′ ∈ [0 , 1], ξ < ζ < ζ ′ < ξ ′ then D ( f − 1 ([0 , ζ ])) ∩ I ( f − 1 ([ ζ ′ , 1])) = ∅ but D ( ˜ f − 1 ([0 , ξ ])) ⊂ D ( f − 1 ([0 , ζ ])) and I ( ˜ f − 1 ([ ξ ′ , 1])) ⊂ I ( f − 1 ([ ζ ′ , 1])) from which it follows D ( ˜ f − 1 ([0 , ξ ])) ∩ I ( ˜ f − 1 ([ ξ ′ , 1])) = ∅ . This genera lization allows us to remov e the clo sure condition in [29, Theor. 3], w hich therefore reads Theorem 3.3 . L et E b e a normal ly pr e or der e d sp ac e and let S b e a su bsp ac e with the pr op erty that if X , Y ⊂ S satisfy D S ( X ) ∩ I S ( Y ) = ∅ then D ( X ) ∩ I ( Y ) = ∅ . Then every c ontinuous isotone funct ion f : S → [0 , 1] c an b e extende d to a c ontinuous isotone function F : E → [0 , 1] . If we are in the discr ete order case a nd S is clos ed, the assumption o n the- orem 3.3 is satisfied, thus o ne recovers Tietze’s extens io n theorem for b ounded functions. W e can now prov e (note that S need not b e closed; see also remark 4.2 for a different pro of ). 8 Theorem 3.4. L et E b e a normal ly pr e or der e d sp ac e and let S b e a c omp act subsp ac e, then any c ontinuous isotone function f : S → [0 , 1] c an b e ext en de d to a c ontinuous isotone function F : E → [0 , 1] . Pr o of. Let X , Y ⊂ S be such that D S ( X ) ∩ I S ( Y ) = ∅ . The se t D S ( X ) is a closed subs e t of the c o mpact s e t S th us it is compa ct which implies that, by P r op. 2.2, d ( D S ( X )) is closed. Analo gously , i ( I S ( Y )) is closed. Moreover, d ( D S ( X )) ∩ i ( I S ( Y )) = ∅ for if y ∈ d ( D S ( X )) ∩ i ( I S ( Y )) there ar e z ∈ D S ( X ) and x ∈ I S ( Y ), such tha t x ≤ y ≤ z which implies x ≤ z or ∅ 6 = i S ( I S ( Y )) ∩ D S ( X ) = I S ( Y ) ∩ D S ( X ), a contradiction. As a co nsequence, D ( X ) ∩ I ( Y ) ⊂ d ( D S ( X )) ∩ i ( I S ( Y )) = ∅ . The de s ired co nclusion follows now from theore m 3.3. Lemma 3.5. (extension and sep ar ation lemma) L et ( E , T , ≤ ) b e a close d pr e- or der e d c omp act sp ac e. L et K b e a (p ossibly empty) c omp act subset of E , let A ⊂ E b e a close d de cr e asing set and let B ⊂ E b e a close d incr e asing set such that A ∩ B = ∅ . L et f : K → [0 , 1] b e a c ontinuous isotone function on K such that A ∩ K ⊂ f − 1 (0) and B ∩ K ⊂ f − 1 (1) . Then t her e is a c ontinuous isotone function F : E → [0 , 1] which ex tends f such that A ⊂ F − 1 (0) and B ⊂ F − 1 (1) . Pr o of. By theorem 2.4 the to polo gical preo rdered space E is a normally pre- ordered s pace. Since A a nd B are clos ed and hence compa ct subsets of E , the set K ′ = A ∪ K ∪ B is a c o mpact subset of E . The function f ′ : K ′ → [0 , 1] defined by f ′ | A = 0, f ′ | K = f , f ′ | B = 1, is is otone. Let us prov e that f ′ is c o n tinuous on K ′ with the induced top olog y . Clearly f ′− 1 ([0 , 1]) is closed in K ′ as it coincide s with K ′ . W e need only to prov e that f ′− 1 ([ α, 1 ]), α > 0, is closed in K ′ , the pro o f for the ca se f ′− 1 ([0 , β ]), β < 1, being a nalogous. The set f − 1 ([ α, 1 ]) be ing a clo sed subset of K is a compact subset of E thus I ( f − 1 ([ α, 1 ])) = i ( f − 1 ([ α, 1 ])). But A ∩ f − 1 ([ α, 1 ]) = ∅ a nd A is decr easing thus A ∩ I ( f − 1 ([ α, 1 ])) = ∅ . Since f is contin uous on K there is some closed s e t C in E such that f − 1 ([ α, 1 ]) = C ∩ K . The clo sed set C ′ = C ∩ I ( f − 1 ([ α, 1 ])) has again the prop erty f − 1 ([ α, 1 ]) = C ′ ∩ K and is disjoint fro m A . Now we can write f ′− 1 ([ α, 1 ]) = B ∪ ( C ′ ∩ K ) = B ∪ ( C ′ ∩ K ′ ) , which prov es that f ′− 1 ([ α, 1 ]) is the union o f tw o closed subsets o f K ′ . W e conclude that f ′ is co n tinuous on K ′ . By theorem 3.4 f ′ can b e extended to a contin uous isotone function F : E → [0 , 1], which is the desired function. Theorem 3 .6. (impr ove d exten s ion and sep ar ation r esult) L et ( E , T , ≤ ) b e a k ω -sp ac e e quipp e d with a close d pr e or der. L et K b e a (p ossibly empty) c omp act subset, let D b e close d de cr e asing and let I b e close d incr e asing, D ∩ I = ∅ . L et f : K → [0 , 1] b e a c ont inuous isotone fun ction on K such that D ∩ K ⊂ f − 1 (0) and I ∩ K ⊂ f − 1 (1) . Then ther e is a c ontinuous isotone funct ion F : E → [0 , 1 ] which ex tends f such t hat D ⊂ F − 1 (0) and I ⊂ F − 1 (1) . 9 Pr o of. Let K i , E = S i K i , b e a n admiss ible sequence according to the definitio n of k ω -space. Without loss o f generality w e c an assume K i ⊂ K i +1 and K 1 = K . By theorem 2.4 a nd prop os itio n 2.5 each subset K i endow ed with the induced top ology a nd preo rder is a nor mally preor dered space. Define f 1 : K 1 → [0 , 1], by f 1 = f . W e make the inductive a ssumption that there is a contin uous isotone function f i : K i → [0 , 1] such that D ∩ K i ⊂ f − 1 i (0) and I ∩ K i ⊂ f − 1 i (1). Applying lemma 3.5 to E = K i +1 with the induced o rder, A = D ∩ K i +1 , B = I ∩ K i +1 , and co mpact subspa ce K i , we get that ther e is a contin uous is otone function f i +1 : K i +1 → [0 , 1] which e x tends f i such that D ∩ K i +1 ⊂ f − 1 i +1 (0) and I ∩ K i +1 ⊂ f − 1 i +1 (1). W e conclude that there is an iso tone function F : E → [0 , 1] defined by F | K i = f i such tha t D ⊂ F − 1 (0) and I ⊂ F − 1 (1). W e r ecall that in a k ω -space with admis s ible sequence K i , K i ⊂ K i +1 , a function g : E → R is contin uous if and only if for every i , g | K i is contin uous in the subspa ce K i . By constructio n, the function f i is c o n tinuous in K i th us F is contin uous. W e can give a sec o nd pro o f to theo r em 2.7. Se c ond Pr o of. Let D ⊂ E be a closed decrea sing subset and let I ⊂ E b e a clos ed increasing subset s uch that D ∩ I = ∅ . Let K = ∅ , then b y theorem 3.6 there is a co ntin uous is otone function F : E → [0 , 1] such that D ⊂ F − 1 (0) and I ⊂ F − 1 (1). The op en sets { x : F ( x ) > 1 / 2 } and { x : F ( x ) < 1 / 2 } , ar e resp ectively op en increasing , op en decr easing, disjoint and containing resp ectively I a nd D , th us ( E , T , ≤ ) is a nor ma lly preor de r ed space. Example 3.7 . W e give an ex ample of no r mally pr eordered space whic h admits a no n-closed subset S , which satisfies the ass umptions of theor em 3.3. Let E = R × S 1 , S 1 = [0 , 2 π ), be equipp ed with the pro duct preo rder ≤ , where R is endow ed with the usual orde r  , a nd S 1 is given the indis crete preor de r . Let the top ology T on E b e the coa rsest top olog y which ma kes the pro jection on the first factor, π : E → R , cont inuous. By theor em 2.7 E is normally preo rdered (or use the rema rk 4 .1 below, and the fact that ( R ,  ) is nor mally or dered). The subset S = [0 , π ] 2 is co mpact, thu s it s atisfies the ass umptions of theor em 3.3, but non-closed, its closure b eing S = [0 , π ] × S 1 . The subset S = (0 , π ) 2 is non-clos ed and non-co mpa ct but it still sa tisfies the a ssumptions of theorem 3.3. 4 The ordered quotien t space Let us introduce the equiv a lence relation x ∼ y on E , given by “ x ≤ y and y ≤ x ”. Let E / ∼ b e the quotient space, T / ∼ the quo tien t top ology , and let . b e defined by , [ x ] . [ y ] if x ≤ y for some r epresentativ es (with some abuse of notation we shall denote with [ x ] b oth a subset of E and a p oint on E / ∼ ). The quotient preorder is by constructio n an o rder. The triple ( E / ∼ , T / ∼ , . ) is a top olog ical ordered space and π : E → E / ∼ is the co n tinuous quo tien t pro jection. 10 R emark 4.1 . T ak ing into a ccount the definition of q uo tien t top olog y w e hav e that ev ery op en (closed) increasing (decreasing) set on E pro jects to a n o pen (resp. clo sed) increa sing (r e sp. decr easing) set on E / ∼ and all the latter sets can b e regar ded a s such pro jectio ns. As a conseque nc e , ( E , T , ≤ ) is a normally preorder ed space (semiclosed preordered space, reg ularly preo rdered spac e ) if and only if ( E / ∼ , T / ∼ , . ) is a normally order ed space (resp. semiclosed or dered space, regular ly o rdered space). R emark 4.2 . An alternative pro of of theorem 3.4 uses the fact tha t E / ∼ is normally or dered and π ( S ) is compact, so that f can be pas sed to the quotient, extended using [29, Theor . 6, Chap. I] a nd then lifted to E . The closed preor dered prop erty do es not pa s s smo othly to the quotient, but in the co mpa ct case and in the k ω -space ca se, using theor em 2.7 (2.4) and theorem 1.1 we obtain. Corollary 4.3. If E is a close d pr e or der e d c omp act sp ac e then E / ∼ is a close d or der e d c omp act sp ac e. If E is a close d pr e or der e d k ω -sp ac e then E / ∼ is a close d or der e d k ω -sp ac e. Pr o of. The first statement is a tr iv ial consequence of theorem 2.4. As for the second sta temen t, by Theor . 1 .1 ( E / ∼ , T / ∼ ) is a k ω -space. Since E is a closed preorder ed k ω -space then it is normally preorde r ed from whic h it fo llows that E / ∼ is a no rmally or dered space and hence a clo s ed or dered s pace. R emark 4.4 . The first statement in the prev ious result is co n tained in [7, Lemma 1] but the pro of is incorr ect ag ain for the tricky Hausdo rff condition which they inadverten tly use in the quo tien t. I ndee d, they a rgument a s follows: they take a net ([ a α ] , [ b α ]) ∈ G ( . ) co n verging to ([ a ] , [ b ]) a nd prove that a subnet ( a β , b β ) ∈ G ( ≤ ) con verges to some pair ( a ′ , b ′ ) ∈ G ( ≤ ). Since π is con tinuous ([ a β ] , [ b β ]) conv erges to ([ a ′ ] , [ b ′ ]) ∈ G ( . ) (and a ls o to ([ a ] , [ b ])) but this does not mean that ([ a ′ ] , [ b ′ ]) = ([ a ] , [ b ]) as the uniqueness of the limit r equires the Hausdorff conditio n and this is ass ured only after it is pr ov ed tha t E / ∼ is a closed ordere d space. R emark 4.5 . In the Haus dorff case, the second statement in corolla ry 4.3 can be pr ov ed using the s trategy contained in [19]. If E is a Hausdorff k ω -space then E / ∼ is a Hausdorff k ω -space [1 9, Prop. 2.3b], then π × π is a quotient map [1 9, Prop. 2 .3a, 2.2] which implies since G ( ≤ ) = ( π × π ) − 1 ( G ( . )) that G ( . ) is close d. One can then work out the pro o f of theor em 2.7 in the or dered framework o f E / ∼ using remar k 4 .1. 5 The existence of c on tin uous utilities Let us write x < y if x ≤ y and y 6≤ x . A utility is a function f : E → R such that “ x ∼ y ⇒ f ( x ) = f ( y ) and x < y ⇒ f ( x ) < f ( y )”. W e say that the preorder admits a re pr esentation by a family of functions F if “ x ≤ y iff ∀ f ∈ F , f ( x ) ≤ f ( y )”. It is easy to prov e that the pr e order of a nor mally preor dered 11 space is repr e s en ted b y the family o f contin uo us iso to ne functions [29, Theor. 1]. It is interesting to inv estigate under which conditions a contin uo us utilit y or, more strong ly , a repres e n tation through co n tinuous utilities exists. This problem has b een tho r oughly inv estigated, esp ecially in the eco nomics literature. The simplest a pproach passes through the as sumption that the preorder e d space under consider ation is norma lly pre o rdered [2 5], although alternative stra tegies based on weak er hypo thesis have also b een investigated [1 3]. The representation of r elations through isotone and utility functions is still a n active field of resea rch [30, 3, 4]. The re ader m ust b e warned that some author s ca ll utility what we ca ll iso- tone function [14, 3, 9] and use the word r epr esentation fo r the existence of just one utility , altho ug h one utilit y do es not allow us to recov er the pr e order. This unfortunate circumstance comes from the fac t that in economics most ter minol- ogy was intro duced in connec tio n with the total pr eorder ca se, that is, befor e the impo rtance of the general preo rder case was reco gnized. Let us reca ll that a p erfe ctly normal ly pr e or der e d space is a s emiclosed pre- ordered spac e s uch that if A is a closed decr easing set a nd B is a clo sed increas ing set with A ∩ B = ∅ then there is a contin uous is otone function f : E → [0 , 1] such that A = f − 1 (0) and B = f − 1 (1). Clear ly , p erfectly normally preorder ed spaces a re no rmally preo rdered spa ces. A closed decreas ing (increa sing) s e t S is functional ly-pr e or der e d close d if there is a contin uous isotone function f : E → [0 , 1] such that S = f − 1 (0) (resp. S = f − 1 (1)). They can also be ca lle d decre asing (increas ing) zer o sets as do ne in [23]. A pair ( A, B ), A ∩ B = ∅ , is functional ly-pr e or der e d close d if there is a contin uous is otone function f : E → [0 , 1] such that A = f − 1 (0) and B = f − 1 (1). The next result is stated without pr o o f in [23]. Prop osition 5.1. If in the p air ( A, B ) , A ∩ B = ∅ , with A close d de cr e asing and B close d incr e asing, A is funct ional ly-pr e or der e d close d and B is fun ctional ly- pr e or der e d close d t hen ( A, B ) is functional ly-pr e or der e d close d. Pr o of. Let A = g − 1 (0) and B = h − 1 (1) with g , h : E → [0 , 1] contin uous isotone functions. Let α : ([0 , 1] × [0 , 1]) \ (0 , 1) → [0 , 1] b e a contin uous function which is isotone a ccording to the pro duct or der on the square . Let α b e also such that α − 1 (0) = { (0 , y ) : y ∈ [0 , 1 ) } and α − 1 (1) = { (( x, 1) : x ∈ (0 , 1] } . A p ossible example is α = 1+ y 2 x (1 − y ) / 2 ; a nother example is α = 1 / [1 + (1 − y ) /x ]. The function f = α ( g , h ) is iso to ne, contin uous, and satisfies f − 1 (0) = g − 1 (0) = A , f − 1 (1) = h − 1 (0) = B . The next result extends a known re s ult for top olog ical spa ces [32, Theor . 16.8]. Prop osition 5.2 . Every r e gularly pr e or der e d Lindel¨ of sp ac e is a normal ly pr e- or der e d sp ac e. Pr o of. Let A and B b e clos ed disjoin t sets which are r e s pectively decreasing and incr easing. Since A ∩ B = ∅ , by pr eorder r egularity for each x ∈ A there is an o pen dec r easing set U x ∋ x such that D ( U x ) ∩ B = ∅ . The c o llection { U x } 12 cov ers A and since the Lindel¨ of prop erty is hereditary with res pect to closed subspaces, there is a c o un table sub collec tio n { U i } with the s ame pro per t y . In the same wa y we find a co un table collection of op en increa sing sets { V i } w hich cov ers B and such that I ( V i ) ∩ A = ∅ . Let us define the sequence of op en decreasing sets W 1 = U 1 , W n +1 = U n +1 \ [ ∪ n i =1 I ( V i )], and the sequence of o pen increasing sets E 1 = V 1 \ D ( U 1 ), E n = V n \ [ ∪ n i =1 D ( U i )]. The open disjoint sets U ′ = ∪ ∞ n =1 W n and V ′ = ∪ ∞ n =1 E n are resp ectively de c reasing and increa sing a nd contain resp ectively A and B . Theorem 5.3. Every se c ond c ountable r e gu larly pr e or der e d s p ac e ( E , T , ≤ ) is a p erfe ct ly normal ly pr e or der e d sp ac e. Pr o of. As second co un tability implies the Lindel¨ of pro p erty , by pro pos ition 5.2 E is normally pre o rdered. Let A be a closed decrea sing set, we hav e only to pr ov e that it is functionally- preordere d clo s ed, the pro o f in the closed inc r easing ca se being analogo us. Suppo se A is op en then E \ A is op en, clos ed and increas ing. Setting for x ∈ A , f ( x ) = 0 and for x ∈ E \ A , f ( x ) = 1 we have finished. Therefore, we c a n assume that A is not op en and hence that A 6 = E . Let U b e a count able base of the top ology of E . Let C ⊂ U b e the subset whose elements, deno ted U i , i ≥ 1 , are such that A ∩ I ( U i ) = ∅ . F o r every U k ∈ C , denote with f k : E → [0 , 1] an isotone contin uous function which separates A a nd I ( U k ), that is, such that f − 1 k (1) ⊃ I ( U k ) and f − 1 k (0) ⊃ A (see [29, Theor. 1]). Let x ∈ E \ A . Applying the preor dered nor mality o f the s pace we can find ˆ U , op en decrea sing sets such that A ⊂ ˆ U ⊂ D ( ˆ U ) ⊂ E \ i ( x ). As E \ D ( ˆ U ) is op en there is some O ∈ U such tha t x ∈ O ⊂ E \ D ( ˆ U ). F urther more, since E \ ˆ U ⊃ O is closed increasing, I ( O ) ⊂ E \ ˆ U which implies A ∩ I ( O ) = ∅ and hence that there is some U k ∈ C such tha t O = U k . Thus there is a co n tinuous isotone function f k such that f − 1 k (1) ⊃ I ( U k ) and f − 1 k (0) ⊃ A . In other words we hav e proved that for each x ∈ E \ A there is some k such that f k ( x ) = 1 . Let us co nsider the function f = ∞ X k =1 1 2 k f k . This function is clearly iso tone, takes v alues in [0 , 1] and it is contin uous b ecause the series c onv e rges uniformly . Since for every k , f − 1 k (0) ⊃ A , the same is true for f . Moreov er, note that if x ∈ E \ A then there is some k ≥ 1 such that f k ( x ) > 0, which implies that f ( x ) > 0 hence f − 1 (0) = A . Lemma 5.4. If a top olo gic al pr e or der e d s p ac e admits a c ountable c ontinu ous isotone fun ct ion r epr esentation, namely if ther e ar e c ontinuous isotone funct ions g k : E → [0 , 1] , k ≥ 1 such that x ≤ y ⇔ ∀ k ≥ 1 , g k ( x ) ≤ g k ( y ) , then ther e is a c oun t able c ont inuous utility re pr esentation, namely ther e ar e c ontinuous utility functions f k : E → [0 , 1 ] , k ≥ 1 such that x ≤ y ⇔ ∀ k ≥ 1 , f k ( x ) ≤ f k ( y ) . 13 Pr o of. Let us consider the function g = ∞ X k =1 1 2 k g k . This function is clearly iso tone, takes v alues in [0 , 1] and it is contin uous b ecause the ser ie s conv erges unifor mly . F ur thermore, if x ≤ y a nd y  x , then g ( x ) < g ( y ) b ecause all the g k are isotone and there is some g ¯ k for which g ¯ k ( x ) < g ¯ k ( y ). Indeed, if it were not so then for every k , g k ( y ) ≤ g k ( x ) which implies y ≤ x , a contradiction. W e conclude that g is a contin uous utility function. Now, define for all k , n ≥ 1, ˜ g kn = (1 − 1 /n ) g k + g /n , so tha t ˜ g kn is a contin uous utility function. W e hav e only to prove tha t if x  y there are some k , n, such tha t ˜ g kn ( x ) > ˜ g kn ( y ) but w e know that there is a ˜ k such that g ˜ k ( x ) > g ˜ k ( y ) (otherwise for every k , g k ( x ) ≤ g k ( y ) which implies x ≤ y , a contradiction). T hus ˜ g ˜ kn ( x ) = (1 − 1 /n ) g ˜ k ( x ) + g ( x ) /n = ˜ g ˜ kn ( y ) + (1 − 1 /n )( g ˜ k ( x ) − g ˜ k ( y )) + ( g ( x ) − g ( y )) /n and thus the desired inequality holds for sufficiently la rge n . Theorem 5. 5. Every se c ond c ou n table r e gularly pr e or der e d sp ac e E admits a c ountable c ontinu ous u t ility r epr esentation, that is, ther e is a c ountable set { f k , k ≥ 1 } of c ontinuous utility functions f k : E → [0 , 1 ] such that x ≤ y ⇔ ∀ k ≥ 1 , f k ( x ) ≤ f k ( y ) . Pr o of. Let G be the s et of contin uous isoto ne functions and for every g ∈ G let G g = { ( x, y ) ∈ E × E : g ( x ) ≤ g ( y ) } . The set G g is closed b e cause of the contin uit y of g . W e alr eady know that the preo r dered s pace ( E , T , ≤ ) is a no rmally preorder ed space (Theor . 5.3), thus ≤ is r epresented by G (by [29, Theor. 1]), namely G ( ≤ ) = T g ∈G G g . As E is second countable, E × E is second countable and hence her editary Lindel¨ of. As a conse quence, the intersection of an arbitr ary family of clos ed sets can b e wr itten as the intersection of a countable subfamily G ′ ⊂ G . The desired conclusion follows now from lemma 5.4. 6 Conclusions Every Hausdor ff lo cally c o mpact space is a co mpletely regular spac e [3 2, Theor . 19.3] a nd hence a r egular spa ce, and every second countable r egular spa ce ( T 3 - space) is metr izable (Urysohn’s theorem). These results allow us to improve the separability properties o f the space. Unfortunately , they hav e no straightforward analogs in the g eneral preo rdered cas e where a quasi- uniformizable spac e need not be a regularly preor dered space [18, Exa mple 1 ]. One still would like to hav e some res ult which improves the preorder sepa r ability pro p erties of the space, g iv en suitable co mpactness o r co un tability conditions on the top o logy . W e hav e then follow ed a different path pr oving that close d preordered k ω -spaces are normally preorder ed. 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