Computational Models of Certain Hyperspaces of Quasi-metric Spaces

Logical Methods in Computer Science V ol. 7 (4:01) 2011, pp. 1–25 www .lmcs-online.org Submitted Oct. 16, 2010 Published Oct. 26, 2011 COMPUT A TIONAL MODELS OF CE R T AIN HYPERSP A CES OF QUASI-METRIC SP A C ES MAHDI ALI-AKBA RI AND MASS OUD POUR MAHDIAN Department of Mathematics, Semnan Unive rsit y , P .O. Bo x 35195-363, Semnan, Iran and School of Mathematics, I n stitute for Research in F undamental Sciences (IPM), T ehran, I ran. e-mail addr ess : m aliakbari@aut.ac.ir School of Mathematics and Computer Science, Amirk abir U niversit y of T echnolog y , T ehran, I ran and School of Mathematics, Institute for Research in F undamental Sciences (IPM), T ehran, I ran. e-mail addr ess : p ourmahd@ipm.ir Abstra ct. In this paper, for a giv en sequentia lly Y oneda-complete T 1 quasi-metric sp ace ( X, d ), the domain theoretic mo d els of the hyperspace K 0 ( X ) of nonempty compact subsets of ( X , d ) are studied. T o this end, the ω -Plo tkin domain of th e space of formal balls B X , denoted b y CB X is considered. This domain is given as the chain completion of the set of all ﬁnite subsets of B X with respect to the Egli-Milner relation. F urther, a map φ : K 0 ( X ) → CB X is established and p ro ved that it is an embedding whenever K 0 ( X ) is equ ipp ed with the Vietoris topology and res p ectively CB X with the S cott top ology . Moreo v er, if any compact su b set of ( X, d ) is d − 1 -precompact, φ is an embedding with respect to the topology of Hausdorﬀ quasi-metric H d on K 0 ( X ). Therefore, it is conclud ed that ( CB X , ⊑ , φ ) is an ω -computational mo del for the hyp erspace K 0 ( X ) en dow ed with the Vietoris and resp ectively the Hausdorﬀ top ology . Next, an algebraic sequentiall y Y oneda-complete qu asi-metric D on CB X is introduced in such a w a y that t he sp ecialization order ⊑ D is equiva len t to the usual partial order of CB X and, furthermore, φ : ( K 0 ( X ) , H d ) → ( CB X , D ) is an isometry . This shows that ( CB X , ⊑ , φ, D ) is a quantitativ e ω -comput ational mo del for ( K 0 ( X ) , H d ). Introduction In th is pap er, we further con tin ue a pr o je ct carried out to in v estiga te connections b et ween domain theory and qu asi-metric spaces [AHPR09]. Here, we p r o vide some d omain theoretic (computational) mo dels for the hypersp ace of n onempt y compact subsets of quasi-metric spaces. On one hand , the recent app lications of quasi-metric spaces in d iﬀeren t su b jects of computer science, e.g. denotational semanti cs of programming languages, complexit y and d ual-complexit y spaces and complexit y distances b et w een algorithms ([R V08, RS V09, GRS08, RSV03, RS99, R R V08]) and, on the other hand, a new insigh t of the domain 1998 ACM Subje ct Classiﬁc ation: F.1.1. Key wor ds and phr ases: Quasi-metric spaces, Y oneda and Smyth completeness, hyperspace of non-emp ty compact su b sets, ( ω -)computational mo dels, ω -Plotkin domain. The aut hors are partially supp orted by IPM, grants No. 89540064 and No. 890301 20. LOGICAL METHODS l IN COMPUTER SCIENCE DOI:10.216 8/LMCS-7 (4:01) 2011 c  M . Ali-Akbari and M. Pourmahdian CC  Cre ative Com mons 2 M. ALI-A KBARI AND M. POU RMAHDIAN theoretic p oint of view in to the theory of h yp erspaces and its new applicatio ns within mathematics, e.g. discrete dynamical systems, measur e and inte gration theory ([Eda95a, Eda97, Ed a95b]), m otiv ate establishin g compu tational mo dels of these structures. Finding a domain theoretic (computational) m o del for a top ological space ( X, τ ) amounts to pr o viding a suitable partially ordered set ( P , ⊑ ) together w ith a top ologica l em b eddin g φ from ( X , τ ) to ( P , ⊑ ) endo w ed with the Scott top ology , denoted b y σ . This is a v ariant of a fundamen tal pr oblem in domain theory , called the maximal p oin t sp ace problem, whic h demands a homeomorph ism b et w een ( X, τ ) and the space of maximal p oint of ( P , ⊑ ). Th e study of computational mo dels for v arious type of top ological spaces go es back to the works of Edalat and resp ective ly Blanc k [Eda95a, Bla00, Bla97]. Late r, the maximal p oin t space problem wa s explicitly form ulated and b ecame a sub ject of int ensiv e inv estigations b y many authors [La w 97, BL08, Mar98, Rut98]. Some sp ecial cases of this problem ha v e satisfactory solutions [AHP09, K KW04]. The d omain theoretic constr u ction B X of the space of f orm al b alls, introdu ced b y Edalat and Hec kmann, pr o vides a concrete (computational) mo del for a metric space ( X , d ) [EH98]. The imp ortance of th is construction is th at, ﬁr st of all, it connects some metric prop erties of ( X, d ) to the order theoretic prop erties of B X . S econdly , it ties th e ab ov e notion of computational mo del to the n otion of computabilit y for metric space ( X , d ) [ES99, La w 98]. Th e notion of form al balls is also deﬁned in the s ame wa y for a quasi-metric space ( X, d ) and the order theoretic prop erties of B X are tigh tly connected to the top ologica l prop erties of ( X , d ) [AHPR09, R V09, R V10]. In particular, for a T 1 quasi-metric s pace ( X, d ) is sequen tially Y oneda-complete if and on ly if B X is a dir ected complete partially ordered set. Edalat an d Hec kmann also constructed the P lotkin p o werdomain P B X of the sp ace of formal balls of a metric space ( X , d ) and sh o wed that there is a one-to-one corresp ondence b et w een the n onempt y compact s ubsets of ( X, d ) and th e maximal elements of the Plotkin p o w erdomain of B X . As an application, a domain th eoretic p ro of was giv en for a classical result of Hutc hinson ([Hut85]) wh ich states that if ( X , d ) is complete, then an y hyp erb olic iterated function system h as a u nique non-empty compact attractor. It can b e shown that this construction is a computational mo del for the hyper s pace of nonemp t y compact subsets of X , denoted by K 0 ( X ), with the Vietoris or equiv alen tly the Hausdorﬀ top ology . Th is fact w as also p ro v ed in a diﬀerent w a y b y Martin in [Mar04 ]. His interesting idea is b ased on the existence of a certain measur emen t, called Leb esgue measurement, on any domain D whic h mo dels the metric space ( X , d ). Subsequently , Liang and Kou in [LK04] generalized these results to con tin u ous dcp o’s whic h ha v e the La wson condition, i.e. the La wson and Scott top ologies coincide on the space of maximal p oints. Indeed, und er the L a w son condi- tion, it is p ro v ed that there is a h omeomorph ism b etw een th e space of nonempt y compact subsets of maximal p oin ts of a con tin uous dcp o D endo wed with the Vietoris top ology and the space of maximal p oin ts of Plotkin p o werdomain D equipp ed with the in d uced Scott top ology . More recen tly , in another line of research, Berger et al. ([Ber10]) show ed that for any T 1 top ological space which is repr esen ted by an ω -domain D , th e hyp erspace of its nonempt y compact subsets can b e repr esen ted by th e Plotkin p o w erd omain P D of domain D . T h is result was made p ossible b y a theorem of Sm yth ([Sm y83]) whic h states that f or an y ω -con tinuous dcp o D the space ( P D , σ ) is homeomorph ic to th e space of D -lenses, i.e. n onempt y compact subsets of d omain D whic h are intersectio n of a closed set and a saturated set, end o w ed w ith the Vietoris top ology . COMPUT A TIONAL MODELS OF CER T AIN HYPERSP ACES OF QU ASI-METRIC SP ACES 3 In the present w ork, we study computational mo dels of th e hyp erspace K 0 ( X ) of a T 1 quasi-metric sp ace ( X , d ) equipp ed with the Vietoris and resp ectiv ely the Hausdorﬀ top ology , pro ving th at when ev er ( X, d ) satisﬁes certain completeness prop erties, e.g. Y oneda and r esp ectiv ely Smyth completeness, this sp ace has a compu tational mo del. It is worth men tioning that the sp ace of formal balls of a qu asi-metric space do es not generally satisfy the La wson nor count able based conditions. Therefore, the r esults of Liang and Kou [LK04 ] and Berger et al. [Ber10] do n ot app ly to the present cont ext. Also, un lik e the metric case, there is no natural candid ate for a m easuremen t on the space of formal b alls of a quasi- metric space and hence the method used by Martin in [Mar04] cann ot b e app lied here either. Edalat and Hec k m ann used the Plotkin p o w er d omain P B X giv en as the ideal comple- tion of the abstract basis of ﬁnite subset of B X , P f in B X , with resp ect to the Egli-milner relation, ≺ E M , to pr esen t a computational mo d el of K 0 ( X ), for ev ery metric space ( X , d ). T o th is end, they emplo y ed the sym metry axiom of metric d , to get a k ey fact that an y maximal ideal has a coﬁnal ω -chain. In the case of quasi-metric spaces, the Plotkin p o w er- domain P B X can also b e deﬁned, though, the lac k of symm etry for the quasi-metric ( X, d ) prev en ts u s f rom ﬁ nding coﬁnal ω -chains in maximal id eals. That is why w e p refer to wo rk directly w ith the ω -c hains and this leads us to th e c h ain-completion construction instead. So, for a T 1 quasi-metric space ( X , d ), we consider the space B X of formal balls and let CB X b e the c hain completion of ( P f in B X, ≺ E M ). T his construction is called the ω -Plotkin domain. By the general construction of chain completion, C B X is a con tin uous ω -dcp o, i.e. a con tinuous p oset in whic h ev ery ω -chain has a least upp er b ound. Now, to ac h iev e our purp ose in ﬁnd ing a computational mo del, w e deﬁne a one-to-one map φ : K 0 ( X ) → CB X , whic h is an em b edding if w e consider the Vietoris top ology on K 0 ( X ) and assu me that ( X, d ) is a sequent ially Y oneda-co mplete T 1 quasi-metric space. Moreo v er, φ is an emb edding with resp ect to the top ology of the Hausdorﬀ quasi-met ric H d on K 0 ( X ) if an y compact sub set of X is d − 1 -precompact. Therefore, ( CB X , φ ) serv es as an ω -computational mo del for K 0 ( X ) endo w ed with the men tioned top ologies. Although it is not known w hether CB X is a dcp o and therefore a co mputational mod el of K 0 ( X ), nev ertheless, th an k s to F act 1.1 and T heorem 2.5, the ideal completion of C B X giv es a computational mo del for K 0 ( X ). In section 5, we tak e another w ell-kno wn notion of computational mo del, called the quan titativ e ω -computational mod el. Th is is an ω -computational model ( P , ⊑ , φ ) carrying an additional quasi-metric D such that φ is an isometry fr om ( X, d ) into ( P , D ) together with some extra conditions wh ic h capture the order structure of ( P , ⊑ ) (Deﬁnition 4.1). A mo diﬁed ve rsion of this notion can b e found in [R V09, Ru t98, Sc h03, W as06]. W e prov e that in fact CB X is a quantitat iv e ω -computational mo del for ( K 0 ( X ) , H d ), b y constructing a quasi-metric D on CB X . T o this end, we consider a qu asi-metric q deﬁned by Romaguera and V alero ([R V09]) on B X . T he primary reason to choose this quasi-metric on B X is that ( B X, q ) is a quantita tiv e computational mo del for ( X , d ). Therefore, its sp ecializ ation order ⊑ q is equiv alen t to the partial order of B X . C onsequent ly , one could n aturally extend q to the Hausdorﬀ qu asi-metric H q on P f in B X of the ﬁn ite subsets of B X , whose main prop erty is that it indu ces the Egli-Milner relation on P f in B X . Sub sequen tly , the quasi-metric H q can b e lifted up to a quasi-metric D on CB X in suc h a wa y that the ord er ed stru ctures ( CB X, ⊑ D ) and ( CB X, ⊑ ) coincide. O nce D is established one can show that ( CB X , D ) is a Y oneda-complete space and in fact Y oneda-completion of ( P f in B X, H q ). This mak es ( CB X, ⊑ , φ, D ) a qu an titati v e ω -computational m o del for ( K 0 ( X ) , H d ). 4 M. ALI-A KBARI AND M. POU RMAHDIAN W e, ﬁ nally , conclud e this p ap er by comparing th e Plotkin p o w er d omain and th e ω - Plotkin domain constru ctions. W e prov e th at if ( X, d ) is either, Smyth-complete and all of its compact subsets are d − 1 -precompact, or an ω -algebraic Y oneda-co mplete space, then the Plotkin P o werdomain P B X is order -isomorp h ic to C B X . 1. Preliminaries W e assume the reader is familiar with th e b asic d eﬁnitions and facts a b out domain theory whic h can b e f ound in ([AJ94, GHK + 03]), though, w e brieﬂy explain some of the deﬁnitions and f acts wh ic h are more crucial in this note. Let ( P , ⊑ ) b e a p artially ordered set (abbr. b y p oset). The b in ary relation ≺ is called an auxiliary relation on the p oset ( P , ⊑ ) if (1) p ≺ p imp lies p ⊑ p , (2) p ⊑ s ≺ r ⊑ q implies p ≺ q and (3) satisﬁes the in terp olation pr op erty , i.e. for any ﬁ nite sub set M of P and p ∈ P , if for ev ery m ∈ M , m ≺ p then there exists some q ∈ P suc h that m ≺ q ≺ p , for ev ery m ∈ M . The pair ( P , ≺ ) is called an abstr act b asis , if ≺ is a transitiv e relation whic h also satisﬁes the inte rp olation pr op ert y . A nonempt y dir ected lo w er s u bset I of P is called a r ound i de al if for an y x ∈ I there is y ∈ I suc h that x ≺ y . The set of all round ideals of P partially ordered b y ⊆ is called the ide al c omp letion of P , denoted b y I dl ( P ). Let ↑ ↑ p = { q : p ≺ q } and ↓ ↓ p = { q : q ≺ p } . An auxiliary relation is called appr oxima ting if ↓ ↓ p ⊆ ↓ ↓ q implies p ⊑ q . On e can see that the set {↑ ↑ p : p ∈ P } forms a basis for a top ology called the pseudoSc ott top olo gy on P , denoted by P σ . The follo wing f act is needed f or the pro of of Th eorem 3.12. F act 1.1. Let ( P , ⊑ ) b e a p oset with an au x iliary r elation ≺ . Then (1) ( I dl ( P ) , ⊆ ) is a con tinuous dcp o. (2) If ≺ is appr oximati ng on P , then the map j : P → I dl ( P ) deﬁned b y j ( p ) = ↓ ↓ p is an em b edding of ( P , P σ ) into ( I dl ( P ) , σ ) where σ denotes the Scott top ology . (3) If ≺ is approxima ting and all ≺ -directed sets of P ha v e upp er b ound s, then j (max P ) = max I dl ( P ). Pr o of. See [KKW04], Theorem 2.3. Belo w, we ﬁx the key notion of a compu tational mo del for a T 0 top ological space. Before that, recall an y T 0 top ology τ on a space X ind uces a p artial order ⊑ τ , called the sp ecialization order, whic h is deﬁn ed by x ⊑ τ y ⇔ x ∈ cl τ y , for all x, y ∈ X . cl τ y stands for the closure of y w ith r esp ect to τ . Also, a partially ordered set ( P , ⊑ ) is an ω -dcp o if eve ry ⊑ -ascendin g sequence has a least up p er b ound (see [Kn i91]). Deﬁnition 1.2. A triple ( P , ⊑ , φ ) is a ( ω -)c omp utational mo del for ( X , τ ) whenev er (1) ( P , ⊑ ) is a con tin u ous ( ω -)dcp o. (2) φ is a top ological em b edding f r om ( X , τ ) into ( P , ⊑ ) end o wed w ith the Scott top ology . (3) φ ( M ax ( X, ⊑ τ )) = M ax ( P, ⊑ ). Blanc k in [Bla00 ] consid ered th is d eﬁnition as a domain r epresen tation for ( X , τ ) w ith - out men tioning the third co ndition. If we restrict ourselves to T 1 top ological spaces, then the ab o v e deﬁn ition coincides with the usual d eﬁnition of compu tational m o del in wh ic h φ deﬁnes a homeomorphism from ( X, τ ) on to the sp ace of maximal elemen ts of ( P , ⊑ ) [KKW04, MMR02]. COMPUT A TIONAL MODELS OF CER T AIN HYPERSP ACES OF QU ASI-METRIC SP ACES 5 Next, we deﬁne th e notion of a quasi-metric s pace. F or m ore details the reader ma y consult the references [FL82, Kel63, K ¨ un95, K¨ u 02, KS02]. A quasi-metric d on a set X is a f u nction d : X × X → [0 , ∞ ) suc h that for an y x , y , z ∈ X : (1) x = y iﬀ d ( x, y ) = d ( y , x ) = 0, (2) d ( x, z ) ≤ d ( x, y ) + d ( y , z ). If we drop the if part of condition (1), d is called a quasi-pseudom etric . The pair ( X , d ) is called quasi-(pseudo)metric space. Eac h quasi-metric d on the set X ind uces a T 0 top ology on X , denoted by τ d , w hose b ase is the set of all balls of the form N ǫ ( x ) = { y ∈ X : d ( x, y ) < ǫ } , for any x ∈ X and ǫ > 0. The topology τ d is T 1 if and only if the condition (1) can b e replaced b y: x = y ⇔ d ( x, y ) = 0. T h e qu asi-metric d generates another quasi- metric d − 1 on the set X , called the c onjugate of d , d eﬁned by d − 1 ( x, y ) = d ( y , x ). Also, the function d ∗ can b e deﬁned on X × X by d ∗ ( x, y ) = max { d ( x, y ) , d − 1 ( x, y ) } wh ic h is a metric on X . The qu asi-metric space ( X , d ) is p oint symmetric if τ d ⊆ τ d − 1 . F or example, any compact T 1 quasi-metric space ( X, d ) is p oin t sym metric ([W es57 ], Lemma 2). A sequence ( x n ) n> 0 is calle d Cauchy ( biCauchy ) sequence if for ev ery ǫ > 0 there is N > 0 such th at d ( x n , x m ) < ǫ wheneve r m ≥ n ≥ N ( m, n ≥ N ). An elemen t x ∈ X is called a Y one da limit of the sequence ( x n ) n> 0 , if for any y ∈ X , d ( x, y ) = inf n sup m ≥ n d ( x m , y ) . The quasi-metric space ( X , d ) is se quential ly Y one da-c omplete if ev ery Cauc h y sequence has a Y oneda limit. It is easy to see that the Y oneda limit is uniqu e if it exists. A p oin t e ∈ X is called ﬁnite if for any Cauch y sequ ence ( x n ) n> 0 in X with the Y oneda limit x , d ( e, x ) = sup n inf m ≥ n d ( e, x m ) . The quasi-metric space ( X , d ) is called algebr aic if eac h elemen t of X is the Y oneda limit of a C auc h y sequence of ﬁnite el emen ts. The quasi-metric space ( X , d ) is Smyth-c omplete if an y Cauch y sequ ence ( x n ) n> 0 con verges strongly in X , i.e. there is a p oin t x ∈ X suc h that ( x n ) n> 0 con verges to x in the top ology of the m etric d ∗ . Finally , we review some basic deﬁnitions from the hypers pace theory [CR06, RR02]. Let ( X, d ) b e a b ound ed quasi-metric space and K 0 ( X ) denote the set of all nonempty compact subsets of X . T h e upp er Hausdorﬀ quasi-pseudometric H + d and the lo w er Hausdorﬀ qu asi- pseudometric H − d on K 0 ( X ) are d eﬁ ned as follo ws: H + d ( A, B ) = sup b ∈ B d ( A, b ) , H − d ( A, B ) = sup a ∈ A d ( a, B ) for all A, B ∈ K 0 ( X ), wh ere d ( A, x ) = in f a ∈ A d ( a, x ) and d ( x, A ) = inf a ∈ A d ( x, a ). Th e Hausdorﬀ qu asi-pseudometric H d is deﬁn ed as H + d ∨ H − d or equiv alen tly H d ( A, B ) = max { sup b ∈ B d ( A, b ) , su p a ∈ A d ( a, B ) } for all A, B ∈ K 0 ( X ). It is kno w n that H + d , H − d and H d are quasi-pseudometrics on K 0 ( X ). F or a T 1 quasi-metric space ( X , d ), H d is a quasi-metric. F urthermore, for an y A, B ∈ K 0 ( X ), H d ( A, B ) = 0 if an d only if B ⊆ A ⊆ cl τ d B . ( ∗ ) In [AP10], the authors present an example wh ic h shows that ( K 0 ( X ) , H d ) ma y not b e a T 1 space, even though ( X, d ) is a T 1 quasi-metric space. Ho w ev er, one can infer from ( ∗ ) 6 M. ALI-A KBARI AND M. POU RMAHDIAN that ( K 0 ( X ) , H d ) is T 1 if ( X , d ) is Hausd orﬀ (more generally K C-space in whic h all compact subsets are closed). Recall th at a subset K of a quasi-metric space ( X , d ) is d - precompact if for an y ǫ > 0, there is a ﬁnite su bset F of K suc h that for an y k ∈ K , d ( x, k ) < ǫ , f or some x ∈ F . Unlik e the metric spaces, a compact subset of a qu asi-metric space ( X, d ) is not necessarily d − 1 - precompact. The follo wing theorem sho ws that if we imp ose this extra condition to ( X , d ), then th e Smyth-complete ness of ( X , d ) can b e lifted up to ( K 0 ( X ) , H d ). This th eorem is used in section 5, Lemma 4.14. Theorem 1.3. L et ( X , d ) b e a Smyth-c omplete quasi-metric sp ac e su c h that any c omp act subset of X is d − 1 -pr e c omp act. Then ( K 0 ( X ) , H d ) i s Smyth-c ompl ete. Pr o of. See [AP10], Theorem 3.7. There are other top ologies on the hyper s pace K 0 ( X ). The most famous of these top olo- gies is the Vietoris top ology τ V whic h is the supremum of lo w er Viet oris topology and upp er Vietoris top ology . Th e low er Vietoris top ology τ − V is generated by all sets of the form ♦ V = { K ∈ K 0 ( X ) : K ∩ V 6 = ∅} whereas the u pp er Vietoris top ology τ + V is generated by all sets of the form  V = { K ∈ K 0 ( X ) : K ⊆ V } for op en V . In general, this top ology is coarser than the top ology of Hausdorﬀ qu asi-pseu dometric H d on the hyp erspace K 0 ( X ). Ho wev er, wh enev er an y compact subset of X is d − 1 -precompact, these top ologies co incide [RR02]. 2. The sp ace of formal balls and its ω -Plotkin do main The space of formal b alls of a metric sp ace ( X , d ), denoted by B X , was deﬁned by Edalat and Heckma nn in [EH98]. T his constru ction giv es a concrete compu tational mo del for metric spaces in which the order -theoretic p rop erties of B X are closely connected with the metric p rop erties of ( X , d ). In [Rut98], Rutten was probably the ﬁ r st who stud ied the space of formal balls f or quasi- metric spaces via co-Y oneda em b edding. More recen tly , Ali-Akbari et al. in [AHPR09] and Romaguera and V alero in [R V09, R V10] studied the set of formal balls for quasi-metric spaces in the spirit of Edalat and Heckmann’s work. Deﬁnition 2.1. F or a quasi-metric space ( X, d ), the s p ace of formal b al ls is the pair ( B X, ⊑ ) where B X = { ( x, r ) : x ∈ X and r ≥ 0 } , and ( x, r ) ⊑ ( y , s ) if and only if d ( x, y ) ≤ r − s, for an y ( x, r ) , ( y , s ) ∈ B X . It is easy to see that ( B X, ⊑ ) is a p oset. An elemen t ( x, r ) of B X is called a f ormal ball. One can d eﬁ ne an auxiliary relation ≺ on B X as f ollo ws: ( x, r ) ≺ ( y , s ) if and only if d ( x, y ) < r − s. It can b e sh o w n that the r elation ≺ satisﬁes the int erp olation prop ert y and therefore ( B X, ≺ ) forms an abstract b asis. The f ollo wing theorem sh ows some interesting pr op erties of the p oset of formal balls. Theorem 2.2. L et ( X, d ) b e a quasi-metric sp ac e . (1) The function ι : ( X, τ d ) → ( B X , P σ ) deﬁne d by ι ( x ) = ( x, 0) is an emb e dding. COMPUT A TIONAL MODELS OF CER T AIN HYPERSP ACES OF QU ASI-METRIC SP ACES 7 (2) If ( X, d ) is T 1 and se quential ly Y one da-c omplete, then ( B X, ⊑ ) is a dcp o. In addition, if ( X , d ) is also algebr aic, ( B X, ⊑ ) is a doma in (c ontinuous dcp o). (3) If ( X, d ) is Smyth-c omplete, then ( B X , ⊑ ) is a domain. M or e over, the auxiliary r elation ≺ c oincides with the way-b elow r elation of B X . Pr o of. See Th eorem 3.12, Corollary 3.13 an d Theorem 3.17 in [AHPR09]. In the lig h t of the ﬁ rst part of the ab ov e Theorem, to ease our notatio ns, we iden tify the set X with the set ι ( X ) = { ( x, 0) : x ∈ X } . I n p articular, any compact subset of X is iden tiﬁed with a compact subs et of th e set ι ( X ). No w, we review the deﬁnition of c hain completion of an abstract basis ( P , ≺ ). The set of all ω -c hains, i.e. ≺ -ascending sequences, of P is denoted by C P . Deﬁnition 2.3. F or t w o ω -c hains ( x n ) n> 0 and ( y m ) m> 0 in C P deﬁne ( x n ) n> 0 ⊑ ( y m ) m> 0 ⇔ ∀ n ∃ m x n ≺ y m , ( x n ) n> 0 ∼ ( y m ) m> 0 ⇔ ( x n ) n> 0 ⊑ ( y m ) m> 0 & ( y m ) m> 0 ⊑ ( x n ) n> 0 . The chain c ompletion of the abstract basis ( P , ≺ ) is deﬁned to b e th e partially ord ered set ( C P , ⊑ ) where C P = C P / ∼ and [( x n ) n> 0 ] ⊑ [( y m ) m> 0 ] ⇔ ( x n ) n> 0 ⊑ ( y m ) m> 0 , for an y [( x n ) n> 0 ] and [( y m ) m> 0 ] in C P . It is a well- kno wn fact th at ( C P , ⊑ ) is a con tin uous ω -dcp o [Kni91]. T he wa y-b elo w relatio n is giv en by: [( x n ) n> 0 ] ≪ [( y m ) m> 0 ] ⇔ ∃ m ∀ n x n ≺ y m . ( x n ) n> 0 is called a r epr esentation of [( x n ) n> 0 ] ∈ C P . By abuse of notation, for an y equiv a- lence class I of C P , we wr ite x ∈ I if x is an ele men t of one of the sequ ences repr esenting I . Deﬁnition 2.4. F or subs ets A and B of the abstract b asis ( P , ≺ ), deﬁne (1) A ≺ U B ⇔ ∀ b ∈ B ∃ a ∈ A a ≺ b , (2) A ≺ L B ⇔ ∀ a ∈ A ∃ b ∈ B a ≺ b , (3) A ≺ E M B if an d only if A ≺ U B and A ≺ L B . The relatio n ≺ E M stands for Egli- Milner relation. Let P f in P b e the set of all non-empty ﬁnite subsets of P . It is ea sy to see that ( P f in P , ≺ E M ) is an abs tr act basis. Since there is no danger of confusion, for b revit y , we drop the subs cript E M . The c hain completion of ( P f in P , ≺ ) is called ω - P lotkin domain of P whic h is denoted by C P . In particular, for a qu asi-metric space ( X , d ), we consider the ω -Plotkin d omain CB X of the abstract basis ( B X , ≺ ). F or F ∈ P f in B X and I ∈ CB X , deﬁne r F = max { r : ( x, r ) ∈ F } and r I = inf { r F : F b elongs to a representati on of I } . Note that I ⊑ J implies that r I ≥ r J . Also from F ≺ G , rF > r G follo ws. Therefore, if I ≪ J then r I > r J . Belo w, an imp ortant prop ert y of this s tr ucture is highlighte d. Theorem 2.5. L et ( X , d ) b e a qu asi-metric sp ac e. Then ω -Plotkin domain ( CB X , ⊑ ) is a c ontinuous ω -dcp o and mor e over, any ≪ -dir e cte d subset of it has a le ast upp er b ound. 8 M. ALI-A KBARI AND M. POU RMAHDIAN Pr o of. As w e men tioned earlier, the ﬁrst part is kno wn . F or the second p art, le t  b e the partial ord er relation generated by ≪ , i.e. I  J if and only if I = J or I ≪ J , for every I , J ∈ C B X . W e ﬁrst ve rify that ( CB X ,  ) is a d cp o. By a w ell-kno wn fact from [Mark78] it s uﬃces to examine that ev ery  -c hain of C B X has a least u pp er b ound . Let A = ( I α ) α ∈ I b e a  -c h ain in CB X . Without loss of generalit y , we ma y assume that A has no maxim um elemen t. T hen ( r I α ) α ∈ I is a strictly d ecreasing chain in the set of nonnegativ e real n um b ers and therefore has an inﬁm um, sa y r . Fix I 1 and inductive ly for any n ≥ 2, c h o ose I n − 1 ≪ I n suc h that r < r I n < r + 1 n . W e cl aim that ( I n ) n> 0 is a coﬁn al subs equ ence of A in CB X . Let I α b e an arbitrary elemen t of A . Then c ho ose I n in su c h a wa y that r I n < r I α . Now since any t w o elemen ts of A are co mparable and r I n < r I α , it follo ws that I α ≪ I n . But ( CB X, ⊑ ) is an ω -dcp o. Ther efore, ( I n ) n> 0 has the least upp er b ound I in ( CB X , ⊑ ), whic h is also the least upp er b oun d for A . No w, let D b e a ≪ -directed subset of CB X . Then it is easy to see that D is also  -directed. Th erefore, it has the  -least u pp er b oun d I . It is, then, straigh tforw ard to sho w that I is also the ≪ -least upp er b ound of D . Although we are not able to show that CB X is a dcp o, the ab ov e Theorem give s a crucial feature of C B X w hic h will help us in obtaining Theorem 3.12. W e introd u ce the follo wing abbr eviations whic h will b e useful for a num b er of later pro ofs. Note 2.6. Let F , G ∈ P f in B X and ǫ > 0 b e given. (1) F + ǫ = { ( x, r + ǫ ) : ( x, r ) ∈ F } . (2) If F ≺ G , then put δ ( F , G ) = min { ( r − s ) − d ( x, y ) : ( x, r ) ∈ F , ( y , s ) ∈ G , ( x, r ) ≺ ( y , s ) } . Remark 2.7. The follo wing pr op erties of the ab ov e n otations are straigh tforw ard. (1) F or 0 < ǫ ′ < ǫ , F + ǫ ≺ F + ǫ ′ ≺ F . (2) F or any ǫ < δ ( F , G ), ( x, r ) ∈ F and ( y , s ) ∈ G , with ( x, r ) ≺ ( y , s ), we ha v e ( x, r ) ≺ ( y , s + ǫ ). Hence F ≺ G + ǫ . 3. Embedding of K 0 ( X ) into CB X In this section, w e app ly the tec hniques used by Edalat and Hec kmann in [EH98] to the ω -Plotkin domain of the sp ace of formal balls, leading us to ﬁnd a computational mo del for the space K 0 ( X ) of the n onempt y compact subsets of a quasi-metric space ( X, d ). More precisely , for a sequenti ally Y oneda-complete T 1 quasi-metric s pace ( X , d ), we construct the ω -Plotkin domain of the abstract basis ( B X , ≺ ), as introdu ced in Deﬁnition 2.1, and sho w that the hyp erspace K 0 ( X ) equ ipp ed with the Vietoris top ology τ V can b e emb edded in CB X equipp ed with the S cott top ology . Moreo v er, this embed d ing ser ves as an ω - computational mo del of ( K 0 ( X ) , τ V ). F rom now on, w e assume that ( X , d ) is a sequen tially Y oneda-complete T 1 b ound ed quasi-metric sp ace. COMPUT A TIONAL MODELS OF CER T AIN HYPERSP ACES OF QU ASI-METRIC SP ACES 9 Deﬁnition 3.1. Let K b e a n onempt y compact subset of X . Since K is compact, it is d -precompact. So, for any n > 0, one can choose x n 1 , . . . , x n m n of K suc h that for any x ∈ K , d ( x n i , x ) < 1 2 n 2 for some x n i . Pu t F n = { ( x 1 1 , 1 n ) , . . . , ( x 1 m 1 , 1 n ) , . . . , ( x n 1 , 1 n ) , . . . , ( x n m n , 1 n ) } . It can b e easily c hec k ed th at ( F n ) n> 0 is an ω -c h ain in ( P f in ( B X ) , ≺ ). Call ( F n ) n> 0 a standar d rep r esen tatio n of K . Although a compact subset K of X migh t h a ve s everal stand ard r epresen tations, we sho w th at all standard represen tatio ns of K are ∼ -equiv alen t. The follo wing auxiliary lemmas w ill b e usefu l in sev eral pro ofs. Lemma 3.2. F or any x ∈ K and any standar d r epr esentatio n ( F n ) n> 0 of K , ther e is a se quenc e (( x n , r n )) n> 0 such th at ( x n , r n ) ∈ F n and d ∗ ( x n , x ) → 0 . Pr o of. First note that for any n > 0, F n ≺ K . Hence for any n > 0, there is ( x n , r n ) ∈ F n suc h that ( x n , r n ) ≺ ( x, 0). As r F n → 0, it implies d ( x n , x ) → 0. Now, since K is compact and th erefore d is p oin t symmetric on K , it follo ws d ( x, x n ) → 0. Therefore d ∗ ( x n , x ) → 0. Let I b e an element of C B X . W e can obtain a n onempt y compact satur ated su bset of B X , d enoted b y I + , as I + = \ n> 0 ↑ F n , where ( F n ) n> 0 is a r epresen tation of I . In fact, since I + is a ﬁltered int ersection of nonempty compact saturated su bsets of ( B X , P σ ) and B X is a dcp o, similar to Theorem 7.2.27 in [AJ94], it can b e prov ed that the pseudoScott top ology P σ is sob er . So I + is a nonempty compact sat urated subset of B X . Also, I + is indep endent of the c hoice of its repr esen ta- tions. On e can easily sho w that for any represent ations ( F n ) n> 0 and ( E m ) m> 0 of I , \ n ↑ F n = \ m ↑ E m . Lemma 3.3. L e t I b e in CB X with r I = 0 . Then f or any standar d r epr esentation ( F n ) n> 0 of ι − 1 ( I + ) and any G, H ∈ P f in B X with G ≺ H ≺ I + , G ⊑ F n for some n > 0 . Pr o of. Let ( y , s ) ∈ G . Since G ≺ I + , there is ( x, 0) ∈ I + suc h that ( y , s ) ≺ ( x, 0). No w as ( x, 0) ∈ T n> 0 ↑ F n , by L emm a 3.2, there is a sequence (( x n , r n )) n> 0 suc h that ( x n , r n ) ∈ F n and d ∗ ( x n , x ) → 0. F or ǫ = s − d ( y , x ), take n > 0 su c h th at d ( x, x n ) < ǫ/ 2 and r n < ǫ/ 2. One can readily see that ( y , s ) ≺ ( x n , r n ). So, for any ( y , s ) ∈ G , th ere is F n and ( x n , r n ) ∈ F n suc h that ( y , s ) ≺ ( x n , r n ). Assume that F in ( F n ) n> 0 is an upp er b ound for all F n , arisen in this w a y . Now, G ⊑ L F is straigh tforw ard. Next, put δ = δ ( G, H ), as deﬁned in Notatio n 2.6. Cho ose F n suc h that r F n < δ and F ≺ F n . T ak e ( x, r ) ∈ F n . Then ( x, 0) ∈ I + . Ther efore, there exists ( z , t ) ∈ H su c h that ( z , t ) ≺ ( x, 0). Also, G ≺ U H implies that there exists ( y , s ) ∈ G suc h that ( y , s ) ≺ ( z , t ). It follo ws that d ( y , x ) ≤ d ( y , z ) + d ( z , x ) ≤ ( s − t ) − δ + t < s − r . Hence G ≺ U F n . No w from G ⊑ L F and F ≺ F n , it follo ws that G ⊑ L F n and consequen tly G ⊑ F n . 10 M. ALI-A KBARI AND M. POU RMAHDIAN Prop osition 3.4. F or any K ∈ K 0 ( X ) , al l standar d r epr esentations of K ar e ∼ -e qui v alent. Pr o of. Let ( F n ) n> 0 and ( E m ) m> 0 b e t w o stand ard represent ations of K . F or any m > 0, since E m ≺ E m +1 ≺ K , it follo ws from L emm a 3.3 th at ther e is n > 0 su ch that E m ≺ F n . Therefore ( E m ) m> 0 ⊑ ( F n ) n> 0 . Similarly , it can b e pro v ed that ( F n ) n> 0 ⊑ ( E m ) m> 0 and therefore ( E m ) m> 0 and ( F n ) n> 0 are in an equiv alence class. In th e ligh t of th e ab o v e prop osition, the follo win g deﬁnition is established. Deﬁnition 3.5. Denote the equiv alence class of a stand ard r epresent ation ( F n ) n> 0 of a nonempt y compact su bset K by K ∗ and let φ : K 0 ( X ) → CB X b e φ ( K ) = K ∗ . W e pr ov e some prop erties of this map . Prop osition 3.6. L et K, L ∈ K 0 ( X ) and I ∈ CB X with r I = 0 . Then (1) K = ( K ∗ ) + . (2) F or any r epr esentation ( F n ) n> 0 of I ; I + = { G n a n : a n ∈ F n , ( a n ) n> 0 is an asc ending se quenc e } . (3) K ∗ ⊑ L ∗ implies L ⊆ K ⊆ cl τ d L , wher e cl τ d L is the closur e of L in τ d . Pr o of. (1) It is r outine to c hec k that K ⊆ ( K ∗ ) + . F or the opp osite inclusion, sup p ose that there is ( x, 0) ∈ ( K ∗ ) + \ K . F or any ( y , 0) ∈ K , put s y = 1 2 d ( y , x ). Usin g compactness of K , choose a ﬁnite s ubset G 0 of { ( y , 1 2 s y ) : ( y , 0) ∈ K } suc h that G 0 ≺ K . Select F ∈ K ∗ with r F < m in { 1 2 s y : ( y, 1 2 s y ) ∈ G 0 } . W e claim that G ≺ U F , where G = { ( y , s y ) : ( y , 1 2 s y ) ∈ G 0 } . T o pro v e the claim, let ( z , t ) ∈ F . So ( z , 0) ∈ K and therefore there is ( y , 1 2 s y ) ∈ G 0 suc h that ( y , 1 2 s y ) ≺ ( z , 0). Since t < 1 2 s y , it follo ws ( y , s y ) ≺ ( z , t ) and consequen tly G ≺ U F . No w, by d eﬁnition of ( K ∗ ) + , it is clear that F ≺ U ( K ∗ ) + . So there is ( a, u ) ∈ F with ( a, u ) ≺ ( x, 0). Since G ≺ U F , there is ( y , s y ) in G such that ( y , s y ) ≺ ( a, u ). Hence ( y , s y ) ≺ ( x, 0) or equiv alen tly d ( y , x ) < s y , which is a con tradiction. (2) Clearly the su premum of any ascending sequ ence ( a n ) n> 0 , where a n ∈ F n , b elongs to I + . Let ( a, 0) ∈ I + . Pu t G n = { ( x, r ) ∈ F n : ( x, r ) ≺ ( a, 0) } . F or any ( x, r ) a nd ( y , s ) in S n G n , deﬁn e ( x, r ) R ( y , s ) if and only if for some n > 0, ( x, r ) ∈ G n , ( y , s ) ∈ G n +1 and ( x, r ) ≺ ( y , s ). T he bin ary relation R deﬁn es a lo cally ﬁnite directed grap h on the inﬁ nite set G = S n G n with at most | F 1 | -connected comp onents. So G has an inﬁn ite connected comp onent and therefore, in the ligh t of K¨ onig’s Lemm a, there is an ascend in g sequence (( x n , r n )) n> 0 suc h that F n ( x n , r n ) = ( b, 0) ⊑ ( a, 0). Sin ce d is T 1 , b = a follo w s and the pro of is complete. (3) The assumption implies ( L ∗ ) + ⊆ ( K ∗ ) + . So in the ligh t of the ﬁr st part, L ⊆ K . F or K ⊆ cl τ d L , let x ∈ K . Sup p ose that ( F n ) n> 0 and ( G n ) n> 0 are standard representat ions of K and L , resp ectiv ely . By the second part, there is an ascendin g sequence (( x n , r n )) n> 0 , ( x n , r n ) ∈ F n , suc h that F n ( x n , r n ) = ( x, 0). So d ( x n , x ) → 0. Since K is compact and therefore d is p oin t symmetric on K , d ( x, x n ) → 0. F rom K ∗ ⊑ L ∗ , it follo ws that for any F n , there is G m n and ( y m n , s m n ) ∈ G m n suc h that F n ⊑ G m n and ( x n , r n ) ≺ ( y m n , s m n ). Th us d ( x, y m n ) → 0, whic h means x ∈ cl τ d L . COMPUT A TIONAL MODELS OF CER T AIN HYPERSP ACES OF QU ASI-METRIC SP ACES 11 F rom the ﬁ rst part of th e ab o v e prop osition, it follo ws that the map φ : K 0 ( X ) → CB X is one-to -one. Moreo v er, in the follo wing we p ro v e that the map φ give s a one-to- one corresp onden ce b et w een th e maximal element of K 0 ( X ) with resp ect to the sp ecialization order ⊑ H d and the maximal elemen t of the partially ordered s et C B X . Before pro ving th is, w e need the follo wing lemma. Lemma 3.7. F or any maximal element I of CB X , I = I ∗ wher e I = ι − 1 ( I + ) . Pr o of. Note that I + is a nonemp t y compact subset of B X . Since ι : ( X , τ d ) → ( B X , P σ ) is an embedd ing (Theorem 2.2), I = ι − 1 ( I + ) is a compact set in ( X, τ d ). Th us I ∗ is w ell-deﬁned. No w, b y maximalit y of I , it suﬃces to pr o v e that I ⊑ I ∗ . Let ( F n ) n> 0 b e a r epresen- tations of I and ( G m ) m> 0 b e a standard representat ion of I . T ake F n in ( F n ) n> 0 . Since F n ≺ F n +1 ≺ I + , according to Lemma 3.3, F n ⊑ G m for some m > 0. This shows I ⊑ I ∗ . Prop osition 3.8. F or any ma ximal element K of K 0 ( X ) with r esp e ct to the sp e cialization or der ⊑ H d , K ∗ is maximal in ( CB X, ⊑ ) . Conversely, any maximal element I of CB X is of the form K ∗ for some maximal e lement K of ( K 0 ( X ) , ⊑ H d ) . Pr o of. Let K b e a maximal elemen t of ( K 0 ( X ) , ⊑ H d ) and K ∗ ⊑ I . Without loss of gener- alit y , w e assume that I is maximal. So I = I ∗ , where I = ι − 1 ( I + ). W e h a ve K ∗ ⊑ I ∗ and therefore b y the thir d p art of Prop osition 3.6, I ⊆ K ⊆ cl τ d I . Hence H d ( K, I ) = 0. By maximalit y of K , we conclude that K = I and K ∗ = I . No w, let I b e a maximal elemen t of CB X . Lemma 3.7 implies that for an y maximal elemen t I of C B X , I = I ∗ , where I = ι − 1 ( I + ) is a non emp t y compact subset of ( X , d ). T o complete the pro of, we ha v e to sho w that I is maximal in ( K 0 ( X ) , ⊑ H d ). Let I ⊑ H d J for some J ∈ K 0 ( X ). Thus H d ( I , J ) = 0 and consequently J ⊆ I ⊆ cl τ d J . W e s h o w that I ∗ ⊑ J ∗ . Su pp ose ( F n ) n> 0 and ( G m ) m> 0 are s tand ard representa tions of I and J , resp ectiv ely . Sin ce J ⊆ I ⊆ cl τ d J , it follo ws that F n ≺ F n +1 ≺ J , for any n > 0. By Lemma 3.3, F n ⊑ G m for some m > 0. So I ∗ ⊑ J ∗ and by maximalit y of I ∗ it follo ws that I = J . Therefore I is m aximal in ( K 0 ( X ) , ⊑ H d ). Belo w we examine diﬀeren t top ologies on K 0 ( X ) for which the m ap φ b ecomes an em b edding. Theorem 3.9. The map φ is an emb e dding fr om the hyp ersp ac e K 0 ( X ) e quipp e d with the Vietoris to p olo gy into C B X with the Sc ott top olo gy. Pr o of. Let ⇑ I = {J ∈ CB X : I ≪ J } b e a basic op en s et of CB X in the Scott top ology and K ∈ φ − 1 ( ⇑ I ). So K ∗ ∈ ⇑ I or equiv alen tly I ≪ K ∗ . Let ( F n ) n> 0 b e a standard represen tation of K ∗ . By deﬁnition of the w a y-b elow r elation, there is N > 0 suc h that for an y elemen t G ∈ I , G ≺ F N − 1 . Deﬁne an op en sub s et V of ( K 0 ( X ) , τ V ) as V = ( T ( x,r ) ∈ F N ♦ V x ) ∩  V , w here V x = N 1 N ( x ) and V = S ( x,r ) ∈ F N N 1 N ( x ). Clearly K ∈ V . W e show that V ⊆ φ − 1 ( ⇑ I ). Let B ∈ V . First, w e prov e that F N ≺ B . T ak e ( x, 1 N ) ∈ F N . F r om B ∈ ♦ V x , it follo ws that there is b ∈ B ∩ V x . Th us d ( x, b ) < 1 N or equiv alen tly ( x, 1 N ) ≺ ( b, 0) and ther efore F N ≺ L B . Finally F N ≺ U B follo ws from B ∈  V . No w, by Lemm a 3.3, for an y standard represen tati on of B , there is an elemen t H in this representa tion such that F N − 1 ≺ H . So for any ele men t G ∈ I , G ≺ H . T hat means I ≪ B ∗ . On the other hand, w e pro v e that the image of an y upp er (resp. lo w er) Vietoris op en subset u nder the map φ is op en in the relati v e Scott top ology on φ ( K 0 ( X )). Let  V 12 M. ALI-A KBARI AND M. POU RMAHDIAN b e an upp er Vietoris op en subset of K 0 ( X ) and K ∗ ∈ φ (  V ). T ak e ǫ > 0 suc h that S x ∈ K N ǫ ( x ) ⊆ V . Assu me that ( F n ) n> 0 is a s tand ard r epresent ation of K . Cho ose N > 0 suc h that r F N < ǫ and pu t I K = [( E n ) n> 0 ], w here E n = F N + 1 n (See Notation 2.6.). Remark 2.7(1), indicates that the sequence ( E n ) n> 0 is ≺ -ascending and therefore ( E n ) n> 0 is an ω -c hain. Clearly I K ≪ K ∗ . W e pro ve ⇑ I K ∩ φ ( K 0 ( X )) ⊆ φ (  V ) . ( ∗ ) Let B ∗ ∈ ⇑ I K . W e sh o w that B ∗ ∈ φ (  V ) or equiv alen tly B ⊆ V . L et b ∈ B . Because of I K ≪ B ∗ , there is an elemen t H ∈ B ∗ suc h that E n ≺ H , for any n > 0. So there are ( z , t ) ∈ H and ( x n , r n + 1 /n ) ∈ E n , for any n > 0, with ( x n , r n + 1 /n ) ≺ ( z , t ) ≺ ( b, 0). Since the set { ( x n , r n ) : ( x n , r n + 1 /n ) ≺ ( b, 0) } ⊆ F N is ﬁnite, it follo w s that th ere is ( x, r ) ∈ F N suc h that d ( x, b ) ≤ r < ǫ . Th us from x ∈ K , b ∈ V follo w s and consequen tly B ∗ ∈ φ (  V ). No w, let ♦ V b e a lo w er Vietoris op en su bset of K 0 ( X ). S upp ose K ∗ ∈ φ ( ♦ V ) and ( F n ) n> 0 is a standard rep resen tation of K ∗ . Select x ∈ K ∩ V and put F ′ n = F n ∪ { ( x, 1 /n ) } . One can readily see th at ( F ′ n ) n> 0 is equiv alen t to ( F n ) n> 0 and therefore b elongs to K ∗ . Cho ose N > 0 suc h that N 1 N ( x ) ⊆ V . Deﬁne I K = [( E n ) n> 0 ], where E n = F ′ N + 1 n . F or an y n > 0, E n ≺ F ′ N . Thus K ∗ ∈ ⇑ I K . W e pr o v e ⇑ I K ∩ φ ( K 0 ( X )) ⊆ φ ( ♦ V ) . ( ∗∗ ) Assume that I K ≪ B ∗ . There is H ∈ B ∗ suc h that for an y n > 0, E n ≺ H ≺ B . Hence, one can ﬁnd ( b, 0) ∈ B with ( x, 1 N ) ≺ ( b, 0). Thus b ∈ N 1 N ( x ) and b ∈ V . Th at im p lies B ∗ ∈ φ ( ♦ V ). It is kno wn that for any quasi-metric sp ace ( X, d ) whose compact su bsets are d − 1 - precompact, the Vietoris topology on K 0 ( X ) coi ncides with the top ology of the Hausdorﬀ quasi-pseudometric H d ([RR02], Theorem 5). So in the light of Theorem 3.9, und er this assumption, the map φ : ( K 0 ( X ) , H d ) → ( CB X , σ ) is an embed d ing. In the follo wing, we present an alternativ e pro of which a voids this well-kno wn r esult. Theorem 3.10. L et ( X , d ) b e a se quential ly Y one da-c omplete T 1 quasi-metric sp ac e such that any c omp act subset of ( X , d ) is d − 1 -pr e c omp act. Then the map φ : ( K 0 ( X ) , H d ) → ( CB X, σ ) is an emb e dding. Pr o of. Let ⇑ I b e a basic op en set of CB X in the Scott top ology . Let K ∗ 0 ∈ ⇑ I and ( F n ) n> 0 b e a representa tion of K ∗ 0 . By d eﬁnition of the wa y-b elo w r elation, there is a natural n um b er N > 0 such that f or any G ∈ I , G ≺ F N . Put ǫ = 1 2 δ ( F N , F N +1 ). T o complete the pro of of con tinuit y of φ , it s u ﬃces to s h o w that N ǫ ( K 0 ) ⊆ φ − 1 ( ⇑ I ) . Let K ∈ N ǫ ( K 0 ). Deﬁne F ′ = F N +1 + ǫ . By R emark 2. 7(2), F N ≺ F ′ . W e pro v e that F ′ ≺ K . F or th is, tak e ( y , s + ǫ ) ∈ F ′ . Hence ( y , s ) ∈ F N +1 and by F N +1 ≺ K 0 , there is an elemen t x ∈ K 0 suc h that ( y , s ) ≺ ( x, 0). Since H d ( K 0 , K ) < ǫ , there is an x ∗ ∈ K su c h that d ( x, x ∗ ) < ǫ . Thus d ( y , x ∗ ) ≤ d ( y , x ) + d ( x, x ∗ ) < s + ǫ. Consequent ly ( y , s + ǫ ) ≺ ( x ∗ , 0) and F ′ ≺ L K . A similar argumen t sho ws that F ′ ≺ U K and th erefore F ′ ≺ K is established. No w, b y L emma 3.3, since F N ≺ F ′ ≺ K , there is H ∈ K ∗ suc h that F N ⊑ H . Thus for any G ∈ I , G ≺ H and I ≪ K ∗ , as required . COMPUT A TIONAL MODELS OF CER T AIN HYPERSP ACES OF QU ASI-METRIC SP ACES 13 Next, in order to sho w that the map φ is an em b edding, w e p ro v e that φ ( N ǫ ( K )) is a relativ e Scott op en, for an y basic op en set N ǫ ( K ). By the assump tion, K is d − 1 -precompact. Hence there is a ﬁn ite subs et { k 1 , . . . , k m } of K such that for any k ∈ K , d ( k , k i ) < ǫ/ 4, for some 1 ≤ i ≤ m . Assu me that ( F n ) n> 0 is a standard representat ion of K ∗ . Cho ose suﬃcien tly large N > 0 such that r F N < ǫ/ 4 and at the same time f or an y k i ∈ { k 1 , . . . , k m } there is ( x i , r i ) ∈ F N so that d ∗ ( k i , x i ) < ǫ/ 4. Th e latter prop erty can b e ac hiev ed b y L emm a 3.2. Set I = [( E n ) n> 0 ], w here E n = F N + 1 n . Clearly K ∗ ∈ ⇑ I . No w, for p r o ving ⇑ I ∩ φ ( K 0 ( X )) ⊆ φ ( N ǫ ( K )) , tak e B ∗ ∈ ⇑ I and show th at H d ( K, B ) < ǫ . Let k ∈ K . Select k i , ( x i , r i ) ∈ F N and (similar to the pro of of ( ∗∗ ) in the p receding theorem) ( b, 0) ∈ B suc h th at d ( k, k i ) < ǫ/ 4, d ( k i , x i ) < ǫ/ 4 and ( x i , r i ) ⊑ ( b, 0). Therefore d ( k , b ) ≤ d ( k , k i ) + d ( k i , x i ) + d ( x i , b ) < ǫ 4 + ǫ 4 + ǫ 4 = 3 ǫ 4 . In other w ords, H − d ( K, B ) < ǫ . Next, for H + d ( K, B ) < ǫ , we p ic k up b ∈ B . One can use the same argument used in the pr o of of ( ∗ ) in the pr eceding theorem to ﬁ nd ( x, r ) ∈ F N suc h that d ( x, b ) ≤ r < ǫ/ 3. Since x ∈ K , H + d ( K, B ) < ǫ follo ws . Roughly sp eaking, the ab o v e theorems s tate that un der certain conditions on ( X , d ) th e h yp erspace K 0 ( X ) with resp ect to the Vietoris or Hausdorﬀ top ologies can b e em b edded in a suitable con tinuous ω -d cp o. Hence, in th e ligh t of Deﬁnition 1.2, the follo wing theorem is obtained. Theorem 3.11. L et ( X, d ) b e a se quential ly Y one da-c omplete T 1 quasi-metric sp ac e. Then (1) The p air ( CB X, φ ) gives an ω -c omputational mo del for ( K 0 ( X ) , τ V ) . (2) If, in addition, any c omp act subset of ( X , d ) is d − 1 -pr e c omp act, then the p air ( CB X, φ ) gives an ω -c omputational mo del f or ( K 0 ( X ) , H d ) . Also, in the light of F act 1.1 and Th eorem 2.5, since th e w a y-b elow relation of any con tin u ous p oset is ap p ro ximating, the ideal completio n of ( CB X, ⊑ ) with the auxiliary relation ≪ giv es a computational m o del of K 0 ( X ). S o the follo wing theorem is established . Theorem 3.12. L et ( X, d ) b e a se quential ly Y one da-c omplete T 1 quasi-metric sp ac e. Then (1) ( K 0 ( X ) , τ V ) has a c omputational mo del. (2) If, in addition, any c omp act subset of ( X, d ) is d − 1 -pr e c omp act, ( K 0 ( X ) , H d ) has a c omputational mo del. The follo wing corollary is a trivial consequ en ce of the ab ov e Theorems. See also [EH98, LK04, Mar04]. Corollary 3.13. L et ( X, d ) b e a c omplete metric sp ac e. Then the hyp ersp ac e ( K 0 ( X ) , H d ) has a c omputational mo del. It could b e readily seen that the Vietoris and resp ectiv ely Hausdorﬀ top ologies are T 1 if and only if the φ -image of K 0 ( X ) is a subs et of the maximal elements of CB X . It is kno wn that b oth these top ologies are not necessarily T 1 . Therefore the φ -image of K 0 ( X ) ma y not lie in the maximal elements of CB X . As it was noted b efore, if ( X , d ) is Hausdorﬀ (more generally K C), these top ologies are T 1 . The f ollo wing examples giv e an application of Th eorem 3.12. 14 M. ALI-A KBARI AND M. POU RMAHDIAN Example 3.14. (1) Let R be the set of real num b ers and let d b e a T 1 quasi-metric deﬁned on R b y d ( x, y ) = y − x if x ≤ y an d d ( x, y ) = 1 if x > y . Then the top ology τ d is the Sorgenfrey top ology on R and R l = ( R , d ) calle d the Sorgenfrey lin e. It is ea sy to see that ( R , d ) is a Ha usdorﬀ (sequen tially) Y oneda-complete space. It is a well-kno wn f act that an y compact subset of Sorgenfrey line is compact with resp ect to the u sual top ology of R and hence it is d − 1 -precompact. (2) Let Σ b e a n on-empt y s et and Σ ∞ b e the set of ﬁ n ite and in ﬁnite sequ ences o ver Σ. Deﬁne the relation  on Σ ∞ as x  y ↔ x is a p r eﬁx of y . F or x, y ∈ Σ ∞ , we d enote the longest common preﬁx of x and y by x ⊓ y . Also the length of an ele men t x ∈ Σ ∞ , is denoted b y l ( x ) ∈ N ∪ {∞} . Deﬁne q b : Σ ∞ × Σ ∞ → [0 , 1], giv en a s: q b ( x, y ) = 2 − l ( x ) − 2 − l ( y ) if x  y , q b ( x, y ) = 1 otherwise . W e adopt the con v entio n 1 ∞ = 0. It is pro v ed in [RR V08] th at (Σ ∞ , q b ) is Hausdorﬀ and ( q b ) − 1 -righ t K -sequen tially complete. F ur thermore, it is quite straigh tforward to complete the argument in [RR V0 8] to p r o v e this space is in fact sequ entially Y oned a-complete. (3) Let d b e the r estriction of the Sorgenfrey metric deﬁn ed in Example 1, to [0 , 1]. Put X = [0 , 1] [0 , 1] as the set of all contin uous fun ctions f : [0 , 1] → [0 , 1] with r esp ect to the quasi-metric sp ace ([0 , 1] , d ). F or f , g ∈ X , let D ( f , g ) = sup x ∈ [0 , 1] d ( f ( x ) , g ( x )) . Note that D ( f , g ) < 1 forces f ≤ g . Hence any C auc h y sequence ( f n ) n> 0 should b e ev entually increasing. No w, for such a sequence tak e f = sup n ≥ n 0 f n , where n 0 is the index f rom w h ic h th e s equence is increasing. Then, f is the Y oneda-limit of ( f n ) n> 0 and hence ( X , D ) is a T 1 sequen tially Y oneda-complete space. I n fact, it can b e r eadily seen that ( X , D ) is a Hausd orﬀ sp ace. (4) Let C = [0 , 1] ω . F or p ∈ [1 , ∞ ) the function q p : C → [0 , 1] is deﬁned b y q p ( f ) =  ∞ X n =0 (2 − n f ( n )) p  1 p . Then usin g an argumen t in [RS V03], Theorem 1, one can sho w that the fu nction f q p : C × C → [0 , 1] giv en by f q p ( f , g ) = q p ( f − g ) if f ≥ g , f q p ( f , g ) = 1 otherwise , deﬁnes a Hausdorﬀ quasi-metric on C . F urthermore, by adopting the pro ofs of T h eorems 3 and 4 in [RSV03], one can also show that ( C , f q p ) is sequent ially Y oneda-complete. (This qu asi-metric is in fact the conju gate of the qu asi-metric e d p giv en in [RSV03].) Corollary 3.15. L et ( X, d ) b e one of the examples giv en in 3.14. Then the sp ac e ( K 0 ( X ) , τ V ) is T 1 and has a c omputational mo del. F urthermo r e in the c ase of Sor genfr ey line R l , the sp ac e ( K 0 ( R l ) , H d ) i s T 1 and ha s a c omputational mo del. COMPUT A TIONAL MODELS OF CER T AIN HYPERSP ACES OF QU ASI-METRIC SP ACES 15 4. A quantit a tive ω -co mput a tional model of K 0 ( X ) So far we ha v e sho wn that, u nder certain circumstances, ( K 0 ( X ) , H d ) can b e embed d ed in CB X a nd therefore ( CB X, φ ) provides an ω -compu tational mo del for ( K 0 ( X ) , H d ). In th is section, we tak e the we ll-kno wn quan titativ e ( ω -)computational mo del appr oac h, pro ving that CB X is also a quantit ativ e ω -computational mo del for ( K 0 ( X ) , H d ) and so the results of the preceding section are strengthened. Ou r d eﬁ nition of qu an titat iv e ( ω -)computational mo del f ollo ws Rutten ([Rut98], Section 7). Deﬁnition 4.1. A quantitative ( ω -)c omputational mo del of a quasi-metric space ( Y , d ) is a quadrup le ( L, ⊑ , D , φ ) w here ( L, ⊑ ) is a cont in uous ( ω -)dcp o, D is an algebraic sequen tially Y oneda-complete quasi-metric on L and φ : Y → L is a map s u c h that: (1) The sp ecialization p artial order ⊑ D is equiv alen t to th e partial order of L . (2) φ is an isometry from ( Y , d ) into ( L, D ). (3) φ ( M ax ( Y , ⊑ d )) = M ax ( L, ⊑ ). It is worth mentioning that the ab ov e deﬁn ition is wea k er than th e deﬁnition of Ro- maguera and V alero (Deﬁnition 1 in [R V09]) in whic h ( L, D ) is considered to b e Smyth- complete and the top ology τ D coincides with th e Scott top ology σ L . W e pr efer to tak e Rutten’s notion of quant itativ e ω -computational mo del and then stud y a sp ecial case where ( L, D ) satisﬁes the extra conditions of Deﬁnition 1 in [R V09] (Th eorems 4.16). In [R V09], Romaguera and V alero follo w ed the w ork of Hec kmann [Hec99] for a complete w eigh ted quasi-metric space ( X , d ) and deﬁned a complete partial quasi-metric Q on the space of formal balls. A quasi-metric q on B X is then deriv ed from Q whic h ind uces the same top ology as Q on B X and moreov er, ( B X, q ) is Smyth-complete. It is useful to note that the quasi-metric q can b e deﬁn ed directly on B X for an y quasi-metric s p ace without the existence of a partial metric. Deﬁnition 4.2. Let ( X , d ) b e a quasi-metric space. F or ( x, r ) , ( y , s ) ∈ B X , deﬁne q (( x, r ) , ( y , s )) = max { d ( x, y ) , | r − s |} + ( s − r ) . It is easy to see that q deﬁnes a quasi-metric on B X . The next lemma s ho ws th at ( B X, q ) in herits Smyth-co mpleteness and sequen tially Y oneda-completeness from ( X , d ). The p ro of is more or less th e same as the pro of of Theorem 4.1 of [R V09]. Lemma 4.3. L et ( X, d ) b e a se quential ly Y one da-c omplete (r esp e ctively Smyth-c omplete) quasi-metric sp ac e. Then ( B X , q ) is also se quential ly Y one da-c omplete (r esp e ctively Smyth- c omplete). The f ollo wing theorem generalizes Theorem 5.1 of [R V09]. Theorem 4.4. Each algebr aic se quential ly Y one da-c omplete T 1 quasi-metric sp ac e has a quantitative c omputa tional mo del. Pr o of. Let ( X, d ) b e an algebraic sequen tially Y on ed a-complete T 1 quasi-metric sp ace and q b e the qu asi-metric deﬁn ed in Deﬁnition 4.2 on B X . By Theorem 2.2, B X is a conti n uous dcp o. Also, b y the ab ov e lemma ( B X, q ) is sequenti ally Y oneda-co mplete. A straigh tfor- w ard co mputation sho ws that for any ﬁnite elemen t x ∈ X , ( x, r ) is ﬁnite in ( B X , q ) and moreo ver the set of such elements f orms a base for ( B X, q ). Therefore ( B X , q ) is algebraic. The other parts follo w easily from the p r o of of Theorem 4.1 of [R V09]. 16 M. ALI-A KBARI AND M. POU RMAHDIAN No w, we turn to the main topic of this section. W e wish to deﬁn e a quasi-metric D on CB X and sho w that ( C B X, D ) together with th e map φ : K 0 ( X ) → CB X whic h is deﬁned as φ ( K ) = K ∗ , form a quanti tativ e computational mo d el for ( K 0 ( X ) , H d ). T o emphasize, we ﬁx ( X , d ) to b e a sequentia lly Y oneda-complete T 1 quasi-metric space, though ( X , d ) need not b e algebraic. Deﬁnition 4.5. L et q b e the qu asi-metric deﬁn ed in Deﬁnition 4.2 on B X . Recall that H q on P f in B X is d eﬁned by H q ( F , G ) = max { sup ( x,r ) ∈ F inf ( y, s ) ∈ G q (( x, r ) , ( y , s )) , sup ( y, s ) ∈ G inf ( x,r ) ∈ F q (( x, r ) , ( y , s )) } , for any F , G ∈ P f in B X . P ut D on CB X as follo ws: D ( I , J ) = sup F n inf G m H d ( F n , G m ) , for all I , J ∈ CB X and representa tions ( F n ) n> 0 , ( G m ) m> 0 of I and J , r esp ectiv ely . Next, w e show that D is indep end en t from any particular choic e of r ep resen tations. Lemma 4.6. D is wel l-deﬁne d on CB X . Pr o of. Let ( F n ) n> 0 and ( F ′ k ) k > 0 b e tw o diﬀeren t representa tions for I . Since ( F n ) n> 0 ∼ ( F ′ k ) k > 0 , for an y F n there is F ′ k suc h that F n ≺ F ′ k . So H q ( F n , F ′ k ) = 0 and therefore H q ( F n , G m ) ≤ H q ( F ′ k , G m ) . So by taking inﬁmum on G m , w e hav e inf G m H q ( F n , G m ) ≤ in f G m H q ( F ′ k , G m ) . Th us inf G m H q ( F n , G m ) ≤ sup F ′ k inf G m H q ( F ′ k , G m ) , sup F n inf G m H q ( F n , G m ) ≤ sup F ′ k inf G m H q ( F ′ k , G m ) . Similarly , we can pro v e that sup F ′ k inf G m H q ( F ′ k , G m ) ≤ sup F n inf G m H q ( F n , G m ) . Prop osition 4.7. D is a quasi-metric on CB X . In add ition, the sp e cialization or der ⊑ D is e quivalent to the p artial or der ⊑ deﬁne d on CB X . Pr o of. Th e tr iangular inequ alit y is straigh tforw ard. So we only c hec k that D ( I , J ) = D ( J , I ) = 0 imp lies I = J . Let D ( I , J ) = 0 and ( F n ) n> 0 and ( G m ) m> 0 b e repr esen tatio ns for I and J , resp ectiv ely . Fix n > 0 and pu t δ = δ ( F n , F n +1 ). F rom the d eﬁ nition of D , there is m > 0 su c h that H q ( F n +1 , G m ) < δ . W e pr o v e F n ≺ G m . Let ( x, r ) ∈ F n . Th ere is ( x ′ , r ′ ) ∈ F n +1 with ( x, r ) ≺ ( x ′ , r ′ ). Also, by H q ( F n +1 , G m ) < δ , there is ( y , s ) ∈ G m suc h that q (( x ′ , r ′ ) , ( y , s )) < δ . T his means that d ( x ′ , y ) < r ′ − s + δ and consequently d ( x, y ) ≤ d ( x, x ′ ) + d ( x ′ , y ) < ( r − r ′ ) − δ + ( r ′ − s ) + δ = r − s. COMPUT A TIONAL MODELS OF CER T AIN HYPERSP ACES OF QU ASI-METRIC SP ACES 17 Th us ( x, r ) ≺ ( y , s ) and therefore F ≺ L G . F ≺ U G can b e sho wn in a similar fashion. Hence F ≺ G is established and therefore I ⊑ J . Now D ( J , I ) = 0 also implies that J ⊑ I . F or the second part, clearly b y the ab o ve argument, D ( I , J ) = 0 implies I ⊑ J . Con v ers ely , le t I ⊑ J . So for an y F ∈ I , th ere is G ∈ J with F ≺ G . Thus H q ( F , G ) = 0 whic h implies D ( I , J ) = 0. Prop osition 4.8. The domain CB X e quipp e d with the quasi-metric D is se quential ly Y one da-c omp lete. Pr o of. Let ( I n ) n> 0 b e a Cauc hy sequ en ce in ( CB X, D ) and for an y n > 0, ( F n m ) m> 0 b e a representat ion for I n . F or an y n > 0, there is a natural num b er N n > n such that N n > N n − 1 for n > 1 and for an y l ≥ k ≥ N n , D ( I k , I l ) < 1 2 n +1 , i.e. ∀ F k r ∈ I k ∃ F l s ∈ I l H q ( F k r , F l s ) < 1 2 n +1 . Deﬁne ( G ij ) i,j > 0 as f ollo ws: Fix F N 1 1 ∈ I N 1 and p ut G 11 = F N 1 1 . In d uctiv ely for any k ≥ 2, choose G 1 k ∈ ( F N k m ) m> 0 suc h that H q ( G 1 k − 1 , G 1 k ) < 1 2 k . In a similar w a y , for any i ≥ 2, p ut G i 1 = F N 1 i and in ductiv ely ﬁnd ( G ik ) k > 0 suc h that for an y i < j , G ik ≺ G j k and H q ( G ik − 1 , G ik ) < 1 2 k . Let L k = G k k + 1 2 k − 1 . Th e follo wing tw o claims complete the pr o of. Claim 1. I = [( L k ) k > 0 ] ∈ CB X , i.e. ( L k ) k > 0 is an ascending sequence in P f in B X . Let k > 0 and ( x, r + 1 2 k − 1 ) ∈ L k . Hence ( x, r ) ∈ G k k . Sin ce H q ( G k k , G k +1 k +1 ) ≤ H q ( G k k , G k +1 k ) + H q ( G k +1 k , G k +1 k +1 ) < 0 + 1 2 k +1 there is a ( y , s ) ∈ G k +1 k +1 suc h that q (( x, r ) , ( y , s )) < 1 2 k +1 . Hence d ( x, y ) < r − s + 1 2 k +1 . By a simple calculation, d ( x, y ) < ( r + 1 2 k − 1 ) − ( s + 1 2 k ) . Therefore ( x, r + 1 2 k − 1 ) ≺ ( y , s + 1 2 k ), wh ere ( y , s + 1 2 k ) ∈ L k +1 . This means that L k ≺ L L k +1 . Similarly , we can pro v e L k ≺ U L k +1 and consequentl y L k ≺ L k +1 . Claim 2. I is the Y oneda limit of ( I n ) n> 0 . First, we p ro v e that D ( I n , I ) → 0. Let ǫ > 0. Since ( I n ) n> 0 is a Cauc h y sequence, there is N > 0 such that for an y r ≥ s ≥ N , D ( I s , I r ) < ǫ 3 . Fix n > N and F ∈ I n = [( F n m ) m> 0 ]. Cho ose N k > n from the coﬁnal sequence ( N n ) n> 0 constructed abov e suc h th at 1 2 k − 1 < ǫ 3 . Since D ( I n , I N k ) < ǫ 3 , there is F N k m 0 ∈ I N k suc h that H q ( F , F N k m 0 ) < ǫ 3 . Cho ose G lk ∈ I N k with F N k m 0 ≺ G lk and l > k . No w, H q ( G lk , G ll ) < 1 2 k +1 + · · · + 1 2 l < 1 2 k < ǫ 3 implies that H q ( F , L l ) ≤ H q ( F , F N k m 0 ) + H q ( F N k m 0 , G lk ) + H q ( G lk , G ll ) + H q ( G ll , L l ) < ǫ 3 + 0 + ǫ 3 + ǫ 3 = ǫ. 18 M. ALI-A KBARI AND M. POU RMAHDIAN No w, we sho w that I is the Y oneda limit of ( I n ) n> 0 . Because of D ( I n , I ) → 0, it su ﬃ ces to pro v e that D ( I , J ) ≤ inf n sup k >n D ( I k , J ), for an y J ∈ CB X . Put inf n sup k >n D ( I k , J ) = s . W e pro v e that for any ǫ > 0, D ( I , J ) ≤ s + ǫ . Let L ∈ I . Since inf n sup k >n D ( I k , J ) = s , there is N > 0 su c h that for an y n > N , D ( I n , J ) ≤ s + ǫ/ 2. T ak e k > 0 s uc h that L ≺ L k , 1 2 k − 1 < ǫ/ 2 and N k > N . So, for any represen tation ( G m ) m> 0 of J , there is m > 0 su ch that H q ( L, G m ) ≤ H q ( L, L k ) + H q ( L k , G k k ) + H q ( G k k , G m ) < 0 + 1 2 k − 1 + s + ǫ / 2 < s + ǫ. This complete the pr o of. Belo w, w e sho w that the qu asi-metric sp ace ( CB X , D ) is in fact the s equen tial Y oneda completion of ( P f in B X, H d ). First, we recall the deﬁ n ition of sequentia l Y oneda completion [KS02]. Deﬁnition 4.9. Let ( Y , d ) b e a quasi-metric sp ace and b Y = { ( x n ) n> 0 : ( x n ) n> 0 is a Cauch y sequence } . Deﬁne the quasi-pseu dometric b d and the equiv alence relation ≈ as follo ws : b d (( x n ) n> 0 , ( y m ) m> 0 ) = inf n sup k ≥ n sup m inf p ≥ m d ( x k , y p ) , ( x n ) n> 0 ≈ ( y m ) m> 0 ⇔ b d (( x n ) n> 0 , ( y m ) m> 0 ) = b d (( y m ) m> 0 , ( x n ) n> 0 ) = 0 . Put Y = b Y / ≈ and deﬁne the qu asi-metric d by d ([( x n ) n> 0 ] , [( y m ) m> 0 ]) = b d (( x n ) n> 0 , ( y m ) m> 0 ) . The p air ( Y , d ) is called the sequent ial Y oneda completion of ( Y , d ). Prop osition 4.10. ( CB X , D ) is the se quential Y one da c omp letion of ( P f in B X, H d ) . Pr o of. Let ( F n ) n> 0 b e a C auc hy sequence in P f in B X an d H d b e the quasi-metric completion of H d (Deﬁnition 4.9). Without loss of generali t y , w e ca n assume th at for any 0 < n ≤ m , H d ( F n , F m ) < 1 2 n +1 , since there is an H d -equiv alen t s u bsequence of ( F n ) n> 0 , satisfying this prop erty . No w, w e sh o w that there is an ascendin g chain ( L k ) k > 0 in P f in B X which is H d -equiv alen t. F or any n > 0, d eﬁne I n = [( E n m ) m> 0 ] where E n m = F n + 1 m . One can readily chec k that ( I n ) n> 0 is a Cauc h y sequence. F or k > 0, put L k = E k k + 1 2 k − 1 . A sim ilar argumen t as used in Claim 1 of Prop osition 4.8, sho ws that [( L k ) k > 0 ] is a ≺ -ascending c hain. In ord er to show H d (( F n ) n> 0 , ( L m ) m> 0 ) = 0 , w e hav e to verify the follo wing: ∀ ǫ > 0 ∃ n ∀ k ≥ n ∀ m ∃ p ≥ m H d ( F k , L p ) < ǫ. Let ǫ > 0 b e giv en. Find n > 0 so that 1 2 n +1 < ǫ 2 . Next, for an y k ≥ n and m > 0, tak e p ≥ k , m such that 1 p + 1 2 p − 1 < ǫ 2 . S ince L p = F p + ( 1 p + 1 2 p − 1 ), it easily follo ws that H d ( F k , L p ) < ǫ . The equalit y H d (( L m ) m> 0 , ( F n ) n> 0 ) = 0 can b e prov ed in a similar wa y . So, the map ψ : ( P f in B X , H d ) − → ( CB X, D ) [( F n ) n> 0 ] 7− → [( L k ) k > 0 ] . deﬁnes a bijectiv e isometry , as r equired. COMPUT A TIONAL MODELS OF CER T AIN HYPERSP ACES OF QU ASI-METRIC SP ACES 19 Corollary 4.11. The domain CB X e quipp e d with the quasi-metric D is al gebr aic. Pr o of. By Corollary 23 of [KS02 ], any Y oneda completion is algebraic. Hence, according to Prop osition 4.10 , ( CB X, D ) is algebraic. Prop osition 4.12. The ma p φ is an isometry b etwe en ( K 0 ( X ) , H d ) and ( CB X, D ) . Pr o of. Let K 1 , K 2 ∈ K 0 ( X ) and ( F n ) n> 0 and ( G m ) m> 0 b e standard rep resen tations for K 1 and K 2 , resp ectiv ely . Assume th at H d ( K 1 , K 2 ) = d and ǫ > 0. W e sh o w that D ( K ∗ 1 , K ∗ 2 ) ≤ d + ǫ . Fix n > 0 and let δ = δ ( F n , F n +1 ). W e wish to ﬁnd m > 0 such that H q ( F n , G M ) < d + ǫ. Let ( x, r ) ∈ F n . Since F n ≺ F n +1 ≺ K 1 , there are ( x ′ , r ′ ) ∈ F n +1 and ( z , 0) ∈ K 1 suc h that d ( x, x ′ ) < ( r − r ′ ) − δ and d ( x ′ , z ) < r ′ . Because of H d ( K 1 , K 2 ) = d , there is y ∈ K 2 suc h that d ( z , y ) ≤ d + ǫ . By Lemma 3.2, th ere is a sequence (( y m , s m )) m> 0 suc h that ( y m , s m ) ∈ G m and d ∗ ( y m , y ) → 0. S elect y m suc h that d ( y , y m ) < δ 2 and s m < δ 2 . Then d ( x, y m ) ≤ d ( x, x ′ ) + d ( x ′ , z ) + d ( z , y ) + d ( y , y m ) < ( r − r ′ ) − δ + r ′ + d + ǫ + δ 2 = r + d + ǫ − δ 2 Th us f or any ( x, r ) ∈ F n , there is ( y m x , s m x ) ∈ G m x suc h that d ( x, y m x ) < r − s m x + d + ǫ . T ak e M ≥ max { m x : ( x, r ) ∈ F n } with r G M < δ 2 . Hence, b y the ab ov e calculation, for an y ( x, r ) ∈ F n there exists ( y , s ) ∈ G M suc h that q (( x, r ) , ( y , s )) < d + ǫ . Similarly , for any ( y , s ) ∈ G M one can ﬁ nd ( x, r ) ∈ F n with q (( x, r ) , ( y , s )) < d + ǫ. So H q ( F n , G M ) < d + ǫ and therefore D ( K ∗ 1 , K ∗ 2 ) ≤ H d ( K 1 , K 2 ). On the other hand, w e show that H d ( K 1 , K 2 ) ≤ D ( K ∗ 1 , K ∗ 2 ). Let D ( K ∗ 1 , K ∗ 2 ) = d and x ∈ K 1 . Again, Lemma 3.2 imp lies the existence of a sequence (( x n , r n )) n> 0 suc h that ( x n , r n ) ∈ F n and d ∗ ( x n , x ) → 0. F or ǫ > 0, c h o ose ( x N , r N ) suc h that d ( x, x N ) < ǫ 3 and r N < ǫ 3 . S ince D ( K ∗ 1 , K ∗ 2 ) = d , there is G m N and ( y m N , s m N ) ∈ G m N suc h that d ( x N , y m N ) < r N − s m N + d + ǫ 3 . The follo win g inequalities d ( x, y m N ) ≤ d ( x, x N ) + d ( x N , y m N ) < ǫ 3 + r N − s m N + d + ǫ 3 < d + ǫ, imply in f y ∈ K 2 d ( x, y ) ≤ d and therefore sup x ∈ K 1 inf y ∈ K 2 d ( x, y ) ≤ d. No w, in a similar w a y , sup y ∈ K 2 inf x ∈ K 1 d ( x, y ) ≤ d . Hence H d ( K 1 , K 2 ) ≤ d and consequ en tly H d ( K 1 , K 2 ) ≤ D ( K ∗ 1 , K ∗ 2 ). 20 M. ALI-A KBARI AND M. POU RMAHDIAN No w, th e main theorem of this section is stated. Theorem 4.13. ( CB X, D ) to g ether with the map φ : K 0 ( X ) → CB X form a quantitative ω -c omputa tional mo del for ( K 0 ( X ) , H d ) . Pr o of. By the ab o v e prop ositions, ( CB X, D ) is an algebraic sequent ially Y oneda-complete quasi-metric space, the sp ecialization partial order ⊑ D is equiv alent to th e partial ord er of CB X and the map φ is an isometry b et w een ( K 0 ( X ) , H d ) and ( CB X , D ). The co ndition φ ( M ax ( K 0 ( X ) , ⊑ H d )) = M ax ( CB X , ⊑ ) follo ws f rom p r op ositions 3.8 and 4.7. Next, we imp ose some extra conditions on ( X , d ) u nder wh ic h this space has a quant i- tativ e ω -computational mo del in the sense of Romaguera and V alero [R V09]. W e, hereby , supp ose that ( X , d ) is Smyth-complete and any compact subset of ( X, d ) is d − 1 -precompact. These conditions guarantee that any Cauc h y sequence in ( P f in B X, H q ) is biCauch y . Lemma 4.14. L et ( X, d ) b e a Smyth-c omplete quasi-metric sp ac e of which al l of its c omp act subsets ar e d − 1 -pr e c omp act. Then any ω -chain in P f in B X is biCauchy. Pr o of. First note, one can easily sh o w that an y compact subs et of ( B X, q ) is q − 1 -precompact. No w, let ( F n ) n> 0 b e an ω -chai n in P f in B X . F or an y n < m , F n ≺ F m . Hence H q ( F n , F m ) = 0 and ( F n ) n> 0 is a Cauc h y sequence. According to Lemma 4.3 , ( B X, q ) is S m yth-complete. So by Theorem 1.3, ( K 0 ( B X ) , H q ) is Smyth-complete. Therefore P f in B X as a subsp ace of K 0 ( B X ) is Smyth-completa ble and so any Cauc h y sequence in ( P f in B X, H q ) is biCauc hy . Hence ( F n ) n> 0 is a biCauch y sequence. The f ollo wing auxiliary lemma will b e useful in sev eral p ro ofs. Lemma 4.15. L et G ≺ H ⊑ M and H q ( M , L ) < δ ( G, H ) . Then G ≺ L . Pr o of. Let ( x, r ) ∈ G and δ = δ ( G, H ). Th en there are ( y , s ) ∈ H , ( z , t ) ∈ M an d ( a, u ) ∈ L suc h that ( x, r ) ≺ ( y , s ) ⊑ ( z , t ) and q (( z , t ) , ( a, u )) < δ. Then the follo wing inequalities follo w: d ( x, a ) ≤ d ( x, y ) + d ( y , z ) + d ( z , a ) < ( r − s ) − δ + ( s − t ) + ( t − u ) + δ = r − u. Hence ( x, r ) ≺ ( a, u ) and consequen tly G ≺ L L . A similar calculation sho ws that G ≺ U L and th erefore G ≺ L as required . Theorem 4.16. Under the assumptions of L emma 4.14, (1) ( CB X, D ) is Smyth-c omplete. (2) The to p olo gy τ D induc e d by D c oincides with the Sc ott top olo gy of th e do main CB X . Pr o of. (1) Let ( I n ) n> 0 b e a Cauc h y sequen ce in ( CB X , D ) and for any n > 0, ( F n m ) m> 0 b e a represen tation for I n . No w, ﬁx the natural sequence ( N n ) n> 0 , the double sequence ( G ij ) i,j > 0 and I = [( L k ) k > 0 ] whic h are constru cted in th e pro of of P rop osition 4.8. W e sho w that ( I n ) n> 0 con verges strongly to I . Because of D ( I n , I ) → 0, it suﬃces to v erify D ( I , I n ) → 0. Let ǫ > 0. Since ( L k ) k > 0 is a Cau ch y ω -c hain in ( P f in B X, H q ), so, by Lemma 4.14, it is a biCauch y sequ ence. Cho ose k > 0 such that 1 2 k − 1 < ǫ/ 4 and for any r , s ≥ k , H q ( L s , L r ) < ǫ/ 4. W e sho w that for an y n > N k , D ( I , I n ) < ǫ . COMPUT A TIONAL MODELS OF CER T AIN HYPERSP ACES OF QU ASI-METRIC SP ACES 21 T ak e n > N k and L i ∈ I . Recall that the sequence ( N j ) j > 0 has the follo win g prop erties: ∀ r ≥ s ≥ N j , D ( I s , I r ) < 1 2 j +1 . P articularly D ( I N k , I n ) < 1 2 k +1 . So, th ere is F n m ∈ I n suc h that for G k k ∈ I N k , H q ( G k k , F n m ) < ǫ/ 4. No w, if i ≤ k , then H q ( L i , L k ) = 0. O therwise, since ( L k ) k > 0 is a biCauch y sequen ce, H q ( L i , L k ) < ǫ 4 . Thus H q ( L i , F n m ) ≤ H q ( L i , L k ) + H q ( L k , G k k ) + H q ( G k k , F n m ) < ǫ 4 + 1 2 k − 1 + ǫ 4 < 3 ǫ 4 . (2) Supp ose that ⇑ I is a b asic op en subset of CB X and J ∈ ⇑ I . Let ( G n ) n> 0 b e a represent ation of J . S o, there is n > 0 suc h that F ≺ G n , for any F ∈ I . Set δ = δ ( G n , G n +1 ). W e pr o ve that N δ ( J ) = {L : D ( J , L ) < δ } ⊆ ⇑ I Let L ∈ N δ ( J ) and ( L m ) m> 0 b e one of its representat ions. F rom D ( J , L ) < δ , there is L m suc h that H q ( G n +1 , L m ) < δ . Also, according to th e Lemm a 4.15, G n ≺ G n +1 ⊑ G n +1 and H q ( G n +1 , L m ) < δ imp lies that G n ≺ L m . Finally , since for eve ry F ∈ I , F ≺ G n it follo ws that f or any F ∈ I , F ≺ L m . Therefore L ∈ ⇑ I . No w, let N δ ( I ) b e a b asic op en su bset in τ D . Assu m e that ( F n ) n> 0 is a represent ation of I . According to Lemma 4.14, ( F n ) n> 0 is a biCauc h y sequ en ce. So there is N > 0 suc h that ∀ n, m ≥ N H q ( F n , F m ) < δ 2 . Put J = ( G n ) n> 0 , where G n = F N + 1 n . Clearly J ≪ I . W e prov e that ⇑ J ⊆ N δ ( I ) . Let L ∈ ⇑ J and ( L m ) m> 0 b e a represen tatio n for L . Sin ce J ≪ L , there is L m suc h that for any n > 0, G n ≺ L m . F or an y F n ∈ I , if n < N then H q ( F n , F N ) = 0. Otherwise H q ( F n , F N ) < δ 2 . T ak e k > 0 w ith 1 k < δ 4 . Then we ha v e H q ( F n , L m ) ≤ H q ( F n , F N ) + H q ( F N , G k ) + H q ( G k , L m ) < δ 2 + 1 k + 0 < 3 δ 4 . Th us D ( I , L ) < δ or equiv alen tly L ∈ N δ ( I ). Remark 4.17. Th e isometry of φ (Proposition 4. 12) and the ab o v e Theorem, imply that for any Smyth-co mplete T 1 quasi-metric space w hose compact sub sets are d − 1 -precompact, the map φ is an em b edding. T h is result was already p ro v ed in Theorem 3.10 . 5. Plotkin powerdomain vs . ω -Plotkin domain In this secti on, we co mpare the Plotkin P ow erdomain and ω -Plokin constructions of B X . The Plotkin p o w erdomain of B X , d enoted by P B X , is the ideal completion of the abstract basis ( P f in B X, ≺ E M ). In the follo wing, w e sh o w that for any T 1 quasi-metric sp ace ( X , d ) if ( X , d ) is either Smyth-co mplete and all of its compact su bsets are d − 1 -precompact, or ω -algebraic Y oneda-complete, then P B X and CB X are order -isomorp h ic. 22 M. ALI-A KBARI AND M. POU RMAHDIAN Theorem 5.1. L et ( X , d ) b e a T 1 quasi-metric sp ac e. Assume also that either of the fol lowing c onditions hold. (1) ( X, d ) is a Smyth-c omplete sp ac e al l of whose c omp act su bsets ar e d − 1 -pr e c omp act. (2) ( X, d ) is an ω -algebr aic Y one da-c omplete sp ac e. Then P B X and CB X ar e or der-isomorph ic. Pr o of. (1) Assu me ﬁ r st that ( X, d ) satisﬁes condition 1 ab o v e. Let I b e a r ound id eal in P f in B X . W e claim that I has a coﬁn al ≺ E M -ascending sub sequence ( F n ) n> 0 . Note that I can b e co nsidered as a Cauc h y net in ( P f in B X, H q ), since if G ≺ E M H ∈ I , then H q ( G, H ) = 0. Un der the ab ov e assumptions, similar to th e argumen t used in Lemma 4.14, ( P f in B X, H q ) is Smyth-co mpletable. Hence I is biCauch y and h as a biCauc h y su bsequence ( F n ) n> 0 , satisfying th e follo wing prop ert y: ∀ H , G ∈ I , F n ≺ E M H , G ⇒ H q ( H , G ) < 1 n . ( ∗ ) Without loss of generalit y , we m a y a ssume that ( F n ) n> 0 is ≺ E M -ascending. No w, we sh ow that this sequence is ≺ E M -coﬁnal in I . Let G ∈ I . N > 0 must b e fou n d suc h that G ≺ E M F N . Cho ose H ∈ I with G ≺ E M H and put δ = δ ( G, H ). T ake N > 1 suc h that 1 N − 1 < δ and let M ∈ I w ith F N , H ≺ E M M . T herefore b y F N − 1 ≺ E M F N ≺ E M M and ( ∗ ), w e ha v e H q ( M , F N ) < 1 N − 1 < δ . Finally , according to the Lemma 4.15, G ≺ E M H ≺ E M M and H q ( M , F N ) < δ implies G ≺ E M F N . No w, it is easy to c hec k that the mapping ψ : P B X − → CB X I 7− → [( F n ) n> 0 ] deﬁnes an ord er isomorph ism. (2) No w consider that ( X, d ) is an ω -algebraic Y oneda-complete space. Let X 0 b e a coun table algebraic sub set of X . Then the set B Q X 0 = X 0 × Q + and resp ectiv ely the set P f in B Q X 0 of all ﬁn ite subs ets of B Q X 0 form a counta ble basis for B X and for P f in B X resp ectiv ely . No w, in the light of Theorem 6.2.3 of [AJ94], P B X is giv en by the ideal completion of ( P f in B Q X 0 , ≺ E M ). F urth ermore, since the s et P f in B Q X 0 is coun t- able, by Pr op osition 2.2.3 in [AJ94], ev er y round ideal in P f in B Q X 0 has a coﬁnal ≺ E M - ascending sub sequence and hen ce follo wing the same p ro of as in (1), one can pr o ve that ( P B Q X 0 , ≺ E M ) is order-isomorphic to ( CB Q X 0 , ≺ E M ). O n th e other hand, one can eas- ily show that an y ≺ E M -ascending subsequence ( F n ) n> 0 in P f in B X is ∼ E M -equiv alen t to a ≺ E M -ascending sub sequence ( G n ) n> 0 in P f in B Q X 0 . Hence ( C B Q X 0 , ≺ E M ) is order- isomorphic to ( CB X , ≺ E M ). Ther efore, the pro of is established. Corollary 5.2. L et ( X, d ) b e a c omplete metric sp ac e. Th en the Plotkin p ower doma in P B X and ω -Plotkin CB X ar e isomorp hic. 6. Future W ork In th is pap er, v arious compu tational m o dels of th e h yp erspace K 0 ( X ) of the non-empt y compact subsets of a quasi-metric space ( X, d ) were studied. It was sho wn ho w to use a sp ecial computational m o del B X of ( X, d ) to get the corresp onding ω -computational mo del CB X of K 0 ( X ). The ab ov e construction would h a v e b een more satisfactory if a computational mo del for the K 0 ( X ) could hav e b een deﬁned, starting fr om an arbitrary COMPUT A TIONAL MODELS OF CER T AIN HYPERSP ACES OF QU ASI-METRIC SP ACES 23 computational mo del of ( X, d ). This id ea is already dev elop ed for the case of metric spaces b y Martin [Mar04], by app ealing to the notion of a measuremen t, and for th e spaces with coun table based mo dels by Berger et. al. in [Ber10]. Therefore, an inte resting sub j ect of researc h is to ﬁ nd a fairly general framew ork und er whic h the ideas from [Mar04] and [Ber10] can b e generalized to the p resen t cont ext. An other topic of researc h is to stu dy the eﬀectiv eness of K 0 ( X ). So one could ask whether CB X su pp orts an eﬀectiv e b ase whenever ( X, d ) is an eﬀectiv e quasi-metric. Section 5 inv olv es a generalization of th e r esults obtained earlier, sh o wing that K 0 ( X ) has a q u an titativ e ω -computational mo del. One would desire to ha v e a quan titativ e compu - tational mo del for K 0 ( X ). This, for example, requires to generate another computational mo del f rom ( CB X , ⊑ , D , φ ), similar to what we obtained in Th eorem 3.12. One wa y to ac hieve this is to employ the Y oneda-co mpletion of ( CB X, D ), whic h serv es as a natural generalizat ion o f the ideal completion. In section 6, the Plotkin p ow erdomain and ω -Plotkin domain of B X we re compared and the situations in whic h b oth constructions are order-isomorphic were observ ed. No w, the question of ﬁndin g an example for whic h th ese constru ctions are n ot isomorph ic is imp osed. A cknowledgement The authors would like to thank R. Hec km an n and O. V alero for their us efu l corresp on- dence. Th e authors wo uld also like to thank t w o anon ymous referees for their constru ctiv e commen ts wh ich enabled us to correct some of the results and impro v e the presenta tion of the pap er. Referen ces [AJ94] S. 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Computational Models of Certain Hyperspaces of Quasi-metric Spaces

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