Topological completeness of the provability logic GLP

T op ological completeness of the pro v abilit y logic GLP Lev Beklemi shev ∗ Steklov Institut e of Mathema tics, Moscow Da vid Gab ela ia Razmadze Institut e of Mat hemati cs, Tbili si No v ember 5, 2018 Abstract Pro v abilit y logic GLP is we ll-kno wn to b e incomplete w.r.t. Kripke seman tics. A natural top ologica l seman tics of GLP in terprets mo dal- ities as deriv ativ e op erators of a p olytop ological space. Suc h spaces satisfying all the axioms of GLP are called GLP-sp aces. W e de- v elop some constructions to build n on trivial GLP-sp aces and show that GLP is complete w.r.t. the class of all GLP-spaces. Key words: prov a bilit y logic, s cattered spaces, GLP 1 In tro d uction This pap er con tinues the study of top ological seman tics of an imp ortant p oly- mo dal pro v ability log ic GLP initiated in [6, 5]. This sys tem, in tro duced by Japaridze [11, 12], describ es in the st yle of prov a bilit y logic all the univ ersally v alid sc hemata f or the reflection principles of restricted log ical complexit y in arithmetic. Th us, it is complete with resp ect to a v ery natural kind of pro of- theoretic seman tics. The logic GLP has b een extensiv ely studied in the early 1990s b y Ignatiev and Bo olos who simplified and extended Japaridze’s work (see [8]). More recen tly , intere sting applications of GLP ha v e b een f ound in pro of theory and ordinal analysis of arithmetic. In pa rticular, GLP giv es rise to a natural system of ordinal notations fo r the ordinal ε 0 . Based on the use of GLP , the first author o f this pap er gav e a pro of-t heoretic analysis of P eano arithmetic, whic h s tim ulated further intere st to w ards GLP (see [1, 2] for a detailed surv ey). ∗ Suppo rted by the Russian F oundation for Basic Resear ch (RFBR), Russian P residen- tial Co uncil for Suppor t of Lea ding Scientific Schools, a nd the Swiss–Russian c o op eration pro ject STCP–CH–RU “ Computational pro of theory .” 1 The main obstacle in the study o f GLP is t hat it is incomplete w.r.t. an y class of Kripk e frames. Ho wev er, a more general top olog ical seman tics for the G ¨ odel–L¨ ob pro v abilit y logic GL has b een know n since the w ork of Simmons [1 3] and Esakia [9]. In the sense of this seman t ics, the diamond mo dality is in terpreted as the top o lo gical deriv ativ e op erator acting on a scat- tered top ological space. The idea to extend this a pproac h to the p olymo dal logic GLP comes quite na turally . 1 The language of GLP has den umerably man y mo dalities eac h of whic h individually b ehav es lik e the one of GL and can therefore b e in terpreted as a deriv ativ e op erato r of a p olytop ological space ( X , τ 0 , τ 1 , . . . ). The a dditio nal axioms of GLP imply certain dep endencies b et w een the scattered top ologies τ i , whic h lead the authors o f [6] to the concept o f GLP-sp a c e . Th us, GLP- spaces provide an adequate top ological seman t ics fo r GLP . The question of completeness o f GLP w.r.t. this seman t ics turned out t o b e more difficult. The main con tribution of [6 ] was to show that the frag- men t o f GLP with only tw o mo dalities was top ologically complete. Ho w- ev er, a lr eady for the fra gmen t with three mo dalities the question remained op en. The presen t pap er answ ers this question p ositiv ely for the language with infin itely man y mo dalit ies and sho ws that GLP is complete w.r.t. the seman tics of GL P-spaces. 2 Preliminaries GLP is a prop ositional mo da l logic form ulated in a language with infinitely man y mo dalities [0], [1], [2], . . . . As usual, h n i ϕ stands for ¬ [ n ] ¬ ϕ , and ⊥ is the logical constan t ‘false’. GLP is give n b y the follo wing axiom sc hemata and inference rules. Axioms: (i) Bo olean tautolog ies; (ii) [ n ]( ϕ → ψ ) → ([ n ] ϕ → [ n ] ψ ); (iii) [ n ]([ n ] ϕ → ϕ ) → [ n ] ϕ (L¨ ob’s axiom); (iv) [ m ] ϕ → [ n ] ϕ , for m < n ; (v) h m i ϕ → [ n ] h m i ϕ , for m < n . Rules: (i) ⊢ ϕ, ⊢ ϕ → ψ ⇒ ⊢ ψ (mo dus p o nens); (ii) ⊢ ϕ ⇒ ⊢ [ n ] ϕ , for each n ∈ ω (nec essitation). In other words, for eac h mo dality , GLP con tains the axioms and infer- ence rules of the G ¨ odel-L¨ ob L ogic GL . Axioms (iv) and (v) relate differen t mo dalities t o o ne another. Neighb orho o d sem a ntics for mo dal logic can b e seen b oth as a generaliza- tion of Kripk e seman tics and as a particular kind of alg ebraic semantics . Let 1 Leo Esakia raised this question several times in conv er sations with the first author. 2 X b e a nonempt y set and let δ n : P ( X ) → P ( X ), for eac h n ∈ ω , b e some unary op erators acting on the b o olean a lgebra o f a ll subsets of X . Suc h a structure X will b e called a neighb orho o d fr am e . A valuation on X is a map v : V ar → P ( X ) from the set of prop ositional v ariables to the p ow erset of X , whic h is extended to all formulas in the langua g e of GLP as follows : • v ( ϕ ∨ ψ ) = v ( ϕ ) ∪ v ( ψ ), v ( ¬ ϕ ) = X \ v ( ϕ ), v ( ⊥ ) = ∅ , • v ( h n i ϕ ) = δ n ( v ( ϕ )), v ([ n ] ϕ ) = ˜ δ n ( v ( ϕ )) , where ˜ δ n ( A ) := X \ δ n ( X \ A ), for an y A ⊆ X . A form ula ϕ is valid in X , denoted X  ϕ , if v ( ϕ ) = X for all v . The lo gic of X is the set Log( X ) of all formulas v alid in X . Next w e observ e that an y neighbor ho o d fra me o f GLP is, essen tially , a p olytop olo gical space, in whic h all op erators δ n can b e in terpreted as t he deriv ed set op erators. Supp ose ( X , τ ) is a to p ological space. The derive d set op er ator on X is the map d τ : P ( X ) → P ( X ) asso ciating with each A ⊆ X its set of limit p oin ts, denoted d τ ( A ). In other w ords, x ∈ d τ ( A ) iff ev ery op en neighborho o d of x con tains a p oint y 6 = x suc h that y ∈ A . W e shall write dA for d τ ( A ) whenev er the top ology τ is giv en from the context. A top ological space ( X , τ ) is called sc atter e d if ev ery nonempty subspace A ⊆ X has an isolated p oin t. A p olytop olog ical space ( X , τ 0 , τ 1 , . . . ) is called a GLP-sp ac e (cf. [6]) if the follo wing conditions hold, for eac h n < ω : • τ n is scattered; • τ n ⊆ τ n +1 ; • d τ n ( A ) is τ n +1 -op en, for eac h A ⊆ X . This concept is justified b y the basic observ ation tha t GLP-spaces are equiv- alen t to the neigh b or ho o d frames v alidating all the axioms of GLP . Th us, to each GLP -space w e asso ciate a neigh b orho o d frame ( X , d 0 , d 1 , . . . ) where d n = d τ n , for eac h n < ω . Then the following prop osition holds. Prop osition 2.1. (i) If ( X , τ 0 , τ 1 , . . . ) is a GLP-sp a c e, then i n t he as- so ciate d neighb orho o d fr ame al l the the o r ems of GLP ar e valid: ( X , d 0 , d 1 , . . . )  GLP . (ii) Supp ose ( X , δ 0 , δ 1 , . . . ) is a neigh b orho o d fr ame such that X  GLP . Then ther e a r e n atur al ly d efine d top ol o gies τ 0 , τ 1 , . . . on X such that δ n = d τ n , for e ac h n < ω . Mor e over, ( X , τ 0 , τ 1 , . . . ) is a GLP-sp ac e. A pro of of this prop osition builds up on the ideas of H. Simmons [13] and L. Esakia [9, 10], whic h b y now ha v e b ecome almost folklore, but it is somewhat lengthy . F or the reader’s conv enience we give this pro of in the App endix. By Prop o sition 2.1, the study of neighborho o d semantics for GLP b e- comes the s tudy of G LP-spaces. Since GLP is w ell-known t o b e incomplete w.r.t. any class of Kr ipke frames the fo llowing question naturally ar ises: 3 • Is GLP c omplete w.r.t. ne igh b orho o d seman tics? In other w ords, w e ask whether there is a suitable class of neigh b orho o d frames C suc h that an y f orm ula is v alid in all fra mes in C iff it is prov able in GLP . Equiv alen tly , this problem w as stated in [6] as the question whether GLP is the logic o f the class of all GLP-spaces. This question w as p ositively answ ered for the la ng uage with only tw o mo dalities in [6]. Ho wev er, for the case of three or mor e mo dalities ev en a more basic problem was o p en: • Is there a GLP -space in whic h a ll the top o logies ar e non-discre te? Some difficulties surrounding these pr o blems are exp osed in the pap ers [6, 5, 3 ]. G iv en a scattered space ( X , τ ) we can define a new top ology τ + on X as the coarsest to p ology containing τ ∪ { d τ ( A ) : A ⊆ X } . Then ( X , τ , τ + , τ ++ , . . . ) b ecomes a GLP-space whic h w e call a G LP-space natu- r al ly gener ate d f r o m ( X, τ ). As a fundamental example, one can consider the class of GLP-spaces naturally generated from the standard order top ology τ < on the o r dina ls. W e call them or dinal GLP-sp ac es . Quite unexp ectedly , these spaces turned out to ha v e some deep relations with set theory , in particular, with stationary reflection. F or example, it can b e sho wn that t he first limit p oin t o f τ + < is the cardinal ℵ 1 , whereas the first limit p oint of τ ++ < is the so-called doubly r efle cting c ar dinal . The existen ce of this (relativ ely w eak) large cardinal is, ho w ev er, indep enden t f r om the axioms o f ZF C . Th us, it is indep enden t from ZF C whether τ ++ is discrete on any ordinal G LP-space. In spite of the ab o v e, the presen t pap er giv es p ositiv e answ ers to b o t h questions formulated a b o v e while firmly standing on the gro unds of ZF C . This is a chiev ed b y dev eloping new top olog ical tec hniques related to the study of maximal rank preserving extensions of scattered to p ologies. In particular, w e in t ro duce a certain class of to p ologies w e call ℓ -maximal and sho w that they are sufficien tly w ell-b eha v ed w.r.t. the op eratio n τ 7→ τ + . As another ingredien t of the top ological completeness pro of , w e in t r o duce an op era t io n on scattered spaces called d -pr o duct . It can b e seen as a gener- alization of the usual m ultiplicatio n op eratio n on the ordinals (considered as linear o r derings) to ar bit r a ry scattered spaces. W e think that this op eratio n could b e of some in terest in its own right. The pap er is organized as fo llo ws. In Section 3 w e in tro duce some useful standard notions relat ed to scattered spaces and prov e a few facts a b out the Can to r–Bendixon r a nk function. Maximal rank preserving and ℓ -maximal spaces are in tro duced in Section 4 . In Section 5 we show ho w this tec h- niques allo ws one to build a non-discrete GLP-space. Section 6 essen tially deals with logic and con tains a reduction o f the top ological completeness theorem to some statemen t of purely top olog ical and com binatorial nature (main lemma). The r est of the pap er is dev oted to a pro of of t his lemma. In Section 7 the d -pro duct o p eration is in tro duced and a few basic prop erties of this op eration a re established. Using d - pro ducts, as w ell as the tech niques 4 of Sections 4 and 5, t w o ba sic constructions on GLP- spaces are presen ted in Section 6. Finally , Se ction 8 contains a proof of the main lemma. 3 Scattered spaces, ranks and d -maps Giv en a scattered space X = ( X , τ ) one can define a transfinite Cantor– Bendixon se quenc e o f closed subsets d α X of X , for any ordinal α , as follows: • d 0 X = X ; d α +1 X = d ( d α X ) and • d α X = T β <α d β X if α is a limit ordinal. Since X is a scattered space, d α +1 X ⊂ d α X is a strict inclusion unless d α X = ∅ . Therefore, from cardinalit y considerations, fo r some ordinal α we m ust ha v e d α X = ∅ . Call the least suc h α the Cantor–Bendixon r an k of X a nd denote it b y ρ ( X ). The r ank function ρ X : X → On is defined b y ρ X ( x ) := min { α : x / ∈ d α +1 ( X ) } . Notice that ρ X maps X on to ρ X ( X ) = { α : α < ρ ( X ) } . Also, ρ X ( x ) ≥ α iff x ∈ d α X . W e omit t he subscript X whe nev er there is no danger of confusion. Example 3.1 . Let Ω b e an ordinal equipp ed with its left top olo gy , that is, a subset U ⊆ Ω is op en iff ∀ α ∈ U ∀ β < α β ∈ U . Then ρ ( α ) = α , f o r all α . Example 3.2 . Let Ω b e an ordinal equipped with its or der top olo gy generated b y { 0 } and the in terv als ( α, β ], for all α < β ≤ Ω. Then ρ is the function r defined b y r (0) = 0; r ( α ) = β if α = γ + ω β , f o r some γ , β . By the Can tor normal form t heorem, for an y α > 0, suc h a β is uniquely defined. A map f : X → Y b et w een top ological spaces is called a d-map if f is con t inuous, op en and p ointwise discr ete , that is, f − 1 ( y ) is a discrete sub- space of X for each y ∈ Y . d -maps a r e w ell-kno wn to satisfy the pro p erties expresse d in the following lemma (see [7]). Lemma 3.1. (i) f − 1 ( d Y ( A )) = d X ( f − 1 ( A )) , for any A ⊆ Y ; (ii) f − 1 : ( P ( Y ) , d Y ) → ( P ( X ) , d X ) is a homomorphism of mo dal algebr as; (iii) If f is onto, then Log( X ) ⊆ Log ( Y ) . In fact, (i) is easy to ch ec k directly; (ii) follows from (i) and (iii) from (ii). F ro m (i) w e easily obtain t he follo wing corollar y by tra nsfinite induction. Corollary 3.2. Supp os e f : X → Y is a d -map. Th en, for e ach or dinal α , d α X X = f − 1 ( d α Y Y ) . 5 The following lemma states that the rank function, when the ordinals are equipped with their left top ology , becomes a d -map. It is a lso uniquely c har a cterized by this pro p ert y . Lemma 3.3. L et Ω b e the or dinal ρ ( X ) taken with its left t op olo gy. Then (i) ρ X : X ։ Ω is an on to d -map; (ii) If f : X → λ is a d -ma p, w h er e λ is an or dinal with its left top olo gy, then f ( X ) = Ω and f = ρ X . Pro of. Let ρ de note ρ X . (i) ρ is con tin uo us, b ecause t he set ρ − 1 [0 , α ) = X \ d α X is op en. ρ b eing op en means that, for eac h op en U ⊆ X , whenev er α ∈ ρ ( U ) and β < α o ne has β ∈ ρ ( U ). Fix an x ∈ U suc h that ρ ( x ) = α . Consider the set X β := ρ − 1 ( β ) = d β X \ d ( d β X ). F or an y subset A of a scattered space w e hav e d ( A ) = d ( A \ dA ), hence dX β = d ( d β X ) ⊆ d α X . Since ρ ( x ) = α it follo ws that x ∈ dX β . Hence U ∩ X β 6 = ∅ , t ha t is, β ∈ ρ ( U ). ρ b eing p oin t wise discrete means X α = ρ − 1 ( α ) is discrete, for each α . In fact, X α = d α X \ d ( d α X ) is the set of isolated p oin ts of d α X . Thu s, it cannot help b eing dis crete. (ii) Since f is a d -map, b y Corollary 3.2 w e obtain t ha t f − 1 [ α, λ ) = d α X , for eac h α < λ . Hence, f − 1 ( α ) = ρ − 1 ( α ), f o r eac h α < λ , that is, f = ρ a nd f ( X ) = ρ ( X ) = Ω. ⊣ Corollary 3.4. If f : X → Y is a d -map, then ρ X = ρ Y ◦ f . Pro of. Clearly , ρ Y ◦ f : X → Ω is a d -map. Statemen t (ii) of the previous lemma yields the result. ⊣ Note that if U ∈ τ is op en, then the image of U under the map ρ is alw ay s a left w ards closed inte rv al of ordinals and thus is itself an ordinal, whic h w e denote ρ ( U ). W e denote the complemen t of a set d α X by O α ( X ) or simply O α when there is no danger of confusion. 4 Maximal and ℓ -maximal to p ologie s First we intro duce tw o notions: t ha t of a r ank pr eservin g extensi o n of a scattered top olo g y , and a more restrictiv e notion of an ℓ -extension . The first one is quite natur a l and it will help us to build a non-discrete GLP- space. The second is the one w e actually need for the pro of of the top olog ical completeness theorem. Definition 4.1. Let ( X , τ ) b e a scattered space. • A to p ology σ on X is called a rank preserving extension of τ , if σ ⊇ τ and ρ σ ( x ) = ρ τ ( x ), for all x ∈ X . 6 • σ is an ℓ -extension of τ , if it is a rank-preserving extension o f τ and the iden tity function id : ( X , τ ) → ( X, σ ) is con t inuous at all p oints of success or rank, that is, ( ℓ ) for a n y U ∈ σ and any x ∈ U with ρ ( x ) / ∈ Lim there exists V ∈ τ suc h that x ∈ V ⊆ U . W e note that b o th notions a re transitive and, in fact, define par tial o r ders on the set of all scattered top ologies on X . The follo wing observ ation will b e rep eatedly used b elo w. Lemma 4.2. σ is a r ank pr eservin g ex tens i o n of τ iff ρ τ : ( X , σ ) ։ ρ τ ( X ) is an op en map iff ρ τ ( U ) is leftwar ds clos e d, fo r e ach U ∈ σ . This statement f ollo ws from Lemma 3.3. W e are in terested in the maximal rank preserving and maximal ℓ - extensions. These are naturally defined a s follo ws. Definition 4.3. (i) ( X , τ ) is m a ximal 2 if ( X , τ ) do es not hav e an y prop er rank-preserving extensions, in other w ords, if ∀ σ ( σ % τ ⇒ ∃ x ρ σ ( x ) 6 = ρ τ ( x )) . (ii) ( X , τ ) is ℓ -maxima l if ( X , τ ) do es not hav e any pro p er ℓ -extensions. It is w orth noting t hat any maximal top olo g y is ℓ -maximal, but not con- v ersely . Lemma 4.4. (i) Any ( X , τ ) has a maximal extension ; (ii) Any ( X , τ ) has an ℓ -maximal ℓ -extension. Pro of. Consider the set of all ( ℓ -)extens ions o f a giv en top ology τ ordered b y inclusion. W e ve rify , for each o f the t w o orderings, that ev ery chain in it has an upp er b ound. The result then follows b y Z o rn’s lemma. Supp ose ( τ i ) i ∈ I is a linear c hain of extensions. Then the top olo gy σ gen- erated b y the union υ = S i ∈ I τ i is a pparen tly a scattered top ology containing τ . Note that υ is clos ed under finite in tersections and thus serv es as a base for σ . Let ρ : X ։ Ω b e the common rank function o f eac h of the τ i . In order to apply Lemma 4.2 w e c hec k that ρ is op en w.r.t. σ . In fact, any basic U ∈ υ is op en in the sense of some τ i , and hence ρ ( U ) must b e op en in Ω. Lemma 4.2 show s that ρ is the rank function o f σ . Hence (i) holds. Supp ose no w that ( τ i ) i ∈ I is a c hain of ℓ -extensions. Since any ℓ -extension is an extension, σ (defined as ab ov e) is an extension of τ . T o chec k the condition ( ℓ ) supp o se U ∈ σ is giv en and x ∈ U is suc h that ρ ( x ) / ∈ Lim. Since σ is generated by the ba se υ , there exists U ′ ∈ υ with x ∈ U ′ ⊆ U . It follow s that U ′ ∈ τ i for some i . As τ i is an ℓ -extension o f τ , there exists V ∈ τ suc h that x ∈ V ⊆ U ′ . Since U ′ ⊆ U , w e are done. ⊣ 2 In the sta nda rd ter minology used in general to po logy , maximal or maximal sc atter e d would mean something entirely differen t than defined here. Throughout this pap er we use the term maximal as a shortha nd for maximal sc att er e d with the given r ank function . 7 Next w e pro v e a work able c haracterization of ℓ -maximal t o p ologies. Lemma 4.5. L et ( X , τ ) b e a sc atter e d sp ac e and ρ its r ank function. Then X is ℓ -maximal iff the fol lowing c ondition holds . ( l m ) F or a ny x ∈ X with r ank λ = ρ ( x ) ∈ Lim and a ny op en V ⊆ O λ , either V ∪ { x } ∈ τ or ther e is a neig h b orho o d U of x such that ρ ( V ∩ U ) < λ . In tuitively , condition ( l m ) means that in the neigh b orho o d of a p oint x of limit r a nk any op en set V is either v ery large (con ta ins a punctured neigh b orho o d of x ), o r relativ ely small (there is a punctured neighborho o d whose in tersection with V has b ounded rank). Pro of. (only if ) Supp ose the condition ( lm ) is not met. Thus , t here exists an x ∈ X with ρ ( x ) = λ ∈ Lim and an op en V 0 ⊆ O λ suc h that V := V 0 ∪ { x } is not op en and ρ ( U ∩ V 0 ) = λ , for any neigh b orho o d U of x . Let us generate a new top ology σ by adding V to τ . W e claim that σ is an ℓ -extension of τ . F irst, we observ e that t he neigh b or ho o d filter at any p oin t z ∈ X , z 6 = x , did not change. In fact, any σ - neigh b orho o d W of z either con tains a τ -neigh b orho o d of z or con tains a subset o f the form V ∩ U where U ∈ τ and z ∈ V ∩ U = ( V 0 ∩ U ) ∪ { x } . In the fo rmer case w e ar e done. In the latter case, if z 6 = x , w e ha v e z ∈ V 0 ∩ U ∈ τ and V 0 ∩ U ⊆ W . F ro m this observ ation we conclude that id : ( X , τ ) → ( X , σ ) is con tinuous at all the p oints z 6 = x , in particular, condition ( ℓ ) holds. W e sho w that ρ σ = ρ b y applying Lemma 4.2. T o c heck that ρ : ( X , σ ) → Ω is op en it is sufficien t to show that ρ ( W ) is a neigh b orho o d of λ = ρ ( x ) (in the left top olo gy) for an y σ -neigh b orho o d W of x . F or all the other p oints the statemen t is ob vious b y the previous obse rv ation. W e kno w that W c on tains a set of the fo rm V ∩ U with x ∈ U ∈ τ . Clearly , V ∩ U = ( V 0 ∩ U ) ∪ { x } . By the c ho ice of V 0 , w e ha v e ρ ( V 0 ∩ U ) = λ and hence ρ ( W ) ⊇ ρ ( V ∩ U ) = [0 , λ ] is a neighborho o d of λ , as required. Th us, σ is a prop er ℓ - extension of τ , hence X is not ℓ - maximal. (if ) Supp ose X is not ℓ -maximal and let σ b e its prop er ℓ - extension. Then the map id : ( X , τ ) → ( X , σ ) is no t contin uous at certain points. Let x ∈ X b e suc h a p oint with t he least rank ρ ( x ) = λ . It follow s from condition ( ℓ ) that λ ∈ Lim. Since the map id is not con t inuous at x , there exists a σ -op en neigh b orho o d V of x w hic h con tains no τ -op en neigh b o r ho o d of x . Denote V 0 := V ∩ O λ . It is clear that V 0 ∈ σ . It follows from the minimalit y o f λ that V 0 ∈ τ . F rom t he discon tinuit y of id at x w e may conclude that V 0 ∪ { x } 6∈ τ . Ho w ev er, { x } ∪ V 0 = V ∩ ( { x } ∪ O λ ) ∈ σ , hence, f o r a ny τ -neigh b orho o d U of x w e ha v e ( U ∩ V 0 ) ∪ { x } = U ∩ ( V 0 ∪ { x } ) is a σ -neigh b orho o d of x . It follo ws that ρ ( U ∩ V 0 ) = λ . Thus x and V 0 witness that t he condition ( lm ) is violated fo r τ . ⊣ Our next ob jectiv e is to sho w tha t whenev er f : X → Y is an o nto d -map and Y ′ is an y ℓ -maximal ℓ -extension of Y , one can alw ays find a suitable ℓ -maximal ℓ -extension X ′ of X so that f : X ′ → Y ′ is still a d -map. W e need an auxiliary lemma. 8 Lemma 4.6. L et f : X → Y b e a d -map b etwe en a sc atter e d sp a c e X = ( X , τ ) a n d an ℓ -maxim al sp ac e Y = ( Y , σ ) . L et X ′ = ( X, τ ′ ) b e any ℓ - extension of X . Then f : X ′ → Y i s also a d -map. Pro of. That f : X ′ → Y is c on tin uous and point wise discrete follo ws from the f act that τ ′ ⊇ τ . W e only hav e to sho w that f : X ′ → Y is o p en. F or the sak e o f con tradiction supp ose f is not. Then there exists a p oin t x ∈ X ′ and a neigh b orho o d U ∈ τ ′ of x suc h that f ( U ) do es not con ta in a neigh b orho o d of y = f ( x ). W e can tak e suc h an x o f the minimal p ossible rank λ . This ensures that the restriction of f to the subspace O λ ( X ′ ) is op en, hence a d -map. (Since X ′ is a rank preserving exten sion of X , the set O λ = O λ ( X ) is the same as O λ ( X ′ ).) Since id : X → X ′ is con tin uous at the p oints of non-limit ranks and f : X → Y is a d - map, w e observ e that λ ∈ Lim. Otherwise, for a sufficien tly small τ -neigh b orho o d V of x w e w ould ha v e V ⊆ U , and then f ( V ) ⊆ f ( U ) w ould b e a σ -neighbor ho o d of f ( x ). Since O λ ∈ τ , we ma y assume that the se lected neigh b orho o d U has the form U = U 0 ∪ { x } where U 0 ⊆ O λ and U 0 ∈ τ . Th us, ρ ( x ) = λ ∈ Lim, V 0 := f ( U 0 ) is op en, and V := f ( U ) = V 0 ∪ { y } is not op en in Y . Since Y is ℓ -maximal, b y Lemma 4.5 we obtain an open neighborho o d W of y suc h that β := ρ ( V 0 ∩ W ) < λ . W e notice that f ( U 0 ∩ f − 1 ( W )) = V 0 ∩ W . Hence, ρ ( U 0 ∩ f − 1 ( W )) = β . Since f − 1 ( W ) ∈ τ and U ∈ τ ′ w e obtain that U 1 := U ∩ f − 1 ( W ) = ( U 0 ∩ f − 1 ( W )) ∪ { x } is a τ ′ -op en neigh b orho o d of x . Therefore, on the one hand, ρ ( U 1 ) = ρ τ ′ ( U 1 ) = [0 , λ ], as τ ′ is a rank preserving extens ion of τ . How ev er, on the other ha nd, ρ ( U 1 ) = ρ (( U 0 ∩ f − 1 ( W )) ∪ { x } ) = β ∪ { λ } , a con tradiction. ⊣ Lemma 4.7. L et X = ( X , τ ) and Y = ( Y , σ ) b e sc atter e d sp ac es, let Y ′ = ( Y , σ ′ ) b e an ℓ -max imal ℓ -extension of Y and let f : X → Y b e a d -ma p . Then ther e exists an ℓ -maximal ℓ -extensio n X ′ = ( X, τ ′ ) of X s uch that f : X ′ → Y ′ is a d -map. X d / / lm   Y lm   X ′ d / / Y ′ Pro of. It is easily se en that the collection θ = { f − 1 ( U ) : U ∈ σ ′ } qualifies for a top ology on X . Since f : X → Y is con tin uous, θ con tains τ . It is readily seen that f : ( X , θ ) → ( Y , σ ′ ) is a d -map. Th us θ is a rank preserving extension of τ . T o see that the condition ( ℓ ) is met, tak e any x ∈ X o f successor rank and an y f − 1 ( V ) ∋ x suc h that V ∈ σ ′ . Since f ( x ) is of the same ra nk as x , b y condition ( ℓ ) applied to σ ′ , there exists U ∈ σ with f ( x ) ∈ U ⊆ V . It follo ws that x ∈ f − 1 ( U ) ⊆ f − 1 ( V ) and f − 1 ( U ) ∈ τ . Therefore, θ is an ℓ -extension of τ . T ak e a ny ℓ -maximal ℓ - extension τ ′ of θ . By Lemma 4.6 we obta in that f : ( X ′ , τ ′ ) → Y ′ is an onto d -map. Since τ ′ is also an ℓ -maximal ℓ - extension of τ , the pro of is finished. ⊣ 9 5 Buildin g a n on-disc rete GLP-space Recall that the next top olo gy τ + on X is generated b y τ and { d ( A ) : A ⊆ X } . Let X + denote the space ( X, τ + ). The follo wing lemma giv es a useful c har a cterization of the next top o logy for ℓ -maximal spaces. Lemma 5.1. S upp ose ( X , τ ) is ℓ -maximal. Then τ + is gener ate d by τ and the sets { d β +1 ( X ) : β < ρ ( X ) } . Pro of. Let ( X , τ ) b e ℓ -maximal and let τ ′ denote the top ology generated b y τ and the sets { d β +1 ( X ) : β < ρ ( X ) } . It is clear that eac h set d β +1 ( X ) = d ( d β X ) is o p en in τ + . W e sho w the con v erse. Let A ⊆ X , we sho w that d ( A ) is op en in τ ′ . Consider an y x ∈ d ( A ) and let α = ρ ( x ). If α is no t a limit ordinal, { x } is op en in τ ′ . In fact, since ρ is a d -map, ρ − 1 ( α ) is discrete as a subspace of ( X , τ ). Moreo v er, ρ − 1 ( α ) = d α ( X ) \ d α +1 ( X ), hence it is clop en in τ ′ . It follows that x is isolated in τ ′ . Supp ose α ∈ Lim and let C denote the in terior of O α \ A . Since x ∈ dA w e ha v e { x } ∪ C / ∈ τ . Hence, by condition ( l m ), there is an op en U ∈ τ with x ∈ U and a β < α suc h that U ∩ C ⊆ O β . Consider V := U ∩ d β +1 X . Since U is op en in τ , V is op en in τ ′ . Moreov er, x ∈ V . Th us, we only ha v e to sho w that V ⊆ dA . Supp ose the contrary that z ∈ V \ dA for some z . Then there exists an op en set U z ∪ { z } suc h that U z ∩ A = ∅ and U z ⊆ O α . It follow s that U z ⊆ C and hence U z ∩ U ⊆ O β . Since z ∈ V ⊆ U , w e hav e that U ′ := ( U z ∩ U ) ∪ { z } = ( U z ∪ { z } ) ∩ U is a n op en neigh b orho o d of z . As ρ is an op en map, ρ ( U ′ ) must b e left w ards closed. W e ha v e ρ ( z ) ≥ β , since z ∈ d β +1 X , how ev er ρ ( U z ∩ U ) ⊆ ρ ( O β ) ⊆ β , a contradiction. ⊣ Lemma 5.2. S upp ose ( X , τ ) is ℓ -maximal and f : X → Y a d -map. Then f is a d -map b etwe en X + and Y + . Pro of. W e only ha ve to sho w that f : X + → Y + is op en. F rom the previous lemma we know that τ + is generated by τ and d β +1 X X for β < α . Consider a τ + -op en set of the f o rm A ∩ d β +1 X X . Since f − 1 ( d β +1 Y Y ) = d β +1 X X ( f is rank preserving), we ha v e f ( A ∩ d β +1 X X ) = f ( A ) ∩ d β +1 Y Y , whic h is op en in Y + . ⊣ R emark 5.1 . In general, t he ‘next top o logy’ op eration is non-monotonic: There is a space X such tha t X + is discrete while ( X ′ ) + is not, where X ′ is some maximal extension of X . Let Ω denote an ordinal with its left top olo gy . It is easy to c hec k (see [6]) that Ω + coincides w ith the usual order top ology on Ω. Let r denote its rank function (se e a b ov e). In gene ral, for a n a r bit r a ry scattered space X let ρ + X denote the rank function of X + . Corollary 5.3. If X is ℓ -maximal, then ρ + X = r ◦ ρ X . 10 Pro of. Let Ω := ρ ( X ) b e the rank of X . Consider the d -map ρ : X ։ Ω. By Lemma 5.2, ρ : X + ։ Ω + is a d -map. Since r is the rank function of Ω + , r : Ω + → Ω is a lso a d -map. Hence, r ◦ ρ : X + → Ω is a d -map a nd coincides with the rank function of X + . ⊣ R emark 5.2 . F or an a rbitrary scattered space X w e only hav e ρ + X ≤ r ◦ ρ X . No w we are ready to sp ecify a suitable class of G LP-spaces whic h will b e used for the top ological completeness pro o f. Definition 5.4. Let ( X , τ ) b e a scattered space. A p oly-top olog ical space ( X , τ 0 , τ 1 , . . . ) is called an lme-sp ac e b ase d on τ if 3 τ 0 is an ℓ - maximal ℓ - extension of τ and, for each n , τ n +1 is an ℓ -maximal ℓ -extension o f τ + n . Clearly , any lme-space is a G LP-space. ( X , τ 0 , τ 1 , . . . ) is called an or dinal lme-sp ac e if X is an ordinal (or an inte rv al of the o r dina ls) a nd τ is the order top ology o n X . Giv en an lme-space X , let ρ n denote the ra nk function of τ n . Lemma 5.5. ρ n +1 = r ◦ ρ n . Pro of. τ n +1 has the same rank function as τ + n , b eing its ℓ - extension, hence ρ n +1 = ρ + n . By Coro llary 5.3, ρ + n = r ◦ ρ n . ⊣ No w we can giv e an example of a GLP-space in whic h all topolo gies are non-discrete. T ake any scattered space ( X , τ ) whose rank Ω satisfies ω Ω = Ω, for example, X = ε 0 with the order t o p ology . Generate some lme-space ( X , τ 0 , τ 1 , . . . ) based on τ . Then clearly ρ n ( X ) = r n ( ρ 0 ( X )) = r n (Ω) = Ω, for eac h n . In pa r t icular, an y top olo gy τ n is non-discrete. Th us, we ha v e pro v ed Theorem 5.6. Ther e is a c ountable GLP-sp ac e ( X , τ 0 , τ 1 , . . . ) such that e ach τ n is no n-discr ete. 6 T op ological c o mpletene s s of GLP In this section w e reduce the construction of a poly-top ological space whose logic is GLP to a tec hnical lemma. The rest o f the pap er is dev oted to a pro of of this lemma. Our pro of of t o p ological completeness will mak e use of a subsystem of GLP in tro duced in [4] and denoted J . This logic is defined by w eak ening axiom (iv) of GLP to the following axioms (vi) and (vii) bo th of whic h are theorems of GLP : (vi) [ m ] ϕ → [ n ][ m ] ϕ , for n ≥ m ; (vii) [ m ] ϕ → [ m ][ n ] ϕ , for n > m . J is the logic of a simple class of fr a mes, which is established by standard metho ds [4, Theorem 1]. 3 The abbreviation lme stands for limit maximal extens ion . 11 Lemma 6.1. J is sound a nd c omplete w ith r esp e ct to the class of (finite) fr ames ( W , R 0 , R 1 , . . . ) such that, for al l x, y , z ∈ W , 1. R k ar e tr ansitive and dual ly wel l-founde d binary r elations; 2. If xR n y , then xR m z iff y R m z , for m < n ; 3. xR m y and y R n z imply xR m z , for m < n . Let R ∗ n denote the transitiv e closure of R n ∪ R n +1 ∪ . . . , and let E n denote the reflexiv e, symmetric, transitive closure of R ∗ n . Ob viously , eac h E n +1 refines E n . W e call each E n equiv alence class a n -she et . By 2 ., all p oin ts in an n -sheet are R m incomparable, for m < n . But R n defines a natural ordering o n n + 1-sheets in the following sense: if α and β are n + 1- sheets, then αR n β , iff ∃ x ∈ α ∃ y ∈ β xR n y . By the standard tec hniques, one can impro v e o n Lemma 6.1 to show that J is complete fo r suc h frames, in whic h the set of n + 1-sheets con ta ined in each n -shee t is a tree under R n , and if αR n β then xR n y for all x ∈ α , y ∈ β (see [4, Theorem 2 and Corollary 3.3]). Ev ery suc h structure is auto matically a J- f rame, w e call suc h frames tr e e-like J-fr ames . As sho wn in [4], GLP is reducible to J in the fo llo wing sense. Let M ( ϕ ) := ^ i n . It is obvious that X ω 2 ϕ . ⊣ The to p ological completeness theorem can also b e stated in a stronger uniform w a y . Recall that ε 0 is the suprem um of the coun t able ordinals ω k recursiv ely defined b y ω 0 = 1 and ω k +1 = ω ω k . Theorem 6.10. Th er e is an or din a l lme-sp ac e X = ( ε 0 , τ 0 , τ 1 , . . . ) such that Log( X ) = GLP . Pro of. Let ϕ 0 , ϕ 1 , . . . b e an en umeration of all the formu las of L ω . Using Theorem 6.9 select o rdinal lme-spaces X i = ([1 , λ i ] , τ i 0 , τ i 1 , . . . ) in suc h a w a y that X i 2 ϕ i , for eac h i < ω . W e can assume that λ i < ε 0 , for eac h i < ω . Consider the ordinal λ := P i<ω λ i . The interv al [1 , λ ) is naturally iden tified with the disjoint union F i<ω [1 , λ i ]. Hence, w e can define the top olog ies τ i on [1 , λ ) in suc h a wa y that X = ([1 , λ ) , τ 0 , τ 1 , . . . ) is isomorphic to the top ological sum F i<ω X i . Then clearly λ ≤ ε 0 and each formula ϕ such that GLP 0 ϕ is refutable on X . Hence, Log ( X ) = GLP . In fact, λ m ust coincide with ε 0 . Assume λ < ω n . Then f o r the top olog y τ n w e hav e ρ n ( X ) ≤ r n +1 ( ω n ) = 0 by Theorem 5.6. How eve r, this con tradicts the fact that the unpro v able form ula [ n ] ⊥ is refutable in X . Therefore, λ = ε 0 and X is isomorphic to an o r dinal lme-space based on ε 0 . ⊣ In order to pr ov e the main lemma we in tro duce the notion of d-pr o duct of scattered spaces. 7 d -pro duct Definition 7.1. Let ( X , τ X ) and ( Y , τ Y ) b e an y top o lo gical spaces. W e define their d -pr o duct sp ac e ( Z , τ Z ), denoted X ⊗ d Y , as follows. Notice that Y is a union of its isolated p oin ts a nd limit p oin ts, Y = iso ( Y ) ∪ d ( Y ). F or all y ∈ iso ( Y ), let X y denote pairwise disjoin t copies of X , and let i y : X → X y b e the asso ciated ho meomorphism maps. Let Z 0 b e the top ological sum of { X y : y ∈ iso ( Y ) } , that is, Z 0 := F y ∈ i so ( Y ) X y . Z 0 can also be de fined as the cartesian pro duct X × iso ( Y ) of X and the discrete space iso ( Y ). Pro jection π 0 : Z 0 ։ X is defined in a natural w a y , that is, π 0 ( i y ( x )) = x , for eac h y ∈ iso ( Y ). Let Z 1 b e a copy of the set d Y disjoint from Z 0 , and π : Z 1 → d Y the asso ciated bijection. P ut Z := Z 0 ∪ Z 1 . W e set π 1 ( x ) := y , if x ∈ X y 15 and y ∈ iso ( Y ), and π 1 ( x ) := π ( x ), if x ∈ Z 1 . It is a lso conv enien t to let X y := { y } , if y ∈ d Y , thus, X y = π − 1 1 ( y ), for eac h y ∈ Y . Let a to p ology τ Z on Z b e generated b y the one inherited from Z 0 (with the basic op en sets { i y ( V ) : V ∈ τ X , y ∈ iso ( Y ) } ) and b y all sets { π − 1 1 ( U ) : U ∈ τ Y } . W e note that, fo r eac h y ∈ iso ( Y ) a nd U ⊆ Y , the set π − 1 1 ( U ) ∩ X y is either empt y or coincides with X y . Hence, the ab ov e basic op en sets fo rm a base of top ology τ Z . It follows t ha t an y o p en set of τ Z has the fo r m V ∪ π − 1 1 ( U ), where V is op en in Z 0 and U ∈ τ Y . (P ay atten tion that this unio n nee d no t b e disjoin t.) It also fo llo ws that the to p ologies induced from Z on Z 0 and Z 1 are homeomorphic to tho se of the pro duct X × iso ( Y ) and Y , resp ectiv ely . As a t ypical example, consider the d -pro duct of t wo compact ordinal spaces [1 , λ ] and [1 , µ ] tak en with their in terv al top ologies. W e claim that [1 , λ ] ⊗ d [1 , µ ] is isomorphic to [1 , λµ ] (with the interv a l to p ology). Indeed, ev ery α ∈ [1 , λµ ] either has the fo r m λβ with β ∈ Lim , or b elongs to a (clop en) interv al I β +1 := [ λβ + 1 , λ ( β + 1)] isomorphic to [1 , λ ]. In the former case, α = λβ corr esp o nds to a limit p oint β ∈ [1 , µ ]. In the latter case, α b elongs to a copy of [1 , λ ] corresp onding to an isolated p oin t β + 1 o f [1 , µ ]. The describ ed bijection is, in fact, a homeomorphism: a n in terv al of the form ( δ , α ], where δ < α ≤ λµ is a neigh b orho o d of α in the d -pr o duct top ology . This is clear if α ∈ I β +1 . If α = λγ with γ ∈ Lim, then for all sufficien tly la r ge β < γ , I β ⊆ ( δ, α ], if β ∈ Suc, and λβ ∈ ( δ , α ], if β ∈ Lim; hence, the claim. The conv erse is also clear: a neigh b orho o d of α in the d -pro duct top ology con tains a suitable in terv a l of the form ( δ , α ]. Lemma 7.2. (i) π 0 : Z 0 ։ X is a d -map; (ii) The map π 1 : Z ։ Y is c ontinuous an d op en. Pro of. (i) This follow s from the fact tha t Z 0 is homeomorphic to the pro d- uct X × iso ( Y ) with iso ( Y ) disc rete. (ii) The con tin uit y of π 1 is clear. T o sho w that it is op en, w e chec k that π 1 ( U ) is op en in Y , f o r eac h basic op en set U of Z . If U is π − 1 1 ( V ) f o r a set V ∈ τ Y , w e are done. If U = i y ( V ), for some nonempt y V ∈ τ X and y ∈ iso ( Y ) , then π 1 ( U ) = { y } ∈ τ Y as w ell. ⊣ The f ollo wing observ a tions will also be helpful. Lemma 7.3. (i) Supp os e x ∈ Z 1 . Then U is a punctur e d neighb orho o d of x in τ Z iff { y ∈ Y : X y ⊆ U } is a punctur e d neighb orho o d of π 1 ( x ) in τ Y . (ii) L et A ⊆ Z , x ∈ Z 1 . Then, x ∈ d Z ( A ) iff π 1 ( x ) ∈ d Y { y ∈ Y : A ∩ X y 6 = ∅ } . Clearly , X ⊗ d Y is scattered if so are X and Y . Let us compute the rank function of X ⊗ d Y . 16 Lemma 7.4. (i) If x ∈ Z 0 then ρ Z ( x ) = ρ X ( π 0 ( x )) . (ii) If x ∈ Z 1 then ρ Z ( x ) = ρ ( X ) + ρ d Y ( π 1 ( x )) . (Obvio usly, 1 + ρ d Y ( y ) = ρ Y ( y ) .) Pro of. F or (i), w e just notice that ρ Z ( x ) = ρ Z 0 ( x ), since Z 0 is op en in Z . Since π 0 : Z 0 → X is a d - map, w e ha v e ρ Z 0 ( x ) = ρ X ( π 0 ( x )). F or (ii) w e first pro v e that Z 1 ⊆ d β Z ( Z ), for eac h β < ρ ( X ). This go es b y transfinite induction on β . The cases when β = 0 or β ∈ Lim are easy . Supp ose the claim is true for all α ≤ β . W e prov e tha t Z 1 ⊆ d β +1 Z ( Z ) = d Z ( d β Z ( Z )). By (i), if β < ρ ( X ) t hen d β Z ( X y ) = i y ( d β X ( X )) 6 = ∅ , for all y ∈ iso ( Y ). Hence, an y y ∈ Z 1 is a limit p oin t of d β ( Z 0 ), hence of d β Z ( Z ), a s required. As a consequence we obtain that d ρ ( X ) Z ( Z ) = Z 1 . Hence, d ρ ( X )+ α Z ( Z ) = d α Z ( Z 1 ) = π − 1 1 ( d 1+ α Y ( Y )), for eac h α . ⊣ Next w e w ould lik e to sho w that d -pro duct is we ll-b ehav ed w.r.t. ℓ - extensions. Lemma 7.5. Supp ose X ′ , Y ′ ar e ℓ -extensions of X , Y , r esp e ctively. T hen X ′ ⊗ d Y ′ is an ℓ -extension of X ⊗ d Y . Pro of. The rank function is preserv ed b y the previous lemma. W e only ha ve to c hec k that the iden tity function id : X ⊗ d Y → X ′ ⊗ d Y ′ is con tin uous at the p oints x of success or rank. Let Z = X ⊗ d Y . If x ∈ Z 0 , the claim follo ws from the hypothesis ab out X ′ . Supp ose x ∈ Z 1 . By Lemma 7.4 ρ Y ( π 1 ( x )) is not a limit. Consider a basic op en neigh b or ho o d V ′ of x in Z ′ = X ′ ⊗ d Y ′ . V ′ has the form π − 1 1 ( U ′ ), where U ′ is a Y ′ -neigh b orho o d of π 1 ( x ). Since Y ′ is an ℓ -extension of Y , there is a Y - neighbor ho o d U ⊆ U ′ suc h that π 1 ( x ) ∈ U . Then x ∈ π − 1 1 ( U ) ⊆ V ′ , a s req uired. ⊣ Lemma 7.6. Supp ose X and Y ar e ℓ -maxima l and ρ ( X ) ∈ Suc . Then X ⊗ d Y is ℓ -maximal. Pro of. W e use Lemma 4.5 . Let Z = X ⊗ d Y and supp ose x ∈ Z and ρ Z ( x ) = λ ∈ Lim. Consider any op en V ⊆ O λ ( Z ) = { z ∈ Z : ρ Z ( z ) < λ } . W e show t ha t either V ∪ { x } is op en, or there is a op en neigh b orho o d U x of x suc h that ρ Z ( V ∩ U x ) < λ . Case 1: x ∈ Z 0 . In this case, V ⊆ O λ ( Z ) ⊆ Z 0 b y Lemma 7.4 (i). Also, Z 0 is ℓ -maximal as a to p ological sum of ℓ -maximal spaces. Hence, the claim follo ws from ℓ -maximality of Z 0 . Case 2: x ∈ Z 1 . In this case we represen t V as a union W ∪ π − 1 1 ( U ), where W is o p en in Z 0 and U in Y . Let y := π 1 ( x ) and let µ := ρ Y ( y ). By Lemma 7.4 (ii) w e ha v e ρ ( X ) + µ ′ = λ whe re µ = 1 + µ ′ . Since λ is a limit ordinal, so is µ (unless µ ′ = 0 and λ = ρ ( X ), but then ρ ( X ) w o uld b e a limit). Hence, w e can use the ℓ - maximalit y of Y for y , µ , and U ⊆ O µ ( Y ). Supp ose ρ Y ( U ∩ U y ) = β < µ , for some op en neighborho o d U y of y in Y . Let U x := π − 1 1 ( U y ). Then U x ∩ π − 1 1 ( U ) = π − 1 1 ( U ∩ U y ) is a neighbor ho o d of x 17 (b y the con tin uit y of π 1 ). W e also hav e ρ Z ( V ∩ U x ) ≤ ρ ( X ) + ρ d Y ( U ∩ U y ) ≤ ρ ( X ) + β < λ . If, on the other hand, U ∪ { y } is open in Y , then π − 1 1 ( U ) ∪ { x } is op en in Z , b y the con tin uit y of π 1 . Hence, so is V ∪ { x } = W ∪ π − 1 1 ( U ) ∪ { x } . ⊣ Consider now t w o spaces X = [1 , λ ] and Y = [1 , µ ] equipped with the in t erv al top ologies. Notice that since X is compact there is an ordinal α ∈ X whose rank is maximal. Then ρ ( X ) = r ( α ) + 1 ∈ Suc. Let X ′ and Y ′ b e an y ℓ -maximal ℓ -extensions of X a nd Y , resp ectiv ely . Com bining the previous t w o lemmas we obta in the following corollary . Corollary 7.7. X ′ ⊗ d Y ′ is an ℓ -maximal ℓ -extension of [1 , λµ ] taken with the interval top olo gy. Next, w e in v estigate ho w d -pro duct top o lo gy b ehav es w.r.t. the plus op- eration, for the case of ℓ -maximal spaces. Lemma 7.8. Supp ose X and Y ar e ℓ -maxima l and ρ ( X ) ∈ Suc . Then ( X ⊗ d Y ) + ≃ ( X + × iso ( Y )) ⊔ ( d Y ) + . Here ⊔ denotes the top ological sum and iso ( Y ) comes with the discrete top ology . Also notice that X + × i s o ( Y ) is homeomorphic to Z + 0 , and that ( d Y ) + is homeomorphic to the restriction of Y + to the set d Y . (Any set d A on Y is con tained in d Y .) Pro of. Let Z = X ⊗ d Y and let W denote ( X + × i s o ( Y )) ⊔ ( d Y ) + . W e can assume that Z and W hav e t he same underlying set. By Lemma 5.1 the top ology of W is generated b y sets of the f o rm 1. i y ( V ), where y ∈ iso ( Y ), V ∈ τ X or V = d α +1 X with α < ρ ( X ); 2. π − 1 1 ( U ∩ d Y ) for U ∈ τ Y and π − 1 1 ( d β +1 Y ) with β < ρ ( Y ). T o prov e the inclusion of τ W in t o τ + Z w e chec k that all these basic op en sets are op en in Z + . If V ∈ τ X then i y ( V ) ∈ τ Z , hence it is op en in Z + . If V = d α +1 X then i y ( V ) = X y ∩ d α +1 Z , whic h is op en in Z + as the intersec tion of tw o op en sets. If U ∈ τ Y , then π − 1 1 ( U ∩ d Y ) = π − 1 1 ( U ) ∩ Z 1 is op en in Z + . In f a ct, Z 1 = d ρ ( X ) Z is op en in Z + , since ρ ( X ) ∈ Suc. If U = π − 1 1 ( d β +1 Y ), then U = d β +1 Z Z 1 = d ρ ( X )+ β +1 Z Z whic h is op en in Z + . No w we c hec k that τ + Z is included in τ W . Since X ⊕ d Y is ℓ - maximal, τ + Z is generated by τ Z and sets of the form d α +1 Z for α < ρ ( Z ). By L emma 7.4 d α +1 Z = ( d α +1 Z 0 ∪ Z 1 , if α < ρ ( X ) π − 1 1 ( d β +1 Y ) , if α = ρ ( X ) + β . In b oth cases it is clearly op en in W . On the o t her ha nd, op en sets in Z are generated b y i y ( V ) with V ∈ τ X , in whic h case w e are done, and π − 1 1 ( U ) with U ∈ τ Y . Let U 0 := U ∩ iso ( Y ) and U 1 := U ∩ d Y . Notice that π − 1 1 ( U 0 ) = S y ∈ U 0 X y is op en in Z 0 , and hence in W , whereas π − 1 1 ( U 1 ) is op en in d Y , hence in ( d Y ) + and W . Hence, τ Z is included in τ W and we are done. ⊣ 18 8 Some op e rations on lme-s paces Recall that ( X , τ 0 , . . . , τ n ) is an lme- s p ac e based on a scattered to p ology τ if τ 0 = τ ′ and τ i +1 = ( τ + i ) ′ , f or each i < n , where σ ′ denotes an y ℓ - maximal ℓ -extension of σ . Obviously , any suc h space is a GLP n -space. W e call ( X , τ 0 , . . . , τ n ) an or dinal lme-sp ac e if X is an ordinal a nd τ is the interv al top ology on X . W e sp ecify t w o constructions on lme-spaces. First, we extend the op eratio n of d - pro duct to G L P- spaces. Definition 8.1. Supp ose ( X, τ 0 , . . . , τ n ) and ( Y , σ 0 , . . . , σ n ) ar e t w o G LP n - spaces. Let ( Z , θ 0 ) b e the d -pro duct ( X , τ 0 ) ⊗ d ( Y , σ 0 ). F or eac h i = 1 , . . . , n w e sp ecify a top olog y θ i on Z as the sum of the to p ologies τ i on X y , for each y ∈ iso ( Y ) , a nd of σ i on d Y , where iso ( Y ) and d Y refer to the space ( Y , σ 0 ). In other w ords, θ i consists of the sets of the for m [ y ∈ i so ( Y ) i y ( U y ) ∪ π − 1 1 ( V ∩ d Y ) where U y ⊆ X , U y ∈ τ i and V ∈ σ i . W e no t e t ha t the functions π 0 : ( Z 0 , θ i ↾ Z 0 ) ։ ( X, τ i ) and π 1 : ( Z 1 , θ i ↾ Z 1 ) ։ ( d Y , σ i ↾ d Y ) are d -ma ps, for i = 1 , . . . , n . Lemma 8.2. ( Z , θ 0 , θ 1 , . . . , θ n ) is a GLP n -sp ac e. Pro of. W e mak e use of the f a ct that the plus opera t io n on top ologies dis - tributes ov er top ological sums. Hence, θ + i ⊆ θ i +1 on Z , for all i = 1 , . . . , n − 1. Th us, w e only hav e t o show that θ + 0 ⊆ θ 1 . Consider an y A ⊆ Z . By Lemma 7.3 d Z ( A ) = π − 1 1 ( d Y { y : A ∩ X y 6 = ∅ } ) ∪ d Z 0 ( A ∩ Z 0 ) . In fact, any x ∈ d Z ( A ) ∩ Z 0 m ust b elong to d Z 0 ( A ∩ Z 0 ), since Z 0 is op en in Z , hence the claim. How ev er, b oth π − 1 1 ( d Y { y : A ∩ X y 6 = ∅ } ) and d Z 0 ( A ∩ Z 0 ) are op en in θ 1 . This is because d Y { y : A ∩ X y 6 = ∅ } is op en in ( d Y , σ 1 ) and d Z 0 ( A ∩ Z 0 ) is open in ( Z 0 , θ 1 ). ⊣ Lemma 8.3. Supp ose ( X , τ 0 , . . . , τ n ) and ( Y , σ 0 , . . . , σ n ) ar e lme-sp ac es b ase d on τ and σ , r esp e ctively, such that b oth ρ ( X, τ ) a n d ρ ( Y , σ ) a r e suc c essor or dinals. Then X ⊗ d Y is an lme-sp ac e b as e d o n ( X , τ ) ⊗ d ( Y , σ ) . Mor e over, ρ (( X, τ ) ⊗ d ( Y , σ )) is a suc c essor or di n al. Pro of. Let Z = ( Z , θ 0 , . . . , θ n ) denote X ⊗ d Y . The fa ct that ( Z, θ 0 ) is an ℓ -maximal ℓ -extension o f ( X , τ ) ⊗ d ( Y , σ ) follows from Lemmas 7.5 a nd 7.6. W e sho w tha t ( Z, θ 1 ) is an ℓ -maximal ℓ - extension of ( Z , θ + 0 ). By Lemma 7.8 ( Z , θ + 0 ) ≃ (( X , τ + 0 ) × iso ( Y )) ⊔ ( d Y , σ + 0 ) . On the other hand, b y definition, ( Z , θ 1 ) ≃ (( X , τ 1 ) × iso ( Y )) ⊔ ( d Y , σ 1 ) . 19 W e ha ve that ( X , τ 1 ) is an ℓ -maximal ℓ -extension of ( X, τ + 0 ) and ( d Y , σ 1 ) that of ( d Y , σ + 0 ). This relatio n then holds for the resp ectiv e topolo g ical sums. Finally , w e remark tha t ( Z , θ i +1 ) is an ℓ -maximal ℓ -extens ion of ( Z, θ + i ), for i = 1 , . . . , n , b ecause ( X , τ i +1 ) is an ℓ -maximal ℓ -extension of ( X , τ + i ) and ( d Y , σ i +1 ) is an ℓ -maximal ℓ - extension of ( d Y , σ + i ). These relations then m ust also hold for the resp ective top olog ical sums. ⊣ Corollary 8.4. L et X and Y b e or dinal lme-sp ac es on [1 , λ ] and [1 , µ ] , r e- sp e ctively. Then X ⊗ d Y is an or dinal lme-sp a c e on [1 , λµ ] . W e are going to in tro duce another ke y op eration on lme-spaces called lifting . Before doing it w e state a simple ‘pullbac k’ lem ma. Lemma 8.5. L et ( X , τ 0 , . . . , τ n ) b e an lme- s p ac e b ase d on τ , and let h : ( Y , σ ) → ( X , τ ) b e a d -m ap. T hen ther e is a n lme-sp ac e ( Y , σ 0 , . . . , σ n ) b ase d on σ such that h : ( Y , σ i ) → ( X, τ i ) is a d -map, for e ach i ≤ n . Pro of. This statemen t is prov ed b y a rep eated application of Lemmas 4.7 and 5.2 as indicated in the following diagram. ( Y , σ ) d   lm / / ( Y , σ 0 ) d   ( Y , σ + 0 ) d   lm / / ( Y , σ 1 ) d   . . . ( X , τ ) lm / / ( X , τ 0 ) ( X , τ + 0 ) lm / / ( X , τ 1 ) . . . Here, the arro ws lab eled b y ‘ d ’ indicate d -maps; the ar ro ws la b eled b y ‘ l m ’ indicate ℓ - maximal ℓ -extensions. Dotted arro ws are b eing pro v ed to exist giv en the rest. Th us, the t w o squares represen t the first t wo applications of Lemma 4.7, and the transition f r o m the righ t v ertical arrow of the first square to the left v ertical arrow of the second one is an application of Lemma 5.2. ⊣ Lemma 8.6 (lifting) . Supp ose X = ([0 , λ ] , τ 1 , . . . , τ n ) is an or dinal lme- sp ac e. Th en ther e i s an or di n al lme-sp ac e Y = ([1 , ω λ ] , σ 0 , σ 1 , . . . , σ n ) such that r : ([1 , ω λ ] , σ i ) ։ ([0 , λ ] , τ i ) is a d -map, for e ach i = 1 , . . . , n . Suc h an Y can b e called a lifting of the space X , since it is similar to X w.r.t. higher to p ologies (starting fro m the second one rather than t he firs t). Pro of. T op ology σ 0 , b eing an ℓ - maximal ℓ -extension of the order top olo gy , has the same ra nk function. Therefore, r : ([1 , ω λ ] , σ 0 ) ։ ([0 , λ ] , τ ← ) is a d -map. By Lemma 5.2 w e o bta in that r : ([1 , ω λ ] , σ + 0 ) ։ ([0 , λ ] , τ < ) is a d -map, as well. Since τ 1 is an ℓ -maximal ℓ -extension of the order t o p ology , w e are now in a p osition to apply Lemma 8.5. So, w e obta in an lme-space ([1 , ω λ ] , σ 1 , . . . , σ n ) based o n σ + 0 suc h that r : ([1 , ω λ ] , σ i ) ։ ( [0 , λ ] , τ i ) is a d -map, fo r eac h i = 1 , . . . , n . It follows t ha t Y = ([1 , ω λ ] , σ 0 , σ 1 , . . . , σ n ) is as required. ⊣ 20 9 Pro of of main l emma No w w e prov ide the key construction pro ving Lemma 6.8 a b ov e. Pro of. F or eac h J n -tree ( T , R 0 , . . . , R n ) with a ro ot a w e are going to build an ordinal lme-space X = ( [1 , λ ] , τ 0 , . . . , τ n ) and a J n -morphism f : X ։ T suc h that f − 1 ( a ) = { λ } . Suc h J n -morphisms will b e called suitable . The construction go es by induction on n with a sub o rdinate induction on the R 0 -heigh t of T , whic h is denoted ht 0 ( T ). If n = 0 w e let τ 0 b e the in terv al top olog y and notice that on an y λ < ω ω this top olog y is ℓ - maximal (since there are no p oints of limit rank). F rom the top ological completeness pro ofs for the G¨ odel–L¨ ob log ic it is kno wn (see [7]) that there is an ordinal λ < ω ω and a suitable d -map from [1 , λ ] on to ( T , R 0 ). This ma p is constructed by induction on ht 0 ( T ). If ht 0 ( T ) = 0, then T consists of a single p oin t a . W e put λ = 1 and f (1) = a . If ht 0 ( T ) = m > 0 let a 1 , . . . , a l b e the c hildren of the ro ot a , and let T i denote the subtree generated b y a i , for i ≤ l . By the induction h yp othesis, there are o rdinals κ 1 , . . . , κ l and suitable d -maps g i : [1 , κ i ] ։ T i , for each i = 1 , . . . , l . Let κ := κ 1 + · · · + κ l , then [1 , κ ] can b e iden tified with the top olog ical sum F l i =1 [1 , κ i ]. Let g : [1 , κ ] ։ F l i =1 T i b e defined b y g ( α ) := g i ( β ) , if α = κ 1 + · · · + κ i − 1 + β , β ∈ [1 , κ i ] . Then g is clearly a d - map. W e now let λ := κω and let f : [1 , λ ] ։ T b e defined by f ( α ) := ( g ( β ) , if α = λn + β where n < ω , β ∈ [1 , κ ] , a, if α = λ. It is then easy to v erify that f is, indeed, a suitable d -map. This accoun ts for the case n = 0. F or the induction step suppo se the lemma is tr ue for eac h J k -tree with k < n . Let T = ( T , R 0 , . . . , R n ) b e an J n -tree with the ro ot a . W e prov e our claim by induction o n the R 0 -depth of T . Case 1: ht 0 ( T ) = 0, in other w ords R 0 = ∅ . Let T 1 := ( T , R 1 , . . . , R n ). By the induction h yp othesis there is a suitable J n − 1 -morphism f 1 : X 1 ։ T 1 where X 1 = ([1 , λ 1 ] , τ 1 , . . . , τ n ). W e note that X 1 is isomorphic to ([0 , µ ] , τ 1 , . . . , τ n ), for some µ ( o b viously , µ = λ 1 if λ 1 is infinite). By the Lifting lemma there is an ordinal lme-spac e X = ([1 , λ ] , σ 0 , σ 1 , . . . , σ n ) suc h that λ = ω µ and r : ([1 , λ ] , σ i ) ։ ([0 , µ ] , τ i ) is a d -map, for eac h i ∈ [1 , n ]. It fo llo ws that f := r ◦ f 1 is a suitable J n - morphism. In fact, it is immediate that conditions ( j 1 ), ( j 2 ) are met and that ( j 3 ) , ( j 4 ) a re satisfied for eac h k ≥ 1. Let us consider ( j 3 ) for k = 0. 21 Since R 0 is empt y , the only 1-hereditary ro ot of T is in fact the unique 0-hereditary ro ot a , th us R ∗ 0 ( a ) = T \ { a } . Then clearly f − 1 ( R ∗ 0 ( a )) = [1 , λ ) and f − 1 ( R ∗ 0 ( a ) ∪ { a } ) = [1 , λ ], b oth of whic h a re σ 0 -op en. Th us ( j 3 ) is me t. Condition ( j 4 ) for k = 0 b oils down to the f act that f − 1 ( a ) is discrete. Ho w ev er, f − 1 ( a ) is the singleton { λ } . Th us ( j 4 ) is also met and f : X ։ T is the required J n -morphism. Case 2: ht 0 ( T ) = m > 0. Let a 1 , . . . , a l b e the immediate R 0 -successors of a whic h are hereditary 1 - ro ots. Denote T i = { a i } ∪ R ∗ 0 ( a i ) for i ∈ [1 , l ] and T 0 = { a } ∪ R ∗ 1 ( a ). Note tha t T = S l i =0 T i . F urthermore, for eac h i ∈ [1 , l ] the subframe T i of T is a J n -tree of R 0 -depth less than m . By the induction hypothesis there exist ordinal lme-spaces S i = ([1 , κ i ] , ξ i 0 , . . . , ξ i n ) and suitable J n -morphisms g i : S i → T i . Let κ := κ 1 + · · · + κ l , then [1 , κ ] can b e identified with the disjoint union F l i =1 [1 , κ i ]. Let ξ 0 , . . . , ξ n b e the top ologies of the corresp onding to p ological sum, that is, ξ j = F l i =1 ξ i j , and let g : [1 , κ ] → F l i =1 T i b e t he disjoin t union of g i , i.e. g = F l i =1 g i . Notice t ha t F l i =1 T i is iden tified with R 0 ( a ). It is easy to see tha t X = ([1 , κ ] , ξ 1 , . . . , ξ n ) is an ordinal lme-space and that g : [1 , κ ] → R 0 ( a ) is a J n -morphism. No w consider the 1-sheet ( T 0 , R 1 , . . . , R n ). By the induction h yp othesis (for n ) there is an ordinal lme-space Y 0 = ([1 , λ 0 ] , τ 1 , . . . , τ n ) and a suitable J n − 1 -morphism g 0 : Y 0 ։ T 0 . Let Y = ([1 , ω λ 0 ] , σ 0 , σ 1 , . . . , σ n ) b e an ordinal lme-space defined as in Case 1 and let h : Y ։ ( T 0 , ∅ , R 1 , . . . , R n ) b e the corresp onding suitable J n -morphism. W e no w consider the d -pro duct Z := X ⊗ d Y of these ordinal lme-spaces. Note that iso ( Y ) = { α + 1 : α < κ 0 } a nd d Y = L im ∩ [1 , κ 0 ]. Hence, w e can iden tify Z with an ordinal lme-space ( [1 , λ ] , θ 0 , . . . , θ n ) where λ := κ · ω λ 0 , X α +1 = [ κα + 1 , κ ( α + 1)], f or all α < κ 0 := ω λ 0 . Hence, Z 0 = F α<κ 0 X α +1 and Z 1 = { κλ : λ ∈ Lim , λ ≤ κ 0 } . The associated pro jection maps π 0 : Z 0 ։ X and π 1 : Z ։ Y are defined by formulas π 1 ( κλ ) = λ and π 0 ( κα + β ) = β , where λ ∈ Lim, λ ≤ κ 0 , β ∈ [1 , κ ], α < κ 0 . W e define the required J n -morphism f : Z ։ T as follows: f ( z ) := ( g ( π 0 ( z )) , if z ∈ Z 0 , h ( π 1 ( z )) , if z ∈ Z 1 . W e hav e to chec k that f satisfies ( j 1 )–( j 4 ). R ecall t hat for k ≥ 1 the space ( Z , θ k ) is homeomorphic to the top olog ical sum of Z 0 ≃ F α<κ 0 ( X , ξ k ) and Z 1 ≃ ( Y , σ k ). Then b oth π 0 : ( Z 0 , θ k ↾ Z 0 ) ։ ( X , ξ k ) and π 1 : ( Z 1 , θ k ↾ Z 1 ) ։ ( Y , σ k ) a r e d -maps. Since b oth g and h are J n -morphisms, it f o llo ws t hat conditions ( j 1 )–( j 4 ) are satisfied f or all k ≥ 1. W e must o nly c heck ( j 2 )–( j 4 ) for k = 0 . Recall that the to p ology θ 0 on a d - pro duct X ⊗ d Y is generated by the base of op en sets { i y ( V ) : V ∈ τ X , y ∈ iso ( Y ) } and { π − 1 1 ( U ) : U ∈ τ Y } . Hence, in order to c hec k ( j 2 ) it is sufficien t to show that the image under f of an y suc h basic op en set is open. Since i y ( V ) ⊆ Z 0 and π 0 ( i y ( V )) = V w e obtain that f ( i y ( V )) = g ( V ) is op en ( g is a J n -morphism). On the o ther hand, if U is non-empt y , then f ( π − 1 1 ( U )) = h ( U ) ∪ g ( X ) = h ( U ) ∪ R 0 ( a ). 22 This holds b ecause ev ery nonempt y op en subset of Y , in particular U , has a p oin t y of rank 0. Then X y ⊆ π − 1 1 ( U ) and hence f ( π − 1 1 ( U )) ⊇ f ( X y ) = g ( X ). Clearly , both h ( U ) and R 0 ( a ) are op en in T . Hence, f satisfies ( j 2 ). Condition ( j 3 ) follows fro m the fact tha t b oth π 0 and π 1 are con tin uous. Indeed, if w is a hereditary 1- ro ot of T , then either w = a or w ∈ R 0 ( a ). In the former case R ∗ 0 ( w ) = T \ { a } and hence f − 1 ( R ∗ 0 ( w )) = [1 , λ ) is op en. Similarly , f − 1 ( R ∗ 0 ( w ) ∪ { w } ) = Z is op en. If w ∈ R 0 ( a ) then b oth R ∗ 0 ( w ) and R ∗ 0 ( w ) ∪ { w } are con tained in R 0 ( a ). Since g is a J n -morphism, g − 1 ( R ∗ 0 ( w )) is op en. Then f − 1 ( R ∗ 0 ( w )) = π − 1 0 ( g − 1 ( R ∗ 0 ( w )) is op en, b y the contin uit y of π 0 . The argumen t for R ∗ 0 ( w ) ∪ { w } is similar. Hence, condition ( j 3 ) is me t. T o chec k condition ( j 4 ) assume w is a hereditary 1-ro o t of T . If w = a then f − 1 ( w ) is the singleton { λ } . If w ∈ R 0 ( a ) then g − 1 ( w ) is discrete as a subspace of X , since g is a J n -morphism. W e know that π 0 : Z 0 ։ X is b o th con t inuous and p oin t wise discrete. Hence, f − 1 ( w ) = π − 1 0 ( g − 1 ( w )) is dis crete in Z 0 and thereb y in Z ( Z 0 is op en in Z ). This show s ( j 4 ). Th us, w e hav e che c k ed that f : Z ։ T is a suitable J n -morphism, whic h completes the pro of o f Lemma 6.8 and thereby o f Theorem 6.6. ⊣ A App en d ix: Pro of of Pro p osition 2.1. The corresp ondence b et w een Magar i frames and scattered to p ological spaces is essen tia lly due to Esakia. A frame ( X , δ ) is called a Magari fr ame if it satisfies the following identities , for any A, B ⊆ X : (i) δ ( A ∪ B ) = δ A ∪ δ B ; δ ∅ = ∅ ; (ii) δ A = δ ( A \ δ A ). It is w ell-kno wn and easy to see that ( X , δ ) is Magari iff ( X , δ ) v alidates the axioms of G¨ odel–L¨ ob logic GL (corresp onding to Axioms ( i) –(iii) of GLP ). W e notice that an y suc h op erator δ is monotone, that is, A ⊆ B implies δ A ⊆ δ B . In addition, δ δ A ⊆ δ A holds in any Magari frame, since the form ula ✸✸ p → ✸ p is a theorem of GL . Lemma A.1. If ( X , τ ) is a sc atter e d top olo gic a l s p ac e then ( X , d τ ) is a Ma- gari fr ame. Pro of. The v alidit y of (i) is ob vious, whereas (ii) means that any limit p oint of A is a limit p oint of the set iso ( A ) of isolated p oin ts of A . Let x ∈ d τ A and let U b e an op en neigh b orho o d of x . U ∩ A \ { x } is not empty , hence it has an isolated p oin t y . Then y ∈ iso ( A ) as we ll. ⊣ Supp ose ( X , τ 0 , τ 1 , . . . ) is a GLP-space. T o pro v e P art (i) of Propo sition 2.1 observ e that Axioms (i)–(iii) of GLP are satisfied in ( X , d 0 , d 1 , . . . ) b y the previous corollary . Axiom (iv) is clearly v alid s ince τ n ⊆ τ n +1 . 23 T o c heck Axiom (v) consider a set of the form d n ( A ). Since X is a GLP-space, d n ( A ) is op en in τ n +1 . Hence, ev ery x ∈ d n ( A ) cannot b e a τ n +1 - limit p oin t of X \ d n ( A ), that is, x ∈ ˜ d n +1 ( d n A ). In other w ords, d n ( A ) ⊆ ˜ d n +1 ( d n A ), f o r any A , that is, Axiom (v) is v alid. T o prov e P a rt (ii) of Prop osition 2.1 we first remark that, if ( X, δ ) is a Magari frame, then the op erator c ( A ) := A ∪ δ A satisfies the Kuratow ski axioms of the top ological closure. This defines a top olog y on X in whic h an y set A is closed iff A = c ( A ) iff δ A ⊆ A . (Alternat ively , one can c hec k that the collection of all se ts U satisfying U ⊆ ˜ δ U is a top ology .) Lemma A.2. Supp ose ( X , δ ) is Magari. T h en, for al l x ∈ X , (i) x / ∈ δ ( { x } ) ; (ii) x ∈ δ A ⇐ ⇒ x ∈ δ ( A \ { x } ) . Pro of. (i) By Ax iom (iii) δ { x } ⊆ δ ( { x } \ δ { x } ). If x ∈ δ { x } then δ ( { x } \ δ { x } ) ⊇ δ ( { x } \ { x } ) = δ ∅ = ∅ . Hence, δ { x } = ∅ , a contradiction. (ii) x ∈ δ A implies x ∈ δ (( A \ { x } ) ∪ { x } ) = δ ( A \ { x } ) ∪ δ { x } . By (i), x / ∈ δ { x } , hence x ∈ δ ( A \ { x } ). The other implication follo ws from the monotonicit y of δ . ⊣ Lemma A.3. S upp ose ( X , δ ) is Magari and τ is the asso ciate d top olo gy. Then δ = d τ . Pro of. Let d = d τ ; w e sho w tha t, fo r a ny set A ⊆ X , dA = δ A . Notice that, for any B , cB = dB ∪ B = δ B ∪ B . Assume x ∈ δ A then x ∈ δ ( A \ { x } ) ⊆ c ( A \ { x } ) ⊆ d ( A \ { x } ) ∪ ( A \ { x } ) . Since x / ∈ A \ { x } , w e obtain x ∈ d ( A \ { x } ). By the monotonicit y of d , x ∈ dA . Similarly , if x ∈ dA then x ∈ d ( A \ { x } ). Hence, x ∈ c ( A \ { x } ) = δ ( A \ { x } ) ∪ ( A \ { x } ) . Since x / ∈ A \ { x } w e o btain x ∈ δ A . ⊣ Lemma A.4. S upp ose ( X , δ ) is Magari and τ is the asso ciate d top olo gy. Then ( X, τ ) is sc atter e d. Pro of. Since δ is L¨ ob w e kno w that δ = d τ . W e s ho w that an y nonempt y subspace A ⊆ X ha s an isolated p oin t. Supp ose not, then is o ( A ) = A \ δ A = ∅ . Then δ A = δ ( A \ δ A ) = δ ∅ = ∅ . Then A = A \ δ A = ∅ . ⊣ 24 No w w e prov e Part (ii). Let ( X , δ 0 , δ 1 , . . . ) b e a neigh b orho o d frame satisfying GLP . Then each of the frames ( X , δ n ) is M agari, hence it defines a scattered t op ology τ n on X for whic h δ n = d τ n . Recall that U ∈ τ n iff U ⊆ ˜ δ n ( U ). W e only hav e to sho w that the last tw o conditions of a GLP- space are met. Supp ose U ∈ τ n , then U ⊆ ˜ δ n ( U ) ⊆ ˜ δ n +1 ( U ) b y Axiom (iv). Hence, U ∈ τ n +1 . Th us, τ n ⊆ τ n +1 . 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