A New Causal Topology and Why the Universe is Co-compact

A NEW CAU SAL TOPO L OGY AND WHY THE UNIVERSE IS CO-C OMP A CT MAR TIN MARIA KO V ´ AR Abstra ct. W e sho w that there exists a canonical top ology , naturally connected with the causal site of J. D. Christensen and L. Crane, a p ointl ess algebraic structure motiv ated b y quantum gra vit y . T aking a causal site compatible with Minko wski space, on ever y compact subset our top ology b ecame a reconstruction of th e original top ology of the spacetime (only from its causal structu re). F rom the global p oint of view, the reconstructed top ology is the de Gro ot du al or co-compact with respect t o the original, Eu clidean top ology . The result indicates that th e causalit y is th e p rimary structure of the spacetime, carrying also its top ological in formation. 1. Introduction The b elief th at the causal structure of sp acetime is its most fun damen tal underlying structur e is almost as old as the idea of the relativistic spacetime itself. But ho w it is related to the top ology of spacetime? By tr adition, there are n o doubts regarding the top ology of spacetime at least lo cally , since it is considered to b e lo cally homeomorphic with the cartesian p o wer of th e real line, equipp ed with th e Eu clidean top olog y . But m ore r ecently , there app eared concepts of discrete and p oin tless mo dels of spacetime in which the causal structur e is int ro duced axiomatical ly and so indep enden tly on th e lo cally Euclidean mo dels. Is, in these cases, the axiomatic causal structure ric h enough to carry also the full top ological information? And, after all, ho w th e top ology that w e p erceive around us and which is essentiall y and implicitly at the bac kground of man y physica l ph en omena, m a y arise? In this pap er w e introduce a general construction, suitable for equipp ing a set of ob jects with a top olog y-lik e str u cture, using the inner, natural and in tuitiv e r elationships b et wee n them. W e use the construction to sh o w that another algebraic structure, motiv ated by the research in quant um geome- try and gra vitatio n – the causal site of J. D. Christensen and L. Crane – v ery n aturally generates a compact T 1 top ology on itself. T esting the con- struction on Minko wski sp ace we s ho w th at coming out from its causalit y structure, the un iv erse – in its first appr o ximation represented b y Minko wski space – naturally h as so called co-compact top ology (also called the de Gro ot dual top olog y) wh ic h is compact, sup er conn ected, T 1 and non-Hausdorff. Key wor ds and phr ases. Causal site, de Groot dual, Minko wski space, qu antum gra vity . 1 2 MAR TIN MARIA KO V ´ AR The co-compact top ology on Mink o ws ki space coincides with th e Eu clidean top ology on all compact sets – in the more physically related terminology , at th e fin ite distances. T herefore, the studied construction has probably n o impact to the description of lo cal physical p henomena, but it changes the global view at the un iv erse. P erhaps it could h elp to explain how th e top ol- ogy that we p erceiv e “around us” (in an y wa y – by our everyda y exp erience, as well as b y exp eriments, measurements and other physica l phenomena) ma y arise from causalit y . 2. Ma thema tical Prer equisites Throughout this p ap er, we mostly use the usual terminology of general top ology , for whic h the reader is r eferr ed to [4] or [7], with one exception – in a consensus with a mo d ern app roac h to general top ology , w e no longer as- sume the Hausd orff sep aration axiom as a part of the definition of compact- ness. This is esp ecially affected by some recen t motiv ations from computer science, but also the cont en ts of the pap er [12] confir ms that suc h a mo difi - cation of the definition of compactness is a relev an t idea. Thus w e sa y that a top ological space is c omp act , if ev ery its op en co v er has a fin ite s ub cov er, or equiv alen tly , if ev ery cen tered system of closed sets or a closed filter base has a non-empt y intersecti on. Note that b y th e w ell-known Alexander’s subbase lemma, the general closed sets ma y b e replaced by more s p ecial element s of an y closed su bbase for the top olog y . W e hav e already m en tioned the co-compact or the de Gro ot dual top ol- ogy , wh ic h was first systematically stud ied pr obably at th e en d of the 60’s b y J. d e Gro ot and his co w ork ers, J. M. Aarts, H. Herrlic h , G. E. S trec k er an d E. W attel. The in itial pap er is [10]. Ab out 20 y ears later the co-co mpact top ology again came to the cen ter of in terest of some top olo gists and theoret- ical computer scien tists in connection with their researc h in domain th eory . During discussions in the comm un it y th e original definition due to de Gro ot w as sligh tly change d to its cur r en t form, inserting a w ord “saturated” to the original defi n ition (a set is s aturated, if it is an in tersection of op en sets, so in a T 1 space, all sets are saturated). Let ( X, τ ) b e a top olog ical sp ace. The top ology generated b y the f amily of all compact saturated sets used as the base f or the closed sets, w e denote by τ G and call it c o-c omp act or de Gr o ot d ual with resp ect to the original top olog y τ . In [17] J. La wson and M. Mislo ve stated question, whether the s equ ence, con taining the iterated du als of the original top ology , is infin ite or the pr o cess of taking d uals terminates after fi nitely man y steps with top ologies that are dual to eac h other. In 2001 the au th or solv ed the question and pr o ved that only 4 differen t top ologies ma y arise (see [14]). The follo wing theorem s ummarizes the previously men tioned facts im- p ortant for understand ing the main results, conta ined in Section 4. The theorem itself is not new, und er sligh tly d ifferen t terminology the reader A NEW CAUSAL TOPOLOGY AND WHY THE UNIVERSE IS CO-COM P A CT 3 can essen tially fin d it in [10]. A more general result, equiv alen tly char- acterizing the top olo gies satisfying τ = τ GG , the reader ma y find in th e second author’s pap er [15]. F or our pu r p oses, the reader ma y replace a gen- eral non-compact, lo cally compact Hausdorff space b y the Minko wski sp ace equipp ed with the Eu clidean top ology . The p ro of we present here only for the reader’s con v enience, without an y claims of originalit y . F or the pro of we need to use the f ollo wing notion. Let ψ b e a f amily of sets. W e sa y that ψ has the finite inte rsection p r op erty , or briefly , th at ψ has f.i.p. , if for every P 1 , P 2 , . . . , P k ∈ ψ it follo w s P 1 ∩ P 2 ∩ · · · ∩ P k 6 = ∅ . In some literature (for example, in [4]), a collection ψ with th is pr op erty is called c e nter e d . Theorem 2.1. L et ( X , τ ) b e a non-c omp act, lo c al ly c omp act Hausdorff top o- lo gic al sp ac e . Then (i) τ G ⊆ τ , (ii) τ = τ GG , (iii) ( X, τ G ) is c omp act and sup er c onne cte d, (iv) the top olo gies induc e d fr om τ and τ G c oincide on every c omp act subset of ( X, τ ) . Pr o of. Th e top ology τ G has a closed b ase wh ic h consists of compact sets. Since in a Hausdorff space all compact sets are closed, we h av e (i). Let C ⊆ X b e a closed set with resp ect to τ , to show th at C is compact with r esp ect to τ G , let us tak e a non-empty family Φ of compact sub sets of ( X, τ ), such that the family { C } ∪ Φ has f.i.p. T ake some K ∈ Φ. Then the family { C ∩ K } ∪ { C ∩ F | F ∈ Φ } also h as f .i.p. in a compact set K , so it has a non-empt y intersect ion. Hence, also the inte rsection of { C } ∪ Φ is non-empty , whic h means that C is compact with resp ect to τ G . Consequently , C is closed in ( X, τ GG ), w hic h means that τ ⊆ τ GG . The top ology τ GG has a closed base consisting of sets whic h are compact in ( X , τ G ). T ak e su ch a set, sa y H ⊆ X . Let x ∈ X r H . Since ( X, τ ) is lo cally compact and Hausdorff, for ev er y y ∈ H there exist U y , V y ∈ τ su c h that x ∈ U y , y ∈ V y and U ∩ V = ∅ , with cl U y compact. Denote W y = X r cl U y . W e ha v e y ∈ V y ⊆ W y , so the sets W y , y ∈ H cov er H . The complemen t of W y is compact with resp ect to τ , so W y ∈ τ G . The family { W y | y ∈ H } is an op en co ve r of the compact set H in ( X , τ G ), so it has a finite sub co ve r, say { W y 1 , W y 2 , . . . , W y k } . Denote U = T k i =1 U x i . Then U ∩ H = ∅ , x ∈ U ⊆ X r H , whic h means that X r H ∈ τ and H is closed in ( X , τ ). Hence, τ GG ⊆ τ , an together with the previously pr ov ed conv erse inclusion, it giv es (ii). Let us sh o w (iii). T ake any collection Ψ of compact subsets of ( X, τ ) ha ving f.i.p. They are b oth compact and closed in ( X , τ ), so T Ψ 6 = ∅ . Then ( X, τ G ) is compact. Let U, V ∈ τ G and s u pp ose that U ∩ V = ∅ . The complemen ts of U , V are compact in ( X, τ ) as inte rsections of compact closed sets in a Hausd orff sp ace. Then ( X , τ ) is compact as a union of t w o compact sets, wh ich is not p ossible. Hence, it h olds (iii). 4 MAR TIN MARIA KO V ´ AR Finally , take a compact subset K and a closed subset C of ( X , τ ). Th en K ∩ C is compact in ( X , τ ) and h ence closed in ( X , τ G ). Thus the top ology on K in duced from τ G is finer than the top olog y ind uced from τ . T oget her with (i), we get (iv).  3. How to Topol ogize Ever ything As it has b een recen tly noted in [12], the n ature or the p h ysical universe, whatev er it is, has probably n o existing, real p oin ts lik e in the classical Euclidean geometry (or, at least, w e cannot b e absolutely s u re of that). P oin ts, as a useful mathematical abstraction, are infi nitesimally small and th us cann ot b e m easured or detected by an y p h ysical wa y . But what we can b e sur e th at r eally exists, there are v arious locations, con taining concrete physic al ob jects. In this pap er we w ill call these locations plac es . V arious places can o v erlap, they can b e merged, emb edded or glued together, so the theoretically under s to o d virtual “observ er” can visit m ultiple places si- m ultaneously . F or instance, the Galaxy , the Solar system, the Earth, (the territory of ) Europ e, Brno (a b eautiful cit y in Czec h R ep ublic, the place of author’s residen ce), the ro om in which the reader is present just no w, are simple and natural examples of p laces conceiv ed in our sense. C ertainly , in this sens e, one can b e presen t at man y of these places at the same time, and, also certainly , there exist p airs of places, where the sim ultaneous pres- ence of an y physical ob jects is not p ossible. Or , at least, from our ev eryda y exp erience it seems the nature b eha v es in this wa y . T h us the presence of v arious physical ob jects connects these primarily f ree ob jects – our p laces – to the certain structure, wh ic h we call a fr amework . Note that it d o es not matter that the p laces are, at the first sigh t, de- termined r ather v aguely or w ith some uncertain t y . They are conceiv ed as elemen ts of some algebraic structure, with no any additional geometrical or metric structure and as w e will see later, the “uncertain ty” could b e partially eliminated by the relationships b et ween them. Let’s now give the pr ecise definition. Definition 3.1. L et P b e a set, π ⊆ 2 P . We say that ( P , π ) is a fr amework. The elements of P we c al l plac es, the set π we c al l fr amolo gy. Although ev ery top ological space is a framewo rk by the defin ition, the elemen tary in terpretation of a framework is very differen t fr om the usual in terpretation of a top ological sp ace. The elements of the fr amology are not primarily considered as n eigh b orho o ds of places, although this seems to b e also v ery natural. If P con tains all th e places that are or can b e obs er ved, the framology π con tains the list of observ ations of the fact that the virtu al “observ er” or some physica l ob ject that “really exists” (whatev er it means), can b e present at some p laces simultaneo usly . Th e structure w hic h ( P , π ) represent s arises f r om these observ ations. A NEW CAUSAL TOPOLOGY AND WHY THE UNIVERSE IS CO-COM P A CT 5 Let us intro d uce some other u seful notions. Definition 3.2. L et ( P , π ) and ( S, σ ) b e fr ameworks. A mapping f : P → S satisfying f ( π ) ⊆ σ we c al l a fr amework morphism. Definition 3.3. L et ( P , π ) b e a fr amework, ∼ an e quivalenc e r elation on P . L et P ∼ b e the set of al l e quivalenc e classes and g : P → P ∼ the c orr esp onding quotient map. Then ( P ∼ , g ( π )) is c al le d the quotient fr amework of ( P , π ) (with r esp e ct to the e quiv alenc e ∼ ). Definition 3.4. A fr amework ( P , π ) is T 0 if for every x, y ∈ P , x 6 = y , ther e exists U ∈ π such that x ∈ U , y / ∈ U or x / ∈ U , y ∈ U . Definition 3.5. L e t ( P , π ) b e a fr amework. Denote P d = π and π d = { π ( x ) | x ∈ P } , wher e π ( x ) = { U | U ∈ π , x ∈ U } . Then ( P d , π d ) is the dual fr amework of ( P , π ) . The plac es of the dual fr amework ( P d , π d ) we c al l abstr act p oints or simply p oints of the original f r amework ( P , π ) . The f ramew ork du ality is a simp le but handy to ol for switc hing b et w een the cla ssical p oint-se t represen tatio n (lik e in topological spaces) and the p oint- less repr esen tatio n, introd uced ab ov e. Some Examples. There is a num b er of n atural examples of mathe- matical stru ctures satisfying the definition of a fr amew ork, including non- orien ted graphs, top ologica l spaces (with op en maps as morph isms), measur- able spaces or texture spaces of M. Diker [5]. Among physica lly motiv ated examples, we may men tion the F eynman diagrams with particles in the r ole of places and inte ractions as the asso ciated abstract p oin ts. V ery likely , certain asp ects of th e string th eory , related to general top olog y , can also b e form ulated in terms of the fr amew ork th eory . It sh ould b e noted that the notion of a framewo rk is a sp ecial case of the notion of the formal c ontext , d ue to B. Gan ter and R. Wille [8], s ometimes also referred as th e Chu space [3]. Recall that a formal conte xt is a triple ( G, M , I ), where G is a s et of ob jects, M is a s et of attributes and I ⊆ G × M is a b inary relation. Thus a framew ork ( P , π ) ma y b e r epresen ted as a formal con text ( P, π , ∈ ), w h ere ob jects are th e p laces and th eir attributes are the abstract p oints. Ev en though the theory and m etho ds of formal concept analysis ma y b e a usefu l to ol also for our pur p oses, w e prefer the top olog y- related terminology that we int ro duced in th is section b ecause it seems to b e more close to the w a y , ho w mathematical p hysics understands to th e notion of spacetime. It also seems that f ramew orks are closely related to the notion of partial metric d ue to S . Matthews [18 ], b ut these relationships will b e studied in a s eparate pap er. Prop osition 3.1. L e t ( P , π ) b e a fr amework. Then ( P d , π d ) is T 0 . Pr o of. Denote S = π , σ = { π ( x ) | x ∈ P } , so ( S, σ ) is the d u al framework of ( P , π ). Let u, v ∈ S , u 6 = v . Sin ce u, v ∈ 2 P are d ifferen t sets, either there 6 MAR TIN MARIA KO V ´ AR exists x ∈ u suc h that x / ∈ v , or there exists x ∈ v , such that x / ∈ u . Then u ∈ π ( x ) and v / ∈ π ( x ), or v ∈ π ( x ) and u / ∈ π ( x ). In b oth cases th ere exists π ( x ) ∈ σ , contai ning one elemen t of { u, v } and not conta ining the other.  Theorem 3.1. L et ( P , π ) b e a fr amework. Then ( P dd , π dd ) is isomorphic to the qu otient of ( P , π ) . M or e over, if ( P , π ) is T 0 , then ( P dd , π dd ) and ( P , π ) ar e isomorphic. Pr o of. W e denote R = P d = π , ρ = π d = { π ( x ) | x ∈ P } , S = R d = ρ , σ = ρ d = { ρ ( x ) | x ∈ R } . Then ( S, σ ) is the double dual of ( P , π ). It remains to show, that ( S, σ ) is isomorphic to some quotient of ( P , π ). F or ev ery x ∈ P , we p ut f ( x ) = π ( x ). Th en f : P → S is a su rjectiv e mapping. It is easy to sh o w, that f is a morph ism . Indeed, if U ∈ π , then f ( U ) = { π ( x ) | x ∈ U } = { π ( x ) | x ∈ P , U ∈ π ( x ) } = { V | V ∈ ρ, U ∈ V } = ρ ( U ) ∈ σ . Therefore, f ( π ) ⊆ σ , whic h means th at f is an epimorphism of the framew ork ( P , π ) on to ( S, σ ). No w, we defin e x ∼ y for every x, y ∈ P if and only if f ( x ) = f ( y ). T hen ∼ is an equiv alence relation on P . F or every equiv alence class [ x ] ∈ P ∼ w e put h ([ x ]) = f ( x ). T he mappin g h : P ∼ → S is correctly defin ed, moreo v er, it is a bijection. Th e verificat ion that h is a framew ork isomorph ism is standard, but, b ecause of completeness, it has its natural place here. Th e quotient framology on P ∼ is g ( π ), where g : P → P ∼ is the quotien t map. The quotien t map g satisfies the condition h ◦ g = f . Let W ∈ g ( π ). T here exists U ∈ π su c h th at W = g ( U ). Then h ( W ) = h ( g ( U )) = f ( U ) ∈ σ . Hence h ( g ( π )) ⊆ σ , whic h m eans that h : P ∼ → S is a fr amew ork morphism. Con v ersely , let W ∈ σ = { ρ ( U ) | U ∈ π } . W e will show that h − 1 ( W ) ∈ g ( π ). By the previous paragraph, ρ ( U ) = f ( U ) for ev ery U ∈ π , so there exists U ∈ π , suc h that W = f ( U ) = h ( g ( U )). Since h is a bijection, it follo ws that h − 1 ( W ) = g ( U ) ∈ g ( π ). Hence, also h − 1 : S → P ∼ is a framewo rk morphism, so the framew orks ( P ∼ , g ( π )) and ( S, σ ) are isomorphic. No w let u s consider the sp ecial case wh en ( P , π ) is T 0 . Supp ose th at f ( x ) = f ( y ) for some x, y ∈ P . Then π ( x ) = π ( y ), whic h is p ossible only when x = y . Then th e relation ∼ is the diagonal r elation, and th e quotient mapping g is an isomorph ism.  Corollary 3.1. Ev ery fr amework arise as dual if and only if it is T 0 . Corollary 3.2. F or ev ery fr amework ( P , π ) , it holds ( P d , π d ) ∼ = ( P ddd , π ddd ) . 4. Topology of Ca usal S ites In this section we show that the n otion of a fr amew ork, in tro duced and studied in the previous section, h as some real utilit y and sense. In a con trast to sim p le examples men tioned ab o ve, from a prop erly defined fr amew ork we will b e able to constru ct a top olog ical structure w ith a real p hysic al meanin g. A NEW CAUSAL TOPOLOGY AND WHY THE UNIVERSE IS CO-COM P A CT 7 Recall that a c ausal site ( S, ⊑ , ≺ ) defined by J. D. Christensen and L. Crane in [2] is a set S of r e gions equipp ed with t w o bin ary relations ⊑ , ≺ , where ( S, ⊑ ) is a partial order h a ving the binary suprema ⊔ and the least elemen t ⊥ ∈ S , an d ( S r {⊥} , ≺ ) is a strict p artial order (i.e. an ti-reflexiv e and transitive ), lin ked together b y the follo wing axioms, wh ich are satisfied for all regions a, b, c ∈ S : (i) b ⊑ a and a ≺ c implies b ≺ c , (ii) b ⊑ a and c ≺ a implies c ≺ b , (iii) a ≺ c and b ≺ c implies a ⊔ b ≺ c . (iv) There exits b a ∈ S , called cutting of a by b , suc h that (1) b a ≺ a and b a ⊑ b ; (2) if c ∈ S , c ≺ a and c ⊑ b then c ⊑ b a . Consider a causal site ( P , ⊑ , ≺ ) and let u s d efine app ropriate framew ork structure on P . W e sa y that a sub set F ⊆ P set is cen tered, if for ev ery x 1 , x 2 , . . . , x k ∈ F there exists y ∈ P , y 6 = ⊥ satisfying y ⊑ x i for ev ery i = 1 , 2 , . . . , k . If L ⊆ 2 P is a chain of centered subsets of P linearly ord ered b y the set inclusion ⊆ , then S L is also a cent ered set. T hen ev ery cent ered F ⊆ P is con tained in some maximal cente red M ⊆ P . Let π b e the family of all maximal cen tered sub sets of P . No w, consider the framework ( P , π ) and its dual ( P d , π d ). Let ( X, τ ) b e the top ological sp ace with X = P d = π and the top ology τ generated by its closed subbase (that is, a subb ase for the closed sets) π d . Theorem 4.1. The top olo gic al sp ac e ( X, τ ) , c orr esp onding to the f r amework ( P d , π d ) and the c ausal site ( P , ⊑ , ≺ ) , is c omp act T 1 . Pr o of. By the w ell-kno wn Alexander ’s su bbase lemma, for proving the com- pactness of ( X , τ ) it is su ffi cien t to sho w, that any subfamily of π d ha ving the f.i.p., h as nonempty intersecti on. Th e sub base for the closed sets of ( X , τ ) has th e f orm π d = { π ( x ) | x ∈ P } , so any su bfamily of π d can b e ind exed by a sub set of P . Let F ⊆ P and su pp ose that for ev ery x 1 , x 2 , . . . , x k ∈ F w e ha v e π ( x 1 ) ∩ π ( x 2 ) ∩ · · · ∩ π ( x k ) 6 = ∅ . Then there exists U ∈ π such that U ∈ π ( x 1 ) ∩ π ( x 2 ) ∩ · · · ∩ π ( x k ), so x i ∈ U for ev ery i = 1 , 2 , . . . , k . Since U is a (maximal) centered family , there exists ⊥ 6 = y ∈ P such that y ⊑ x i for ev ery i = 1 , 2 , . . . , k . Th u s F is a cent ered family , conta ined in some maximal cen tered family M ⊆ P . But then we ha v e M ∈ π , s o M ∈ \ x ∈ M π ( x ) ⊆ \ x ∈ F π ( x ) 6 = ∅ . Hence, ( X, τ ) is compact. Let U, V ∈ X = π , U 6 = V . Since b oth are m aximal cente red su bfamilies of P , n one of them can cont ain the other one. So, there exist x, y ∈ P such that x ∈ U r V and y ∈ V r U . Then U ∈ π ( x ), V / ∈ π ( x ), V ∈ π ( y ), U / ∈ π ( y ). Thus X r π ( x ), X r π ( y ) are op en s ets in ( X , τ ) con taining 8 MAR TIN MARIA KO V ´ AR just one of the p oints U, V . So th e top ological space ( X, τ ) satisfies the T 1 axiom.  The motiv ation for introd ucing and stud y in g th e n otion of a causal s ite lies esp ecia lly in the hop e th at it m a y b e helpful in formulati on and s olution of certain problems in quant um gra v ity . Esp ecia lly in those situations, in whic h the traditional mo d els are less con v enient or ev en may fail (see [2] for more detail). In this situations, p ossib ly v ery differen t from our macro- scopic, ev eryda y exp erience, also th e top ological structur e of sp acetime is an imp ortant and legitimate s ub ject of the researc h. This is one of the p ossible motiv ations for the top ology that we ha v e in tro duced by the wa y describ ed ab o v e and also a go o d motiv ation for Theorem 4.1. Another, p erhaps ev en more imp ortan t motiv ation it is to inv estigate h o w the top ology of space- time, which is p erceive d in the realit y and implicitly is inv olv ed in physical phenomena, arises. So the first question w e s h ould ask it is whether the cor- resp ond ing top ology , constructed from the causal site by the describ ed w a y has an y physic al meaning. But how to d o that? Certainly , first w e must test the construction at those situations that are working and well under s to o d in the scop e of th e classical, traditional mo d els. That is why we c ho ose Mink o wski space and its causal stru cture for the next considerations. If our previous construction is w orth, then the output top ology that w e receiv e should b e closely related to the Euclidean top olog y on M . In [2], the authors sho w that the definition of a causal site is compatible with the inn er structur e of the Minko wski space. Moreo v er, it is also s ho wn that the same is true for the stably causal Lorent zian manifold (for the precise d efinition of stable causalit y see [2]; b y a result of S. Hawking and G. Ellis [11], it is equiv alent to the existence of a global time f unction). Ho wev er, it is easy to c hec k th at the causal site compatible with the stably causal Loren tzian manifold need b e not unique. A s w e will see later, for the pu rp oses of reconstru ction of the top ology from the causal structure we need muc h finer setting for the corresp on d ing causal site, than it is used in the t w o simp le examples of the pap er [2]. Let us denote b y M = R 4 the Mink o ws k i space. Recall th at it h as a natural structure of a real, 4-dimensional ve ctor space, equip p ed with the bilinear form η : M × M → R , called the Min ko wski inn er pr o duct. Th e Mink o wski mod ifi cation of th e inner pro d uct is not p ositiv ely definite as the u sual inn er p r o duct, bu t in the standard basis it is represente d by the diagonal matrix with the diagonal en tries (1 , − 1 , − 1 , − 1). Then a v ector v ∈ M is called timelike , if η ( v , v ) > 0, lightl ik e or n ull if η ( v , v ) = 0 and spacelik e, if η ( v , v ) < 0. F urther, th e v ecto r v is said to b e futu re-orien ted, if its first co ordinate, whic h r epresen ts the time, is p ositiv e. Similarly , v is past-orient ed, if its first co ordinate is negativ e. W e w rite v ≪ w for v , w ∈ M if the vecto r w − v is timelik e and fu ture-orien ted. In [2] the A NEW CAUSAL TOPOLOGY AND WHY THE UNIVERSE IS CO-COM P A CT 9 sets of the f orm D ( p, q ) = { x | x ∈ M , p ≪ x ≪ q } are called diamonds. They are u sed for the construction of an example of a certain causal site. In this setting, diamonds are op en sets in the Eu clidean top olog y , b ounded b y t w o light cones at p oin ts p, q ∈ M . It is n ot difficult to sh o w that op en diamonds form a base for the Eu clidean top ology on M . Ho wev er, for the purp ose of a reconstruction of th e top ology f rom the causal stru cture it is more con venien t to consider the closed v arian t of diamonds (with resp ect to the Euclidean top olog y). W e define p 6 q if the ve ctor q − p is non-past-orien ted and non-spacelik e, that is, if its time coord inate is non-negativ e and η ( q − p, q − p ) ≥ 0. W e also denote 0 = (0 , 0 , 0 , 0). No w, w e put J + ( p ) = { x | x ∈ M , p 6 x } , J − ( p ) = { x | x ∈ M , x 6 p } and J ( p ) = J + ( p ) ∪ J − ( p ) . Let k · k b e the Euclidean norm on M . F or a r eal n um b er ε > 0 and a p oin t x ∈ M , by B ε ( x ) we denote the op en ball B ε ( x ) = { y | y ∈ M , k x − y k < ε } . The Euclidean top ology on M , generated b y the norm k · k an d these op en balls, w e den ote b y τ E . The de Gro ot dual or co-compact top ology on M w e d enote by τ G E . F or our next consider ations we will need s ev er al lemmas, whic h w ill p oin t out some imp ortant pr op erties of the relation 6 and of the cones J ( p ) in M . W e do n ot claim originalit y for these results, only the cont ext in which w e will u s e them – the construction of a certain causal site on M – is new. Although the results can b e essentia lly foun d in the literature, in order to a void problems with different n otation and also for the r eader’s con venience, w e p resen t here the complete pro ofs. Ho wev er, for a more adv anced found a- tions of the conus theory , the reader is referred to the comprehensive pap er [16]. Lemma 4.1. The sets J + ( 0 ) and J − ( 0 ) ar e c lose d with r esp e ct to the op- er ation + of the ve c tor sp ac e ( M , + ) . Pr o of. Let x, y ∈ J + ( 0 ). L et x = s + t , y = r + u , w here r , s, t, u ∈ M and the vecto rs r , s h a v e zero time co ordinate, and the v ecto rs u , t h a ve zero space co ord inates. S ince x ∈ J + ( 0 ), we hav e η ( x, x ) ≥ 0, whic h is equ iv alen t to k t k ≥ k s k . Similarly , from y ∈ J + ( 0 ) we get k u k ≥ k r k . Since the time co ordinates of x , y and so t , u are of the same sign, and only one co ordin ate of t , u can b e non-zero, it follo ws that k t + u k = k t k + k u k ≥ k s k + k r k ≥ k s + r k . Then η ( x + y , x + y ) ≥ 0. Since the time co ordinate of x + y is non-negativ e (as the sum of th e non-negativ e coordin ates of x , y ), we fin aly get x + y ∈ J + ( 0 ). The pro of for J − ( 0 ) is analogous.  10 MAR TIN MARIA KO V ´ AR Lemma 4.2. The binary r elation 6 is a p artial or der on M . Pr o of. Certainly , 6 is r eflexiv e. Supp ose that p 6 q and q 6 r for some p, q , r ∈ M . Then r − p = ( r − q ) + ( q − p ), so if the time co ordinates of q − p and r − q are non-negativ e, th e same holds also for r − p . Since η ( q − p, q − p ) ≥ 0 and η ( r − q , r − q ) ≥ 0, we hav e q − p ∈ J + (0) and r − q ∈ J + (0). By Lemma 4.1, r − p ∈ J + (0). Then 0 6 r − p , which giv es η ( r − p, r − p ) ≥ 0. Thus 6 is also a transitiv e relation.  W e denote ♦ ( p, q ) = J + ( p ) ∩ J − ( q ) , where p, q ∈ M , p 6 q . No w let u s construct a causal site which reflects causalit y and top ological prop erties of Minko wski space M . Denote D = { ♦ ( p, q ) | p, q ∈ Q 4 , p 6 q } . No w, let ( P , ∪ , ∩ ) b e the set lattice generated by the elemen ts of D . Since P can b e represente d by lattice p olynomials (see, e.g. [9 ]), every elemen t of P can b e expressed by unions and in tersectio ns of fin itely many elemen ts of D , it is compact and closed w ith r esp ect to the Euclidean top ology τ E on M . Lemma 4.3. The f amily P is a close d b ase for the c o-c omp act top olo gy on M . Pr o of. Th e co-co mpact top ology τ G E on M is generated b y its op en b ase, whic h is formed by the complemen ts of sets, compact in the Euclidean top ol- ogy τ E . Let K ⊆ M b e compact. Denote U = M r K . T ak e a p oin t x ∈ U . F or ev er y y ∈ K there exist p y , q y ∈ Q 4 , p y 6 q y , suc h that y ∈ int ♦ ( p y , q y ), where the int erior is considered with resp ect to the Euclidean top ology τ E on M , and x / ∈ ♦ ( p y , q y ). Sin ce K is compact, there exist y 1 , y 2 , . . . , y k ∈ K with K ⊆ k [ i =1 in t ♦ ( p y i , q y i ) . Then x ∈ k \ i =1 ( M r ♦ ( p y i , q y i )) = M r k [ i =1 ♦ ( p y i , q y i ) ⊆ U, and the closed set S k i =1 ♦ ( p y i , q y i ) is an elemen t of P . Hence, also ev ery set U , which is op en with resp ect to τ G E , is a u nion of complemen ts of elemen ts of P , whic h are closed in the same top ology . Then P forms a closed base for τ G E .  Finally , we are ready to complete the construction of the causal site on M . Let A, B ∈ P non-empt y . W e put A ≺ B if A 6 = B and for ev ery a ∈ A , b ∈ B , a 6 b . Theorem 4.2. ( P , ⊆ , ≺ ) is a c ausal site. A NEW CAUSAL TOPOLOGY AND WHY THE UNIVERSE IS CO-COM P A CT 11 Pr o of. First of all, w e need to show that ≺ is a transitiv e on the set P r { ∅ } (the anti -reflexivit y of ≺ follo ws directly f r om the defin ition). Supp ose that A ≺ B and B ≺ C , where A, B , C are n on-empt y . Let a ∈ A , c ∈ C . Since B 6 = ∅ , ther e is s ome b ∈ B . The vect ors b − a and c − b are non-spacelik e and non-p ast-oriente d. T hen also the vec tor c − a = ( c − b ) + ( b − a ) is also non-space-lik e and non-past-orien ted. S u pp ose that A = C . T hen A ≺ B and B ≺ A . T aking an y a ′ ∈ A and b ′ ∈ B , we get that b oth ve ctors a ′ − b ′ and b ′ − a ′ are non-spacelik e and non-past-orient ed, whic h giv es a ′ = b ′ . Then A = B is a singleton, but this equalit y con tradicts to the d efinition of the relation ≺ . Thus ≺ is transitiv e. Since ⊆ is the set inclusion, th e axioms (i)-(iii) are satisfied trivially . Let us c hec k the axiom (iv). Let A ∈ P , A 6 = ∅ . Since in the Euclidean top ological stru cture the compact sets are b oun ded, there exists a d iamond D = ♦ ( p 0 , q 0 ) with A ⊆ D . Denote O A = { p | p ∈ D, A ⊆ J + ( p ) } . Since q 0 ∈ O A , O A 6 = ∅ . L et L ⊆ O A b e a non-empt y linearly ordered c hain with r esp ect to 6 . W e will sho w that L has an upp er b ound in O A . Consid er the n et id L ( L, 6 ). S ince D is compact, id L ( L, 6 ) h as a clus ter p oint, sa y p L ∈ D . Su pp ose th at ther e is s ome l ∈ L such that p L / ∈ J + ( l ). Since the set J + ( l ) is closed in M , there exists ε > 0 su ch that B ε ( p L ) ∩ J + ( l ) = ∅ . By the defin ition of the cluster p oint, there exists m ∈ L , l 6 m , suc h that m ∈ B ε ( p L ). Th en m ∈ J + ( m ) ∩ B ε ( p L ), but this is not p ossible since J + ( m ) ⊆ J + ( l ). Hence, p L ∈ T l ∈ L J + ( l ), wh ic h means that p L is an upp er b ound of L in D . It remains to sho w that A ⊆ J + ( p L ). Supp ose con v ersely , that th ere exists some r ∈ A r J + ( p L ). Since J + ( p L ) is closed in M , there exists ε > 0 su ch that B ε ( r ) ∩ J + ( p L ) = ∅ . Since p L is a cluster p oint of the net id L ( L, 6 ), there exists n ∈ L , n ∈ B ε/ 2 ( p L ). Then r ∈ A ⊆ J + ( n ). Denote q = r + ( p L − n ). The vecto r q is the translation of r by the v ector p L − n , and J + ( p L ) is the translation of the cone J + ( n ) b y the same vect or, so q ∈ J + ( p L ). No w , 0 < ε ≤ k r − q k = k n − p L k < ε 2 , whic h is a contradictio n. Th us A ⊆ J + ( p L ), and so p L ∈ O A is the up p er b ound of th e c h ain L . Let M A b e the s et of all maximal elemen ts of O A (with resp ect to the order 6 ). By Zorn’s Lemma, for ev ery p ∈ O A there exists m ∈ M A suc h that p 6 m . W e pu t A ⊥ = [ m ∈ M A J − ( m ) , and for B ∈ P , B 6 = A w e denote B A = B ∩ A ⊥ . T o claim that B A ∈ P , w e need to sh o w that M A is finite. The b oundary of A ∈ P can b e decomp osed in to a fin ite set S A of pieces of th e b oundaries of the cones J ( t ), t ∈ T A , where T A is a p rop er fin ite set. If m ∈ M A , then the b oun dary of J ( m ) must in tersect s ome elemen ts of S , otherwise m cannot b e maximal. Moreo v er, the cone J ( m ) is fully determined by a 12 MAR TIN MARIA KO V ´ AR finite and limited num b er of su c h inte rsections, b eca use the p oin ts of these in tersections must satisfy the equation of the b oundary of J ( m ). But this w ould not b e p ossible for an infinite set M A . T hen B A ∈ P . L et b ∈ B A , a ∈ A . By the defi nition of B A , there exists some m ∈ M A with b ∈ J − ( m ), so b 6 m . W e also hav e a ∈ A ⊆ J + ( m ), so m 6 a . Then b 6 a , w h ic h implies B A ≺ A . Supp ose that C ≺ A , C ⊆ B for some C ∈ P . L et c ∈ C . If a ∈ A , then c 6 a , whic h giv es a ∈ J + ( c ). Therefore, A ⊆ J + ( c ). Then c ∈ O A , so there exists m ∈ M A , such that c 6 m . Then c ∈ J − ( m ) ⊆ A ⊥ . Hence, C ⊆ A ⊥ , whic h together w ith C ⊆ B giv es the r equested inclus ion C ⊆ B A .  No w w e will concentrat e us on the reconstruction of the original top ology on M from the causalit y structur e of ( P , ⊆ , ≺ ). Let π b e the family of all maximal cen tered sub sets of P . Theorem 4.3. The top olo g i c al sp ac e ( X , τ ) c orr esp ond ing to the fr amework ( P d , π d ) is home omor phic to M e qu ipp e d with the c o-c omp act top olo gy. Pr o of. As w e already defin ed b efore, X = P d = π . Note that any p oin t p ∈ M defines a maximal centred sub set of P , say f ( p ) = { C | C ∈ P , p ∈ C } . The f amily f ( p ) ob viously is centered, since P is closed under fi nite in tersections and f ( p ) con tains those elements of P , whose contai n p . Let Q b e another cen tered family such that f ( p ) ⊆ Q ⊆ P . Supp ose that there is some F ∈ Q , su c h that p / ∈ F . The set M r F is op en with r esp ect to the Euclidean top ology τ E , so there exist u, v ∈ Q 4 , u 6 v , such that p ∈ ♦ ( u, v ) ⊆ M r F . But ♦ ( u, v ) ∈ P , so ♦ ( u, v ) ∈ f ( p ) ⊆ Q , while ♦ ( u, v ) ∩ F = ∅ . This cont radicts to the assumption that Q is cen tered. Th us all element s of Q con tain p , wh ich means that Q = f ( p ). No w it is clear that f ( p ) is a maximal centred subfamily of P . Con v ersely , a maximal cen tered sub f amily Q ∈ π has a non emp t y in- tersection, b ecause of compactness of M in the co-compact top ology . If { x, y } ⊆ T F ∈ Q F , where x 6 = y , then th ere exist u, v ∈ Q 4 , u 6 v , su c h that x ∈ ♦ ( u, v ) and y / ∈ ♦ ( u, v ). Then Q ∪ { ♦ ( u, v ) } ⊆ P is an extension of Q whic h is also cente red, wh ic h con tradicts to the m aximalit y of Q . Thus the in tersection of T F ∈ Q F con tains only one element , sa y g ( Q ). Consequently w e h a v e g ( f ( p )) = p and f ( g ( Q )) = Q . Thus the mappin gs f : M → X and g : X → M are bijections inv erse to eac h other. F urther, for A ∈ P we hav e g − 1 ( A ) = { Q | Q ∈ π , g ( Q ) ∈ A } = { Q | Q ∈ π , Q ∈ f ( A ) } = { Q | Q ∈ π , A ∈ Q } = π ( A ), w hic h is a subb asic closed set in ( X, τ ). Then g : X → M is con tinuous. No w, tak e a set π ( B ), where B ∈ P , fr om the closed base π d of τ . Then f − 1 ( π ( B )) = { p | p ∈ M , f ( p ) ∈ π ( B ) } = { p | p ∈ M , B ∈ f ( p ) } . F or every p ∈ f − 1 ( π ( B )), f ( p ) is a maximal cen tered subf amily of P , contai ning the set B (whic h is compact with r esp ect to τ E ). As we h a v e sho wn ab ov e, its in tersection con tains the only element g ( f ( p )) = p . S o f − 1 ( π ( B )) = { p | p ∈ M , p ∈ B } = B . Since B is a compact set with resp ect to the Euclidean A NEW CAUSAL TOPOLOGY AND WHY THE UNIVERSE IS CO-COM P A CT 13 top ology τ E on M , it is closed in the co-compact top ology and the map f : M → X is con tin uous. Hence, the spaces ( X, τ ) and M , equip p ed w ith the co-compact top olo gy , are homeomorph ic.  5. Final Rema rks in H ist orical Context The pr ogress in mathematical and theoretical ph ysics witn esses that v ari- ous applications of top ology in p hysics m a y b e f ar-r eac hin g and illuminating. It could b e v ery difficu lt to trac k d o wn the origins of su c h applications, bu t one of the first attempts m a y b e asso ciated with the y ear 1914, wh en A. A. Robb came with his axiomatic system for Minko wski space M , analogous to the well-kno wn axioms of Euclidean plane geometry . In [19] he essentiall y pro v ed that the geometrical and top olog ical structure of M can b e r econ- structed from the un derlying s et and a certain order relation among its p oints. As it is n oted in [6], some prominent mathematicians and physicists criticized the u se of lo cally Euclidean top olo gy in mathematical mo dels of the spacetime. Perhaps as a reflection of these d iscussions, appr o xim ately at the same time wh en de Gro ot w r ote his pap ers on co-compactness dualit y , there app eared t w o int eresting p ap ers [22] and [23], in whic h E. C. Zeeman studied an alternativ e top ology f or Minko wski space. The Zeeman top ol- ogy , also referred as the fi ne top ology , is the fi nest top ology on M , whic h induces the 3-dimensional Eu clidean top ology on ev ery space-axis and the 1- dimensional Eu clidean top ology on the time-axis. Among other in teresting prop erties, it indu ces the d iscrete top ology on eve ry ligh t ra y . A. Kartsaklis in [13] stud ied connections b et w een top ology and causalit y . He attempted to axiomatiz e causalit y relationships on a p oin t set, equipp ed with thr ee b i- nary relations, satisfying certain axioms, by a stru cture called a c ausal sp ac e . He also in trod uced so called c hr onolo gi c al top olo gy , the coarsest top ology , in wh ic h every n on-empt y in tersectio n of the c hronologica l future and th e c h ronological past of t w o distinct p oin ts of a causal space is op en. In the camp of quant um gra vit y , th ere app eared similar efforts and at- tempts to get s ome gain from stud ying th e underlyin g structure of s pace- time – top ological, geometrica l or discrete – ho w ev er, significan tly later. The p ossible motiv ation is explained, for instance, in [20]. C. R ov elli notes here that the lo op quant um gravit y leads to a view of the geometry struc- ture of spacetime at the s hort-scale lev el extremely differen t f rom that of a smo oth geometry backg round. Also the top olo gy of spacetime at Planc k scales could b e very different from that we meet in our everyda y exp erience and which has b een originally extrap olated from the fun damen tal concepts of the con tin u ous and smo oth mathematics. Thus the usual prop erties and attributes of the spacetime, like its Hausdorffness or metrizabilit y may n ot b e satisfied (for a groundbr eaking pap er, see [12]). The most imp ortant source of inspiration for our pap er w as the w ork [2] of J. D. Christensen and L. Crane. Motiv ated by certain requirements of th eir researc h in quan tum 14 MAR TIN MARIA KO V ´ AR gra vity , these authors d ev eloped a no v el axiomatic system for th e general- ized spacetime, called c ausal site , qualitativ ely different fr om the previous, similar attempts. Th e notion itself is a su ccessful synthesis of tw o other notions, a Grothendiec k site (wh ic h basically is a small category equip p ed with the Gro ethendiec k topology) [1] and a causal set of R. Sorkin [21]. One of the m ost imp ortan t merits of the n ew axiomatic system it is the fact that the causal site is a p ointle ss stru cture, not unlik e to some w ell-kno wn concepts of p oin tless top ology and lo cale theory . The conte n ts of our p ap er can b e considered as a certain kind of a vir- tual exp erimen t. W e constructed a top ology from a general causal site by a purely mathematical, straigh tforw ard and canonical w a y . T aking the causal site giv en by Minko wski space w e did not receiv e th e usual and naturally exp ected Eu clidean top ology on M , but its de Gro ot dual. Th is is su rprisin g, b ecause the receiv ed top ology seems to b e more closely related to the w a y , ho w the philosoph y of physics traditionally u nderstands the infinit y in a con- text of exp ected finiteness of the physical quan tities. As it wa s remarke d b y d e Gr o ot in [10] (and also by J. M. Aarts in oral communication with de Gro ot), fr om the philosophical p oint of view, the co-compact top ology is naturally related to the concept of p oten tial infinity – in a con trast to th e no- tion of actual infinity , w hic h is mostly used in th e traditional m athematical approac h. T o illustrate the difference, consider a counta bly infin ite sequ ence x 1 , x 2 , . . . of p oin ts lying on a straigh t line in space or sp acetime, with the constan t distance b et w een x i and its successor x i +1 . In the usual, Eu clidean top ology , the sequence is d ivergen t and it app roac h es to an impr op er p oint at infi nit y . T o mak e it conv ergen t, one n eed to em b ed the space in to its com- pactification (for instance, the Alexandroff one-p oin t compactification is a suitable one). T he p oin ts completed by the compactificat ion then app ear at the infinite distance from any other p oin t of the sp ace. On the other hand, the co-compact top ology , wh ic h lo cally coincides with th e u sual top ology , is already compact and su p erconnected, so the sequence x 1 , x 2 , . . . is resid- ually in eac h neigh b orho od of ev ery p oint. Since the co-compact top olog y lo cally coincides with the Eu clidean top ology , in m ost cases it p erforms the same j ob , but in a “more elegan t” wa y – with less op en sets. Both top ologies are closely related to eac h other via the de Gro ot dualit y as we d escrib ed in Section 2. W e may close the p ap er by returnin g to the qu estion, that w e stated at the b eginning. T h e result of our virtual exp erimen t certainly is n ot a rigorous pro of of the conjecture that the constru cted causal top ology will fit with the realit y also in more complex and more complicated physical situations. But, at least, it confirms that n otion of causal site of J . D. Christensen and L. Crane is designed correctly . And it giv es a strong reason for the b eliev e, that the causal structure is the pr imary stru cture of the spacetime, whic h also carries its top ological in f ormation. A NEW CAUSAL TOPOLOGY AND WHY THE UNIVERSE IS CO-COM P A CT 15 Referen ces 1. A rtin M., Gr othendie ck T op olo gy , Notes on a Seminar, Harward Universit y Press (1962), pp . 1-95. 2. Christensen J. D., Crane L., Causal Sites as Quantum Ge ometry , J. Math. Phys. 46 (2005), 122502-122523 . 3. 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Lawson J.D., Mislo ve M., Pr oblems in domai n the ory and top olo gy , Op en problems in top ology (v an Mill J., R eed G. M. eds.), North-Holland, Amsterd am (1990), p p. 349-372. 18. Matthews S. G., Partial M etric T op olo gy , Papers on General T opology and Applica- tions, Eight Summer Conference at Qu eens Colleg e (S. And ima et.al. eds.), Ann . N. Y. A cad. Sci. 728 (1994), 183-197. 19. Robb A . A., Ge ometry of Ti me and Sp ac e , Cam bridge U nv ersit y Press, 1936, pp. 1- 408. (the second edition of A the ory of Tim e and Sp ac e , Cam brige Universit y Press, 1914) 20. Rovel li C., Quantum Gr avity , Cam b rid ge Unversi ty Press, 2005, pp . 1-458. 21. Sorkin R. D., Causal Sets: Di scr ete Gr avity , gr-qc/0309009 22. Zeeman E. C., Causality implies the L or entz gr oup , J. Math. Phys. 5 (1964), no.4, 490-493. 23. Zeeman E. C., The top olo gy of Minkowski sp ac e , T op ology 6 (1967), 161-170. Dep ar tme nt of Ma thema tics, F a cul ty of Electrical Engin eering and Com- munica tion, University of Technology, Te chnick ´ a 8, B rno, 616 69, Czech Re- public E-mail addr ess : kovar@feec.v utbr.cz