Dimension on discrete spaces

Preprint SU-GP-93/1-1 January 25, 1993 DIMENSION ON DISCRETE SPACES. Alexander V. Evako Syracuse University, Department of Physics Syracuse N.Y. 13244-3901. Abstract In this paper we develop some combinatorial models for continuous spaces. In this spirit we study the approximations of continuous spaces by graphs, molecular spaces and coordinate matrices. We de ne the dimension on a discrete space by means of axioms, and the axioms are based on an obvious geometrical background. This work presents some discrete models of n-dimensional Euclidean spaces, n-dimensional spheres, a torus and a projective plane. It explains how to construct new discrete spaces and describes in this connection several three-dimensional closed surfaces with some topological singularities It also analyzes the topology of (3+1)-spacetime. We are also discussing the question by R. Sorkin [19] about how to derive the system of simplicial complexes from a system of open covering of a topological space S. GR-QC-9402035 Introduction. A number of workers have been unhappy about applications of the continuum picture of space and spacetime. They have believed that the breakdown of the functional integral at the Plank length shows not merely the failure of the classical eld equations but also indicates that a di erential manifold upon which they are built should be replaced by some nite theory. This was certainly one of the motivations behind Penrose invention [15] of spin networks and recent works by Finkelstain on a novel spacetime microstructure [1]. Isham, Kubyshin and Renteln [2] introduce a quantum theory on the set 1 of all topologies on a given set, and show that for a nite basic set almost all metrics can be obtained by embedding this set into a vector space and then varying the norm of this space. One more approach to a combinatorial model of space and spacetime is studied in work [19] by R.Sorkin. He replaces general topological spaces by a nite ones and describes how to associate a nite space with any locally nite covering of a topological space. He also presents some examples of posets derived from simple spaces. Another way is Regge calculus [18]. Many suggestions for formulating various Regge calculus versions have been made in order to face a number of problems. Regge calculus describes general relativity spacetime by using a simplicial complex. Its fundamental variables are a set of edge lengths and an incidence matrix that describes how they are connected. One approach supposes that the connectivity of a simplicial complex is xed, but the lengths of edges can be varied. Another approach xes the edge length and varies the connectivity of a simplicial complex in order to change the metric of a spacetime. The Regge calculus, however, supposes that there is a continuous underlying spacetime and does not account naturally for the appearance of a minimal length in e ective theories. At the same time in mathematics there exist several quickly developed approaches to discrete spaces in the frame of digital topology which can be useful in physics. Digital topology is the study of topological properties of image arrays. It provides the theoretical foundations for image processing operations such as image thinning, border following and object counting. The paper [12] reviews the fundamental concepts of digital topology, surveys the major theoretical results in this eld and contains the bibliography of almost 140 references. Traditionally a discrete or digital space is considered as a graph whose edges between vertices de ne the nearness and connectivity in the neighborhood of any vertex. This approach was used by Rosenfeld [16,17] who proved the rst version of the Jordan curve theorem by using a graph theoretical model of a digital plane. However, this model does not utilize a topological basis and requires di erent nearness for the curve and its background. An alternative topological approach to the digital topology uses the notion of a connected topology on a totally ordered set Z of integers [9-11,13]. The digital plane Z  Z or the three-dimensional digital space Z  Z  Z are the 2 topological products of two or three such spaces respectively. Using this construction the Jordan curve theorem in two and three dimensions was proven. Another approach to nite topology is o ered by Kovalevsky in [14]. He builds the digital space as a structure consisting of elements of di erent dimensions by using a such well-known in topology element as a cellular complex. Our approach to discrete spaces is based on using three combinatorial tools: a graph a molecular space and a coordinate matrix [3-8]. In this panel the material to be presented below begins with short description of some geometrical background for the de nition of the dimension on a graph. Then we shall show the connection between a graph, a molecular space and a coordinate matrix. We shall de ne the dimension on discrete spaces which is based on some geometrical ground. We shall analyze the dimensions of di erent models of two, three and n-dimensional discrete spaces. We present some examples of three-dimensional discrete closed spaces with strange topological features which do not have direct continuous analogies. Then we prove some theorems showing how to construct closed three dimensional spaces with nonstandard topology. Finally we discuss the topological structure of a (3+1) space-time. Geometrical background for the de nition of the dimension on a graph. We are going to construct now a graph with certain properties which can be thought as a convenient tool for describing the ideas of nearness and continuity by combinatorial methods. This will be done in the rst place by picking out in elementary geometry those properties of nearness which seem to be fundamental and taking them as axioms. To get a glimpse of the intuitive geometrical ground of the dimension consider the following example. Let E n be n-dimensional Euclidean space and p a point in it. The neighbourhood of p is commonly de ned to be any set U such that U contains an open solid n 1 n disk D1 of center p. The boundary of this disk is the sphere S1 . The de nition of neighborhood is formulated in this way so as to be as free as possible from the ideas of size and shape, concepts that play no part in topology. Using this de nition of a neighbourhood of a p in Euclidean space it is easy to see that the family of sets U satisfy the usual topological axioms. 3 1. belongs to any neighbourhood of . 2. If is a neighbourhood of and  , then is a neighbourhood of . 3. If and are neighbourhood of , so as is T . 4. If is a neighbourhood of , then there is a neighbourhood of such that  and is a neighbourhood of each of its points. Taking these properties as axioms in an abstract formulations we can de ne a topological space as a set with a family of subsets of satisfying the four properties listed earlier. We can also de ne a subset of open if for each point in , W is a neighbourhood of . Note that the disk 1 plays crucial role in this de nition. In the continuous case the sphere 1 1 contain in itself an in nite sequence of disks and spheres 1 of center . p p U p U V V p U V p U U V p V U V E E E W p p V W E p D n S n n n Di p Si  1  n D1 n S1 n D2 n S2    1   1 n :::: Di :::: n :::: Si :::: However the situation is di erent in the discrete case where the sequences of disks and spheres can not be in nite and axiom 4 is not realized. Therefore we have a nite series of the form  1  n D1 n S1 n D2 n S2   1   :::: :::: n Dt 1 n St The smallest disk and the smallest sphere 1 can not be reduced in the sense that they do not contain disks and spheres others then themselves (Figure 1). The topological meaning this construction for a graph reveals that the vertex is considered as n-dimensional if its minimal neighbourhood is the sphere 1. The point and the nearest sphere 1 together form the smallest disk of center . Point of a discrete space is considered as one-dimensional if n n Dt St p S n p p S n p D G 4 n its nearest neighbourhood is a zero-dimensional sphere S 0. It is well known that S 0 is a set of two disconnected points. In the other words S 0 is disjoined graph of two points. In one-dimensional discrete sphere S 1 all points are onedimensional. Obviously the minimal number of points required for S 1 is four. For two-dimensional discrete sphere S 2 all its points are two-dimensional. It means that nearest neighbourhood of any point of S 2 should be S 1 and so on. A molecular space and a coordinate matrix of a graph. In order to make this paper self-contained we shall summarize the necessary results from our previous papers. Let E 1 be in nite-dimensional Euclidean space. Take the coordinates of a point x; x 2 E 1, as a sequence of real numbers x = (x1; x2; ::; x ; ::) = [x ]; i 2 N: We de ne unit cube K 2 E 1 in the following way: each x; x 2 K , has coordinates x satisfying conditions presented in [3,4,5,8]: n i i n  x  n + 1; i 2 N; n i i i i integer: Therefore, K is an in nite-dimensional cube with unit edges. In [3,4,5,8] K is called a kirpich. We will use this name in the present paper. The position of K in E 1 is determined by the left vertex coordinates. For the given kirpich we have K = (n1; n2; :::; n ; :::) = [n ]; i 2 N: n i Two kirpiches are called adjacent, if they have common points. The distance d(K1; K2) between kirpiches K1 = [n ] and K2 = [m ] is de ned by using sup norm i d(K1; K2) = maxjn i i m j; i 2 N: i Obviously, two kirpiches are adjacent if their appropriate coordinates distinguish not more then 1, or the distance between them equals 1. Any set of kirpiches in E 1 is called a molecular space and is denoted by M . Clearly, any molecular space can be represented by its intersection graph G(M ). It 5 was shown in [4,8] that any graph G can be represented by a molecular space M (G), such that G = G(M (G)). Clearly, more than one M (G) can be built for the graph G. There exists isomorphism between any two M (G). Let M be a molecular space with a set of kirpiches V = (K1; K2; ::; K ); K1 = [k1 ]; K2 = [k2 ]; :::K = [k ] The matrix [k ] is called the coordinate matrix of the molecular space M and its intersection graph G(M ) and is denoted A(M ) or A(G(M )). This matrix has n rows and in nite columns. n i i n ni pi In fact we shall always use a nite-dimensional Euclidean space. The intuitive background for using the in nite-dimensional unit cube is the attempt to create some universal element not depending on the dimension and suitable for describing elements of di erent dimensions: zero-dimensional points, onedimensional lines, two-dimensional surfaces and so on. Let S be a surface in E . The molecular space M (S ) of S is a set of kirpiches intersecting S . Figure 2 shows the graph G, its molecular space M and its coordinate matrix A. n The dimension and the metric of a discrete space . Our objective now is to de ne the dimension on graphs. Later on we will use the names discrete space and point for a graph and its vertex when we want to emphasize the notion of the dimension on it. Since in this paper we only use induced subgraphs, we shall use the word subgraph for an induced subgraph. We shall also use some symbols, notations and names introduced in our previous works. De nitions Let G; G1 and v be a graph, its subgraph and its point.  The subgraph B (G1) containing G1 is called the ball of G1 if any point of B (G1) is adjacent to at least one point of G1.  The subgraph B (G1) without points of G1 is called the rim of G1 and it is denoted O(G1 ): Obviously B (G1) G1 = O(G1):  If G1 is a point v then B (v) and O(v) are called the ball and the rim of v respectively. 6 ) ( ) = ( 1) T ( 2) T ( n) is called the joint ball of the points 1 2 n.  The subgraph ( 1 2 n) ( 1 2 n) = ( 1) T ( 2) T ( n) is called the joint rim of the points 1 2 n. Let and 1 be a graph and its subgraph with points ( 1 2 n) and ( 1 2 p) respectively. It is clear that [ [ [ ( 1 ) = ( 1) ( 2) ( p) 1 [ [ [ ( 1 ) = ( 1) ( 2) ( p)  A graph n ( 1 2 n) of points is called completely connected or complete if any its two points are adjacent.  A graph n ( 1 2 n) of points is called completely disconnected if any its two points are disjoined.  The join 1  2 of two graphs 1 and 2 is the graph which consists of two graphs 1 and 2 and all edges joining points of 1 with points of 2. To this end we begin by de ning the dimension on a graph in the following way. De nition 1 Zero-dimensional normal space 0 is the graph which consists of two non-adjacent points.  The subgraph ( B v1 ; v2 ; :::vn ; B v1 ; v2 ; :::vn B v B v :::B v ; O v O v :::O v ; v ; v ; :::v O v ; v ; :::v ; O v ; v ; :::v v ; v ; :::v G G v ; v ; :::v v ; v ; :::v O G O v B G B v K v ; v ; :::v n H v ; v ; :::v n G G G O v ::: B v G O v ::: G B v G G G G S De nition 2 A point v of a graph G is called a normal n-dimensional point if its rim ( ) is a normal (n-1)-dimensional space. O v De nition 3 For any integer n, n  1, de ne a normal n-dimensional space to be a connected graph in which any point is n-dimensional normal. According to these de nitions one-dimensional normal space is any circle 4 n Further the denotation ( ) will be used for the dimension of a graph . Figure 3 represents normal zero, one and two-dimensional spheres and one, two and three-dimensional disks. Figure 4 shows normal two-dimensional discrete at spaces and their molecular spaces. 12 is the two-dimensional discrete space in Khalimsky topology [9-13]. C ; n : p G G E 7 Tree-dimensional normal sphere is the graph 3 depicted in Figure 5. It can be veri ed without diculties that the complete (n+1) partite graph (2 2 2) is the minimal graph describing n [3,4,6]. Therefore, the minimal number of elements necessary to describe n is 2n+2. Notice that the same number of points is used by R. Sorkin [19] to describe n in the nitary topology approach. A normal torus 2 and a projective plane 2 are presented in Figure 5. It can be checked directly that the Euler characteristic and the homology groups of all graphs depicted in Figures 3-5 match the Euler characteristic and the homology groups of their continuous counterparts [5,7]. In [3] normal n-dimensional space is called of the type 2. This separation to the di erent types is caused by the fact the normal molecular spaces and graphs of any type n 6= 1 2 have some unusual properties di erent from those of direct discrete models of continuous spaces in m. Our objective now is to de ne a generalization of the dimension which includes the above de nition. It is natural to consider a point as zerodimensional if its neighborhood does not contain any normal space. De nition 4 A point of a graph is called zero-dimensional, ( ) = 0, if ( ) does not contain the normal zero-dimensional sphere 0. De nition 5 A connected graph is called zero-dimensional, ( ) = 0, if any of its points is zero-dimensional. By this de nition in a zero-dimensional connected graph any two points are adjacent. Therefore, this graph is a complete graph on any number of vertices. A disconnected zero-dimensional graph is considered as a zerodimensional sphere 0 if it has exactly two components. It is clear that 0 contains normal zero-dimensional sphere as its subgraph. We will extend this analogy to higher dimensions. De nition 6 A graph is called closed n-dimensional if 1. For any point ( ) . 2. is homotopic to some normal n-dimensional space. S K ; ; ::: S S S T P ; n ; E v v G O v p v S G p G S S G v p v n G De nition 7 A point v is called n-dimensional, p(v ) = n, if 1. ( ) contains a closed (n-1)-dimensional space. O v 8 2. ( ) does not contain any closed n or more-dimensional space. O v De nition 8 A graph G is called n-dimensional, p(G) = n, if 1. contains at least one n-dimensional point 2. For any point () . G v p v  n In de nition 6 we use homotopy of graphs. Two graphs are called homotopic if each of them can be turned into the other by contractible transformations which consist of contractible gluing and deleting of vertices and edges of a graph. It was shown [5-7] that these transformations do not change the Euler characteristic and the homology groups of graphs. Let us look at some examples of n-dimensional (not normal) discrete spaces and their molecular spaces. Spheres 0, 1, their molecular spaces and the molecular space ( 2) of sphere 2 are drawn in Figure 6. ( 2) is a hollow space, it does not contain the central unit cube. These spheres are not normal but satisfy de nitions 6-7. Any sphere depicted in Figure 6 has the same Euler characteristic and homology groups as continuous and can be transformed to the sphere drawn in Figure 3 by contractible transformations. Flat one, two and three-dimensional spaces and their molecular spaces are shown in Figure 7. It is easy to construct three and more dimensional spaces but it is dicult to draw it. For a at three-dimensional space the only molecular space is shown. However a n-dimensional space can be easily described by its coordinate matrix of the0 form 1 11 12 1 C B B 21 22 2 C B CC B ... ... . . . ... C B CC B B A @ 1 2 S S M S S M S S n S S n n x x


x n x x


x n xp xp


xpn











where = 0 1 2 ; = 1 2 3 ; = 1 2 The standard de nition of the distance on a graph can be applied to a discrete space. De nition 9 The distance ( 1 2) between two points 1 and 2 in a discrete space is the length of a shortest path joining them if any; otherwise ( 1 2) = . xik ;  ;  ; ::: i ; ; ; :: d v ;v ; ; :::n: v G d v ;v k 1 9 v Obviously the distance is a metric. The Plank length can be thought as the length of an edge of the graph. Mathematical observations. Before proceeding to the main result of this paper let us pause to describe some mathematical observations relating to this approach. The following surprising facts were revealed.  Suppose that S 1 is a circle of radius R. Let A be a cover of S 1 by arcs whose length is small enough compared with R. Denote G(A) the intersection graph of this cover. This graph is called the circular arc graph. It appears that : 1. Dimension of G(A) is equal to one, p(G(A)) = dim(S 1) = 1. 2. G(A) has the same Euler characteristic and homology groups as S 1. 3. G(A) can be reduced to the cycle graph C4 by contractible transformations [5,6,7] (S11 in Figure 3).  Suppose we have some two-dimensional closed surface, for example, a sphere S 2 of radius R. Consider any tiling A of S 2 by elements (a1; a2; :::an) whose size is small enough relative to the radius R. Construct the intersection graph G(A)(v1; v2; :::vn) in the following way: Two vertices v1 and v2 are adjacent i elements a1 and a2 have at least one common point. In most cases it turns out that 1. Dimension of G(A) is equal to two, p(G(A)) = dim(S 2) = 2. 2. G(A) has the same Euler characteristic and homology groups as S 2. 3. G(A) can be reduced by contractible transformations into the minimal two-dimensional sphere on 6 vertices [5,6,7] (S 2 in Figure 3).  Suppose that P k is a surface in E n ; n = 2; 3 (for spheres n can be any number). Divide E n into a set of cubes with the scale l1 of the cube edge and call the molecular space M1(P k ) of P k the family of cubes intersecting P k . Denote G1 (P k ) the intersection graph of M1(P k ). Change the scale of the cube edge from l1 to l2 and obtain M2(P k ) and G2(P k ) by using the same structure. It is revealed that in most of cases 1. p(G1(P k )) = p(G2 (P k )) = dim(P k ) 2. G1(P k ) and G2(P k ) have the same Euler characteristic and the homology groups as P k . 3. G1(P k ) and G2(P k ) can be transformed from one to the other with four kinds of transformations if the divisions are small enough. 10 These facts allow us to assume that the graph and the molecular space contain topological and perhaps geometrical characteristics of the surface . Otherwise, the molecular space and the graph are the discrete counterparts of a continuous space . P k M G k P Singular spaces. This section describes a method of obtaining new spaces from given ones. We will see that there exist n-dimensional normal spaces with some peculiar properties. These spaces give rise to new discrete structures that have di erent topologies in di erent points. Theorem 1 Let ( 1 2 ) and 2( 1 2) be a n-dimensional normal space and the completely disconnected space on two points respectively. Then 2( 1 2)  ( 1 2 ) is a (n+1)-dimensional normal space. Proof. The proof is by induction. (i) For n=0,1 the theorem is veri ed directly. (ii) Assume that theorem is valid for any n,  . Let ( 1 2 ) be a normal (k+1)-dimensional discrete space. Consider G p ; p ; :::pr H v ;v H v ;v G p ; p ; :::pr n W = ( H2 v1 ; v2 k ) ( G p1 ; p2 ; :::pr G p ; p ; :::pr ) It is necessary to show that is a (k+2)-dimensional discrete normal space. Take any point . With respect to the de nition of a normal space, ( ) in denoted ( )j is a k-dimensional normal space. Therefore, according to the assumption 2  ( ) is (k+1)-dimensional normal space. Hence any point in has the rim which is (k+1)dimensional normal space. The rims of points 1 and 2 in are the (k+1)-dimensional normal space by construction. W pi O pi G O pi G H pi v v O pi W W G ( )j = O pi W H2  (O(pi )jG); i =1 2 ; ; :::n; ( )j = O vk W G; k =1 2 ; Therefore, the rim of any point of is a (k+1)-dimensional normal space, and, by the de nition, is a normal (k+2)-dimensional space. That completes the proof. 2 W W 11 We are now in a position to describe n-dimensional normal spaces with peculiar properties.  Firstly construct a space without singularities. Let be n-dimensional sphere . It means that the rim of any point of is a normal sphere 1, and can be turned into the minimal on 2n+2 points by contractible transformations [3,6,7]. Consider = 2( 1 2) . If 2 then ( )j = 2( 1 2) 1 = . For points 1 and 2 the rim is itself. Therefore, the rim of any point of is sphere , and is a normal (n+1)-dimensional space. It is easy to show that can be reduced to the minimal (n+1)-sphere +1 by contractible transformations and, therefore, = +1 . G S S n S n S W H v ;v v S S n p v n S S n S n n n O p W H n v ;v n Sp n Sp W W W S W = W S n n 2S H n = S +1 n ( )j = 2  1 = 2 ; ( 1 )j = ( 2 )j =  Suppose that is a discrete two-dimensional torus 2 depicted in Figure 5. For any point of 2 ( ) = 1. Therefore, in = 2 ( 1 2)  2 the rim of any point is a two-dimensional sphere 2, ( )j = 2. However, for points 1 and 2 their rims are the torus 2 itself, ( ) = 2, i=1,2. Notice that the dimension of 2 is equal to 2. Hence is a normal threedimensional space in which the rims of points have a di erent topology. For points 1 and 2 the space has torus neighborhood 2, in all other points the neighborhood is spherical, 2. = 2 2 ( )j = 2  1 = 2 2 2; ( 1)j = ( 2)j = 2  Another peculiar three-dimensional space appears when we choose the projective plane 2 (Figure 5) as a basic space . In three-dimensional normal space = 2( 1 2)  2 the neighbourhoods of 1 and 2 are the projective plane 2, the neighbourhoods of all other points are usual spheres 2. O p W n H n Sp Sp ; p S n O v W G p T O p Sp W Sp v H O p T W v ;v S n T Sp O vi T v W T p v O v T W v T S W O p W H Sp Sp ; H p T T O v P O v W T G W v W v H v ;v P P S = 2P 2 p 2 P ; W H 2 ( )j = 2  1 = 2 ( 1 )j = ( 2 )j = 2  In general we can create a number of three-dimensional normal spaces with two singularities by taking discrete models of closed two-dimensional oriented or non-oriented surfaces as a basic space. O p W H Sp Sp ; 12 O v W O v W P The dimensional local structure of a physical discrete (3+1) spacetime. Now we are ready to discuss some general features of the physical (3+1) space-time. We will restrict our consideration by local properties of a point . Theorem 2. (3+1) space-time is four-dimensional non-normal. Proof. We have to prove that in (3+1) space-time the rim of any point is a closed three-dimensional non-normal discrete space. Suppose that a physical object is in point of a three-dimensional discrete space ( ) at a given moment and at either the same or the nearest point 1 at the next moment + (Figure 8a). In (3+1) spacetime ( ) we have two three-dimensional spaces ( ) and ( + ) corresponding to the di erent moments. Obviously these spaces are joined together in the following way. Point on ( ) should be connected with the ball ( ) on ( + ) (Figure 8b). Therefore, in the (3+1) space-time ( ) (Figure 8c) the rim ( )j( ) of point is as shown in Figure 8d. (i) If the rim ( )j ( ) of in ( ) is a non-normal closed twodimensional space, then, for the same reasons as in theorem 1, ( ) in ( ) is a non-normal closed three-dimensional space, and ( ) is a non-normal four-dimensional space. (ii) Suppose that ( ) is a normal three-dimensional space. Then ( ) in ( ) is a normal two-dimensional discrete space. Obviously ( ) in ( ) contains the normal three-dimensional space ( 1 2)  ( )j ( ) where 1 and 2 are in ( + ) and ( ). By theorem 1 it is a normal three-dimensional space. Take 1 in ( + ), ) 1 2 ( )j( ). It is easy to see that in ( + ) 1 2 ( + is adjacent to all points of the rim of this 1 in ( )j( ). Hence ( )j( ) is a non-normal closed three-dimensional space which can be reduced into normal ( 1 2)  ( )j ( ) by contractible transformations. Thus ( ) is a non-normal four-dimensional space-time. 2 v v R t t v t Dt: R; T R t v B v R t O v R t v Dt R t Dt R; T O v R t R; T v R t O v R; T R; T R t O v R t O v O v R; T R t H u ;u u u v R t Dt R t Dt v v R t Dt ; v O v R; T v v O p R; T H u ;u R; T 13 R t O v R t O v R t R; T Dt Dt Acknowledgment The author would like to thank Rafael Sorkin for his useful comments which resulted in several improvements of the presentation of this paper. References 1. D. Finkelstain, First ash and second vacuum, International Journal of Theoretical Physics 28 (1989) 1081-1098. 2. C. J. Isham, Y. Kubyshin and P. Renteln, Quantum norm theory and the quantisation of metric topology, Classical and Quantum Gravity 7 (1990) 1053-1074. 3. A.V. Ivashchenko (Evako), Dimension of molecular spaces, VINITI, Moscow, 6422-84 (1985) 3-11. 4. A.V. Ivashchenko (Evako), Geometrical representation of a graph, Kibernetika 5 (1988) 120-121. 5. A.V. Ivashchenko (Evako), Homology groups on molecular spaces and graphs, Discrete Math., to appear. 6. A.V. Ivashchenko (Evako), Yeong-Nan Yeh, Minimal graphs of a torus, a projective plane and spheres. Some properties of minimal graphs of homology classes, Discrete Math., to appear. 7. A.V. Ivashchenko (Evako), Representation of smooth surfaces by graphs. Transformations of graphs which do not change the Euler characteristic of graphs, Discrete Math., 122 (1993) 219-233. 8. A.V. Ivashchenko (Evako), The coordinate representation of a graph and n-universal graph of radius 1, Discrete Math., 120 (1993) 107-114. 9. E. Halimski (E. Khalimsky), Uporiadochenie Topologicheskie Prostranstva (Ordered Topological Spaces), Naukova Dumka, Kiev, 1977. 10. E. Khalimsky, R. Kopperman, P. Meyer, Computer graphics and connected topologies on nite ordered sets, Topology and Applications 36 (1990) 1-17. 14 11. T. Kong, P. Meyer, R. Kopperman, A topological approach to digital topology, Amer. Math. Monthly 98 (1991) 901-917. 12. T. Kong, A. Rosenfeld, Digital topology: Introduction and survey, Comput. Vision Graphics Image Process 48 (1989) 357-393. 13. R. Kopperman, P. Meyer, R. Wilson, A Jordan surface of three- dimensional digital space, Discrete Comput Geom. 6 (1991) 155-161. 14. V.A. Kovalevsky, Finite topology as applied to image analyses, Comput. Vision Graphics Image Process 46 (1989) 141-161. 15. R. Penrose, Angular momentum: an approach to combinatorial spacetime. in "Quantum theory and beyond", ed. by T. Bastin ( Cambridge University Press, Cambridge, 1971). 16. A. Rosenfeld, Connectivity in digital pictures, J. Assoc. Comput. Mach. 17 (1970) 146-160. 17. A. Rosenfeld, Digital topology, Amer. Math. Monthly 86 (1979) 621630. 18. T. Regge, General relativity without coordinates, Nuovo Cimento 19 (1961) 558-571. 19. R. Sorkin, Finitary substitute for continuous topology, International Journal of Theoretical Physics 30 (1991) 923-947. Figure Captions. Figure 1: Di erence between an in nite and nite number of enclosed disks Dn in continuous and discrete spaces respectively. Figure 2: Graph G, its molecular space M (G) and its coordinate matrix A(G). Figure 3: Zero (S 0), one (S11; S21) and two (S 2) dimensional normal discrete spheres, and one (v1), two (v2; v3) and three (v4) dimensional points. 15 Figure 4: Normal discrete two-dimensional planes and their molecular spaces. E12 is the two-dimensional plane in Khalimsky topology. Figure 5: A discrete normal three-dimensional sphere S 3, a two-dimensional torus T 2, a two-dimensional projective plane P 2. The Euler characteristic and the homology groups of these graphs are consistent with the Euler characteristic and the homology groups of their continuous counterparts. Figure 6: Zero and one-dimensional non-normal spheres S 0 and S 1 and their molecular spaces M (S 0) and M (S 1). M (S 2) is a molecular space of the two-dimensional non-normal sphere S 2. It does not contain the central unit cube. Figure 7: Non-normal discrete one and two-dimensional at spaces E 1 and E 2 and their molecular spaces M (E 1 ) and M (E 2). M (E 3) is a molecular space of a non-normal discrete three-dimensional at space E 3. Figure 8: Theorem 2 for (1+1) space-time. (1+1) space-time is not normal because O(v) is not a normal one-dimensional sphere. 16