Informatica 22 (1998) 287-308 287 Topological Informational Spaces Anton P. Zeleznikar An Active Member of the New York Academy of Sciences Volariceva ulica 8 SI—1111 Ljubljana, Slovenia anton.p.zeleznikar@ijs.si and at home s51em@lea.hamradio.si http://lea.hamradio.si/~s51em/ Keywords: basis, closed system, connectedness, covering, discrete space, exterior, indiscrete space, informational space: operand formula vector, vector distributiveness (orthogonality), metrics (meaning); interior, linkage, neighborhood, open system, subbasis; system of informational formulas, operands and basic transition formulas; topology of systems Edited by: Vladimir A. Fomichov Received: April 28, 1998 Revised: August 18, 1998 Accepted: August 25, 1998 A system, of different informational formulas, ip, can possess various topological structures, £). By this, topological informational spaces of the form D) can be constructed and the question arises: How can the topological structures be introduced reasonably for concrete systems of informational formulas? A topology causes certain other concepts, e.g., those concerning closed topology, connetedness, continuum, interior, exterior, neighborhood, basis, subbasis, metric, space, etc. of systems, and especially the concept of meaning as a kind of the informational accumulation point. The paper treats topologies of three types of informational formula systems: and . An example of bidirectional consciousness shell is presented enabling a complex engine modeling. 1 Introduction A basic problem of topology1 is to define a general space. By topology a mathematical concept (structure, branch) is meant2 giving sense to various intuitive notions. Topological notions can be innovatively extended into realm of the informational, realizing one of the significant features of the so-called informational space. Such a space can be determined also from other points of view concerning, for instance, the distribu-tivity of informational entities (operands)—which can proceed into different concepts of a vector space. In general, a more complete theory of informational space would need concepts of informational subtheories, such of concerning informational topological space, informational vector space, and informational graph theory. Another, mathematically grounded view to the problem of graph is the so-called topological graph theory [14]. The primitive objective of this theory is to draw a graph on a surface, so that no two edges (graph 1This paper is a private author's work and no part of it may be used, reproduced or translated in any manner whatsoever without written permission except in the case of brief quotations embodied in critical articles. 2Otherwise, topology is a science of position and relation of bodies in space. This paper concerns at least the following topological topics: point system (set) topology (general topology), metric space (e.g., meaning topology), and graph topology. arrows representing informational operators) cross, an intuitive geometric problem that can be solved by specifying symmetries or combinatorial side-conditions (surface graph-imbedding). Although potentially interesting for the informational graph theory, this kind of problem is not in the focus of informational graph investigation. Informationally, graph is merely a presentation of formula or formula system potentiality concerning the setting of the parenthesis pairs (parenthesizing3) in a formula or formula system. Introducing the topology on (over) a system of informational formulas means a challenge of logic—both the informational and the philosophical one—which comes close to some known metamathematical problems [30]. Just imagine a topology in the realm of mathematical axiomatism where for a set of axioms a topology of axiomatic statements (e.g., true formulas) is constructed. Although topology is a general mathematical principle in the realm of the set and space theory, it seems nonobviuously to take it as a set with topology on the set of formulas.4 In mathematics, topology may be considered as an 3Parenthesizing (in German, Einklammerung) has also the philosophical meaning in phenomenology, for instance, in Husserl [17]. 4The author believes too that such an idea reaches beyond the conventional horizon of a mathematician. However, he believes that the following discussion will show the appropriateness of such a trait. 288 Informatica 22 (1998) 287-308 A.P. Zeleznikar abstract study of the limit-point concept [16]. Which factors could dictate the introduction of a topology for a given system of informational formulas? In informational cases, different kinds of reasonable topologies, corresponding intuitive ideas what an understanding, interpretation, conception, perception and meaning should be, are coming to the consciousness, that is, into the modeling foreground. In set theory, the concept of a set (collection, class, family, system, aggregate) itself is undefined. Similar holds for an element x of the set X. The phrases like is in, belongs to, lies in, etc. are used. In informational theory, topology may be considered as an abstract study of the concept of meaning [32, 35] (concerning interpretation, understanding, conceptualism, consciousness, etc. of the informational). Here, meaning of something, of some formula or formula system, functions as an informational limit point, to which it is possible to proceed as near as possible by the additional meaning decomposition of something. The concept of a set is replaced by the concept of a system of informational formulas or/and informational formula systems. In this respect, similar notions to those in mathematics can be used, however, considering the informational character of entities (operands) and their relations (operators). Introducing topological concepts in informational theory, the reader will get the opportunity to experience what happens if the informational concepts, priory described by the author (e.g., [31, 32, 33, 34, 35, 36, 37, 38], to mention some of the available sources) are thrown into the realm of a topological informational space. In this view, informational serialism, parallelism, circularism, spontaneism, gestaltism, tran-sitism, organization, graphism, understanding, interpretation, meaning, and consciousness will appear under various topological possibilities, complementing the already previously presented informational properties, structure, and organization. Mathematical topology, as presented for example in [7, 8, 16, 18, 19, 21, 24, 25], roots firmly in the mathematical set theory [5, 6, 20]. In informational theory, the set is replaced by the concept concerning a system of informational formulas (system, informational system or IS, in short). A system is—said roughly—a set of informationally (operandly, through or by operands) connected informational formulas. The question is, which are the substantial differences occurring between the mathematical and the informational conceptualism in concern to topological structure? Elements of a mathematical set are elements determined by a logical expression (defining formula, relation, statement) and, for example, by notation of the form X = {xi,x2,. ■. ,xm} which presents a concrete structure of the set by its elements. In informational theory, instead of a set, there is a system of informational formulas being elements of the system. Formulas are active, emerging, changing, vanishing informational entities (by themselves) which can inform in a spontaneous and circular manner. What does not change is their informational markers distinguishing the entities. Notation of the form $ ( <¿>2; \m where , Vm presents,5 in fact, only an instantaneous description of the parallel system of markers ipi, by a vertical presentation, denoting concrete formulas (or formula systems), and being separated by semicolons. These are nothing else as a special sort of informational operators, e.g. ||=, meaning the parallel informing of formulas of the system Also, there is a substantial difference between the symbols = and ;=±; the second one is read as 'mean(s)' and denotes meaning and not the usual equality. Another notions to be determined informationally are informational union and informational intersection of systems. It has to be stressed that formulas in a system "behave" in the similar manner as the elements in a set in respect to the union and intersection operation. Thus, the same operators can be used as in mathematics, without a substantial conceptual difference. 2 A Mathematical vs. Informational Dictionary The presented dictionary should bring the mathematical feeling into the domain of informational theory. It certainly concerns the topological terms priory. The correspondence between set-theoretical and system-informational terms yields the following comparative table6: Mathematical vs. Informational Topology set X system $: general formula system transition formula system $i(=r;; and operand formula system set braces: {,} system parentheses: (,) sFor the system-conditional formula, i^i,<£2,• • • ,2, • • • , ' {ip 6 (a) for 'ip of a. TOPOLOGICAL INFORMATIONAL SPACES Informatica 22 (1998) 287-308 2 89 vertical snake-form operand-occurrence braces [5, 6] empty set 0 set element x x is an element of X, x belongs to X, x is in X, x 6 X\ negation: x g X subset A A is a subset of X, A is included in X, A C X; negation: A (¡L X powerset of set X, ¥{X) or 2X union of sets, AU B intersection of sets, AHB difference of sets Y and X,Y\X complement of set X, Cx complement of set X, regarding set Y, CyAr open set O topology 0 topological structure O topological space {X, 0), simply, X carrier X point x £ X vertex (apex, in Russ., Bepimraa [39]), v set of vertices, V vertex connection (rib, pe6po) u: arc, loop, link [39] set of ribs, U . edge, unordered pair e = {ui,V2}, or ordered pair («1,^2) set of edges, E (incidence function) path (route, ijent) V1U1V2U2 ■ ■ ■ Un-l^n formula operand occurrence floor brackets: J empty system 0 system formula tp tp is a formula of , tp belongs to tp is in tp e negation: tp $ $ subsystem \fr $ is a subsystem of is included in <5. f c$; negation: $ £ $ powersystem concerning system q3[$J or V[§\ union of systems, $ U or ($; informing of both systems intersection of systems, means, e.g., |=$), parallel informing of all the systems' components difference of systems $ and $ \ $ complement of system c$ complement of system regarding system open system D informational topology D informational topological structure D topological informational space {$,£>), simply, $ informational carrier $ informational point, formula, formula system tp € $ operand, operand point, £ system of operands, operator, operator arrow, marked by |= or by an operator particularization list of operator markers basic transition, £ (= r), with binary operator system of basic transitions informational route, path, formula scheme Mathematical vs. Informational Graph Theory [4, 35, 39] Î1 N 6 1= ' ■ ■ 1= În graph G = (V, E) [2] informational graph, presented by derived from an actual system Informational space shall mean a non-empty formula system which possesses some type of informational structure (and organization), e.g. metaphysicalism, meaning, informational vector, informational metric (in the form of informational distance) and/or informational topology. Within such a possible structure, the elements in an informational space will be called formulas or points and, in a special case operands. 3 Systems and Subsystems of Informational Formulas 3.1 Informationally Linked Formulas in a System Informational linkage of formulas in a formula system deserves a special attention and theoretical treatment. The consequence of formula linkages via common operands makes the difference between formulas of a system on one side, and between the elements of a set on the other one. Definition 1 If in informational formulas 3, that is, A • • • J and 3. Formally, ((<¿>1 3)) =>■ (1 ¥>3) Operator A denotes informational conjunction (in fact, the operator of parallel informing, |f=, or, usually, semicolon ';' [30]) and operator => informational implication [30]. □ 7lt is to stress that a notation <^[01,02,... ,onJ means informational operands 01,02.....Qn occurring in formula tp, and does not represent the so-called functional form, that is, informational Being-of [29] in the form (¿>(01,02,... ,an). Evidently, ¥>Lal!°2.....OnJ ■ 290 Informatica 22 (1998) 287-308 A.P. Zeleznikar Further, there can exist more than one common operand, e.g., cti,aj,ak ... , am in 1 ^ V>2 . \ <~P\ (fii CCj 2\ <¿>3 > l a'ß \ (¥>i V3) Another significant feature follows from the last definition: Theorem 1 Let the linkages in a circular manner tfl (Wi ip2, ip2 <£>3, lPm—l (Vi Vj); i,j e {1,2,... ,m} Within this conditionality also i 6 {1,2,... ,m} holds in a transitive (consequently multiple-linkage) manner. Another evident meaning of the theorem result is tpi tpx,tp2,... , e §1, If € $2 which represents the so-called alternative system, using comma instead of semicolon between formulas [35]. The intersection of two systems $1 and $2, denoted by $1 n means the system ($1 n $2) (f V€l11) y

eT)V (¥>£$!)) TOPOLOGICAL INFORMATIONAL SPACES Informatica 22 (1998) 287-308 2 89 where T functions as a universal system. Evidently, ^ (T \ $i). Usually, in a complex case, the formula system $ has the role of the currently universal system to which its subsystems can be compared.8 4 Topological Informational Spaces 4.1 Definitions 4.1.1 Open Systems Definition 3 Let $ mark a reasonable9 non-empty system of informational formulas. A class10 (short for informational class) 0 of subsystems o/$, 0 C , is a topology on $ iff 0 satisfies the following axioms: (Ti) The union of any number of systems in 0 belongs to 0. (Tn) The intersection of any two systems in 0 belongs to 0. (Tni) Systems $ and 0 belong to 0. The systems of 0 are then called 0-open systems, or simply open systems, and $ together with 0, i.e. the informational pair ($,0), is called the topological informational space. □ As we see, a topological space is defined as an ordered pair between the carrier $ and its topology 0. Let us formulate Def. 3 in another way to get a different experience of the meaning of an informational topology. Topology can be determined by the following four steps too: 1° A basic system $ of formulas tpi, m exists. 2° There exists a type characteristics11 0 6 ^IjPL^JJ-3° The first axiom is: For each system 0', the informational implication (0' eO)=^| jj E^j € 0^ and $ € 0 hold. 4° The second axiom is: For each Ei and each E2, the informational implication (Si e 0; s2 e 0) => ((Ei n s2) e 0) holds. Such a structure family is called the topological structure, and the relation E € 0 can be expressed verbally as: system E is open in topology 0.12 ®A complex system is, for example, that of informational consciousness, in which several complex subsystems are imbedded. However, it does not mean that, in a specific case, a subsystem appears as a kind of universal to which its subsystems can be system-complementally compared. 9 A reasonable system of informational formulas usually concerns a concrete, cyclically structured informational graph [32, 35]. 10A class (family, collection) of subsystems means a system of subsystems. 12For more details see [6], p. 246. Evidently, the openness of a system concerns its topology. Example 1 Topologies Deducible from Standardized Metaphysicalism. Let the following classes of subsystems of the standardized metaphysicalistic system [32, 35] 971 ^ ((pi;4;y5);(2;¥>3 5 ); <¿>5! Ve); ()) For 03, the union ((V?4;^5) U (6)) ^ (<¿>4; <£5; , in which a topology is defined, is said to be the carrier of the topological space ($,0). System $ can be the carrier of more than one topological space. Thus, a system of different topological spaces for $ is, for instance, /(,0i);\ ($,(0i;02;... ;0„)} ;=± \<*,On)J Example 2 The carrier of a topological informational space can be expressed in different ways, with different structural and organizational14 consequences. Conceptualizing an IS, at the first glance, usually, a system of serial and serially circular informational formulas is determined. According to [35], p. 114, such a system includes serial formulas of the type15 La,ax,-- - ,anJ; 1 (93 C £>) that each open system H in D (ii G £>) is a union of some open systems 21, in that is, (J 2lj. Said in a,-693 another way: for each 11 6 D and each point (formula) ¡p € there exists such a system 23 G 03 that

and each ip G U there exists a finite number of systems in 6, for instance, 2Ui,... ,%Bn, such that (pe(w1n...nwny, (m n...nwn) cu Open systems in the given subbasis 6 are called the subbasic open systems of space It is evident that a topology D of space $ is completely determined by the basis or subbasis of D. (5) Topological cover. In general, we say that a family of systems (Et)lS/ is a cover of system if *C(Ut6, 3.). Cover (Et)iS/ of subsystem $ of topological informational space $ is called open, if all Et are open systems in 4.1.3 Interior, Exterior, and Boundary Formulas (Points) According to [13, 20], some further definitions could be useful also for the purposes of informational topology. Definition 4 Let $ be a subsystem of system a in topological space (a,D). (1) A formula ip G $ is called interior formula (point) of $ if ip belongs to an open system i 6 D contained in that is ip E ^ and $ C where $ is open. The system of interior formulas of denoted by $ , is called the interior o/$. (2) A formula ip G $ is called exterior formula (point) of $ if ip belongs to an open system E G O contained in the complement that is ip £ E; EC C$, where E ¿5 open. The system of exterior formulas of denoted by $ , is called the exterior (3) A formula

, its interior exterior and boundary /3L$J, within a system a, where the complement of denoted by C$, appears together with the complement interior C j>, exterior and boundary ^ Example 3 The connection of an informational loop and to it belonging system, that is, circular formula system, with the environment, can be realized by a special formula, usually a simple transition formula, e.g., cij |= in Fig. 4. This formula belongs to but is not an interior formula of that is, of □ Example 4 Let systems $ form a topological space ((($ U B) U as presented in Fig. 2. How Figure 2: Graphical presentation of formula system a ^ where each of the subsystems has its interior B, respectively. can this system informational graph be interpreted in different ways? We must clarify more precisely what subsystems, marked by $,5,$, might represent. The aim of the graph is to explicate the so-called interior and neighborhood regions of subsystems in respect to informational formulas in general, their operands, and the so-called primitive transitions, represented in the form of the graph route16 __HMi h & h & h vi h • • • 16ln literature, different names are given to the route. Informa-tionally, the name informational scheme or, in short, scheme, is used. The name edge denotes the edge (representing an operator) b ^ (B \ (($ n B) u (B n $))) that is, considering both systems $ and On the other side, the exterior of could roughly be understood as the union $ U B and, adequately, $ as f UB. In general, B is the neighborhood system for both $ and 'J. Further, evidently, f ^fUB etc. This example shows the importance of distinguishing the three possible types of topological spaces: (1) a space of circular/serial informational formulas of arbitrary lengths, (2) a space of basic transition formulas of length 1, and (3) a space, of simple informational operands (formula length 0), □ Another example shows the questionableness of a unique determination of topological structures dealing with different types of informational formula systems and Figure 3: Graphical presentation of formula system ($UB)U \fr, including a circular path. Example 5 According to Fig. 3, let formula systems of a graph polyhedron, to which a graph can be transformed. Path instead of the graph route sounds also adequately. 294 Informatica 22 (1998) 287-308 A.P. Zeleznikar tp iff H is a supersystem of an open system $ (E D f ) containing (ft N (ft N (/V Mm h(7Ml)); tp. Thus, ÎCE; SC$ Mi;ft h/^ ft Ma;ft Mi; 1= 75 7 M; ••• ) be given, where aG^ and 7 S What is now evident (or not quite evident) from the topological graph in Fig. 3? Can basic transition formula systems be determined uniquely, and in which way ? The problem occurs at basic transitions crossing the boundaries of systems B and In Fig. 3 such transitions are £1 |= ft, ft (= 771, and 7 (= a. Evidently, in a strict situation, it would be not possible to express the basic transition systems rigorously. Thus, the compromise notation17 î|=t, ^ (N a |= & (= ft; ft (=); !€i=„ ^ (1= ft; ft h ft; ft |= ft; ft l=); é\=v ^ (h ft; ft h vi 1= 7; 7 h) 17The compromise notation is, for example, (= a and 7 [=. Each informational operator |= is a binary operator, dependent on both operands. If one side of the operator is open, it is meant, that the missing operand is not fixed yet. Other possible interpretations of the neighborhood definition are the following: (2) A system E in a topological space (3>,0) is a neighborhood (O-neighborhood) of a point (formula) ip iff E contains an open system to which tp belongs. (3) In a topological space ($,£)), the neighborhood of the point tp £ is called each subsystem EcO, including an open system \£> such that tp e "i and iCS. Then, the neighborhood of subsystem n c is each subsystem Ec$ which includes an open system that is, satisfies ft C and 4CE. Thus, tp es,«'; n,ec$; ïcs; ft c * can be accepted, where the transition 7 |= a comes additionally. Further, ^ (a;£;ft); Bi ^ (ft;ft;ft); ^ (ft;r?;7) In this situation, Fig. 3 shows ($ n ^ 0. □ The last example presents how basic transition formula systems and operand systems derived from general formula systems can offer various informational interpretations. 4.1.4 Informational Neighborhood Informational neighborhood (neighborhood, for short) is both a metaphor and a formalistic structure concerning various possibilities of informational relationships between formulas and formula systems. Definition 5 Neighborhood of system E in a topological space (<5>, D) is called each system which includes an open system including E. Neighborhoods of one-formula system (ip) are said to be also the neighborhoods of formula tp. □ Let us present the last definition by other words. (1) Let yi be a formula (point) of topological space ($,0), that is, 6 A subsystem h of $ (e c is a neighborhood of Evidently, each neighborhood of system S in $ is also a neighborhood of each system f Cs and, in particular, of each formula in E. In turn, let E be the neighborhood of each formula of system and T be the union of all open systems included in E; then T C E, as well as each formula of belong to an open system, included in S, that is, to $ C T; but T is open according to (Ti); consequently, E is the neighborhood of system In particular, the following comes into the foreground: Supposition 1 That a system is the neighborhood of each its formula, it is necessary and sufficient for it to be open.18 □ Let us mark by the system of all neighbor- hoods of formula tp. Systems in LvJ possess the following properties: (Ni) Each subsystem of system including a system o/9t\tp\, belongs to [J are said to be the neighborhood characteristics. Thus: Supposition 2 If to each formula

), a class 6 of open formula subsystems of that is, & C is a subbasis for the topology D on $ iff finite intersections of members of & form a base 23 for D. □ Any class 21 of formula subsystems of a non-empty formula system $ is the subbasis for a unique topology D on Intersections of members of 21 form a base for the topology £> on Example 7 Let $ ^ (y>o; Vi! ■ ■ ■ ; 6))- Finite intersections of members of 21 gives the base © — ((<¿>3; <¿>4); (WA.); (¥>e); (<¿>4); 0; By definition, $ £ 23 follows, since it is the empty intersection of members of 2l-system. Considering unions of members of 03 gives the family £ ^ ((<^3;<^4); (4;¥>5); (ye); (¥>4); 0; (3;4;); (wwrnw); (w, w,vs)) Formula system £> is the topology on $ generated by formula system 21. □ 4.1.6 Informational Accumulation Point Accumulation point (also, limit point) is a well-known term in mathematical topology. We need the notion of informational accumulation point, for example, as a formula or formula system approaching as close as possible to the meaning of something. This means that the final meaning of something can never be reached, although the meaning of something can be expressed by a formula system as close as required. Definition 8 If 21 is a subsystem of a formula system formula cp £ $ is an accumulation point of 21, iff every open system 0 containing

\ 0. Condition 2 says that meaning concerning operands ip and V informs to be fj.[ip\ only and only if ip is the same operand as ip. In this case, also fi[cp,ipj ^ fi\ip\. By Condition 3, since meaning concerning two operands, fi[ip, ?/?J, is a meaning difference between tp and ip, such a difference informs to be n\ip,ip\. Thus, /i|_inJ theme of intention irrelevant, all reference a fiction, etc. (see At-tridge [1] p. 12). That a text for Derrida, especially a literary text, is always situated, read and re-read in a specific place and times makes it 'iterable' or repeatable, the same but always different, and therefore never reducible to an abstraction by theoretical contemplation (Derrida [11] pp. 172-97). A text is unique and repeatable, concrete and abstract simultaneously. This coexistence lies in the heart of deconstruction and reflects the connectedness of the subject and object in the experience of the self as pure consciousness. Li,ii,---,inJ; V Li.il,j and in the circular serial case, ( Li,ii,---.inJ;\ 1* i Li, £1, • • • , inj; 0 I t c . inj 7^ Li.ii.--- .inj; Li.il.--- .inj/ 298 Informatica 22 (1998) 287-308 A.P. Zeleznikar where the asterisked markers ,... , "V" 1} , denote the systems of serial subformulas of lengths 1, ... ,7i — l,n conditionally in respect to operands '">£n in floor parentheses. Namely, a system L?;iii • • • i J includes only and only such basic transitions of the form a* f= ctj (t — 1) which appear in formula ip L£>£ii''" >£«J or formula "+;V° L£> > " ' i in J (as a whole), respectively. Similar concerns lengths I up to value n or n + 1, respectively. A short analysis shows that in the serial and circular serial case the number of all possible subformulas of a given length can be evaluated by simple formulas. Let tsah mark the length of a subformula in a serial formula with the length . Then, evidently, the number of such subformulas in a formula is if is even Tl . = sub C+i if I^ is odd In a circular case there is if e, is even foil if £0 is odd system, each operand in at least one circular formula. The third system, is the representative of all possible situations occurring by all possible parenthesis pairs displacements within the constructed (analyzed and synthesized) system. □ As said, the originally conceptualized system (obtained by the top-down or bottom-up approach or from both of them) is Thus, the remaining two systems, and -n, evidently emerge from that is, and iN? 5 Variants of Informational Topologies A topology D depends on the carrier system that is, on the characteristic forms of its formulas. Which kinds of formulas in $ can be distinguished? The most usual system of formulas is composed of different serial and circular-serial formulas. These formulas emerge during the analysis of an informational case, usually in a kind of top-down and bottom-up decomposition of an initial (top) marker or an end (bottom) marker, carrying implicitly a yet-not-determined concept, proceeding stepwise into a more detail of the case—a progressive case decomposition from different points of view. This approach seems to be the most natural one, seen from the human point of consciousness. Just after of such a case identification more abstract and convenient approach with possibilities can be considered. Definition 13 The constructed system of formulas, <5, can take the following characteristic forms: ^ (^L-.-fi.-.J; (¿4-.-6-.-J; - ■•; ..£„...]); ^ (£l; ■•• ; Cr.;) u $L&limPl'cit operands]; ^ (ii |= 6; 6 N 6; • • •; £„-i h £») u |= ¡implicit basic transitions] The first system, is an authentic, intuitively constructed representation of a real case. The second system, is strictly expressed by all the occurring system operands as the title operands of a circular formula where —> denotes the corresponding derivation approach. On this basis, three different topologies can be determined, as formula, operand and basic-transition topology, respectively. —> $£(=»; is the formal representative of the corresponding informational graph [35]. Now, let us show, how different topologies can be defined on and in a concrete case, and how all they mirror one and the same informational graph, with different possibilities in regard to various parenthesis displacements in formulas of the system. As an example we choose the metaphysicalistic case. 5.1 Topologies of a Simple Metaphysicalism Simple metaphysicalism is a basic scheme of informational invariance which can be further decomposed in greater details during identification of the involved entities, that is, a formula expressed in the metaphysicalistic form. Thus, the graph in Fig. 4 can be understood as a consequence of the circular metaphysicalistic formula system otj J Subscript j concerns the formula system component k.^>J[£,ij,ctj\ of system e.g., j = 1,2,... ,n. Subscript i concerns the operand component & of metaphysicalistic formula system ksp_\.€ij,oij\ £ Subscript kj concerns the parenthesis-pair combination 1 < kj < n"ub . of the formula subsystem system , with altogether i-r tij llvij lij+iUii) possibilities, considering serial (input) and circular serial formulas of a system of formula systems, where denotes the length of the formula in a formula subsystem. 5.1.1 Topologies on the circular formula system ..^Liij^iJ According to the graph in Fig. 4, one of the possible formula systems can be constructed (reconstructed). Let it be the consequent observing type of metaphysicalism for which the extreme left-parenthesis heaping is characteristic, that is, kj = 1. In this case, the graph TOPOLOGICAL INFORMATIONAL SPACES Informatica 22 (1998) 287-308 2 89 formula component component component component informing counterinforming embedding Figure 4: The graph representing the basic metaphysicalism of a formula system >aj\ € component impacted by something (interior and/or exterior) aj. is interpreted by the one of possible formula systems, that is, ( ai \= ; Wo] \ ((((((& h 3Í„)H£íí) M*,Oh ON M «€«) 1= Ha) NAü\ (((£«., hc^heOhOM^; M P«., Hê«) Me«; Wa] (£í.y N Ha) N Ws] V(efü )=Hu) Mí« [ (4, fh i '-P6 must enter 0Vti. This condition delivers together with 0 and a topology which is the power system of (_<3?^J, called the discrete topology i (see Sect. 4.1.2). The precept of this example is that for all formulas ¡p € (v,2 are (2 and ip3, the topology becomes /0; (<¿>1); {; (p*;^); \ (vi; V2; ); Let include nv formulas, ipi,... , i; <¿>2; <£>3); (<£4; <¿>5; )• Further, a characteristic implication is, for instance, ((<<02; V4); (^35^5); (ye) e e (V2; 3; <¿>4; <¿>5); (<¿>2; ¥>4; ye); VV(y3;y5;ye) / / The initial intention (premise of the implication) is to cover explicitly both subloops for 3^. by <¿>2 and aj\ i and looking into the graph in Fig. 4. The result is, from the consequent observational point of view, Hii - (((((((eí.j h in) f= h iíJ N t= he««,) he«,,; fe h ) N ) N ) [= e€