Topological Informational Spaces

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, transitism, 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 = {x1, 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 Φ ­   φ1; φ2; .. φn  , where φ1, φ2, . . . , φm ! φ1, φ2, . . . , φm presents, in fact, only an instantaneous description of the parallel system of markers φi, 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 Φ. 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 systeminformational terms yields the following comparative table: Mathematical vs. Informational Topology set X system Φ: general formula system Φφ; transition formula system Φξ|=η; and operand formula system Φξ set braces: {, } system parentheses: (, ) 5For the system-conditional formula, φ1, φ2, . . . , φm ! φ1, φ2, . . . , φm see the discussion in Sect. 3.1. 6In this dictionary, the ‘formula φ of Φ’ (φ ∈ Φ) is to distinguish from the informational speech convention where the meaning of ‘of’ represents an informational function, that is, the informational Being-of [29], e.g., φ(α) for ‘φ of α’. TOPOLOGICAL INFORMATIONAL SPACES Informatica 22 (1998) 287–308 289 vertical snake-form formula operand occurrence operand-occurrence floor brackets: b, c braces [5, 6] empty set ∅ empty system ∅ set element x system formula φ x is an element of X, φ is a formula of Φ, x belongs to X, φ belongs to Φ, x is in X, x ∈ X; φ is in Φ, φ ∈ Φ; negation: x 6∈ X negation: φ 6∈ Φ subset A subsystem Ψ A is a subset of X, Ψ is a subsystem of Φ, A is included in X, Ψ is included in Φ, A ⊂ X; Ψ ⊂ Φ; negation: A 6⊂ X negation: Ψ 6⊂ Φ powerset of set X, powersystem concerning P(X) or 2 system Φ, PbΦc or PbΦc union of sets, A ∪B union of systems, Φ ∪Ψ; or (Φ;Ψ), informing of both systems intersection of sets, intersection of systems, A ∩B Φ ∩Ψ means, e.g., (Φ;Ψ; Φ |= Ψ; Ψ |= Φ), parallel informing of all the systems’ components difference of sets Y difference of systems Ψ and X, Y X and Φ, Ψ Φ complement of set X, complement of system Φ, {X {Φ complement of set X, complement of system Φ, regarding set Y , {Y X regarding system Ψ, {ΨΦ open set O open system O topology O informational topology O topological structure informational topological O structure O topological space topological informational (X,O), simply, X space 〈Φ, O〉, simply, Φ carrier X informational carrier Φ point x ∈ X informational point, formula, formula system φ ∈ Φ Mathematical vs. Informational Graph Theory [4, 35, 39] vertex (apex, in Russ., operand, operand point, ξ verxina [39]), v set of vertices, V system of operands, Φξ vertex connection operator, operator arrow, (rib, rebro) u: marked by |= or by an arc, loop, link [39] operator particularization set of ribs, U list of operator markers edge, unordered pair basic transition, ξ |= η, e = {v1, v2}, or with binary operator ordered pair (v1, v2) set of edges, E system of basic transitions (incidence function) Φξ|=η path (route, cep~) informational route, path, v1u1v2u2 . . . un−1vn formula scheme ξ1 |= ξ2 |= · · · |= ξn graph G = (V, E) [2] informational graph, presented by Φξ|=η, derived from an actual system Φφ 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 (Φφ, Φξ|=η) and, in a special case (Φξ), 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 φ1 and φ2 a common operand α appears, that is, φ1b. . . , α, . . .c and φ2b. . . , α, . . .c,7 respectively, notation φ1 α ! φ2 or, simply, φ1 ! φ2 will be used and read as formula φ1 informs formula φ2 via operand α or, simply, formula φ1 is informationally linked to formula φ2. This operation is informationally symmetric. Thus, (φ1 α ! φ2) ­ (φ2 α ! φ1) Transitivity of operator ! can exist in the following way: if α is common to φ1 and φ2, and β is common to φ2 and φ3, that is, φ1b. . . , α, . . .c, φ2b. . . , α, β, . . .c and φ3b. . . , β, . . .c, then φ1 is linked informationally with φ3. Formally, ((φ1 α ! φ2) ∧ (φ2 β ! φ3)) =⇒ (φ1 α,β ! φ3) Operator ∧ denotes informational conjunction (in fact, the operator of parallel informing, ||=, or, usually, semicolon ‘;’ [30]) and operator =⇒ informational implication [30]. ¤ 7It is to stress that a notation φbα1, α2, . . . , αnc means informational operands α1, α2, . . . , αn occurring in formula φ, and does not represent the so-called functional form, that is, informational Being-of [29] in the form φ(α1, α2, . . . , αn). Evidently, φ(α1, α2, . . . , αn) =⇒ φbα1, α2, . . . , αnc. 290 Informatica 22 (1998) 287–308 A.P. Železnikar Further, there can exist more than one common operand, e.g., αi, αj , αk . . . , αm in φ1 and φ2 in a transitive manner. In this case, ( φ1 αi,αj ,αk...,αm ! φ2 ) ­   φ1 αi ! φi′ ; φi′ αj ! φj′ ; φj′ αk ! φk′ ; .. φ`′ αm ! φ2   Transitivity of operator ! applies also to the case of more than one common operator through several formulas, and it can be defined from case to case. Because common operands concern informing between formulas, the implication in the last definition can be expressed by means of a parallel system ( φ1 α ! φ2; φ2 β ! φ3 ) =⇒ (φ1 α,β ! φ3) Another significant feature follows from the last definition: Theorem 1 Let the linkages in a circular manner φ1 ! φ2, φ2 ! φ3, .. φm−1 ! φm, φm ! φ1