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In recent years, “mathematical orientations on real-life problems”, which continue to increase, began to make a significant impact. Information systems for many decision-making problems consist of uncertain, incomplete, indeterminate and indiscernible structures and components. Classical set theory and interpretation methods fail to represent, express and solve the problems of these types or cause to make wrong decisions. For this reason, in this study, we provide definitions and methods to present information and problem representations in more detail and precision. This paper introduces three new topologies called covering-based rough fuzzy, covering-based rough intuitionistic fuzzy and covering-based rough neutrosophic nano topology. Some fundamental definitions such as open set, closed set, interior, closure and basis are given. Neutrosophic definitions and properties are mainly investigated. We give some real life applications of covering-based rough neutrosophic nano topology in the final part of the paper and an explanatory example of decision making application by defining core point.

In this article we introduce and study the notion of operational closure: a transitive set d is called operationally closed iff it contains all constants of OST and any operation f∈d applied to an element a∈d yields an element fa∈d, provided that f applied to a has a value at all. We will show that there is a direct relationship between operational closure and stability in the sense that operationally closed sets behave like Σ1 substructures of the universe. This leads to our final result that OST plus the axiom (OLim), claiming that any set is element of an operationally closed set, is proof-theoretically equivalent to the system KP+(Σ1-Sep) of Kripke–Platek set theory with infinity and Σ1 separation. We also characterize the system OST plus the existence of one operationally closed set in terms of Kripke–Platek set theory with infinity and a parameter-free version of Σ1 separation.

Known properties of ''canonical connections'' from database theory and of ''closed sets'' from statistics implicitly define a hypergraph convexity, here called canonical convexity (c-convexity), and provide an efficient algorithm to compute c-convex hulls. We characterize the class of hypergraphs in which c-convexity enjoys the Minkowski-Krein-Milman property. Moreover, we compare c-convexity with the natural extension to hypergraphs of monophonic convexity (or m-convexity), and prove that: (1) m-convexity is coarser than c-convexity, (2) m-convexity and c-convexity are equivalent in conformal hypergraphs, and (3) m-convex hulls can be computed in the same efficient way as c-convex hulls.

This paper gives analogies of Hutton's quasi-uniformities for complete lattices and extensions of Shi's pointwise quasi-uniformities to closed set lattices. CL"G"O"H denotes the category of all complete lattices with generalized order homomorphisms as morphisms and CSL the category of all closed set lattices. HQUnif and SPQUnif denote the categories of Hutton quasi-uniform spaces on complete lattices and Shi pointwise quasi-uniform spaces on closed set lattices, respectively. We prove that both HQUnif and SPQUnif are fibre-complete topological categories over CL"G"O"H and CSL with respect to the expected forgetful functors, respectively. Also, a mistake in W. Kotze [Uniform spaces, in: U. Hohle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, Vol. 3, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999, pp. 553-580] is pointed out.

Machines whose main purpose is to permute and sort data are studied. The sets of permutations that can arise are analysed by means of finite automata and avoided pattern techniques. Conditions are given for these sets to be enumerated by rational generating functions. As a consequence we give the first non-trivial examples of pattern closed sets of permutations all of whose closed subclasses have rational generating functions.

Uncertain conditional propositions or conditional events pose a special problem for deduction due to their three-valued nature - true, false or inapplicable. This three-valued-ness gives rise to several different kinds of deduction between conditionals depending upon the particular deductive relation being employed. There is a hierarchy of 13 deductive relations between conditionals built up from four elementary ones. which can be expressed in terms of Boolean relations on the components of the conditionals. These different, but interrelated deductive relations on conditionals in turn give rise to various deductively closed systems of conditionals. Theorems are proved relating the deductive relations with each other and with their associated deductively closed sets (DCSs). The principal DCS generated by a single conditional using any of the deductive relations is completely characterized. Except for two of the deductive relations, deduction with a finite number of conditionals or with additional conditional information is also completely characterized as the union of associated principal DCSs. Examples of finite sets of deductively closed conditionals are exhibited including many, unlike Boolean algebra, which are finite and yet non-principal, that is, not generated by any one conditional. An introductory section carefully provides motivations and a complete algebraic characterization of the four chosen operations on conditional events. Results are applied to the famous penguin problem.

International audience; We investigate the stability of a class of dynamical systems that switch among a given finite family of hybrid systems. We propose sufficient conditions tailored to this particular type of hybrid systems which guarantee the uniform global pre-asymptotic stability (UGpAS) of a given closed set. We first assume this set to be UGpAS for each system of the family. A slow switching condition is then presented to maintain this property for the overall system. We introduce for this purpose the concept of hybrid dwell time which characterizes the length of the hybrid time intervals between two successive switching instants.

In this paper, we study closed sets of relations that are preserved by hyperoperations. We examine three already known Galois connections between hyperoperations and relations, and show that none of them induces a relational clone. We also introduce the notion of extended co-clone, and prove that there is a Galois connection (rPol, rInv) between extended hyperoperations on A and relations on non-void subsets of A with the property that rInv F is an extended co-clone. Moreover, we show that the three previously known classes of hyperclones can be defined by rPol.

In this paper, a new kind of connectivity called SR-connectedness in L-topological spaces is introduced by means of semiregular closed L-sets. Some fundamental properties of SR-connectedness are obtained. Especially, the famous K.Fan's Theorem can be extended to L-topological spaces for SR-connectedness.

In this paper, we present a closed labeled tree mining algorithm, FBMiner, based on the add-remove principle of closed sets which is newly introduced. Also, we propose a time-decay module to solve stream data mining which gives more attention on the latest data. Compared to the traditional mining algorithms in data stream, FBMiner performs well even that the data is of high complexity. The experiment shows that FBMiner is efficient in data streams mining by reducing consuming dramatically.

In 1966 A. V. Arkhangel'skii posed the following question: Is it true that every regular finally compact symmetrizable space is separable? S. I. Nedev soon showed that a regular finally compact symmetrizable space is hereditarily finally compact. Consequently any counterexample to Arkhangel'skii's conjecture must be an L-space. Applying the technique of iterated forcing we prove that in the axiom systemZFC for set theory it is consistent to assume the existence of a regular (hereditarily) finally compact symmetrizable space X that is nonseparable. Thus it is impossible to prove using the axiom systemZFC that every regular finally compact symmetrizable space is separable. The space X has additional properties as well: it has a basis consisting of open/closed sets (i.e., it is zero-dimensional in the sense ofind, it can be mapped continuously and one-to-one onto a separable metric space, it is α-left and has cardinality ω1. Bibliography: 25 titles.

Since intuitionistic sets are not generally stable – their membership relations are not always closed under double negation – the open sets of a topology cannot be recovered from the closed sets of that topology via complementation, at least without further ado. Dana Scott asked, first, whether it is possible intuitionistically for two distinct topologies, given as collections of open sets on the same carrier, to share their closed sets. Second, he asked whether there can be intuitionistic functions that are closed continuous in that the inverse of every closed set is closed without being continuous in the usual, open sense. Here, we prove that, as far as intuitionistic set theory is concerned, there can be infinitely-many distinct topologies on the same carrier sharing a single collection of closed sets. The proof employs Heyting-valued sets, and demonstrates that the intuitionistic set theory IZF [4, 624], as well as the theory IZF plus classical elementary arithmetic, are both consistent with the statement that infinitely many topologies on the set of natural numbers share the same closed sets. Without changing models, we show that these formal theories are also consistent with the statement that there are infinitely many endofunctions on the natural numbers that are closed continuous but not open continuous with respect to a single topology.For each prime k ∈ ω, let Ak be this ω-sequence of sets open in the standard topology on the closed unit interval: for each n ∈ ω

Let $X$ be a set, $\ka$ be a cardinal number and let $\iH$ be a family of subsets of $X$ which covers each $x\in X$ at least $\ka$ times. What assumptions can ensure that $\iH$ can be decomposed into $\kappa$ many disjoint subcovers? We examine this problem under various assumptions on the set $X$ and on the cover $\iH$: among other situations, we consider covers of topological spaces by closed sets, interval covers of linearly ordered sets and covers of $\real^{n}$ by polyhedra and by arbitrary convex sets. We focus on these problems mainly for infinite $\kappa$. Besides numerous positive and negative results, many questions turn out to be independent of the usual axioms of set theory.

Let Ek := {0,1,. .. , k - 1}, k ges 2, and let Pk be the set of all k-valued logical functions, i.e., maps f : Ek n rarr Ek for n = 1,2,.... Denote by Pk (2) the set of all functions of Pk whose range contains no more than two elements. The set Pk(2) is a class (i.e., a closed set with respect to the usual superposition operations) and all maximal subclasses of Pk(2) are known. In this paper we study the characteristic vectors for each f isin Pk(2) with respect to the maximal classes and give an explicit formula for the total number of the characteristic vectors in terms of the numbers of the equivalence relations on Ek- Then we show that P3(2) has exactly 75 characteristic vectors and 33,678 equivalence classes of bases, and show that every basis of P3 (2) consists of either 3, 4 or 5 functions.

Given a large set of data, extracting frequent itemsets in this set is a challenging job in data mining. Many efficient algorithms have been proposed in the literature. The idea presented in this paper is to extract a condensed representation of the frequent itemsets called rule-free sets, instead of extracting the whole frequent itemsets collection. We show that this condensed representation can be applied to regenerate all frequent itemsets and their exact frequencies without any access to the original data. An algorithm, RFS-MINER, is presented to extract the frequent rule-free sets and practical experiments show that this representation can be extracted very efficiently. We compared it with another representation in the literature called frequent closed sets and in nearly all experiments we have run, the rule free sets have been extracted much more efficiently than frequently closed sets.

Let A be a finite set with mod A mod >2. The authors describe all clones on A containing the set S/sub A/ of all permutations of A among its unary operations. (A clone on A is a composition closed set of finitary operations on A containing all projections.) With a few exceptions such a clone C is either essentially unary or cellular. The exceptions are subclones of Burle's clone or of its variant (provided mod A mod is even). >

In this paper we define a formal model for the discrete representation of spatial objects and characterize its properties. The model and its manipulation primitives are based only on set theory and do not use any metricbased concept. A general characterization for containment and intersection relations is given. The model is based on a mapping from spatial objects to their representations as sets of points of Z 2 such that queries on spatial objects can be answered by applying simple set-theoretic primitives to their corresponding discrete representations. The mapping makes reference to a suitable topology whose underlying set is a canonical decomposition of the real plane in square cells. But the introduced model keep its properties under every homeomorphic topological space such that its underlying set is a countable partition of the plane whose cells are simply connected. 1 . I N T R O D U C T I O N The definition of a formal framework for the representation and manipulation of spatial objects and (*) Work partially supported by the italian MURST 40% Project "Algoritmi, Modelli di Calcolo e Strutture Informative", by the italian CNR Coordinated Project "Modelli e Sistemi per il Trattamento di Dati Ambientali e Territoriali", and by the European Union's Research Network CHOROCHRONOS on SpatioTemporal Databases. "'Pernaission to make digital or hard copies of part or all of this work tbr personal or classroom use is granted without Ice provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice zmd the full citation on the tqrst page. Copyrights for components of this work o~vncd by others than ACM must be honored. Abstracting with credit is permitted. To cop3' otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a tee." (;, 1997 ACM 0-89791-850-9 97 0002 3.50 !44 their relations is a key issue in the theory of spatial databases. Since we want to use computing devices to represent and manipulate spatial objects and their relations it is of the greatest importance to focus on finite formalizations. In this paper we are therefore concerned with the definition of a theory which models spatial relations among geographical object and is, at the same time, based on finite and discrete representations of spatial objects. Many different models for spatial objects and relations have been introduced in the literature. Among the most widely known there is the Egenhofer and Franzosa's 4intersection model [EF91 ] and its extensions ([Her91 ], [CDO93], [ES93]), all based on point-set topology. Other interesting models formalizing spatial relations are based on simplicial complexes ([EFJ89], ['Wor92]) or on partial orders or on a combination of them [KEG]. Considering finite representation of spatial objects, Guibas, Salesin and Stolfi defined Epsilon-Geometry [GSS89, GSS93], which allows to carry out exact geometric computations even when only inaccurate primitives are available. This approach, while very good in dealing with building accurate discrete representation of real objects does not explicitly address issues of spatial relations. Greene and Yao [GY86] introduced a computational geometry framework for the topologically correct computation of intersection (discrete) points in a set of line segments whose endpoints have discrete coordinates. A completely different approach was taken by Giiting and Schneider [GS93]. They introduced the concept of Realm (i.e., a set of points and non-intersecting lines over a discrete domain)which allows to model spatial relations through an algebraic approach. Discrete representations at the Realm level are derived from real objects by means of computational geometry primitives which take into account rounding problems. Representations at the Realm level are then manipulated through a very clean algebraic interface, where spatial relations are defined by means of a algebraic specifications. But the introduction of an intermediate level, while allowing a very good formalizat ion of spatial relations and manipulations, hides the true relations between objects in the real world and their counterparts in the model. In this paper we address these issues by defining a formal model for spatial objects and relations and characterizing its properties. We have already introduced in [CNT96] a discrete representat ion for spatial relations which al lows to test conta inment and intersection relations among convex polygons. We defined a topology preserving mapping from the set of convex polygons to a discrete space. Here we extend and refine that approach. Objects we consider are arbitrary polygons and lines whose vertices are points with discrete-valued coordinates. The discrete space we use as basis for our framework is a countable set of points whose coordinates are suitably taken from the set of rational numbers. We define a correspondence between this discrete space and a partition of the continuous real plane in squares. This specif ic par t i t ion is chosen only for ease of formal iza t ion , s ince our results hold for any decomposition of the real plane that is a partition in cells such that the cells are the underlying set of a topological space homeomorphic to the one we introduce. The manipulation primitives introduced in the model are purely topological and based on set-theory, without using any concept based on metrics. Thus we are able to give a finite characterization of each of the three basic relative positions of spatial objects (namely: containment, intersection, and disjointness). These are the three fundamental relations to be managed by any efficient organization of spatial data. The paper is structured as it follows. In section 2 we introduce the topological space used for the discrete framework and the mapping between it and real objects. In section 3 we characterize the containment relation be tween po lygons . Sec t ion 4 comple t e s the characterization by studying the intersection relation. Section 5 address generalization issues and, finally, section 6 contains conclusive remarks. 2 . B A S I C D E F I N I T I O N S In this section we introduce the topological space used for the discrete representation of objects and the one-toone mapping between it and objects in the reality. Let N, Z, Q, R denote the sets of, respectively, natural, integer, rational, and real numbers. Each set is taken with its usual total order. Let Q a = { p i ~ Q [p i=P i . l+A, 0 2, be a finite set of points of Ga, such that xi-i and xi are adjacent for i=2 ,3 . . . . . k, and xi:g:x j for i~:j. We say y is a chain. @ DEFINITION 5: Let A be a set of points of Ga. We say A is connected if for each couple of distinct points x and y belonging to A a chain 7={xt, x2 . . . . . Xk} exists such that x l=x and xk=y. @ DEFINITION 6: Let ~-={xl, x2 . . . . . xk), k>_4, be a chain such that Xl=X k. We say ~. is a loop. The (possibly I A closed set of a topological space (X,T) is a subset of X such that its complement in x is an open set of (X.T).

Certain separation problems in descriptive set theory correspond to a forcing preservation property, with a fusion type infinite game associated to it. As an application, it is consistent with the axioms of set theory that the circle 𝕋 can be covered by ℵ1 many closed sets of uniqueness while a much larger number of H-sets is necessary to cover it.

In this paper we define and investigate the notions of \(gI\)-closedsets and \(gI\)-open sets in ideal topological spaces. A \(gI\)-closedset is a new generalization of \(I_{g}\)-closed sets due to Dontchevet al. [3].

Two conjectures related to the problem of finding a "large" family of approach curves along which cluster sets can be preassigned are shown to be equivalent to special cases of the normal Moore space conjecture. 1. In [4] the author posed the following problems for real-valued functions f of two real variables. (Here "approach curves" approach the origin and C(/, y) denotes the cluster set of f at 0 along the curve y.) I. Find a '"small" subset Fo of the set F of all approach curves such that for each function f, {C(f, yo): yo e FoI determines C(f, y) for all y in F. II. Find a "large" subset ro of F such that to any collection {C(yo): yo e Fo I of closed sets there corresponds a function f such that C(f, yo) = C(yo) for all yo in Fo. A satisfactory solution of Problem I was found in [4] by generalizing Rosenthal's "Approach Curve Theorem", but Problem II was only partially solved. It was conjectured that "A family F0 is a solution of Problem II if and only if FO has a nonintersecting truncation". Put another way, the conjecture is that a family Fo satisfies the condition of Problem II if and only if it is somehow possible to "clip" the curves of F0 in such a way that theresulting family of "stubs" (minus the origin) is nonintersecting. In this paper it is shown that the above conjecture is equivalent to the normal Moore space conjecture for a certain family of spaces. In these spaces, collectionwise Hausdorff implies collectionwise normality. Consequently the conjecture is consistent with the usual axioms of set theory. Conjecture (1) of [4] is similarly translated and its consistency established. For the definition of a Moore space and the general status of the Moore space conjecture, see [11], [5] [7], [12]-[14]; for the theory of cluster sets, see [3] and [8]. 2. For a family Fo of approach curves, let X(rF) consist of (a) the points of the punctured plane R2 210 and (b) the curves of F0 (thought of Received by the editors September 7, 1974. AMS (MOS) subject classifications (1970). Primary 26A15, 54D15, 54E30, 54E35; Secondary 02K05, 04A30.

The purpose of this note is to prove a cross section theorem for certain equivalence relations on Borel subsets of a Polish space. This theorem is then applied to show that cross sections always exist on countably separated Borel subsets of the dual of a separable C*-algebra. See Auslander-Moore [2], Bourbaki [3], Kuratowski [9], and Mackey [12] for the main results and notation in Polish set theory used in this paper. The main result of this note is the following theorem. THEOREM 1. Let B be a Borel subset of the Polish space X. Let R be an equivalence relation on B such that each R-equivalence class is both a G0, and an Fa in X, and such that the R-saturation of each relatively open subset of B is Borel. Then the quotient Borel space B/R is standard, and there is a Borel cross sectionf: B/R -* Bfor R. Notice that if the R-saturation of each relatively closed subset of B is Borel, then the R-saturation of each relatively open subset of B is Borel, for each relatively open subset of B is the countable union of relatively closed sets. A number of preliminary lemmas are proved first. LEMMA 2. Let (Y, d) be a separable metric space and let R be an equivalence relation on Y such that the R-saturation of each open set is Borel. Then there is a Borel set S whose intersection with each R-equivalence class which is complete with respect to d is nonempty, and whose intersection with each R-equivalence class is at most one point. PROOF. By the proofs (but not the statements) of Theorem 4, p. 206, Bourbaki [3] and Lemme 2, p. 279, Dixmier [4], there exists a decreasing sequence of Borel subsets of Y, say S,, so that S, n R (y) # 0, diameter(Sn n R(y))-*O, and nn,>(Sn n R(y)) = n n,((Sn n R(y)) n R(y)) for each y in Y. Let S = qnn s,> S is a Borel subset of Y, the intersection of S with each complete R-equivalence class is nonempty, and the intersection of S with each R-equivalence class is at most one point. Q.E.D. LEMMA 3. Let Y be a Polish space and D a subset of Y which is both a G0, and Received by the editors November 15, 1976 and, in revised form, July 18, 1977. AMS (MOS) subject classifications (1970). Primary 28A05, 46L05; Secondary 04A15, 54C65.

We present a general technique for showing that many properties of recursive languages are not provable. Here “provable” is taken with respect to a given sound, recursively axiomatized formal system J, such as Peano arithmetic. A representative application (Theorems 6.1–6.2) concerns the property of intractability, i.e., non-membership in the class P. It says that there exists a language E such that E is not in P, but the formal assertion ‘E is not in P’ is independent of J. Moreover, given any recursive language A ∉ P, we can construct E such that also E ⩽mP A. Our techniques strengthen similar results in the literature and lead to several other applications pertaining to P-immune sets, oracle separations, and the Berman-Hartmanis conjecture. We explain the phenomenon of unprovability in terms of both recursive properties of the formal systems J under consideration, and topological properties of complexity classes in a natural space which we call R. Provable properties correspond to closed sets of r. The topology provides geometric intuition for recognizing classes which are not closed in R, such as NPP (unless it is empty). We show how independence results follow immediately for these classes. In conclusion we argue that the type of independence result presented here forms an obstacle for day-to-day work in complexity theory, but does not bear directly on the possible independence of the P = NP question from Peano arithmetic or set theory. However, we believe our tools capable of measuring the link between the structure of a given language E ∉ P and the formal strength needed to prove the assertion ‘E ∉ P.’ Research in this direction has already been initiated by D. Joseph (J. Comput. System Sci. 25 (1983), 205–228).

Pareto surfaces and attainable sets of complexity 1 (i.e., those having a 1-commodity representation and no 0-commodity representation) are treated. An implicit characterization of these sets is given, and various of their properties are derived. In particular, Pareto surfaces of complexity at most 1 are always closed sets.

A subset X X of E 3 {E^3} is said to have vertical order n n if no vertical line contains more than n n points of X X . We prove that each 2 2 -sphere in E 3 {E^3} which has vertical order 5 bounds a 3 3 -cell.

This paper introduces an operator M called the mixed powerdomain which generalizes the convex (Plotkin) powerdomain. The construction is based on the idea of representing partial information about a set of data items using a pair of sets, one representing partial information in the manner of the upper (Smyth) powerdomain and the other in the manner of the lower (Hoare) powerdomain where the components of such pairs are required to satisfy a consistency condition. This provides a richer family of meaningful partial descriptions than are available in the convex powerdomain and also makes it possible to include the empty set in a satisfactory way. The new construct is given a rigorous mathematical treatment like that which has been applied to the known powerdomains. It is proved that M is a continuous functor on bifinite domains which is left adjoint to the forgetful functor from a category of continuous structures called mix algebras. For a domain D with a coherent Scott topology, elements of MD can be represented as pairs (U, V) where U ⊆ D is a compact upper set, V ⊆ D is a closed set and the downward closure of U ⌢ V is equal to V. A Stone dual characterization of M is also provided.

Publisher Summary This chapter discusses normality and collectionwise normality. A topological space is normal if there exist open sets UA and UB that separate disjoint closed sets A and B. Normality implies that any finite number of disjoint closed sets can be simultaneously separated. A convergent sequence with its limit point shows that “finite” cannot be extended to “countable.” A space is defined to be collectionwise normal if any discrete collection of closed sets can be separated. The chapter further presents theorems of Zermelo–Fraenkel set theory (ZFC). One can get from normality to collectionwise normality in ZFC if one or both of two circumstances hold: the canonical cover has a “nice” refinement or the elements of the collection are “nice.”

For an index set I , let S ( I ) be the sequential fan with I spines, i.e., the topological sum of I copies of the convergent sequence with all nonisolated points identified. The simplicity and the combinatorial nature of this space is what lies behind its occurrences in many seemingly unrelated topological problems. For example, consider the problem which ask us to compute the tightness of the square of S ( I ). We shall show that this is in fact equivalent to the well-known and more crucial topological question of W. Fleissner which asks whether, in the class of first countable spaces, the property of being collectionwise Hausdorff at certain levels implies the same property at higher levels. Next, we consider Kodama's question whether or not every Σ-product of Lasnev spaces is normal. The sequential fan again enters the scene as we show S ( ω 2 ) × S ( ω 2 ) × ω 1 , which can be embedded in a Σ-product of Lasnev spaces as a closed set, can be nonnormal in some model of set theory. On the other hand, we show that the Σ-product of arbitrarily many copies of the slightly smaller fan S ( ω 1 ) is normal.