Topological Closure Operator Research Articles

Characterization of neighborhood operators based on neighborhood relationships

Abstract Neighborhood relationships play a pivotal role in rough set theory, addressing the limitations of equivalence relations. This article focuses on defining upper and lower approximation operators using neighborhood relationships and exploring their properties in terms of serialization, inverse serialization, reflexivity, symmetry, transitivity, and Euclidean relations. Furthermore, a necessary and sufficient condition for the upper approximation operator to function as a topological closure operator is derived. Overall, this research sheds light on the significance of neighborhood relationships and their implications within rough set theory.

A Novel Constructing Continuous and Topology Approach to Fuzzy β-covering

Fuzzy β-covering(Fβ-C) plays a key role in processing real-valued data sets and covering plays an important role in the topological spaces. Thus they have attracted much attention. But the relationship between Fβ-C and topology has not been studied. This inspires the research of Fβ-C from the perspective of topology. In this paper, we construct Fβ-C rough continuous and homeomorphism mappings by using Fβ-C operator. We not only obtain some equivalent descriptions of the mappings but also profoundly reveal the relationship of two Fβ-C approximation spaces. We give the classification method of Fβ-C approximation spaces with the help of homeomorphism mapping, propose a new method to construct topology induced by Fβ-C operator and investigate the properties in the topological spaces further. Finally, we obtain the necessary and sufficient conditions for Fβ-C operators to be topological closure operators.

Dynamic Cantor Derivative Logic

Topological semantics for modal logic based on the Cantor derivative operator gives rise to derivative logics, also referred to as $d$-logics. Unlike logics based on the topological closure operator, $d$-logics have not previously been studied in the framework of dynamical systems, which are pairs $(X,f)$ consisting of a topological space $X$ equipped with a continuous function $f\colon X\to X$. We introduce the logics $\bf{wK4C}$, $\bf{K4C}$ and $\bf{GLC}$ and show that they all have the finite Kripke model property and are sound and complete with respect to the $d$-semantics in this dynamical setting. In particular, we prove that $\bf{wK4C}$ is the $d$-logic of all dynamic topological systems, $\bf{K4C}$ is the $d$-logic of all $T_D$ dynamic topological systems, and $\bf{GLC}$ is the $d$-logic of all dynamic topological systems based on a scattered space. We also prove a general result for the case where $f$ is a homeomorphism, which in particular yields soundness and completeness for the corresponding systems $\bf{wK4H}$, $\bf{K4H}$ and $\bf{GLH}$. The main contribution of this work is the foundation of a general proof method for finite model property and completeness of dynamic topological $d$-logics. Furthermore, our result for $\bf{GLC}$ constitutes the first step towards a proof of completeness for the trimodal topo-temporal language with respect to a finite axiomatisation -- something known to be impossible over the class of all spaces.

Closure System and Its Semantics

It is well known that topological spaces are axiomatically characterized by the topological closure operator satisfying the Kuratowski Closure Axioms. Equivalently, they can be axiomatized by other set operators encoding primitive semantics of topology, such as interior operator, exterior operator, boundary operator, or derived-set operator (or dually, co-derived-set operator). It is also known that a topological closure operator (and dually, a topological interior operator) can be weakened into generalized closure (interior) systems. What about boundary operator, exterior operator, and derived-set (and co-derived-set) operator in the weakened systems? Our paper completely answers this question by showing that the above six set operators can all be weakened (from their topological counterparts) in an appropriate way such that their inter-relationships remain essentially the same as in topological systems. Moreover, we show that the semantics of an interior point, an exterior point, a boundary point, an accumulation point, a co-accumulation point, an isolated point, a repelling point, etc. with respect to a given set, can be extended to an arbitrary subset system simply by treating the subset system as a base of a generalized interior system (and hence its dual, a generalized closure system). This allows us to extend topological semantics, namely the characterization of points with respect to an arbitrary set, in terms of both its spatial relations (interior, exterior, or boundary) and its dynamic convergence of any sequence (accumulation, co-accumulation, and isolation), to much weakened systems and hence with wider applicability. Examples from the theory of matroid and of Knowledge/Learning Spaces are used as an illustration.

Alexandroff topologies and monoid actions

Abstract Given a monoid S acting (on the left) on a set X, all the subsets of X which are invariant with respect to such an action constitute the family of the closed subsets of an Alexandroff topology on X. Conversely, we prove that any Alexandroff topology may be obtained through a monoid action. Based on such a link between monoid actions and Alexandroff topologies, we firstly establish several topological properties for Alexandroff spaces bearing in mind specific examples of monoid actions. Secondly, given an Alexandroff space X with associated topological closure operator σ, we introduce a specific notion of dependence on union of subsets. Then, in relation to such a dependence, we study the family 𝒜 σ , X {\mathcal{A}_{\sigma,X}} of the closed subsets Y of X such that, for any y 1 , y 2 ∈ Y {y_{1},y_{2}\in Y} , there exists a third element y ∈ Y {y\in Y} whose closure contains both y 1 {y_{1}} and y 2 {y_{2}} . More in detail, relying on some specific properties of the maximal members of the family 𝒜 σ , X {\mathcal{A}_{\sigma,X}} , we provide a decomposition theorem regarding an Alexandroff space as the union (not necessarily disjoint) of a pair of closed subsets characterized by such a dependence. Finally, we refine the study of the aforementioned decomposition through a descending chain of closed subsets of X of which we give some examples taken from specific monoid actions.

Generalizing Topological Set Operators

It is well-known that topological spaces can be axiomatically defined by the topological closure operator, i.e., Kuratowski Closure Axioms. Equivalently, they can be axiomatized by other set operators reflecting primitive notions of topology, such as interior operator, derived set operator (or dually, exterior operators, co-derived set operators), or boundary operator. It is also known that topological closure operators (and dually, topological interior operators) can be weakened as in generalized closure (interior) systems. What about boundary operator, exterior operator, and derived set (and co-derived set) operator in the weakened systems? Our paper completely answers this question by showing that these six operators can all be weakened in an appropriate way such that their relationships remain essentially the same as in topological spaces. Our results indicate that topological semantics can be fully relaxed to the weakened systems.

GENERALIZED FRÉCHET-URYSOHN SPACES

In this paper, we introduce some new properties of a topological space which are respectively generalizations of -Urysohn property. We show that countably AP property is a sufficient condition for a space being countable tightness, sequential, weakly first countable and symmetrizable, to be ACP, , first countable and semimetrizable, respectively. We also prove that countable compactness is a sufficient condition for a countably AP space to be countably . We then show that a countably compact space satisfying one of the properties mentioned here is sequentially compact. And we show that a countably compact and countably AP space is maximal countably compact if and only if it is . We finally obtain a sufficient condition for the ACP closure operator to be a Kuratowski topological closure operator and related results.

Algebraic Topological Closure Operators

For any closure operator c there is a T o -closure operator whose lattice of closed subsets are isomorphic to that of c. A correspondence between algebraic topological (T o ) closure operators on a nonempty set X and pre-orderes (partial orders) on X is established. Equivalent conditions are obtained for a T o -lattice to be a complete atomic Boolean algebra and for the lattice of closed subsets of an algebraic topological closure operator to be a complete atomic Boolean algebra. Further it is proved that a complete lattice is an algebraic T o -lattice if and only if it is isomorphic to the lattice of closed subsets of some algebraic topological closure operator on a suitable set.

Neighbourhood lattices – a poset approach to topological spaces

In this paper neighbourhood lattices are developed as a generalisation of topological spaces in order to examine to what extent the concepts of “openness”, “closedness”, and “continuity” defined in topological spaces depend on the lattice structure of P(X), the power set of X.A general pre-neighbourhood system, which satisfies the poset analogues of the neighbourhood system of points in a topological space, is defined on an ∧-semi-lattice, and is used to define open elements. Neighbourhood systems, which satisfy the poset analogues of the neighbourhood system of sets in a topological space, are introduced and it is shown that it is the conditionally complete atomistic structure of P(X) which determines the extension of pre-neighbourhoods of points to the neighbourhoods of sets.The duals of pre-neighbourhood systems are used to generate closed elements in an arbitrary lattice, independently of closure operators or complementation. These dual systems then form the backdrop for a brief discussion of the relationship between preneighbourhood systems, topological closure operators, algebraic closure operators, and Čech closure operators.Continuity is defined for functions between neighbourhood lattices, and it is proved that a function f: X → Y between topological spaces is continuous if and only if corresponding direct image function between the neighbourhood lattices P(X) and P(Y) is continuous in the neighbourhood sense. Further, it is shown that the algebraic character of continuity, that is, the non-convergence aspects, depends only on the properites of pre-neighbourhood systems. This observation leads to a discussion of the continuity properties of residuated mappings. Finally, the topological properties of normality and regularity are characterised in terms of the continuity properties of the closure operator on a topological space.

Verallgemeinerte topogene Ordnungen

Topogenous orders in the sense of Csaszar are a common generalization of proximity and topology. Cech closures are a generalization of the topological closure operators in the sense of Kuratowski. We show that the topogenous orders as well as the Cech closures are special cases of the so called compressed operators. Moreover, the now defined categoryCOM (in germanBAL) of compress spaces and compress faithful maps is a properly fibred topological category in the sense of Herrlich which is weakly cartesian closed, that means the product map of two quotient maps inCOM is a quotient map inCOM. Therefore by results of L. D. Nel it is possible to construct a cartesian closed properly fibred topological category in whichCOM can be nicely embedded. Further it turns out that the compressed operators be in a natural connexion with the uniform convergence structures in the sense of Cook and Fischer and in addition with the limit structures in the sense of Fischer. For principal ideal uniform convergence structures we prove that they are precompact and complete iff the properly constructed compressed operator is compact.

Completion of merotopic spaces and extension of uniformly continuous maps

A general Riesz merotopic space ( X, ν) determines a not necessarily topological closure operator c ν on X. The space ( X, ν) is said to be complete if every cluster on ( X, ν) is contained in an adherence grill on ( X, c ν ). We discuss a method of obtaining a large class of completions of a given Riesz merotopic space with induced T 1 closure space. As special cases we get the simple completion, which induces a simple closure space extension, and the strict completion, which induces a strict closure space extension. We show that the category of complete separated T 1 Riesz merotopic spaces is epireflective in the category of separated T 1 Riesz merotopic spaces, the reflection of an object being the simple completion. Similarly the category of complete clan-covered quasi-regular T 1 Riesz merotopic spaces is epireflective in the category of clan-covered quasi-regular T 1 Riesz merotopic spaces, the reflection of an object being the strict completion.

On a problem of Čech

Assuming CH (but we have stronger results) we partially solve a problem posed by E. Čech. Kuratowski gave the axioms which a topological closure operator, ƒ, must satisfy. If we do not ask that ƒ be idempotent (i.e. that for all X,(ƒ(X))=ƒ(X), then ƒ is known as a closure operator. E. Čech asked if there is a nontrivial closure operator which is onto, that is, for which in some sense every subset of our ‘space’ is ‘closed’. We build such functions. The problem is also well motivated when presented in purely set theoretic terms.

Weak* derived sets of sets of linear functionals

For a Banach space X the w*-sequential closure operator in the adjoint space is, in general, not the topological closure operator. That is, it may happen that the w*-sequential closure of a subspace T of X* is not w*-sequentially closed. The possible length of the chain of repeated w*-sequential closures of a subspace of X* in dependence on the dimension of X**/X is investigated.

On the hull operator

Hull operators, such as the closure, the linear, the convex, the positive hull operator, can be studied starting from a few fundamental properties. This paper deals with an operator j h deduced from a hull operator h, which e.g. characterizes the isolated points in case h is the topological closure operator and the extreme points in case h is the convex hull operator. By a slight generalization a comparable relation can be established between the isolated components and the extreme rays, if the hull operator is respectively the topological closure operator or the positive hull operator. § 1 is introductory. In § 2 j h is studied and § 3 deals with the above-mentioned generalization. ℘(U) will be used in the subsequent paragraphs to denote the set of subsets of the set U.