Pretopological Spaces Research Articles

The language of knowledge spaces in pre-topology1

The theory of knowledge spaces (KST) which is regarded as a mathematical framework for the assessment of knowledge and advices for further learning. Now the theory of knowledge spaces has many applications in education. From the topological point of view, we discuss the language of the theory of knowledge spaces by the axioms of separation and the accumulation points of pre-topology respectively, which establishes some relations between topological spaces and knowledge spaces; in particular, we show that the language of the regularity of pre-topology in knowledge spaces and give a characterization for knowledge spaces by inner fringe of knowledge states. Moreover, we study the relations of Alexandroff spaces and quasi ordinal spaces; then we give an application of the density of pre-topological spaces in primary items for knowledge spaces, which shows that one person in order to master an item, she or he must master some necessary items. In particular, we give a characterization of a skill multimap such that the delineated knowledge structure is a knowledge space, which gives an answer to a problem in [14] or [18] whenever each item with finitely many competencies; further, we give an algorithm to find the set of atom primary items for any finite knowledge space.

On L-Fuzzy Topological Spaces Determined by Implicators

This paper is towards the study of [Formula: see text]-fuzzy topological spaces, [Formula: see text]-fuzzy pretopological spaces and their corresponding operators determined by general implicators. We show that the results regarding one-to-one correspondence between [Formula: see text]-fuzzy reflexive approximation spaces and [Formula: see text]-fuzzy pretopological spaces as well as [Formula: see text]-fuzzy preorder approximation spaces and [Formula: see text]-fuzzy topological spaces heavily depend on implicators. Finally, we establish a relationship between category of [Formula: see text]-fuzzy pretopological spaces and category of [Formula: see text]-fuzzy topological spaces by using category of [Formula: see text]-fuzzy approximation spaces.

Application on Similarity Relation and Pretopology

Due to the tremendous use of computers, the need for dealing with digital information continues to grow. This has led to the need for constructing a structure for saving information in a way that eases data retrieval and processing. An information system is a table form that contains all the data needed for a user to reach a decision. Information systems are the most common form used for knowledge and data representation, and attribute reduction plays an important role in data processing. This paper is devoted to proposing a strategy for attributes reduction based on the similarity relation and pretopological concepts. Moreover, new types of pretopological spaces are to be constructed from the information system. Furthermore, a comparison between pretopologies and their pre-interiors constructed from the information system, as well as the different types of pretopological spaces, is investigated. Finally, the concept of the cover pretopology is applied to the information system.

On the category of ordered pre-topological spaces

On the category of ordered pre-topological spaces

Multi-instance learning of pretopological spaces to model complex propagation phenomena: Application to lexical taxonomy learning

This paper addresses the problem of learning the concept of propagation in the theoretical formalism of pretopology, and then applying this methodology for the well-known problem of learning Lexical Taxonomy. The theory of pretopology, among others, aims at modeling complex relations between sets of entities. The use of such fine-grained modeling implies limitations in terms of scalability. However, it allows for a more accurate capture of real-world relationships, such as the hypernymy relation, by modeling the task of relation extraction as a propagation model under certain structuring constraints, as opposed to traditional approaches that are limited to detecting relations between pairs of elements without considering knowledge on the expected structuring.Our proposal is to define the pseudo-closure operator (modeling the concept of propagation) as a logical combination of heterogeneous neighborhoods, or sources. It allows the learning of models that exploit, for example, the knowledge acquired by both statistical and numerical approaches. We show that the learning of such an operator falls into the Multiple Instance (MI) framework, where the learning process is performed on bags of instances instead of individual instances. Although this framework is well suited for this task, using it for learning a pretopological space leads to a set of bags whose size is exponential. To overcome this problem, we propose a learning method (LPSMI) based on a low estimate of the bags covered by a concept under construction.We first propose an experimental validation of our method, through the simulation of percolation processes (typically forest fires) learned with pretopological propagation models. It reveals that the proposed MI approach is particularly efficient on propagation model recognition task. We then provide a real-world contribution to the Lexical Taxonomy learning task, by modeling this task as a complex (semantic) propagation problem. We propose a very generic framework for training models combining various existing methods for learning Lexical Taxonomies (statistical, pattern-based and embedding-based).

M-Fuzzifying Approximation Spaces, M-Fuzzifying Pretopological Spaces and M-Fuzzifying Closure Spaces

In category theory, Galois connection plays a significant role in developing the connections among different structures. The objective of this work is to investigate the essential connections among several categories with a weaker structure than that of [Formula: see text]-fuzzifying topology, viz. category of [Formula: see text]-fuzzifying approximation spaces based on reflexive [Formula: see text]-fuzzy relations, category of [Formula: see text]-fuzzifying pretopological spaces and the category of [Formula: see text]-fuzzifying interior (closure) spaces. The interrelations among these structures are shown via the functorial diagram.

Functors among Relational Variants of Categories Related to L-Fuzzy Partitions, L-Fuzzy Pretopological Spaces and L-Fuzzy Closure Spaces

Various types of topological and closure operators are significantly used in fuzzy theory and applications. Although they are different operators, in some cases it is possible to transform an operator of one type into another. This in turn makes it possible to transform results relating to an operator of one type into results relating to another operator. In the paper relationships among 15 categories of modifications of topological L-valued operators, including Čech closure or interior L-valued operators, L-fuzzy pretopological and L-fuzzy co-pretopological operators, L-valued fuzzy relations, upper and lower F-transforms and spaces with fuzzy partitions are investigated. The common feature of these categories is that their morphisms are various L-fuzzy relations and not only maps. We prove the existence of 23 functors among these categories, which represent transformation processes of one operator into another operator, and we show how these transformation processes can be mutually combined.

A new representation of α-openness, α-continuity, α-irresoluteness, and α-compactness in L-fuzzy pretopological spaces

Abstract This paper presents a new representation of α-openness, α-continuity, α-irresoluteness, and α-compactness based on L-fuzzy α-open operators introduced by Nannan and Ruiying [1] and implication operation. The proposed representation extends the properties of α-openness, α-continuity, α-irresoluteness, and α-compactness to the setting of L-fuzzy pretopological spaces based on graded concepts. Moreover, we introduce and establish the relationships among the new concepts.

A categorical approach to the maximum theorem

Berge's maximum theorem gives conditions ensuring the continuity of an optimised function as a parameter changes. In this paper we state and prove the maximum theorem in terms of the theory of monoidal topology and the theory of double categories.This approach allows us to generalise (the main assertion of) the maximum theorem, which is classically stated for topological spaces, to pseudotopological spaces and pretopological spaces, as well as to closure spaces, approach spaces and probabilistic approach spaces, amongst others. As a part of this we prove a generalisation of the extreme value theorem.

T0-reflection and some separation axioms in PRETOP

We give the T0-reflection in the category of pretopological spaces with p-continuous maps as arrows. After that we will study some separation axioms in this category.

Alexandroff pretopologies for structuring the digital plane

We explore the possibility of employing Alexandroff pretopologies as structures on the digital plane Z2 convenient for the study of geometric and topological properties of digital images. These pretopologies are known to be in one-to-one correspondence with reflexive binary relations so that graph-theoretic methods may be used when investigating them. We discuss such Alexandroff pretopologies on Z2 that possess a rich variety of digital Jordan curves obtained as circuits in a natural graph with the vertex set Z2. Of these pretopologies, we focus on the minimal ones and study their quotient pretopologies on Z2, which are shown to allow for various digital Jordan curve theorems. We also develop a method for identifying Jordan curves in the minimal pretopological spaces by using Jordan curves in their quotient spaces. Utilizing this method, we conclude the paper with proving a digital Jordan curve theorem for the minimal pretopologies.

A short note on L-fuzzy approximation spaces and L-fuzzy pretopological spaces

In this note, we investigate the relationship between L-fuzzy pretopological spaces and L-fuzzy approximation spaces based on the reflexive L-fuzzy relations from a categorical viewpoint.

L-fuzzy N-convergence structures

Based on a completely distributive lattice L, the concept of L-fuzzy N-convergence structures is introduced. It is shown that the category of L-fuzzy topological spaces can be embedded in the category of L-fuzzy N-convergence spaces. It is also proved the category of (topological) pretopological L-fuzzy N-convergence spaces is isomorphic to the category of (topological) L-fuzzy neighborhood spaces and the former is a bireflective subcategory of the category of L-fuzzy N-convergence spaces.

Peritopological Spaces and Bisimulations

Generalizing ordinary topological and pretopological spaces, we introduce the notion of peritopology where neighborhoods of a point need not contain that point, and some points might even have an empty neighborhood. We briefly describe various intrinsic aspects of this notion. Applied to modal logic, it gives rise to peritopological models, a generalization of topological models, a spacial case of neighborhood semantics. (In a last section, the relation between the latter and the former is discussed, cursorily). A new cladding for bisimulation is presented. The concept of Alexandroff peritopology is used in order to determine the logic of all peritopological spaces, and we prove that the minimal logic K is strongly complete with respect to the class of all peritopological spaces. We also show that the classes of T0, T1 and T2-peritopological spaces are not modal definable, and that D is the logic of all proper peritopological spaces. AMS subject classification : 03B45, 54A05

Measures of compactness in L-fuzzy pretopological spaces

In this paper, we introduce the concepts of the degrees of compactness, countable compactness and Lindelof property in L-fuzzy pretopological spaces by means of implication operator. These definitions do not rely on the structure of the basis lattice L and no distributivity in L is required. The notions of pre-compactness and semi-compactness in L-fuzzy topological spaces can be viewed as special cases of compactness in L-fuzzy pretopological spaces. Their properties are investigated. Further when L is completely distributive, their characterizations are presented.

CONVERGENCE APPROACH SPACES AND APPROACH SPACES AS LATTICE-VALUED CONVERGENCE SPACES

We show that the category of convergence approach spaces is a simultaneously reective and coreective subcategory of the category of latticevalued limit spaces. Further we study the preservation of diagonal conditions, which characterize approach spaces. It is shown that the category of preapproach spaces is a simultaneously reective and coreective subcategory of the category of lattice-valued pretopological spaces and that the category of approach spaces is a coreective subcategory of a category of lattice-valued topological convergence spaces

The Concept of Morphospaces in Evolutionary and Developmental Biology: Mathematics and Metaphors

Formal spaces have become commonplace conceptual and computational tools in a large array of scientific disciplines, including both the natural and the social sciences. Morphological spaces (morphospaces) are spaces describing and relating organismal phenotypes. They play a central role in morphometrics, the statistical description of biological forms, but also underlie the notion of adaptive landscapes that drives many theoretical considerations in evolutionary biology. We briefly review the topological and geometrical properties of the most common morphospaces in the biological literature. In contemporarygeometricmorphometrics,thenotionofamorphospace is based on the Euclidean tangent space to Kendall’s shape space, which is a Riemannian manifold. Many more classical morphospaces, such as Raup’s space of coiled shells, lack these metric properties, e.g., due to incommensurably scaled variables, so that these morphospaces typically are affine vector spaces. Other notions of a morphospace, like Thomas and Reif’s (1993) skeleton space, may not give rise to a quantitative measure of similarity at all. Such spaces can often be characterized in terms of topological or pretopological spaces. The typical language of theoretical and evolutionary biology, comprising statements about the “distance” among phenotypes in an according space or about different “directions” of evolution, is not warranted for all types of morphospaces. Graphical visualizations of morphospaces or adaptive landscapes may tempt the reader to apply “Euclidean intuitions” to a morphospace, whatever its actual topology might be. We discuss the limits of metaphors such as the developmental hourglass and adaptive landscapes that ensue from the geometric properties of the underlying morphospace.

Lattice-valued fuzzy interior operators

A category of lattice-valued fuzzy interior operator spaces is defined and studied. Axioms are given in order for this category to be isomorphic to the category whose objects consist of all the stratified, lattice-valued, pretopological convergence spaces.

Axioms for Sequential Convergence

Centro de Matematica da Universidade de Coimbra/FCT; Unidade de Investigacao e Desenvolvimento Matematica e Aplicacoes da Universidade de Aveiro/FCT

Injective Convergence Spaces and Equilogical Spaces via Pretopological Spaces

Sierpinski space Ω is injective in the category Top of topological spaces, but not in any of the larger cartesian closed categories Conv of convergence spaces and Equ of equilogical spaces. We show that this negative result extends to all sub-cccs of Equ and Conv that are closed under subspaces and contain Top. On the other hand, we study the category PrTop of pretopological spaces that lies in-between Top and Conv/Equ, identify its injective spaces, and show that they are also injective in Conv and Equ.