Fuzzy topology on fuzzy sets: Product fuzzy topology and fuzzy topological groups

The notion of product fuzzy topology in the case of fuzzy topology on fuzzy sets is introduced and the product invariance of fuzzy Hausdorffness, compactness, connectedness are examined. The product fuzzy topology is used to define fuzzy group topology on a fuzzy subgroup of a group G and some properties of fuzzy topological groups are obtained.     

On Inverse Limits and Direct Limits for Fuzzy Topological Spaces

Li has introduced the concepts of inverse system and direct system for fuzzy topological spaces and studied inverse limits and direct limits on such spaces by presenting the explicit constructions of these limits.In this paper some important concepts of fuzzy topology,such as,product fuzzy topology,quotient fuzzy topology,fuzzy continuity etc.,are used for further study of inverse limits and direct limits for fuzzy topological spaces.     

模糊拓扑空间的m—收敛理论

本文在模糊拓扑空间中引进了模糊网和模糊滤子的ｍ－收敛概念，探讨了模糊网的ｍ－收敛与模糊滤子的ｍ－收敛之间的关系，得到了Ｈａｕｓｄｏｒｆｆ拓扑与模糊网的ｍ－收敛和模糊滤子的ｍ－收敛之间的关系以及刻画Ｈａｕｓｄｏｒｆｆ拓扑几何特征的定理。     

模糊粗糙近似算子的拓扑性质

考虑论域上一二元关系所决定的模糊粗糙近似算子的拓扑性质,证明了任一自反二元关系可以决定一模糊拓扑.并且,当二元关系自反对称时,该模糊拓扑中的元是开集当且仅当它是闭集;当二元关系自反传递时,该模糊拓扑的闭包与内部算子恰为模糊粗糙上、下近似算子.     

基于二元关系的直觉模糊近似空间与直觉模糊拓扑

讨论在一般二元关系下直党模糊近似空间诱导的直党模糊拓扑空问的若干性质;由直觉模糊拓扑空间诱导直觉模糊近似空同所需的TC条件及其所得近似空间的近似算子若干性质.     

代数结构上的模糊拟伪$b$-度量

研究代数结构上的模糊(拟)伪$b$-度量. 主要结论有: (1)设$G$是一个抽象群, $\tau$是$G$上一个左不变模糊拟伪$b$-度量(伪$b$-度量)诱导的拓扑,如果$(G,\tau)$是右拓扑群,那么$(G,\tau)$是一个仿拓扑群(拓扑群); (2)设$S$是一个半群,如果$\tau$是$S$上一个不变模糊拟伪$b$-度量诱导的拓扑,那么$(S,\tau)$是一个拓扑半群.     

[0,1]-拓扑空间中的Starplus-连通性

本文在I-拓扑空间中引入Starplus-连通性的概念,并且研究它的性质.借助于水平拓扑,得到了下面结果:Starplus-连通性是一般拓扑学中连通性的好推广而且被连续映射保持.一个Starplus-连通的模糊集一定是O-连通的.另外,模糊单位区间和模糊实直线是Starplus-连通的.     

Regularly open sets in fuzzy topological spaces

This paper is devoted to the study of the role of fuzzy regularly open sets. We prove some properties of fuzzy almost continuous mappings and define fuzzy almost open mappings. We prove that under a fuzzy almost continuous and fuzzy almost open map, the inverse image of a fuzzy regularly open set is fuzzy regularly open. Further we define a new type of fuzzy separation axioms, fuzzy almost separation axioms. It is interesting that there are some deviations in the behaviour of these axioms as compared to those in general topology. For example, in a fuzzy almost T₁ space not every fuzzy singleton is δ-closed. Also a fuzzy space which is fuzzy almost as well as fuzzy almost T₀ is fuzzy almost regular. While in general topology we have to take an almost T₂ space in place of almost T₀ space.     

RELATIVE COMPACTNESS AND COMPACTNESS OF GENERAL SUBSETS IN AN I- TOPOLOGICAL SPACE

Abstract

In a previous work, we introduced a form of compactness applicable to general fuzzy sets in an I-topological space. It was shown that many of the standard results for compactness in general topology remain valid in the fuzzy setting. In this paper we continue our investigations into the behaviour of compact fuzzy subsets. We also introduce the notion of a relatively compact fuzzy subset and obtain results very similar to those of general topology. Many of our results are in the setting of fuzzy neighbourhood space and fuzzy uniform spaces. In particular, a number of criteria for compactness, already known for the whole space, are extended to arbitrary fuzzy subsets in a fuzzy neighbourhood space.     

模糊偏序集上理想完备的幂等性

主要讨论模糊偏序集上理想完备性的本质.并得到以下结论:模糊偏序集的理想完备是幂等的当且仅当理想完备上的广义Scott拓扑与Alexandroff拓扑是一致的.     

On Q-sobriety

The study of fixed-basis variety-based topology was initiated by S.A. Solovyov (in 2008), which, among other things, generalizes fuzzy topology. We extend within this framework, an earlier result due to Srivastava et al. (in 1998), which showed that the category of sober fuzzy topological spaces is the epireflective hull of the fuzzy Sierpinski space in the category of T₀-fuzzy topological spaces.     

On fuzzy compact-open topology

The concept of fuzzy compact-open topology is introduced and some characterizations of this topology are discussed.     

Fuzzy minimal structure and fuzzy minimal vector spaces

This paper deals with fuzzy minimal spaces as a new generalization of the notion of fuzzy topology. Moreover, we present the continuity of fuzzy functions in this setting. We also introduce and investigate the class of fuzzy minimal vector spaces as a generalization of the class of fuzzy topological vector spaces.     

Fuzzy topological spaces and fuzzy compactness

It is the purpose of this paper to go somewhat deeper into the structure of fuzzy topological spaces. In doing so we found we had to alter the definition of a fuzzy topology used up to now. We shall also introduce two functors $\text{}\text{?}$gw and $\text{}\text{?}$gi which will allow us to see more clearly the connection between fuzzy topological spaces and topological spaces. Finally we shall introduce the concept of fuzzy compactness as the generalization of compactness in topology. It will be shown in a following publication that contrary to the results obtained up to now, the Tychonoff-product theorem is safeguarded with fuzzy compactness.     

Fuzzy序列紧性,可数Fuzzy紧性和Fuzzy列紧性

本文引进了Fuzzy序列紧性、可数Fuzzy紧性和Fuzzy列紧性,它们是一般拓扑学中相应概念的“良扩张”(R. Lowen意义下),文中讨论了这些fts的主要性质,以及它们之间的联系。     

Fuzzy topological vector spaces I

Some of the properties of fuzzy topological vector spaces are investigated. Also, there are given necessary and sufficient conditions for a family of fuzzy sets, in a vector space E, to be the family of all neighborhoods of zero for a fuzzy linear topology.     

POWERSET OPERATOR BASED FOUNDATION FORPOINT-SET LATTICE-THEORETIC (POSLAT) FUZZY SET THEORIES and TOPOLOGIES

Abstract

This paper sets forth in detail point-set lattice-theoretic or poslat foundations of all mathematical and fuzzy set disciplines in which the operations of taking the image and pre-image of (fuzzy) subsets play a fundamental role; such disciplines include algebra, measure and probability theory, and topology. In particular, those aspects of fuzzy sets, hinging around (crisp) powersets of fuzzy subsets and around powerset operators between such powersets lifted from ordinary functions between the underlying base sets, are examined and characterized using point-set and lattice-theoretic methods. The basic goal is to uniquely derive the powerset operators and not simply stipulate them, and in doing this we explicitly distinguish between the “fixed-basis” case (where the underlying lattice of membership values is fixed for the sets in question) and the “variable-basis” case (where the underlying lattice of membership values is allowed to change). Applications to fuzzy sets/logic include: development and justification/characterization of the Zadeh Extension Principle [36], with applications for fuzzy topology and measure theory; characterizations of ground category isomorphisms; rigorous foundation for fuzzy topology in the poslat sense; and characterization of those fuzzy associative memories in the sense of Kosko [18] which are powerset operators. Some results appeared without proof in [31], some with partial proofs in [32], and some in the fixed-basis case in Johnstone [13] and Manes [22].     

Fuzzy Closure Spaces vs. Fuzzy Rough Sets

This paper investigates the relationship among fuzzy rough sets, fuzzy closure spaces and fuzzy topology. It is shown that there exists a bijective correspondence between the set of all fuzzy reflexive approximation spaces and the set of all quasi-discrete fuzzy closure spaces satisfying a certain extra condition. Similar correspondence is also obtained between the set of all fuzzy tolerance approximation spaces and the set of all symmetric quasi-discrete fuzzy closure spaces satisfying a certain extra condition.     

Fuzzy metric spaces

In this paper, fuzzy metric spaces are redefined, different from the previous ones in the way that fuzzy scalars instead of fuzzy numbers or real numbers are used to define fuzzy metric. It is proved that every ordinary metric space can induce a fuzzy metric space that is complete whenever the original one does. We also prove that the fuzzy topology induced by fuzzy metric spaces defined in this paper is consistent with the given one. The results provide some foundations for the research on fuzzy optimization and pattern recognition.     

Lower and upper fuzzy topological subhypergroups

This paper provides a new connection between algebraic hyperstructures and fuzzy sets. More specifically, using both properties of fuzzy topological spaces and those of fuzzy subhypergroups, we define the notions of lower (upper) fuzzy topological subhypergroups of a hypergroup endowed with a fuzzy topology. Some results concerning the image and the inverse image of a lower (upper) topological subhypergroup under a very good homomorphism of hypergroups (endowed with fuzzy topologies) are pointed out.