ON CERTAIN INITIAL COMPLETIONS IN FUZZY TOPOLOGY

Abstract

Following a lead given by I.W. Alderton, it is shown that the MacNeille completion and the universal initial completion coincide for the categories of zero-dimensional fuzzy T₀-topological spaces, T₀-fuzzy closure spaces, 2T ₀-fuzzy bitopological spaces, and T ₁-fuzzy topological spaces and that these turn out to be respectively the categories of zero-dimensional fuzzy topological spaces, fuzzy closure spaces, fussy bitopological spaces, and fuzzy R ₀ topological spaces.     

基于k阶二元关系的直觉模糊粗糙集

利用k阶二元关系定义直觉模糊粗糙集,讨论了分别为串行、自反、对称、传递关系时所对应的上、下近似算子的性质。在有限论域U中,研究了任一自反二元关系所诱导的直觉模糊拓扑空间中直觉模糊闭包、内部算子与相对应的上、下近似算子的关系。     

Separation Properties of Bitopological Space

ＳｅｐａｒａｔｉｏｎＰｒｏｐｅｒｔｉｅｓｏｆＢｉｔｏｐｏｌｏｇｉｃａｌＳｐａｃｅ￥ＴｉａｎＬｉｑｉｎ（ＬｕｏｙａｎｇＡｇｒｉｃｕｌｔｒａｌＣｏｌｌｅｇｅ）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒｗｅｓｔｕｄｙａｂｉｔｏｐｏｌｏｇｉｃａｌｓｐａｃｅｗｉ...     

Completely Continuous Functions in Intuitionistic Fuzzy Topological Spaces

In this paper, after giving the basic results related to the product of functions and the graph of functions in intuitionistic fuzzy topological spaces, we introduce and study the concept of fuzzy completely continuous functions between intuitionistic fuzzy topological spaces.     

Weaker forms of the Menger property in bitopological spaces

Abstract

In this paper we continue previous investigations on the weaker forms of the Menger property in bitopological spaces. We introduce weakly Menger property and study some topological properties of almost and weakly Menger bitopological spaces. We also consider the almost Hurewicz spaces in a bitopological context.     

Intuitionistic <Emphasis Type="Italic">I</Emphasis>-fuzzy topological spaces

The main purpose of this paper is to introduce the concept of intuitionistic I-fuzzy quasi-coincident neighborhood systems of intuitiostic fuzzy points. The relation between the category of intuitionistic I-fuzzy topological spaces and the category of intuitionistic I-fuzzy quasi-coincident neighborhood spaces are studied. By using fuzzifying topology, the notion of generated intuitionistic I-fuzzy topology is proposed, and the connections among generated intuitionistic I-fuzzy topological spaces, fuzzifying topological spaces and I-fuzzy topological spaces are discussed. Finally, the properties of the operators Iω, ι are obtained.     

关于生成I(L)-fuzzy拓扑空间问题的研究

以生成I(L)拓扑空间为基础,首先引入了Ω_L,I_L等算子及I(L)-fuzzy拓扑空间的概念,其次研究了该拓扑空间的基本性质及算子的相应运算规则,最后讨论了此拓扑空间与I(L)-fuzzy拓扑空间的关系.得到了生成I(L)-fuzzy拓扑空军的相应性质.     

Insertion of Continuous Real Functions on Spaces,Bispaces, Ordered Spaces and Pointfree Spaces—A Common Root

We characterize normal and extremally disconnected biframes in terms of the insertion of a continuous real function in between given lower and upper semicontinuous real functions and show this to be the common root of several classical and new insertion results concerning topological spaces, bitopological spaces, ordered topological spaces and locales.     

A Note on L-double Gradation Fuzzy Topological and L-double Fuzzy Closure Spaces

We prove the existence of both final L-double gradation fuzzy topological spaces and final L-double fuzzy closure spaces. From this fact, we define and study the notions of the L-double quotient spaces and the sum of L-double fuzzy closure spaces. Finally, we study the additivity of two kinds of L-double fuzzy closure spaces.     

A bitopological point-free approach to compactifications

We study structures called d-frames which were developed by the last two authors for a bitopological treatment of Stone duality. These structures consist of a pair of frames thought of as the opens of two topologies, together with two relations which serve as abstractions of disjointness and covering of the space. With these relations, the topological separation axioms regularity and normality have natural analogues in d-frames. We develop a bitopological point-free notion of complete regularity and characterise all compactifications of completely regular d-frames. Given that normality of topological spaces does not behave well with respect to products and subspaces, probably the most surprising result is this: The category of d-frames has a normal coreflection, and the Stone-?ech compactification factors through it. Moreover, any compactification can be obtained by first producing a regular normal d-frame and then applying the Stone-?ech compactification to it. Our bitopological compactification subsumes all classical compactifications of frames as well as Smyth?s stable compactification.     

Double convergence and products of Fréchet spaces

The paper is devoted to convergence of double sequences and its application to products. In a convergence space we recognize three types of double convergences and points, respectively. We give examples and describe their structure and properties. We investigate the relationship between the topological and convergence closure product of two Fréchet spaces. In particular, we give a necessary and sufficient condition for the topological product of two compact Hausdorff Fréchet spaces to be a Fréchet space.     

Bitopologies on Totally Ordered Sets

We study the category of ray bispaces, that is, the category whose objects are totally ordered sets with two topologies, each having a subbase of rays and so that the resulting bitopological space is pairwise weakly symmetric, and whose morphisms are the pairwise continuous functions. In contrast with the purely topological results of [5], we show that, (1) such spaces are utterly normal and hence monotonically normal (in the sense of [6]), and (2) (Intermediate Value Theorem) the pairwise continuous image of a pairwise connected bitopological space in a selective ray bispace is an interval. We also obtain conditions for the equality of the de Groot dual (see [4]) and the ray dual (see [5]) of a ray topology and show that a selective ray topology is compact if and only if it is skew compact.     

Bitopological spaces

The paper is, in essence, a monograph devoted to the theory of bitopological spaces and its applications. Not exhausting the entire subject, it reflects basic ideas and methods of the theory. The Introduction gives an idea of the origins of the basic notions, contents, methods, and problems both of the classical (in the spirit of Kelly) and of the general theory of bitopological spaces. The classical theory is described rather schematically in Chapter I, only the theory of extensions of topological and bitopological spaces and the theory of completion of uniform spaces are presented in more detail. The main focus is on the general theory of bitopological spaces (Chapter II). Notions, methods, and results presented here have no analogues in the classical theory. As applications, foundations of the theory of bitopological manifolds, in particular, bitopologically represented piecewise linear manifolds (Chapter III), and the foundations of the theory of bitopological groups are presented (Chapter IV). Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 242, 1997, pp. 7–216. Translated by A. A. Ivanov.     

Almost compactness in fuzzy topological spaces

We introduce and study almost compactness for fuzzy topological spaces. We show that the almost continuous image of an almost compact fuzzy topological space is almost compact. Moreover, we show that generally almost compactness for fuzzy topological spaces is not product-invariant, but if X and Y are almost fuzzy topological spaces and X is product related to Y, then their fuzzy topological product is almost compact.     

On concretely both reflective and coreflective subconstructs of Chang fuzzy topological spaces

In this paper, the concretely both reflective and coreflective subconstructs of the construct CFTS of Chang fuzzy topological spaces are characterized and their relationship to such subconstructs of stratified fuzzy topological spaces is investigated. Properties of a particular such subconstruct of CFTS, the construct of weak fuzzy neighborhood spaces, are discussed in detail. And it is proved that there exists exactly c such subconstructs between the subconstruct of weakly induced spaces and CFTS.     

直觉模糊度量空间上的半群作用

讨论在直觉模糊度量空间上的拓扑半群作用,引入诸如拓扑传递性,点传递性及点稠传递性等概念.考虑非敏感性与传递性,等度连续等动力学性质的相互关系.     

FUZZY SYNTOPOGENEOUS STRUCTURES

Abstract

The author gives a detailed analysis of the relation between the theories of realcompactifications and compactifications in the category of ditopological texture spaces and in the categories of bitopological spaces and topological spaces.     

L-拓扑空间的弱局部强F紧性及单点强F紧化

以R.Lowen的强F紧性为基础,定义了L-拓扑空间的弱局部强F紧性及单点强F紧化,推广了有关弱局部紧拓扑空间和拓扑空间的单点紧化的若干结果,证明了L-拓扑空间的弱局部强F紧性是拓扑空间的弱局部紧性的L-推广。     

Fuzzy拓扑线性空间的Fuzzy赋准范化

引进（ＱＬ）型Ｆｕｚｚｙ拓扑线笥空间的优基坯概念,在此基础上得到了分离的Ｆｕｚｚ６ｙ拓扑线性空间可Ｆｕｚｚｙ赋准范化的一个充分必要条件,进上步阐明了Ｆｕｚｚｙ准范线性空间与（ＱＬ）型Ｆｕｚｚｙ拓扑线笥空间的关系。     

The notion of exactness of bitopological extensions

The notion of exactness of topological extensions is generalized to bitopological spaces, and some bitopological notions of similar type are considered. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 242, 1997, pp. 230–236. Translated by N. Yu. Netsvetaev.