T_2 Separation Subsets and T_2 Separateness in L-fuzzy Topological Spaces

The concept of α-CT2 separation L-fuzzy subsets in L-fuzzy topological spaces is presented by taking the stratiform structure of L-fuzzy subsets as the point of departure,and its basic characterizations and some topological properties are discussed,and the relation between it and other separateness is exposed,and the action is studied of α-CT2 separateness in N-compact spaces and N-paracompact spaces.

Almost Fuzzy Compactness in L-fuzzy Topological Spaces

In this paper, the notion of almost fuzzy compactness is dened in L-fuzzy topological spaces by means of inequality, where L is a completely distributiveDeMorgan algebra. Its properties are discussed and many characterizations of it arepresented.

On Fuzzy Regularly Closed Filters in Michálek Fuzzy Topological Space

C.L. Chang’s introduction of fuzzy topology in 1981 opened up new avenues for parallel theories in topology. However, Chang’s work appears to focus more on the topology of fuzzy sets rather than fuzzy topology itself. In 1975, Michálek presented a functional definition of ordinary topology and later developed fuzzy topology as a distinct extension of this idea, setting it apart from Chang’s approach. While there has been significant research on Chang’s fuzzy topology, Michálek’s version has not received as much attention. This paper introduces the concept of fuzzy regularly closed filters, or FRCM filters, within Michálek’s fuzzy topological space and explores some properties of FRCM ultrafilters.

HB-Continuous Mappings in L-Topological Space

In this paper, we introduce and study the notion of HB-closed sets in L-topological space. Then, HB-convergence theory for L-molecular nets and L-ideals is established in terms of HB-closedness. Finally, we give a new definition of fuzzy H-continuous [1] which is called HB-continuity on the basis of the notion of H-bounded L-subsets in L-topological space. Then we give characterizations and properties by making use of HB-converges theory of L-molecular nets and L-ideals.

Inclusive Regular Separation in L-Fuzzy Topological Spaces

In this paper, we first introduce the concept of the inclusive regular separation in L-fuzzy topological spaces. Then we compare the inclusive regular separation with pointed regular separation and regular separation, and discuss the implicative and non-implicative relations among the above three separations. Finally, we illustrate that the inclusive regular separation is harmonic with the inclusive normal separation and inclusive completely regular separation.

Space Topologies and Their Dual Space Topologies for Conventional Functional Space Topologies

In this paper, we have studied the topology of some classical functional spaces. Among these spaces, there are standard spaces, spaces that can be metrizable and others that cannot be metrizable. But they are all topological vector spaces and it is in this context that we have chosen to present this work. We are interested in the topology of its spaces and in the topologies of their dual spaces. The first part, we presented the fundamental topological properties of topological vector spaces. The second part, we studied Frechet spaces and particularly the space S(Rn) of functions of class C∞ on Rn which are as well as all their rapidly decreasing partial derivatives. We have also studied its dual S'(Rn) the space of tempered distributions. The last part aims to define a topological structure on an increasing union of Frechet spaces called inductive limit of Frechet spaces. We study in particular the space D(Ω) of functions of class C∞ with compact supports on Ω as well as its dual D' (Ω) the space distributions over the open set Ω.

Locomotion Paradigms of Topological Space in Shakespeare’s Double Falsehood

In the 16th and 17th century,Britain was constantly strained by sporadic plagues,famines,fires and political conflicts.As a response in literature,Shakespeare has constructed an annular locomotion paradigm in Fletcher and his collaboration work Cardenio.Be more specific,the locomotion routes can be exemplified in Möbius strip,a rhizome system,a reciprocating juxtaposition between the foreground and the background,either of which finally runs to an annular schema.The annular in narration corresponds with The Globe theater and the round world.Shakespeare may express his expectation and prospect for the uprising bourgeoise and the newly found world by the resurrection theme in literature.

EQUILIBRIUM PROBLEMS WITH LOWER AND UPPER BOUNDS IN TOPOLOGICAL SPACES

By employing a fixed point theorem due to Ding, Park and Jung, some existence theorems of solutions for equilibrium problems with lower and upper bounds are proved in noncompact topological spaces. These results further answer the open problem raised by Isac, Sehgal and Singh under much weaker assumptions.

Fractal analysis of polyferric chloride-humic acid(PFC-HA) flocs in different topological spaces

The fractal dimensions in different topological spaces of polyferric chloride-humic acid （PFC-HA） flocs, formed in flocculating different kinds of humic acids （HA） water at different initial pH （9.0, 7.0, 5.0） and PFC dosages, were calculated by effective densitymaximum diameter, image analysis, and N2 absorption-desorption methods, respectively. The mass fractal dimensions （De） of PFC-HA floes were calculated by bi-logarithm relation of effective density with maximum diameter and Logan empirical equation. The Df value was more than 2.0 at initial pH of 7,0, which was 11% and 13% higher than those at pH 9.0 and 5.0, respecively, indicating the most compact flocs formed in flocculated HA water at initial pH of 7.0. The image analysis for those flocs indicates that after flocculating the HA water at initial pH greater than 7.0 with PFC flocculant, the fractal dimensions of D2 （logA vs. logdL） and D3 （logVsphere vs. logdL） of PFC-HA floes decreased with the increase of PFC dosages, and PFC-HA floes showed a gradually looser structure. At the optimum dosage of PFC, the D2 （logA vs. logdL） values of the flocs show 14%-43% difference with their corresponding Dr, and they even had different tendency with the change of initial pH values. However, the D2 values of the floes formed at three different initial pH in HA solution had a same tendency with the corresponding Df. Based on fractal Frenkel-Halsey-HiU （FHH） adsorption and desorption equations, the pore surface fractal dimensions （Ds） for dried powders of PFC-HA flocs formed in HA water with initial pH 9.0 and 7.0 were all close to 2.9421, and the Ds values of flocs formed at initial pH 5.0 were less than 2.3746. It indicated that the pore surface fractal dimensions of PFC-HA floes dried powder mainly show the irregularity from the mesopore-size distribution and marcopore-size distribution.

μ-Separations in generalized topological spaces

This paper takes some investigations on generalized topological spaces with some μ- separations. Some characterizations of μTi-spaces for i = 0, 1, 2, 3, 4, μTn-spaces and μR0-spaces are obtained and some relations among these spaces are established.

Topological degree for 1-set-contractive fields in M-PN spaces and its applications

In this paper, we define the topological degree for 1-set-contractive fields in PN spaces. Based on this, we obtain some new fixed point theorems for 1-set-contractive operators. As an application, we study the existence of solutions for a kind of nonlinear Volterra integral equations in Z-M-PN space.

Separation axioms in topological ordered spaces

The Ti-axiom,the Ti-ordered axiom and Ti-pairwise axiom(i = 0,1,2,3,4) of topological ordered space are discussed and proved that they are equivalence under the certain conditions.

An Introduction to Fuzzy Topological Spaces

Topology has enormous applications on fuzzy set. An attention can be brought to the mathematicians about these topological applications on fuzzy set by this article. In this research, first we have classified the fuzzy sets and topological spaces, and then we have made relation between elements of them. For expediency, with mathematical view few basic definitions about crisp set and fuzzy set have been recalled. Then we have discussed about topological spaces. Finally, in the last section, the fuzzy topological spaces which is our main object we have developed the relation between fuzzy sets and topological spaces. Moreover, this article has been concluded with the examination of some of its properties and certain relationships among the closure of these spaces.

γ and β Approximations via General Ordered Topological Spaces

In this paper, we introduce the concepts of g and b approximations via general ordered topological approximation spaces. Also, increasing (decreasing) g, b boundary, positive and negative regions are given in general ordered topological approximation spaces (GOTAS, for short). Some important properties of them were investigated. From this study, we can say that studying any properties of rough set concepts via GOTAS is a generalization of Pawlak approximation spaces and general approximation spaces.

General Type-2 Fuzzy Topological Spaces

In this paper, a presented definition of type-2 fuzzy sets and type-2 fuzzy set operation on it was given. The aim of this work was to introduce the concept of general topological spaces were extended in type-2 fuzzy sets with the structural properties such as open sets, closed sets, interior, closure and neighborhoods in topological spaces were extended to general type-2 fuzzy topological spaces and many related theorems are proved.

Generalizations of Rough Functions in Topological Spaces by Using Pre-Open Sets

Functions are a means to link or transport from a world to another world may be similarly or completely different from the other world. In this paper we addressed the issue of rough functions and the possibility of transfer it from the real line to the topological abstract view that can be applied to intelligent information systems. The rough function approach has not been studied much specially from a topological point of view. Here we developed a new type of topological generalizations of rough functions with reference to how it is used in medical applications. Considering that the function is in the original a relation can be based on a review of all circular functions from the perspective of relations. Accordingly, the dream that the generalizations of rough functions are transferred to all papers prior to a comprehensive computer application.

On Irresolute Topological Vector Spaces

In this paper, our focus is to investigate the notion of irresolute topological vector spaces. Irresolute topological vector spaces are defined by using semi open sets and irresolute mappings. The notion of irresolute topological vector spaces is analog to the notion of topological vector spaces, but mathematically it behaves differently. An example is given to show that an irresolute topological vector space is not a topological vector space. It is proved that: 1) Irresolute topological vector spaces possess open hereditary property;2) A homomorphism of irresolute topological vector spaces is irresolute if and only if it is irresolute at identity element;3) In irresolute topological vector spaces, the scalar multiple of semi compact set is semi compact;4) In irresolute topological vector spaces, every semi open set is translationally invariant.

Generalized Bi quasi variational Inequalities in Locally Convex Topological Vector Spaces

In this paper we investigate generalized bi quasi variational inequalities in locally convex topological vector spaces. Motivated and inspired by the recent research work in this field,we establish several existence theorems of solutions for generalized bi quasi variational inequalities,which are the extension and improvements of the earlier and recent results obtained previously by many authors including Sun and Ding [18],Chang and Zhang [23] and Zhang [24].

A Ubi Sunt Motif of Life Philosophy:Construction of Three Topological Spaces in Shakespeare’s Merchant of Venice

Topology,as a geometrical branch,studies spaces,whose essence lies in its topological equivalence.As far as literary criticism is concerned,the author’s writing purpose can be regarded as an open set in topology,which contains a number of different topological spaces.No matter how these spaces transform,they remain fundamental properties as long as their essence is invariant.From the perspective of topology,Shakespeare constructed three equivalent metaphorical topological spaces:one-pound-flesh agreement,marriage casket“lottery”,and wedding ring trick.The present paper purports to explore how Shakespeare constructed the three equivalent topological spaces to express the theme of literature:the ubi sunt1 motif of life philosophy.

Category Theoretic Properties of Topological Structures and Proximity Structures on L-fuzzy Space

This paper formulates the category of L-fuzzy spaces and fuzzy functions.It shows that the category of topological spaces and continuous fuzzy functions is a direct generalization of TOP and LTOP Moreover,it defines the concept of proximity space on L-fuzzy space and introduces its fundamental properties.A comparison between the classical case and the ordinary case has been outlined.