On B*c – adherent points and its properties

In this article, we introduce the notion of the complementary soft neighborhood and present three kinds of covering soft rough set (mathcal {CSR}) models. The basic properties of these models are investigated. The relationships among these models are also discussed. Moreover, we establish the topological approach to mathcal {CSR} say, varDelta -topological spaces (i.e., varDelta -TS). Hence, the topological properties for varDelta -TS models such as varDelta -open sets, varDelta -closed sets, varDelta -interior, varDelta -closure, varDelta -boundary, varDelta -neighborhood and varDelta -limit point are studied and the relationships between them are given. Finally, we make use of an algorithm for these proposed models to deal with uncertainties for solving the MGDM problems using the constructed topologies.

The significance fore supra topological spaces as a subject of study cannot be overstated, as they represent a broader framework than traditional topological spaces. Numerous scholars have proposed extension to supra open sets, including supra semi open sets, supra per open and others. In this research, a notion for ⱨ-supra open created within the generalizations of the supra topology of sets. Our investigation involves harnessing this style of sets to introduce modern notions in these spaces, specifically supra ⱨ - interior, supra ⱨ - closure, supra ⱨ - limit points, supra ⱨ - boundary points and supra ⱨ - exterior of sets. It has been examining the relationship with supra open. The research was also enriched with many of characteristics of each concept. Building upon this set classification, we introduced several kinds of maps like supra ⱨ - continuous, supra ⱨ - open, supra ⱨ - tentative, supra ⱨ -globally and supra ⱨ - homeomorphism. Additionally, we have proven a collection of useful relationships for the aforementioned of functions. Furthermore, the research was enhanced with illustrative and refuting examples.

The neutro-fine topological space is a space that contains a combination of neutrosophic and fine sets. In this study, the various types of open sets such as generalized open and semi-open sets are defined in such space. The concept of interior and closure on semi-open sets are defined and some of their basic properties are stated. These definitions extend the concept to generalized semi-open sets. Moreover, the minimal and maximal open sets are defined and some of their properties are studied in this space. As well as, discussed the complement of all these sets as its closed sets. The basic properties of the union and intersection of these open sets are stated in some theorems. Only a few sets satisfy this postulates, and others are disproved as shown in the counterexamples. The converse of some theorems is proved in probable examples.

An approach to solving a complementarity problem entails regularizing/perturbing the problem by adding to the given mapping another mapping multiplied by a small positive parameter. We study properties of the limit point of the solution to the regularized problem. We also derive local error bounds on the distance from the solution to its limit point, expressed in terms of the regularization parameter.

eberhard hopfLet fl be a bounded open set in ordinary w-space.Denote by p a point of that space.A closed subset of the boundary of A is called accessible from within fl if fl contains a semi-open Jordan curve p{t), 0^f< , whose points of accumulation, for t-><, precisely constitute that boundary part.In this note we prove the following theorem.

We provide a concrete isometric description of all the preduals of $\ell_1$ for which the standard basis in $\ell_1$ has a finite number of $w^*$-limit points. Then, we apply this result to give an example of an $\ell_1$-predual $X$ such that its dual $X^*$ lacks the weak$^*$ fixed point property for nonexpansive mappings (briefly, $w^*$-FPP), but $X$ does not contain an isometric copy of any hyperplane $W_{\alpha}$ of the space $c$ of convergent sequences such that $W_\alpha$ is a predual of $\ell_1$ and $W_\alpha^*$ lacks the $w^*$-FPP. This answers a question left open in the 2017 paper of the present authors.

In recent years, some generalized structures of topology were introduced. Supra topology was one the most important of those generalizations. To contribute in this orientation, we devoted this work to studying limit points and separation axioms on supra topological spaces by using supra $b$-open sets. We define them in a similar way of their counterparts on topological spaces. In general, we demonstrate their main properties and investigate the sufficient conditions for some equivalent relations between them. Some novel and interesting examples are provided.

ON FINITE T0-SPACES

The authors’ primary goal in this paper is to enhance the study of $T_0$ topological spaces by using the order of specialization of a $T_0$ -space to introduce the lower topology (with a subbasis of closed sets $\mathord{\uparrow } x$ ) and studying the interaction of the original topology and the lower topology. Using the lower topology, one can define and study new properties of the original space that provide deeper insight into its structure. One focus of study is the property R, which asserts that if the intersection of a family of finitely generated sets $\mathord{\uparrow } F$ , $F$ finite, is contained in an open set $U$ , then the same is true for finitely many of the family. We first show that property R is equivalent to several other interesting properties, for example, the property that all closed subsets of the original space are compact in the lower topology. We then find conditions under which these spaces are compact, well-filtered, and coherent, a weaker variant of stably compact spaces. We also investigate what have been called strong $d$ -spaces, develop some of their basic properties, and make connections with the earlier considerations involving spaces satisfying property R. Two key results we obtain are that if a dcpo $P$ with the Scott topology is a strong $d$ -space, then it is well-filtered, and if additionally the Scott topology of the product $P\times P$ is the product of the Scott topologies of the factors, then the Scott space of $P$ is sober. We also exhibit connections of this work with de Groot duality.

The main objective of this paper is to introduce a new generalized form of topological vector spaces, namely, almost $\\alpha$-topological vector spaces by using the concepts of $\\alpha$-open sets, regular open sets, and almost $\\alpha$-continuous mappings. In the present paper, some important examples of almost $\\alpha$-topological vector spaces with basic properties are characterized.

In this paper, we introduce the concepts of j-ideal lower and j-ideal upper approximations as a generalization of approximation by Pawlak by using j-ideal open sets where j ∈ {α, S, P, γ, β, ⋀β}. Some of their basic properties with the aid of examples are proven. Furthermore, we study the relationships between these approximations to get the best of them.

Topological characterization of filter topological MV-algebras

In this paper, we introduce the concept of fuzzy r-quasi open sets which are generalizations of fuzzy r-open sets, and obtain some basic properties of such fuzzy sets. Also we introduce and study the concepts of fuzzy r-quasi continuous mapping and fuzzy r-quasi open(closed) mapping.

We give a systematic treatment of caloric measure null sets on the essential boundary $\partial_eE$ of an arbitrary open set $E$ in ${\bf R}$. We discuss two characterisations of such sets and present some basic properties. We investigate the dependence of caloric measure null sets on the open set $E$. Thus, if $D$ is an open subset of $E$ and $Z\subseteq\partial_eE\cap\partial_eD$, we show that $Z$ is caloric measure null for $D$ if it is caloric measure null for $E$. We also give conditions on $E$ and $Z$ which imply that the reverse implication is true. We know from \cite{watson2011} that any polar subset of $\partial_eD$ is caloric measure null for $D$, but the reverse implication is not generally true. In our final result we show that, for subsets of a certain component of $\partial_eD$, caloric measure null sets are necessarily polar.

This book is concerned with the differential and integral calculus of functions of several variables. For this purpose one needs first to know some basic properties of euclidean space of arbitrary finite dimension n. We begin this chapter with a brief review of the real numbers and the elements of vector algebra and geometry of such spaces. Later in the chapter the concepts of neighborhood, open set, and closed set are introduced. These constitute the basis for studying what are called topological properties of n-dimensional space.