Separation Axioms Research Articles

On fuzzy intense separation axioms in fuzzy ideal topological space

The research is based on three stages in the construction process. The first stage is to give new definitions for three different types of Fuzzy sets that we called FI-set, FC-set and SFI-set. The second stage is to use what was defined in the previous stage in order to construct also three new definitions of internal, external and boundary fuzzy sets. Finally, the new concepts and separation axioms were defined using the definitions from the previous two stages.

On the soft Ds -dense and Dc -dense in soft ideal topological spaces

The research is based on two basic principles: soft local function and soft Ψ-operation to construct the new concepts of density called Ds-dense and Dc-dense. Through these concepts, we defined a new type of separation axiom, which we called soft ST 1-space.

On Semi Feebly Separation Axioms

The goal of this work is to present some new separation axioms based on the concept of defining new types of open sets namely semi-feebly open set. we investigate their fundamental features.

On nominal sets with support-preorder

Each nominal set 𝑋 can be equipped with a preorder relation ⪯ defined by the notion of support, so-called support-preorder. This preorder also leads us to the support topology on each nominal set. We study support-preordered nominal sets and some of their categorical properties in this paper. We also examine the topological properties of support topology, in particular separation axioms.

Intuitionistic Fuzzy Hypersoft Separation Axioms

In the present paper, we introduce the notion of $T_{i}$ $(i=0,1,2,3,4)$ separation axioms in intuitionistic fuzzy hypersoft topological spaces and discuss some of its properties. By using this notions, we also give some basic theorems of separation axioms in intuitionistic fuzzy hypersoft topological spaces. Finally, we present hereditary property of intuitionistic fuzzy hypersoft topological spaces.

On Intuitionistic Fuzzy Structure Space On Γ-Ring

In this research article, we investigate and study the intuitionistic fuzzy structure space of a $\Gamma$-ring $M$ set up by the class of intuitionistic fuzzy prime ideals of $M$ called the intuitionistic fuzzy prime spectrum of $\Gamma$-ring. Apart from studying basic properties of this structure space, we explore separation axioms, compactness, irreducibility and connectedness in this structure space.

Soft somewhat open sets: soft separation axioms and medical application to nutrition

Soft somewhat open sets: soft separation axioms and medical application to nutrition

Generalized h-Closed Sets in Topological Space

This study introduce a new type of closed sets in topology called Generalized h-closed sets (briefly, gh-closed) define as follow: E ⊆ χ be gh-closed set if CLh (E)⊆ U whenever E ⊆ U and U is open set in (χ,τ). The relation between gh-closed set and other classes of closed sets ( h-closed, g-closed, g -closed, g-closed and αg-closed) are studied. Also, the notion of gh-continuous mapping on topological space is introduce and some properties are proved. Finally, the separation axioms have been studied.

Quantification on the Generalization Performance of Deep Neural Network with Tychonoff Separation Axioms

Learning in Deep Neural Network (DNN) identifies the geometric structure of data. Hence there is a need for topological measures for quantifying this geometric structure. Based on the transformations in DNN, a metric is introduced with certain conditions to generate a base for a suitable topology. The locally indiscrete nature of the topology is applied to quantify the generalization gap in DNN approximations based on Tychonoff separation axioms. It develops a new measure T2 complexity that quantifies the model’s ability to distinguish classes irrespective of the class imbalance and the threshold applied. The efficiency of the new measure on a real dataset for binary and multiclass classification problems compared with commonly used classification metrics Accuracy, Precision, Recall, F1 score, Matthews Correlation coefficient, AUC-ROC score, and AUC-PR score. The experimental analysis shows that the proposed measure gives a necessary condition for an optimal classification model. The study shows that the measure T2 complexity can be included as a performance metric to analyze the model’s ability to distinguish classes on any binary and multiclass classification problem when the existing measures show misleading results due to class imbalance.

Soft Pre Separation Axioms and Function with Soft Pre Closed Graph

Several notions on soft topology are studied and their basic properties are investigated by using the concept of soft pre open sets and soft pre closure operator which are derived from the basics of soft set theory established by Molodtsov [1]. In this paper we introduce some soft separation axioms called Soft pre R0 and soft pre R1 in soft topological spaces which are defined over an initial universe with a fixed set of parameters. Many characterizations and properties of these spaces have been demonstrated. Necessary and sufficient conditions for a soft topological space to be a soft pre Ri space for i = 0, 1 were also presented. Furthermore, the concept of functions with soft pre closed graph and soft pre cluster set are defined. Many results on these two concepts are proved. Also, it is proved that a function has a soft pre closed graph if and only if its soft pre cluster set is degenerate.

Comprehensive note on characterization of neutrosophic soft P-open sets in neutrosophic soft quad-topological space

This paper concern the study of the notions of some new definitions and results. Three new definitions are given namely, neutrosophic soft quad semi-open, neutrosophic soft quad pre-open and neutrosophic soft quad *b open sets in neutrosophic soft quad-topological spaces. Among these one of the interesting neutrosophic soft quad generalized open set known as neutrosophic soft quad pre-open set is chosen then on the basis of this definition some fundamentals are generated. These are including, neutrosophic soft quad interior, neutrosophic soft quad boundary, neutrosophic soft quad exterior and neutrosophic soft quad closer. In continuation, attention has been focused on neutrosophic soft separation axioms which are defined in terms of neutrosophic soft p-open sets with respect to soft points then on the basis of definitions and results given, neutrosophic soft separation axioms are discussed mostly in terms of neutrosophic soft closer of the sets. In addition, some more results are addressed in neutrosophic soft quad-opological spaces with respect to soft points. Stress has been given on the neutrosophic soft quad topological property. Lastly, results on product of neutrosophic soft quad topological spaces, Bolzano-Weierstrass Property, compactness, countably compactness and sequentially compactness are addressed in terms of neutrosophic soft p-open sets in neutrosophic soft quad topological spaces.

Topological Data Analysis with Spherical Fuzzy Soft AHP-TOPSIS for Environmental Mitigation System

The idea of spherical fuzzy soft set (SFSS) is a new hybrid model of a soft set (SS) and spherical fuzzy set (SFS). An SFSS is a new approach for information analysis and information fusion, and fuzzy modeling. We define the concepts of spherical-fuzzy-soft-set topology (SFSS-topology) and spherical-fuzzy-soft-set separation axioms. Several characteristics of SFSS-topology are investigated and related results are derived. We developed an extended choice value method (CVM) and the AHP-TOPSIS (analytical hierarchy process and technique for the order preference by similarity to ideal solution) for SFSSs, and presented their applications in multiple-criteria group decision making (MCGDM). Moreover, an application of the CVM is presented in a stock market investment problem and another application of the AHP-TOPSIS is presented for an environmental mitigation system. The suggested methods are efficiently applied to investigate MCGDM through case studies.

Separation Axioms on Spatial Topological Space and Spatial Data Analysis

Separation Axioms on Spatial Topological Space and Spatial Data Analysis

Results on Topological Lattice Effect Algebras

In this paper, we define the notion of topological lattice effect algebras and investigate some of their properties. By using Sasaki arrows and F-balls, we construct two topology on lattice effect algebras. Then we study separation axioms on lattice effect algebras. Specifically, we find some conditions under which a topological lattice algebra is a $T_{0}$, $T_{1}$, and Hausdorff space. Finally, by using a strong filter and a quotient lattice effect algebra constructed by it, we investigate under what conditions this quotient lattice effect algebra will be a topological lattice effect algebra.

Some Properties of Weak Separation Axioms in Coc-Compact Sets

In this paper, we introduce some separation axioms in coc-compact set, namely coc-T0-space, coc-T 1/4- space, coc-T 3/8- space, cocT1/2- space, coc-T5/8- space, coc-Di- space, coc-Ri- space for i = 0, 1, weak coc-D1- space and weak coc-R0- space, and we study some relations between them, also we prove that some of these separation axioms have ”hereditary property”.

On (Λ, p)-closed Sets and the Related Notions in Topological spaces

This article deals with the concepts of Λp-sets and (Λ, p)-closed sets which are defined by utilizing the notions of preopen sets and preclosed sets. We also introduce and characterize some new low separation axioms. Characterizations of Λp-R0 spaces are given. Moreover, we introduce the concept of weakly (Λ, p)-continuous functions. In particular, several characterizations of weakly (Λ, p)-continuous functions are established.

On initial and final linear fuzzifying uniformities

The main purpose of this paper is to study initial and final Hutton type of linear fuzzifying uniformities. The detailed characterizations of initial and final linear fuzzifying uniformities are obtained. In addition, the boundedness, complete boundedness and T2 separation axiom of initial linear fuzzifying uniform spaces are investigated. Some examples with respect to initial and final linear fuzzifying uniformities are also provided.

SOME PROPERTIES OF T0 FUZZY SOFT TOPOLOGICAL SPACES IN QUASI-COINCIDENCE SENSE

In this paper, we have introduced and studied some new notions of T0 separation axiom in fuzzy soft topological spaces using quasi-coincident relation for fuzzy soft points. We have shown a relationship between ours and other counterparts and observed that all these notions satisfy good extension, hereditary, productive, and projective properties. Moreover, we have also shown that these notions are preserved under one-one, onto, and fuzzy soft continuous mappings. Finally, initial and final soft topologies are studied also.

PARA-COMPACTNESS CONCEPT IN INTUITIONISTIC FUZZY TOPOLOGICAL SPACES

This paper aims to establish the para-compactness concept in intuitionistic fuzzy topological space. Here we give three new notions related to para-compactness and one new notion of IF--compactness in intuitionistic fuzzy topological space. Also, we discuss separation axioms in intuitionistic fuzzy para-compactness and some of its features. Furthermore, using some provisos we will find a relation among second countable, para-compactness, and IF--compactness in intuitionistic fuzzy topological spaces.

Spherical Fuzzy Soft Topology and Its Application in Group Decision-Making Problems

The spherical fuzzy soft set is a generalized soft set model, which is more realistic, practical, and accurate. It is an extended version of existing fuzzy soft set models that can be used to describe imprecise data in real-world scenarios. The paper seeks to introduce the new concept of spherical fuzzy soft topology defined on spherical fuzzy soft sets. In this work, we define some basic concepts including spherical fuzzy soft basis, spherical fuzzy soft subspace, spherical fuzzy soft interior, spherical fuzzy soft closure, and spherical fuzzy soft boundary. The properties of these defined set are also discussed and explained with an appropriate examples. Also, we establish certain important results and introduce spherical fuzzy soft separation axioms, spherical fuzzy soft regular space, and spherical fuzzy soft normal space. Furthermore, as an application, a group decision-making algorithm is presented based on the TOPSIS (Technique of Order Preference by Similarity to an Ideal Solution) method for solving the decision-making problems. The applicability of the proposed method is demonstrated through a numerical example. The comprehensive advantages of the proposed work have been stated over the existing methods.