Topological Spaces Research Articles

On Soft Submaximal and Soft Door Spaces

This paper is divided into two parts. The first part deals with soft submaximal spaces, where we present new theorems and some basic facts. Further, we successfully find a requirement that indicates the soft one-point compactification of a soft topological space is soft submaximal. In the second part, we study soft door spaces. We notice that every soft door space is a soft submaximal space, but a soft submaximal space need not be soft door. The class of soft door spaces is hereditary. We give couterexamples showing that this class is neither additive nor productive. We further show that images of soft door spaces under certain soft functions are also soft door spaces. After that, we characterize certain soft topological spaces in terms of soft limit points and the Krull dimension. At last, we discuss when the soft one-point compactification of a soft topological space is soft door.

The Univalence Principle

The Univalence Principle is the statement that equivalent mathematical structures are indistinguishable. We prove a general version of this principle that applies to all set-based, categorical, and higher-categorical structures defined in a non-algebraic and space-based style, as well as models of higher-order theories such as topological spaces. In particular, we formulate a general definition of indiscernibility for objects of any such structure, and a corresponding univalence condition that generalizes Rezk’s completeness condition for Segal spaces and ensures that all equivalences of structures are levelwise equivalences. Our work builds on Makkai’s First-Order Logic with Dependent Sorts, but is expressed in Voevodsky’s Univalent Foundations (UF), extending previous work on the Structure Identity Principle and univalent categories in UF. This enables indistinguishability to be expressed simply as identification, and yields a formal theory that is interpretable in classical homotopy theory, but also in other higher topos models. It follows that Univalent Foundations is a fully equivalence-invariant foundation for higher-categorical mathematics, as intended by Voevodsky.

On η-Local Functions in Ideal Topological Spaces

This study introduces and investigates a new local function called \( \eta \)-local function in ideal topological space \( (X, \tau, I) \) by using the notion of \( \eta \)-open sets in topological space \( (X, \tau) \). The operator \( (\cdot)_\eta^* : \mathcal{P}(X) \rightarrow \mathcal{P}(X) \) is defined as \((\cdot)_\eta^* (A) = A_\eta^* = \big\{ x \in X \mid A \cap U \notin I \ \text{for every} \ U \in \eta\text{-}O(x) \big\}\) for each \( A \subseteq X \), where \( \eta \)-\( O(x) \) is the set of all \( \eta \)-open subsets of \( X \) containing \( x \). This study establishes some properties of \( A_\eta^* \) including its relationships to the local function and local function \( \Gamma^* \) in ideal topological space \( (X, \tau, I) \). This study also introduces a new type of closure called the \( \eta \)-local closure in ideal topological space \( (X, \tau, I) \) which is denoted by \( Cl_\eta^*(A) \) for each \( A \subseteq X \). Furthermore, this study establishes some properties of the \( \eta \)-local closure.

Low-dimensional controllability of brain networks.

Identifying the driver nodes of a network has crucial implications in biological systems from unveiling causal interactions to informing effective intervention strategies. Despite recent advances in network control theory, results remain inaccurate as the number of drivers becomes too small compared to the network size, thus limiting the concrete usability in many real-life applications. To overcome this issue, we introduced a framework that integrates principles from spectral graph theory and output controllability to project the network state into a smaller topological space formed by the Laplacian network structure. Through extensive simulations on synthetic and real networks, we showed that a relatively low number of projected components can significantly improve the control accuracy. By introducing a new low-dimensional controllability metric we experimentally validated our method on N = 6134 human connectomes obtained from the UK-biobank cohort. Results revealed previously unappreciated influential brain regions, enabled to draw directed maps between differently specialized cerebral systems, and yielded new insights into hemispheric lateralization. Taken together, our results offered a theoretically grounded solution to deal with network controllability and provided insights into the causal interactions of the human brain.

On The Topological Space of Some n- Refined Neutrosophic Real Intervals and Its Open Sets For 𝟒≤𝒏≤𝟓

This paper is dedicated to studying for the first time the building of a topological space based on the intervals defined over 4-refined neutrosophic real numbers and 5-refined neutrosophic real numbers, where we define a special partial order relation on these rings, and we use it to study the structure of the corresponding intervals generated from this relation. Also, we characterize the formula of open sets through these two topological spaces with some illustrated examples.

A Note on Two-Fold Neutrosophic and Fuzzy Topological Space Based on Real Numbers

The objective of this paper is to introduce for the first time the concept of two-fold neutrosophic and fuzzy topological space defined over real numbers, where we combine the two-fold neutrosophic sets with real numbers to get a novel topological space based on them. Also, we present many of its elementary properties and special subsets such as two-fold neutrosophic open sets, two-fold neutrosophic closed sets, and two-fold neutrosophic closure. Many examples and theorems will be provided to clarify the validity of our approach.

On Some Topological Spaces Based On Symbolic n-Plithogenic Intervals

In this paper, we present the topological space of intervals based symbolic m-plithogenic real numbers of orders between 2 and 5, where we clarify how m-plithogenic real intervals can be expressed according to the symbolic plithogenic partial order relation, and we use these intervals to build a topological space. On the other hand, many illustrated and related examples on open and closed sets will be provided to explain the validity of our approach.

On The Topological Spaces of Neutrosophic Real Intervals

In this paper, we present the topological space of intervals based neutrosophic real numbers , where we clarify how neutrosophic real intervals can be expressed according to the neutrosophic partial order relation, and we use these intervals to build a topological space. On the other hand, we use a similar argument to build a topological space over the intervals of refined neutrosophic numbers, with many illustrated and related examples on open and closed sets.

Robust Diabetic Retinopathy Detection and Grading using Neutrosophic Topological Vector Space on Fundus Imaging

Diabetic retinopathy (DR) is an eye disorder triggered by diabetes that might result in loss of sight. Earlier diagnosis of DR is critical since it might cause loss of sight. Manual diagnoses of DR severity by ophthalmologists are time-consuming and challenging. As a result, there has been considerable attention on designing an automatic technique for DR detection using fundus photographs. In medical science, prognosis and diagnosis are the most challenging tasks due to the presence of fuzziness in medical images and the restricted subjectivity of the experts. Neutrosophic Set (NS) in medical image analysis provides an understanding of the NS concepts, together with knowledge of how to collect, handle, interpret, and analyze clinical images using NS techniques. The neutrosophic set (NS), which is a generality of fuzzy set, provides the overcoming prospect of the restriction of fuzzy-based models for the analysis of medical images. This manuscript develops a Robust Diabetic Retinopathy Detection and Grading using Neutrosophic Topological Vector Space (DRDG-NSTVS) technique on fundus images. The DRDG-NSTVS technique begins with Median Filter (MF) noise removal to optimize the clarity of fundus photographs by successfully eliminating noises. Later, the InceptionV3 is used to perform feature extraction for identifying complicated features and patterns related to DR. The parameter tuning is performed by the moth flame optimization (MFO) technique to ensure superior performance of the model. The final diagnoses and classification of DR are accomplished utilizing the NSTVS classifiers that easily perform the uncertainties inherent in medicinal statistics. The simulation was conducted on a benchmark dataset to examine the proposed model performance. This combined method gives a greatly reliable and accurate solution for the earlier diagnosis and detection of DR

Continuous functions on primal topological spaces induced by group actions

<p>If $ G $ is a group acting on a set $ X $, then for any $ a\in G $, the restriction $ \phi_a:X\to X $ of the action to $ a $ induces a topology $ \tau_a $ for $ X $, called the primal topology induced by $ \phi_a $. First, we obtain a characterization of the normal subgroups in terms of the primal topologies. Later, we prove that some commutative relations among elements on the group $ G $ determine the continuity of maps among different primal spaces $ (X, \tau_{ \phi_x}) $. In particular, we prove the continuity of some maps when $ a, b, q\in G $ satisfy a quantum type relation, $ ba = qab $, as is in the quaternion and Heisenberg groups.</p>

Cubic Spherical Linguistic Neutrosophic Topological Space

In this article, we introduce and establish a novel concept called ’cubic spherical linguistic neutrosophic topological spaces’ by employing cubic spherical linguistic neutrosophic sets and topological frameworks. Various foundational definitions, theorems, and properties are provided along with illustrative examples.

SoftWeak θ-Continuity and Preservations of Soft Hyperconnectedness and Soft Near Compactness

In this paper, we introduce a new weak form of soft continuity called soft weak θ-continuity in soft topological spaces and investigate the relationships between soft weak θ-continuity and θ-continuity (resp. soft weak continuity and soft δ-continuity). We obtain several characterizations of soft weak θ-continuity. Also, we give sufficient conditions for the equivalence between soft weak θ-continuity and soft θ-continuity (resp. soft δ-continuity). Moreover, we investigate the link between soft weak θ-continuity and weak θ-continuity in classical topology. Furthermore, via soft weak θ-continuity, we obtain preservation theorems of soft hyperconnectedness and soft near compactness. Finally, we obtain soft restriction, soft product, and soft graph theorems of soft weak θ-continuity.

Significant Features with M-Polar Neutrosophic Topological Spaces and Grey Wolf Optimization Algorithm for Bankruptcy Prediction Model

Significant Features with M-Polar Neutrosophic Topological Spaces and Grey Wolf Optimization Algorithm for Bankruptcy Prediction Model

On some properties of Alexandroff space

The generalized definition of topology is based on the properties of standard Euclidean topology. The goal of this paper is to study spaces that have topologies, which satisfies the stronger condition namely arbitrary intersection of open sets are open. The topological space with this strong property is known as Alexandroff space. With this restriction we lose some important spaces such as Euclidean spaces, but the specialized spaces in turn display interesting properties that are not necessary for a standard Euclidean topology.

SOME KINDS OF SEPARATION AXIOMS ON APPROXIMATION SPACES

This paper investigates topological features of rough sets and deals with approximation spaces through topological approaches.The traditional separation axioms for topological spaces are extended to approximation spaces. We investigate the connections between the separation axioms for approximation spaces and topological spaces, as well as the approximation spaces they imply.

STATISTICAL CONVERGENCE IN TOPOLOGICAL SPACE CONTROLLED BY MODULUS FUNCTION

The notion of \(f\)-statistical convergence in topological space, which is actually a statistical convergence's generalization under the influence of unbounded modulus function is presented and explored in this paper. This provides as an intermediate between statistical and typical convergence. We also present many counterexamples to highlight the distinctions among several related topological features. Lastly, this paper is concerned with the notions of \(s^{f}\)-limit point and \(s^{f}\)-cluster point for a unbounded modulus function \(f\).

Coupling Multi‐Space Topologies in 2D Ferromagnetic Lattice

AbstractTopology can manifest topological magnetism (e.g., skyrmion and bimeron) in real space and quantum anomalous Hall (QAH) state in momentum space, which has changed the modern conceptions of matter phase. While the topologies in different spaces are widely studied separately, their coexistence and coupling in a single phase are seldom explored. Here, a novel phenomenon is reported that arises from the interaction of topological magnetism and band topology, the multi‐space topology, in 2D ferromagnetic lattice. Based on continuum theory and tight‐binding model, it is revealed that the interconnection between skyrmion/bimeron and QAH state generates distinctive localized chiral bound states (CBSs). With moderating topological magnetism through a magnetic field, the multi‐space topologies accompanied with different CBSs can be reversed, facilitating the coupling of multi‐space topologies. By performing first‐principles and atomic spin model simulations, such multi‐space topologies and their coupling in monolayer Cr2NSb are further demonstrated. These results represent an important step toward the development of multi‐space topological phenomena in 2D lattice.

On Single-Valued Neutrosophic Soft Filter Convergence

The concept of single-valued neutrosophic soft filters plays a crucial role in the study of topological spaces. Extensive research has led to several generalizations of these filters, highlighting their significance in neutrosophic theory. This paper introduces a novel approach to single-valued neutrosophic soft filters, along with the idea of single-valued neutrosophic soft quasi-coincident neighborhood spaces, which are characterized by a unique interaction between the filters and quasi-coincident neighborhood structures. Additionally, we explore advanced neutrosophic theories, focusing on the properties and convergence of single-valued neutrosophic soft filters in soft topological spaces. Finally, we demonstrate the existence of product fuzzy soft filters.

Nash equilibrium and minimax theorems via variational tools of convex analysis

In this paper, we first provide a simple variational proof of the existence of Nash equilibrium in Hilbert spaces by using optimality conditions in convex minimization and Schauder's fixed-point theorem. Then applications of convex analysis and generalized differentiation are given to the existence of Nash equilibrium and extended versions of von Neumann's minimax theorem in locally convex topological vector spaces. Our analysis in this part combines generalized differential tools of convex analysis with elements of fixed point theory revolving around Kakutani's fixed-point theorem and related issues.

Exact Solutions to the Oberbeck–Boussinesq Equations for Describing Three-Dimensional Flows of Micropolar Liquids

The article proposes several classes of exact solutions to the Oberbeck–Boussinesq equations to describe convective flows of micropolar fluids. The possibility of using families of exact solutions for convective flows of classical incompressible fluids to micropolar incompressible fluids is discussed. It is shown that the three-dimensional Oberbeck–Boussinesq equation for describing steady and unsteady flows of micropolar fluids satisfies the class of Lin–Sidorov–Aristov exact solutions. The Lin–Sidorov–Aristov ansatz is characterized by a velocity field with a linear dependence on two spatial coordinates. These coordinates are called horizontal or longitudinal. The coefficients of the linear forms of the velocity field depend on the third coordinate (vertical or transverse) and time. The pressure field and the temperature field are described using quadratic forms. Generalizations of the Ostroumov–Birikh class are considered a special case of the Lin–Sidorov–Aristov family for describing unidirectional flows and homogeneous shear flows. An overdetermined system of Oberbeck–Boussinesq equations is investigated for describing non-homogeneous shear flows of non-trivial complex topology in 3D metric space. A compatibility condition is obtained in the Lin–Sidorov–Aristov class. Finally, a class of exact solutions with a vector velocity field that is nonlinear in part of the coordinates is presented in our analysis; such partially invariant solutions correspond to theoretical findings regarding symmetric/asymmetric properties of flow fields in solutions topology in a part of the existence appropriate for symmetry for the obtained invariant solutions.