Topological Spaces Research Articles

Separation axioms on fuzzy neutrosophic bitopological spaces using fuzzy neutrosophic open sets

AbstractIndeterminacy, non-membership and membership, have been studied in a neutrosophic set (NS). Using the notion of NS, neutrosophic topological space (NTS) and neutrosophic bitopological spaces have been developed. The well-known properties (separation axioms) of neutrosophic $${\text{T}}_{i}$$ T i -topological and bitopological spaces $$(i=0, 1, 2)$$ ( i = 0 , 1 , 2 ) are different types of neutrosophic spaces with different characteristics. Here, we define the idea of fuzzy neutrosophic $${\text{T}}_{i}$$ T i -bitopological space $$(i=0, 1, 2)$$ ( i = 0 , 1 , 2 ) through the neutrosophic bitopological space and look into its various characteristics. From neutrosophic $${\text{T}}_{i}$$ T i -bitopological space ($$i=0, 1, 2$$ i = 0 , 1 , 2 ), we demonstrate some intriguing findings with examples about the neutrosophic separation axioms.

Theorem of Alternative and Scalarization of Optimization Problems in Topological Vector Spaces

In this paper, we introduce the definitions of affinelike, preaffinelike, generalized affinelike, and generalized preaffinelike functions by use of "pointed convex cone", and prove that those definitions of generalized affine functions are all different. We discuss optimization problems in topological vector spaces and obtain a theorem of alternative and a scalarization theorem. Our inequalities are given by partial order relations. Our generalized affineness may be used for many discussions in mathematics or applied mathematics wherever the affineness is a condition.

Intuitionistic S4 as a logic of topological spaces

Abstract We design and study various topological semantics for the diamond-free intuitionistic modal logic $\textsf{iS4}$, an intuitionistic analogue of $\textsf{S4}$. Ultimately we prove that ordinary topological spaces can be used as semantics, using the specialization order to interpret intuitionistic implication and the interior for the modality. Some of our soundness and completeness results are mechanised in Coq.

A topological proof of the Riemann–Hurwitz formula

The Riemann–Hurwitz formula is generally given as a result from algebraic geometry that provides a means of constraining branched covers of surfaces via their Euler characteristic. By restricting to the special case of compact Riemann surfaces, we develop an alternative proof of the formula that draws on topology and manifold theory as opposed to more advanced algebraic machinery. We first discuss the foundation in manifold theory, defining Riemann surfaces and providing an example of the complex projective line. We then discuss the local topological structure of holomorphic maps between Riemann surfaces, introducing the notion of a branched cover and of branch points. Next, we discuss triangulations of a topological space and use this to intro- duce the Euler characteristic of Riemann surfaces. Using these definitions, we explicate and prove the Riemann–Hurwitz formula on compact Riemann surfaces. To conclude, we discuss consequences of this formula for adjacent fields such as algebraic topology. We provide visual intuition and examples throughout, drawing primarily on Szameuly’s Galois Groups and Fundamental Groups (2009), as well as Forster’s Lectures on Riemann Surfaces (1981), Guillemin and Pollack’s Differential Topology (1974), and a few other supplementary sources. The main prerequisite for this paper is a background in topology and covering spaces.

Reliable estimation of tree branch lengths using deep neural networks.

A phylogenetic tree represents hypothesized evolutionary history for a set of taxa. Besides the branching patterns (i.e., tree topology), phylogenies contain information about the evolutionary distances (i.e. branch lengths) between all taxa in the tree, which include extant taxa (external nodes) and their last common ancestors (internal nodes). During phylogenetic tree inference, the branch lengths are typically co-estimated along with other phylogenetic parameters during tree topology space exploration. There are well-known regions of the branch length parameter space where accurate estimation of phylogenetic trees is especially difficult. Several novel studies have recently demonstrated that machine learning approaches have the potential to help solve phylogenetic problems with greater accuracy and computational efficiency. In this study, as a proof of concept, we sought to explore the possibility of machine learning models to predict branch lengths. To that end, we designed several deep learning frameworks to estimate branch lengths on fixed tree topologies from multiple sequence alignments or its representations. Our results show that deep learning methods can exhibit superior performance in some difficult regions of branch length parameter space. For example, in contrast to maximum likelihood inference, which is typically used for estimating branch lengths, deep learning methods are more efficient and accurate. In general, we find that our neural networks achieve similar accuracy to a Bayesian approach and are the best-performing methods when inferring long branches that are associated with distantly related taxa. Together, our findings represent a next step toward accurate, fast, and reliable phylogenetic inference with machine learning approaches.

Multiplier rules for Dini-derivatives in a topological vector space

We provide new results of first-order necessary conditions of optimality problem in the form of John's theorem and in the form of Karush–Kuhn–Tucker's theorem. We establish our result in a topological vector space for problems with inequality constraints and in a Banach space for problems with equality and inequality constraints. Our contributions consist in the extension of the results known for the Fréchet and Gateaux-differentiable functions as well as for the Clarke's subdifferential of Lipschitz functions to the more general Dini-differentiable functions. As consequences, we extend the result of B.H. Pourciau in [Modern multiplier rules. Amer Math Monthly. 1980;87(6):433–457, Theorem 6, p. 445] from the convexity to the Dini-pseudoconvexity.

Properties of $s\tilde{\theta}$-open Sets in Generalized Topological Spaces

A study of $\tilde{\theta}$-open sets in generalized topological spaces started in 2011 by Min \cite{wkm}. In this paper, we introduce the concepts of $s\tilde{\theta}$-open sets, $s\tilde{\theta}$-closed sets, $s\tilde{\theta}$-continuous functions and $s\tilde{\theta}$-irresolute functions on generalized topological spaces. We also study some basic propertiesof such sets and functions.

Topological Quantum Field Theories and Homotopy Cobordisms

We construct a category HomCob\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extrm{HomCob}}$$\\end{document} whose objects are homotopically 1-finitely generated topological spaces, and whose morphisms are cofibrant cospans. Given a manifold submanifold pair (M, A), we prove that there exists functors into HomCob\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extrm{HomCob}}$$\\end{document} from the full subgroupoid of the mapping class groupoid MCGMA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{MCG}_{M}^{A}$$\\end{document}, and from the full subgroupoid of the motion groupoid MotMA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Mot}_{M}^{A}$$\\end{document}, whose objects are homotopically 1-finitely generated. We also construct a family of functors ZG:HomCob→Vect\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extsf{Z}}_G:{\ extrm{HomCob}}\\rightarrow {\ extbf{Vect}}$$\\end{document}, one for each finite group G. These generalise topological quantum field theories previously constructed by Yetter, and an untwisted version of Dijkgraaf–Witten. Given a space X, we prove that ZG(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extsf{Z}}_G(X)$$\\end{document} can be expressed as the C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}$$\\end{document}-vector space with basis natural transformation classes of maps from π(X,X0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi (X,X_0)$$\\end{document} to G for some finite representative set of points X0⊂X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_0\\subset X$$\\end{document}, demonstrating that ZG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extsf{Z}}_G$$\\end{document} is explicitly calculable.

Rare Continuity Revisited via δb-open Sets

In this paper we introduce and study the concept of rare δb-continuity in topological spaces as a generalization of rare continuity and weak δb-continuity. We investigate several properties of rarely δb-continuous functions. Rare δb-continuity implied by rare δ-continuity and implies rare b-continuity. We also introduce the concept of I.δb-continuity, which is weaker than δb-continuity and stronger than rare δb-continuity.

A new concept of semistrict quasiconvexity for vector functions

We establish a new concept of semistrict quasiconvexity for vector functions defined on a nonempty convex set in a real linear space X that take values in some real topological linear space Y, partially ordered by a proper solid convex cone C. The so-called semistrict C-quasiconvexity notion recovers the classical concept of semistrict quasiconvexity of scalar functions when Y = R and C = R + . Additionally, analogous to the scalar scenario, if the cone C is closed, a vector function is both semistrictly C-quasiconvex and C-quasiconvex (in the sense of Luc, 1989) if and only if it is explicitly C-quasiconvex (in the sense of Popovici, 2007). Finally, we convey a characterization of semistrictly C-quasiconvex functions by means of scalar semistrictly quasiconvex functions that are compositions of the nonlinear scalarization functions introduced by Gerstewitz (Tammer) in 1983 with the initial vector function. In light of this characterization, the new concept of semistrict C-quasiconvexity seems to be a natural vector counterpart for the scalar concept of semistrict quasiconvexity.

NANO g* sr CLOSED SETS IN NANO TOPOLOGICAL SPACES

The primary focus of this article is to develop, a fresh concept of sets namely Ng*sr closed sets. The goal of this research is to go over the concepts of Ng*sr closed sets and open sets in nano topological spaces. An appropriate illustration is given to study the properties of the above sets. The present relations between some of these sets in nano topological space are provided.

The Machado–Bishop theorem in the uniform topology

The Machado–Bishop theorem for weighted vector-valued functions vanishing at infinity has been extensively studied. In this paper, we give an analogue of Machado’s distance formula for bounded weighted vector-valued functions. A number of applications are given; in particular, some types of the Bishop–Stone–Weierstrass theorem for bounded vector-valued continuous spaces in the uniform topology are discussed; the splitting of C(I×J,X⊗Y) as the closure of C(I,X)⊗C(J,Y) in different senses and the extension of continuous vector-valued functions are studied.

Characterizations of minimal elements of upper support with applications in minimizing DC functions

Abstract In this study, we discuss on the problem of minimizing the differences of two non-positive valued increasing, co-radiant and quasi-concave (ICRQC) functions defined on X X (where X X is a real locally convex topological vector space). For this purpose, we first gave different characterizations of the upper support set’s minimal elements of non-positive co-radiant functions. Then, we presented sufficient and necessary conditions for the global minimizers of the differences of two non-positive ICRQC functions.

Some Operations and Their Closure Properties on Multiset Topological Spaces

In this paper, we introduced some operations on multiset topological spaces to include union, intersection, arithmetic multiplication, Scalar multiplication and Raising to Arithmetic Power. Studies revealed that these operations are closed on the various classes of multiset topological spaces defined.

Universal Finite Functorial Semi-norms

Functorial semi-norms on singular homology measure the “size” of homology classes. A geometrically meaningful example is the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^1$$\\end{document}-semi-norm. However, the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^1$$\\end{document}-semi-norm is not universal in the sense that it does not vanish on as few classes as possible. We show that universal finite functorial semi-norms do exist on singular homology on the category of topological spaces that are homotopy equivalent to finite CW-complexes. Our arguments also apply to more general settings of functorial semi-norms.

Topology-preserved distorted space path planning

Purpose Fast obstacle avoidance path planning is a challenging task for multijoint robots navigating through cluttered workspaces. This paper aims to address this issue by proposing an improved path-planning method based on the distorted space (DS) method, specifically designed for high-dimensional complex environments. Design/methodology/approach The proposed method, termed topology-preserved distorted space (TP-DS) method, mitigates the limitations of the original DS method by preserving space topology through elastic deformation. By applying distinct spring constants, the TP-DS autonomously shrinks obstacles to microscopic areas within the configuration space, maintaining consistent topology. This enhancement extends the application scope of the DS method to handle complex environments effectively. Findings Comparative analysis demonstrates that the proposed TP-DS method outperforms traditional methods regarding planning efficiency. Successful obstacle avoidance tasks in the cluttered workspace validate its applicability on a physical 6-DOF manipulator, highlighting its potential for industrial implementations. Originality/value The novel TP-DS method generates a topology-preserved collision-free space by leveraging elastic deformation and shows significant capability and efficiency in planning obstacle-avoidance paths in complex application scenarios.

Exploration of NαIg-Closed and NαIg-Open Sets in Nano Ideal Topological Spaces with Practical Applications of Nano Ideal Topological spaces

Exploration of NαIg-Closed and NαIg-Open Sets in Nano Ideal Topological Spaces with Practical Applications of Nano Ideal Topological spaces

Compactness and Connectedness in Beta Weakly Semi – Closed Sets in Topological Spaces

Compactness and Connectedness in Beta Weakly Semi – Closed Sets in Topological Spaces

Fixed-Point Theorems for Fuzzy Mappings

Since the 1970s and 1980s, significant contributions have been made by Weiss, Butnariu, Heilpern, Chitra, Subrahmanyam, and others, extending fixed-point theorems to fuzzy mappings and topological spaces. This paper provides two generalizations of two important fixed-point theorems, one provided by Butnariu and the other provided by Chitra. The first generalization ensures that, under certain conditions, an acyclic fuzzy mapping has a fixed point. The second result ensures the existence of a point in the intersection of two or more fuzzy mappings considering contractible finite dimensional ANR spaces, which is a generalization of the statement provided by Chitra.

Generalizations of Rolle’s Theorem

The classical Rolle’s theorem establishes the existence of (at least) one zero of the derivative of a continuous one-variable function on a compact interval in the real line, which attains the same value at the extremes, and it is differentiable in the interior of the interval. In this paper, we generalize the statement in four ways. First, we provide a version for functions whose domain is in a locally convex topological Hausdorff vector space, which can possibly be infinite-dimensional. Then, we deal with the functions defined in a real interval: we consider the case of unbounded intervals, the case of functions endowed with a weak derivative, and, finally, we consider the case of distributions over an open interval in the real line.