Topological Spaces Research Articles

The Collection of Various βws – Closed Sets in Topological Spaces

This research presents an innovative class of Beta weakly semi-closed sets namely, The collection of various βws – closed sets in Topological spaces. Throughout this paper, βws-Semi closure sets, βws-Interior and βws – Neighbourhood sets were examined to get the fundamental facts in the Beta weakly semi-closed sets. In this paper, the notion of countable βws-semi- closed sets and βws-Interior in TS were explored and βws – Neighbourhood (βws-nbd) in TS were also studied to get results. Here, many characterizations were obtained along with some of their feature results.

S STAR p STAR Continuous Maps in Topological Spaces

Goal of this present study seeks to discover and to study the different kind of continuous map and irresolute map is known as semi star pre star continuous map (briefly s*p*continuous) and semi star pre star irresolute map (briefly s*p* irresolute) in topological spaces. As well as discuss a few fundamental characteristics of this continuous map. Further investigate the relationship between the newly defined map and the existing continuous map and irresolute map with suitable examples.

Importance sampling for option pricing with feedforward neural networks

AbstractWe study the problem of reducing the variance of Monte Carlo estimators through performing suitable changes of the sampling measure computed by feedforward neural networks. To this end, building on the concept of vector stochastic integration, we characterise the Cameron–Martin spaces of a large class of Gaussian measures induced by vector-valued continuous local martingales with deterministic covariation. We prove that feedforward neural networks enjoy, up to an isometry, the universal approximation property in these topological spaces. We then prove that sampling measures generated by feedforward neural networks can approximate the optimal sampling measure arbitrarily well. We conclude with a comprehensive numerical study pricing path-dependent European options for asset price models that incorporate factors such as changing business activity, knock-out barriers, dynamic correlations and high-dimensional baskets.

Some Separation Axioms via Soft Somewhat Open Sets

It is commonly known that some topological spaces include structures that may be used to expand abstract notions. somewhat open sets and soft sets is such sort of structures. We obtain several properties and symmetry of the soft somewhat-R0 spaces and soft somewhat-R1 spaces obtained. Furthermore, we present new theorems and results and investigated relation between this concepts and the other structures.

The Role of Embodiment in Learning Representations: Discovering Space Topology via Sensory-Motor Prediction

This paper explores the crucial role of embodiment in learning representations for space topology in robotics. Embodiment, the ability of an agent to interact with its environment and receive sensory feedback, is fundamental to developing accurate and efficient representations. In this work, we investigate this by applying an action-conditional prediction algorithm to data collected from a simulated environment, aiming to learn the topology of the environment through sequences of random interactions. Using a simple mobile-robot-like scenario, by leveraging sensory-motor interactions we demonstrate how the agent can discover the topology of its environment. Our results demonstrate the importance of embodiment in the development of representations and potential applicability in robotic tasks, and a simple but effective method of integrating actions into a learning loop. We suggest that building abstract representations through the use of action-conditional prediction is a step towards unification of the representations used in robotics.

On Some Topological Properties of Ch(X) and of the Dual Operator

The hypo-topology on the algebra C(X) of real-valued continuous functions defined on a Tychonoff space 2X f ∈ C(X) with its hypograph, hypof = {(x,t) ∈ X x R:f(x) ≥ t}. This topology is very useful in the calculus of variations and in optimization theory (e.g. Maximization problems). We denote C(X) with the hypo-topology by Ch(X). Our study deals with fundamental properties of these function spaces, and with the linear operators on them and as well as the characterization of the topological properties of Ch(X) in terms of topological properties of the base space X. We are studying the linear operator between the functional algebras Ch(X) and the Ch(Y). We are primarily concerned with the continuity of the evaluation functional, the general evaluation and the continuity of the characters of Ch(X) before the investigation of the properties of the dual operator f* of a continuous function f:X→Y. This operator is defined by f*:Ch (Y)→Ch (X), where f*(g)=gof for all g ∈ C(Y). The continuity of f* enables us to characterize the continuity of an algebra homomorphism of the type φ:Ch (Y)→Ch (X) for a realcompact space Y. For such a space, we present a type of Riesz theorm with states that an algebra homomorphism φ: Ch(Y)→Ch(X) is continuous if and only if there exists a unique hypo-function f: X→Y such that φ = f*. There after, we give the equivalence between the properties of f and those of f*. The study of the continuous linear functional on Ch(X) helps us to compute the topological dual space of this algebra. We show here that this dual space is useful only when the set of isolated points of X is dense.

On Covering Properties in Intuitionistic Fuzzy Topological Spaces: A survey

We show here some results on covering properties in intuitionistic fuzzy topological spaces. In 1983, K.T. Atanassov proposed a generalization of the notion of fuzzy set: the concept of intuitionistic fuzzy set. D. Çoker constructed the fundamental theory on intuitionistic fuzzy topological spaces, and D. Çoker and other mathematicians studied compactness, connectedness, continuity, separation, convergence and paracompactness in intuitionistic fuzzy topological spaces. Finally, G.-J Wang and Y.Y. He showed that every intuitionistic fuzzy set may be regarded as an L-fuzzy set for some appropriate lattice L. Nevertheless, the results obtained by above authors are not redundant with other for ordinary fuzzy sense.

Computability of Generalized Graphs

We investigate conditions under which a semicomputable set is computable. In particular, we study topological pairs (A,B) which have a computable type, which means that in any computable topological space, a semicomputable set S is computable if there exists a semicomputable set T such that (S,T) is homeomorphic to (A,B). It is known that (G,E) has a computable type if G is a topological graph and E is the set of all its endpoints. Furthermore, the same holds if G is a so-called chainable graph. We generalize the notion of a chainable graph and prove that the same result holds for a larger class of spaces.

Some Results on Neutrosophic I-Statistically Convergence

This paper introduces the neutrosophic -statistical convergent difference sequence spaces defined through a modulus function. Additionally, we establish new topological spaces and examine various topological properties within these neutrosophic -statistical convergent difference sequence spaces.

Regularity of the minmax value and equilibria in multiplayer Blackwell games

AbstractA real-valued function φ that is defined over all Borel sets of a topological space is regular if for every Borel set W, φ(W) is the supremum of φ(C), over all closed sets C that are contained in W, and the infimum of φ(O), over all open sets O that contain W.We study Blackwell games with finitely many players. We show that when each player has a countable set of actions and the objective of a certain player is represented by a Borel winning set, that player’s minmax value is regular.We then use the regularity of the minmax value to establish the existence of ε-equilibria in two distinct classes of Blackwell games. One is the class of n-player Blackwell games where each player has a finite action space and an analytic winning set, and the sum of the minmax values over the players exceeds n − 1. The other class is that of Blackwell games with bounded upper semi-analytic payoff functions, history-independent finite action spaces, and history-independent minmax values. For the latter class, we obtain a characterization of the set of equilibrium payoffs.

The structure and dynamics of the auto parts industry: Product space and complexity perspectives

Economic structures are strong determinants of diversification, growth, innovation, inequality, and many other socio-economic outcomes; however, structural change's effect on specific industries remains unexplored. This study investigates Japan's automotive components industry, analyzing its productive structure (the auto parts product space) and structural change process. We develop exploratory and econometric analyses that unify the measures of product relatedness and complexity. The firm-level data analysis shows that specific structural properties in international and domestic economies also exist in the automobile industry. This fractal structure implies that common principles and regularities underlie the formation of productive structures, notwithstanding the economies’ size, maturity, and industrial scope differences. The exploratory and econometric analyses indicate that the events of a product's appearance are not random but are significantly contingent on the network topology of the product space. Furthermore, network topology significantly impacts the development of complex products with sophisticated capabilities.

Relatively functionally countable subsets of products

A subset A of a topological space X is called relatively functionally countable (RFC) in X, if for each continuous function f:X→R the set f[A] is countable. We prove that all RFC subsets of a product ∏n∈ωXn are countable, assuming that spaces Xn are Tychonoff and all RFC subsets of every Xn are countable. In particular, in a metrizable space every RFC subset is countable.The main tool in the proof is the following result: for every Tychonoff space X and any countable set Q⊆X there is a continuous function f:Xω→R2 such that the restriction of f to Qω is injective.

New ​Results on Difference Paracompactness in Topological Spaces

An interesting area of research in topology is $D-$paracompact spaces. It is a significant type of topological spaces that retain compactness while benefiting from paracompactness, which it is considerd as generalisation of compact spaces. The concept of $D-$paracompactness was introduced and its basic characteristics were examined by the author in $\cite{Q17}$. In this research, we introduce and improve this concept further by using a special type of covering and the difference sets(called as $D-$sets), which contains new and impactful properties. As a result, we obtained several new properties and results. We discuss the concept, characteristics, and theorems that related of $D-$paracompact space. We also studied different characterizations of $D-$paracompact spaces and discussed how they relate to other topological characteristics. We also give numerous instances of $D-$paracompact spaces along with highlighting their applicability in different topological spaces.

A new Outlook on Omega Closed Functions in Bitopological Spaces and Related Aspects

Many studies have employed a variety of techniques to further investigate topological space, particularly the notion of bitopological spaces, due to the significance of topological space in data processing as well as certain implementations. Numerous extended topological structures have been laid out subsequently. Of those abstractions, functions in topology was one of which was most noteworthy. In order to assist in this trend, we focused our research on the idea of open and closed sets, which is one of the strongest techniques available to present scientists for the study of computer graphics and digital topology.New functions, pairwise ω−closed functions, which are strictly weaker than pairwise closed functions, will be introduced in this study. By applying the P˝−space definition, whose is a P−space modification. Additionally, we establish different projection and product theories pertaining to pairwise Lindelöf of and pairwise paracompact spaces utilizing P˝−spaces. We analyze images and inverse images that have been chosen topological attributes for every one of these functions. In the final analysis, we explore several counterexamples that correspond to the offered definitions and theorems.

A Generalization for Somewhere Dense Sets with Some Applications

In this paper, we present a new generalization for somewhere dense of a topological space (Z, τ ), namely ωSWD-open subsets. We introduce the concept of this family and discuss some of their properties with the help of illustrative example. Moreover, we will show if the space (Z, τ ) is anti-locally countable and τ is finer than the cocountable topology then the class of ωSWD-open and somewhere dense subsets of (Z, τ ) will be equivalent. Moreover, we present more properties for the class of somewhere dense subsets of (Z, τ ), the most important of which is a generalization for a theorem in [1]. Furthermore, we finish this work by shedding light on one type of covering properties where we study the notion of almost ωSWD-compact spaces with some of their properties.

Pythagorean Fuzzy Soft Somewhat Continuous Functions

In this work, we introduce the concept of Pythagorean fuzzy soft somewhat open sets utilizing the Pythagorean fuzzy soft interior operator, extending its application to Pythagorean fuzzy soft topological spaces. This study aims to enhance decision-making processes in future-assisted economies by addressing the limitations of existing fuzzy set theories. We investigate the distinctive properties of Pythagorean fuzzy soft somewhat open sets as a subclass of Pythagorean fuzzy soft somewhere dense sets. Additionally, we explore Pythagorean fuzzy soft somewhat metamorphism’s within the context of Pythagorean fuzzy soft somewhat continuous functions, offering new insights into their topological invariant. Through detailed analysis and examples, we demonstrate the applicability of these concepts in various scientific and engineering problems. This work provides a comprehensive framework for understanding and utilizing Pythagorean fuzzy soft sets in complex decision-making scenarios. Finally, we compare various relationships across some generalizations of Pythagorean fuzzy soft continuous functions.

Modifications to Mixed θ(ν1, ν2)-open Sets inGeneralized Topological Spaces

A. Cs´asz´ar and Makai Jr. [5] introduced the concepts of the mixed operation γθ(ν1,ν2) and mixed θ(ν1, ν2)-open sets in generalized topological spaces. In this paper, we extend this framework by introducing the concepts of mixed operation γ˜θ(ν1,ν2) and mixed ˜θ(ν1, ν2)-open sets (briefly, ˜θ(ν1, ν2)- open sets) and investigate their fundamental properties in generalized topological spaces. We explore the relationships among γ˜θ(ν1,ν2), γθ(ν1,ν2), and γθ(ν), as well as the relationships among ˜θ(ν1, ν2)-open sets, θ(ν1, ν2)-open sets, and μ-open sets. Additionally, we introduce the notion of G(ν1, ν2)-regularity in generalized topological spaces. Finally, we provide characterizations of ˜θ(ν1, ν2)-open sets using mixed G(ν1, ν)-regular concept

On a New Operator Based on a Primal and its Associated Topology

This paper aims to introduce and study two new operators $(.)^{\diamond}_{\omega}$ and $cl^{\diamond}_{\omega}(\cdot)$ by utilizing the notion of primal defined by S. Acharjee et al. Also, we investigate some fundamental properties of them. In addition, we showed that the operator $cl^{\diamond}_{\omega}(.)$ satisfied the Kuratowski closure axioms. Therefore, we obtain a new topology denoted by $\tau^{\diamond}_{\omega},$ which is finer than the original one. Moreover, the topology $\tau^{\diamond}_{\omega}$ obtained via the operator $cl^{\diamond}_{\omega}(\cdot)$ is finer than $\tau_{\omega}$, where $\tau_{\omega}$ is the family of all $\omega$-open subsets of a primal topological space $(X,\tau,\mathcal{P}).$ Furthermore, we not only examine the fundamental properties of this class of sets but also provide some counterexamples.

Some New Types of Fuzzy Closed Sets, SeparationAxioms, and Compactness via Double Fuzzy Topologies

In this article, we first defined a stronger form of (r,s)-generalized fuzzy semi-closed sets (briefly, (r,s)-gfsc sets) called (r,s)-g*fsc sets and investigated some of its features. Moreover, we showed that (r,s)-fsc set → (r,s)-g*fsc set → (r,s)-gfsc set, but the converse may not be true. In addition, we explored novel types of fuzzy generalized mappings between double fuzzy topological spaces (U, τ, τ*) and (V, η, η*), and the relationships between these classes of mappings were examined with the help of some illustrative examples. Thereafter, we introduced novel types of higher separation axioms called (r,s)-GFS-regular and (r,s)-GFS-normal spaces with the help of (r,s)-gfsc sets and discussed some topological properties of them. Finally, some novel types of compactness via (r,s)-gfso sets were defined and the relationships between them were introduced.

Upper and Lower Rarely m-I-Continuous Multifunctions

In 1979, Popa [24] first introduced rarely continuous functions. In this paper, we introduce upper and lower rarely m-continuous multifunctions. Moreover, we extend this concept to a multifunction F : (X, τ, I) → (Y, σ), where (X, τ, I) is an ideal topological space.