Topological Spaces Research Articles

The Degree-Topology of CertainTypes of Graphs and PT0-Space

This work aimed to study the degree topology of certain types of graphs namely, the wheel, helm, gear, pan, tadpole, and windmill graphs, respectively. In addition, a new separation axiom known as -space that joins the topological space and paths in the graph is introduced. Also, the relation between -space and the degree-topology of the connected and disconnected graph was stydied. Moreover, many examples were studied in new separation space.

Properties of generalized polyhedral convex multifunctions

This paper presents a study of generalized polyhedral convexity under basic operations on multifunctions. We address the preservation of generalized polyhedral convexity under sums and compositions of multifunctions, the domains and ranges of generalized polyhedral convex multifunctions, and the direct and inverse images of sets under such mappings. Then we explore the class of optimal value functions defined by a generalized polyhedral convex objective function and a generalized polyhedral convex constrained mapping. The new results provide a framework for representing the relative interior of the graph of a generalized polyhedral convex multifunction in terms of the relative interiors of its domain and mapping values in locally convex topological vector spaces. Among the new results in this paper is a significant extension of a result by Bonnans and Shapiro on the domain of generalized polyhedral convex multifunctions from Banach spaces to locally convex topological vector spaces.

A Study of Bases, Continuity and Homeomorphism in Hausdorff Topology

This paper surveys the essential concepts of bases, continuity, and homeomorphism within the context of Hausdorff topology. We examine the characterization of Hausdorff spaces through various types of bases, including minimal generative bases, which serve to illuminate the intrinsic properties of these spaces. The interplay between bases and morphisms is explored, emphasizing how morphisms facilitate continuity of mappings between Hausdorff spaces and their quotient images. We investigate the implications of these relationships for determining homeomorphisms, establishing conditions under which two Hausdorff spaces can be deemed equivalent. Key results include the construction of quotient images using equivalence relations, as well as criteria for continuity and homeomorphism that are uniquely suited to Hausdorff spaces. The findings on bases also demonstrate that a topological space X is Hausdorff if and only if there exists a minimal generative base BM for X such that for any Subfamily B′M of BM that is also a base for X, every element of BM can be expressed as a union of elements from B′M . This result provides an important tool for studying the properties of Hausdorff spaces and their relationships with other topological spaces.

Fuzzy Topological Spaces Generated by Fuzzy Digraphs

In this paper, we defined various fuzzy topological spaces on fuzzy digraph and study interrelation between them. Using adjacency relation on fuzzy vertices and fuzzy edges of fuzzy digraph we defined, namely two types of fuzzy topologies, left(right) fuzzy vertex topology and left(right) fuzzy edge topology on fuzzy digraph, respectively. We have obtained results related to fuzzy topology on fuzzy subdigraph [Formula: see text] of fuzzy digraph [Formula: see text]. Separation axioms [Formula: see text], [Formula: see text] and [Formula: see text] for these fuzzy topological spaces are studied. Some characterization results for fuzzy topological spaces related to isomorphic fuzzy digraphs are given. Further, we have shown that two fuzzy digraphs are isomorphic if and only if their corresponding fuzzy topologies are homeomorphic, this relation on set of fuzzy digraphs forms an equivalence relation. We have also defined and studied fuzzy bitopological spaces on fuzzy digraphs.

More on micro semi – pre-operators in micro topological spaces

The basic objective of this paper is to introduce and investigate the properties of micro semi pre border, micro semi pre kernel and micro semi pre derived set and obtain relation between some of the existing sets.

Topology for discrete mathematics

A series of results is presented on the problem of the existence and the uniqueness of extensions of continuous mappings on topological spaces.

Soft Weakly Quasi-Continuous Functions Between Soft Topological Spaces

As an extension of quasi-continuity in general topology, we define soft quasi-continuity. We show that this notion is equivalent to the known notion of soft semi-continuity. Next, we define soft weak quasi-continuity. With the help of examples, we prove that soft weak quasi-continuity is strictly weaker than both soft semi-continuity and soft weak continuity. We introduce many characterizations of soft weak quasi-continuity. Moreover, we study the relationship between soft quasi-continuity and weak quasi-continuity with their analogous notions in general topology. Furthermore, we show that soft regularity of the co-domain of a soft function is a sufficient condition for equivalence between soft semi-continuity and soft weakly quasi-continuity. Furthermore, we provide several results of soft composition, restrictions, preservation, and soft graph theorems in terms of soft weak quasi-continuity.

On the importance of assessing topological convergence in Bayesian phylogenetic inference.

Modern phylogenetics research is often performed within a Bayesian framework, using sampling algorithms such as Markov chain Monte Carlo (MCMC) to approximate the posterior distribution. These algorithms require careful evaluation of the quality of the generated samples. Within the field of phylogenetics, one frequently adopted diagnostic approach is to evaluate the effective sample size and to investigate trace graphs of the sampled parameters. A major limitation of these approaches is that they are developed for continuous parameters and therefore incompatible with a crucial parameter in these inferences: the tree topology. Several recent advancements have aimed at extending these diagnostics to topological space. In this reflection paper, we present two case studies-one on Ebola virus and one on HIV-illustrating how these topological diagnostics can contain information not found in standard diagnostics, and how decisions regarding which of these diagnostics to compute can impact inferences regarding MCMC convergence and mixing. Our results show the importance of running multiple replicate analyses and of carefully assessing topological convergence using the output of these replicate analyses. To this end, we illustrate different ways of assessing and visualizing the topological convergence of these replicates. Given the major importance of detecting convergence and mixing issues in Bayesian phylogenetic analyses, the lack of a unified approach to this problem warrants further action, especially now that additional tools are becoming available to researchers.

The Standardized Spectroscopic Mixture Model

The standardized spectral mixture model combines the specificity of a physically based representation of a spectrally mixed pixel with the generality and portability of a spectral index. Earlier studies have used spectrally and geographically diverse collections of broadband and spectroscopic imagery to show that the reflectance of the majority of ice-free landscapes on Earth can be represented as linear mixtures of rock and soil substrates (S), photosynthetic vegetation (V) and dark targets (D) composed of shadow and spectrally absorptive/transmissive materials. However, both broadband and spectroscopic studies of the topology of spectral mixing spaces raise questions about the completeness and generality of the Substrate, Vegetation, Dark (SVD) model for imaging spectrometer data. This study uses a spectrally diverse collection of 40 granules from the EMIT imaging spectrometer to verify the generality and stability of the spectroscopic SVD model and characterize the SVD topology and plane of substrates to assess linearity of spectral mixing. New endmembers for soil and non-photosynthetic vegetation (NPV; N) allow the planar SVD model to be extended to a tetrahedral SVDN model to better accommodate the 3D topology of the mixing space. The SVDN model achieves smaller misfit than the SVD, but does so at the expense of implausible fractions beyond [0, 1]. However, a refined spectroscopic SVD model still achieves small (<0.03) RMS misfit, negligible sensitivity to endmember variability and strongly linear scaling over more than an order of magnitude range of spatial resolution.

Numerical study of phase space analysis for the development of subsurface crack detection method using pulsed eddy current

Root-deck cracks, a prominent form of subsurface damage observed in orthotropic steel decks (OSDs), pose significant challenges to maintaining structural integrity. Early detection of these cracks is crucial to prevent further deterioration. This paper introduces novel and robust techniques for processing response signals obtained from the Pulsed Eddy Current Technique (PECT) using phase space analysis. The proposed methods aim to improve sensitivity and noise handling capabilities by employing an appropriate damage index derived from comprehensive phase space analysis, enabling effective detection of subsurface cracks in OSDs. A numerical study demonstrates the effectiveness of the proposed methods in detecting crack length change as small as 1 mm from the bottom of a 12 mm steel plate, with the Change in Phase Space Topology (CPST) identified as a suitable damage index. Furthermore, the newly proposed signal processing techniques allow for a more comprehensive interpretation of response signals for subsurface damage detection, encapsulating the extent of damage within a single value, even in the presence of small crack lengths near the bottom of the steel plate and noise. As a result, the proposed methods overcome the limitations of conventional approaches and offer promising prospects for enhanced subsurface crack detection in OSDs.

On Topologies Induced by Ideals, Primals, Filters and Grills

In this paper, one-to-one correspondences and equivalences between ideals, primals, filters and grills are introduced. It is shown that the local functions and the topological spaces induced by them are the same. From this point of view, the topological properties of one topology can be derived from the topological properties that are valid in the corresponding topology.

ON A WEAK FORM OF SEMI-OPEN FUNCTION BY NEUTROSOPHICATION

In this study, a new perspective in neutrosophic topology is brought to some open set definitions in general topology and previously interpreted in different ways in some other non-standard topological spaces. Then, some types of functions and continuity are introduced using these new open sets. In addition to these, the relations between the types of open sets, which we have brought different perspectives, with the continuity and function types that we reinterpreted in neutrosophic topology, are examined and these relations are clarified by giving examples and counterexamples.

Realizing regular representations of finite groups

Given a regular representation of a finite group G and a positive integer number n, we construct a (finite) topological space X such that its group of homotopy classes of self-homotopy equivalences E(X) and its group of homeomorphisms Aut(X) are isomorphic to G, and the action of G on the n-th homology group Hn(X) is the regular representation. We also discuss other representations.

Causal completion of globally hyperbolic conformally flat spacetimes

In their paper Geroch, Kronheimer, and Penrose, “Ideal points in space-time,” [Proc. R. Soc. London, Ser. A 327, 545–567 (1972)] introduced a way to attach ideal points to a spacetime M, defining the causal completion of M. They established that this is a topological space which is Hausdorff when M is globally hyperbolic. In this paper, we prove that if, in addition, M is simply-connected and conformally flat, its causal completion is a topological manifold with boundary homeomorphic to S × [0, 1] where S is a Cauchy hypersurface of M. We also introduce three remarkable families of globally hyperbolic conformally flat spacetimes and provide a description of their causal completions.

The class C(ω1) and countable net weight

Hart and Kunen in [5] and, independently, Ríos-Herrejón in [14] defined and studied the class C(ω1) of topological spaces X having the property that for every neighborhood assignment {U(y):y∈Y} with Y∈[X]ω1 there is Z∈[Y]ω1 such thatZ⊂⋂{U(z):z∈Z}. It is obvious that spaces of countable net weight, i.e. having a countable network, belong to this class. In this paper we present several independence results concerning the relationships of these two and several other natural classes that are sandwiched between them, thus clarifying some of the main problems that were raised in [14].In particular, we prove that the continuum hypothesis, in fact a weaker combinatorial principle called super stick, implies that every regular space in C(ω1) has countable net weight, answering a question that was raised by Hart and Kunen in both [5] and [6].

Accept/Reject games

Many properties of topological spaces, like compactness, have the form that any cover of the space of a certain type contains a subcover of some other type. We introduce a game theoretic analogue of this selection prototype, the Accept/Reject game, and relate it to classical games and selection principles. Modified versions of this game are shown to be equivalent to standard selection principles games.

Graded pseudo weakly prime spectrum of graded topological modules

In this study, we introduce graded pseudo weakly prime submodules of G-graded R-modules, which are an extension of graded weakly prime ideals over G-graded rings. On the graded spectrum of graded pseudo weakly prime submodules, we investigate the Zariski topology. Different aspects of this topological space are investigated, and they are linked to the algebraic properties of the G-graded R-modules under study.

Graph Contrastive Learning With Adaptive Proximity-Based Graph Augmentation.

Graph neural networks (GNNs) have been successful in a variety of graph-based applications. Recently, it is shown that capturing long-range relationships between nodes helps improve the performance of GNNs. The phenomenon is mostly confirmed in a supervised learning setting. In this article, inspired by contrastive learning (CL), we propose an unsupervised learning pipeline, in which different types of long-range similarity information are injected into the GNN model in an efficient way. We reconstruct the original graph in feature and topology spaces to generate three augmented views. During training, our model alternately picks an augmented view, and maximizes an agreement between the representations of the view and the original graph. Importantly, we identify the issue of diminishing utility of the augmented views as the model gradually learns useful information from the views. Hence, we propose a view update scheme that adaptively adjusts the augmented views, so that the views can continue to provide new information that helps with CL. The updated augmented views and the original graph are jointly used to train a shared GNN encoder by optimizing an efficient channel-level contrastive objective. We conduct extensive experiments on six assortative graphs and three disassortative graphs, which demonstrate the effectiveness of our method.

I-ω^*-Continuous and I-ω^*- Open Functions

In ideal topological space, we define and study a new class of functions known as I-ω^*-continuous functions by using I-ω^*-open sets. This idea is stronger than each of the classes (ω-continuous, ωb-continous, α-I_ω-continuous, pre-I_ω-continuous, β-I_ω-continuous, faintly- ω-continuous, slightly- ω-continuous, 〖 I〗_ω-c-continuous). Furthermore, relationships between this class and other relevant classes of functions are investigated, and some characterizations of this new class of functions are studied. Finally, we introduce the I-ω^*- open function as an extension of the ω- open function through I-ω^*-open sets, and some results are provided.

Study of Topological Spaces that Don’t Rely on Points as their Basis

Traditionally, the concept of points and neighborhood structures has been quite dominant in defining and further analyzing various properties in the study of topological spaces. In pointless topology-or point-free topology, also referred to as locale theory-the key emphasis is shifted from points to lattices of open sets. In addition to gaining further insight into spatial properties, this approach has important implications in fields like theoretical computer science, logic, and categorical theory. It is the purpose of the present work to be mainly interested in issues of the foundation of point-free topology: locales, frames, mutual relationships between them; the ways of interpreting classical topological concepts in the new language and the strong and weak points of such a point-free view. In the process, we focus on some of the significant applications of point-free topology, both within mathematics and beyond. The present paper is an attempt to give a broad overview of pointfree topology, discussing both its theoretical basis and its practical significance.