Topological Notions Research Articles

On Information Orders on Metric Spaces

Information orders play a central role in the mathematical foundations of Computer Science. Concretely, they are a suitable tool to describe processes in which the information increases successively in each step of the computation. In order to provide numerical quantifications of the amount of information in the aforementioned processes, S.G. Matthews introduced the notions of partial metric and Scott-like topology. The success of partial metrics is given mainly by two facts. On the one hand, they can induce the so-called specialization partial order, which is able to encode the existing order structure in many examples of spaces that arise in a natural way in Computer Science. On the other hand, their associated topology is Scott-like when the partial metric space is complete and, thus, it is able to describe the aforementioned increasing information processes in such a way that the supremum of the sequence always exists and captures the amount of information, measured by the partial metric; it also contains no information other than that which may be derived from the members of the sequence. R. Heckmann showed that the method to induce the partial order associated with a partial metric could be retrieved as a particular case of a celebrated method for generating partial orders through metrics and non-negative real-valued functions. Motivated by this fact, we explore this general method from an information orders theory viewpoint. Specifically, we show that such a method captures the essence of information orders in such a way that the function under consideration is able to quantify the amount of information and, in addition, its measurement can be used to distinguish maximal elements. Moreover, we show that this method for endowing a metric space with a partial order can also be applied to partial metric spaces in order to generate new partial orders different from the specialization one. Furthermore, we show that given a complete metric space and an inf-continuous function, the partially ordered set induced by this general method enjoys rich properties. Concretely, we will show not only its order-completeness but the directed-completeness and, in addition, that the topology induced by the metric is Scott-like. Therefore, such a mathematical structure could be used for developing metric-based tools for modeling increasing information processes in Computer Science. As a particular case of our new results, we retrieve, for a complete partial metric space, the above-explained celebrated fact about the Scott-like character of the associated topology and, in addition, that the induced partial ordered set is directed-complete and not only order-complete.

Wave Steering by Relaying Interface States in a Valley-Hall-Derived Photonic Superlattice

Topological notions in physics have become a powerful perspective that leads to the discoveries of topological interface states (TISs). In this work, we present a scheme to achieve negative refraction by leveraging the properties of TISs in a valley photonic crystal (VPC). Due to the chiral characteristics, one type of the TISs deterministically possesses negative dispersion relation, which can cause an obliquely incident wave to undergo a negative lateral shift. By stacking multiple VPC interfaces, the TIS-induced lateral shifts can relay in the transmitted wave towards the far side of incidence. The resultant outgoing wave appears to have undergone negative refraction. This finding is verified in microwave experiments. Our scheme opens new application scenarios for topological systems in bulk wave manipulations.

Topological stereotomic structure with the use of palm rachis applying material physics to architectural design

The use of ancestral construction techniques in wood has evolved throughout history, but consequently many species have been disappearing after indiscriminate logging; but the use of renewable materials is what is needed in the present and in the future after the global pollution and the ecological footprint of materials such as iron or concrete, so it is necessary to glimpse new building materials from a renewable approach. To this end, this research focuses on the application of physicochemistry and physics of materials to establish the feasibility of using Attalca Butyracea palm rachis as a sustainable material with a novel application in the field of architecture. For this purpose, a methodology of morphological configuration is applied that appropriates the identification of specific physical properties together with notions of geometry and topology applying the concept of topological stereotomic configuration from the operational tools and morphological actions of the Lucas Períes fold, resulting in a proposal for a topological stereotomic structure applied to the architectural design of buildings in the municipality of Pelaya, Colombia.

Homeomorphism and Quotient Mappings in Infrasoft Topological Spaces

In this paper, we contribute to infrasoft topology which is one of the recent generalizations of soft topology. Firstly, we redefine the concept of soft mappings to be convenient for studying the topological concepts and notions in different soft structures. Then, we introduce the concepts of open, closed, and homeomorphism mappings in the content of infrasoft topology. We establish main properties and investigate the transmission of these concepts between infrasoft topology and its parametric infratopologies. Finally, we define a quotient infrasoft topology and infrasoft quotient mappings and study their main properties with the aid of illustrative examples.

Topological correspondence between magnetic space group representations and subdimensions

This paper identifies a general class of electronic configurations that in a magnetic background necessarily induce topological structure. The phases additionally feature a correspondence relating antiferromagnetic and ferro/ferrimagnetic realizations. More importantly, the authors find novel types of magnetic topology upon considering this mechanism in all space groups. These phases are exclusively mapped out in this analysis that takes into account planes and lines in the Brillouin zone, prompting a new notion of subdimensional topology that nonetheless has full 3D physical signatures, such as Fermi arcs.

(2011-6279) Structural topology in a category

Several fuzzy topologies are defined and studied by different authors. In this article, we unify five of the most common fuzzy topologies existing in the literature, as well as the standard topology. This is done by introducing the notion of structural topology on objects in a category and proving that topologies on a set as well as fuzzy topologies on fuzzy sets and fuzzy topologies on fuzzy subsets are all structural topologies. We also introduce the notion of structural continuity and we show that the fuzzy continuity defined in the literature in all the above mentioned cases, as well as the standard continuity are structural.

On continuity in generalized topology

In 2002 Á. Császár introduced the notion of generalized topology, which differs from the notion of topology by the lack of the intersection property. Many kinds of generalized continuity may be considered as a continuity in a generalized topology, for example quasicontinuity, precontinuity, porouscontinuity, qualitative continuity, Denjoy property. In the paper we give full characterization of set of points of generalized continuity for functions f:(X,Γ)→(Y,τ), where (X,Γ) is a resolvable generalized topological space and (Y,τ) is a nondiscrete Moore space. We also present some properties of set of points of path continuity with respect to (Γ,T) for functions f:(X,Γ)→(Y,ϱ), where (X,Γ) is a generalized topological space, T is a topology associated with Γ and (Y,ϱ) is a nondiscrete metric space. Some relevant properties of continuity and path continuity are discussed.

On Murray-von Neumann algebras—I: topological, order-theoretic and analytical aspects

For a countably decomposable finite von Neumann algebra \({\mathscr {R}}\), we show that any choice of a faithful normal tracial state on \({\mathscr {R}}\) engenders the same measure topology on \({\mathscr {R}}\) in the sense of Nelson (J Funct Anal 15:103–116, 1974). Consequently it is justified to speak of ‘the’ measure topology of \({\mathscr {R}}\). Having made this observation, we extend the notion of measure topology to general finite von Neumann algebras and denominate it the \({\mathfrak {m}}\)-topology. We note that the procedure of \({\mathfrak {m}}\)-completion yields Murray-von Neumann algebras in a functorial manner and provides them with an intrinsic description as unital ordered complex topological \(*\)-algebras. This enables the study of abstract Murray-von Neumann algebras avoiding reference to a Hilbert space. Furthermore, it makes apparent the appropriate notion of Murray-von Neumann subalgebras, and the intrinsic nature of the spectrum and point spectrum of elements, independent of their ambient Murray-von Neumann algebra. In this context, we show the well-definedness of the Borel function calculus for normal elements and use it along with approximation techniques in the \({\mathfrak {m}}\)-topology to transfer many standard operator inequalities involving bounded self-adjoint operators to the setting of (unbounded) self-adjoint operators in Murray-von Neumann algebras. On the algebraic side, Murray-von Neumann algebras have been described as the Ore localization of finite von Neumann algebras with respect to their corresponding multiplicative subset of non-zero-divisors. Our discussion reveals that, in addition, there are fundamental topological, order-theoretic and analytical facets to their description.

On the Menger and almost Menger properties in locales

<p>The Menger and the almost Menger properties are extended to locales. Regarding the former, the extension is conservative (meaning that a space is Menger if and only if it is Menger as a locale), and the latter is conservative for sober T<sub>D</sub>-spaces. Non-spatial Menger (and hence almost Menger) locales do exist, so that the extensions genuinely transcend the topological notions. We also consider projectively Menger locales, and show that, as in spaces, a locale is Menger precisely when it is Lindelöf and projectively Menger. Transference of these properties along localic maps (via direct image or pullback) is considered.</p>

Tunable fragile topology in Floquet systems

We extend the notion of fragile topology to periodically-driven systems. We demonstrate driving-induced fragile topology in two different models, namely, the Floquet honeycomb model and the Floquet $\pi$-flux square-lattice model. In both cases, we discover a rich phase diagram that includes Floquet fragile topological phases protected by crystalline rotation or mirror symmetries, Floquet Chern insulators, and trivial atomic phases, with distinct boundary features. Remarkably, the transitions between different phases can be feasibly achieved by simply tuning the driving amplitudes, which is a unique feature of driving-enabled topological phenomena. Moreover, corner-localized fractional charges are identified as a ``smoking-gun'' signal of fragile topology in our systems. Our work paves the way for studying and realizing fragile topology in Floquet systems.

On Unique and Nonunique Fixed Points and Fixed Circles in M v b -Metric Space and Application to Cantilever Beam Problem

We introduce M v b -metric to generalize and improve M v -metric and unify numerous existing distance notions. Further, we define topological notions like open ball, closed ball, convergence of a sequence, Cauchy sequence, and completeness of the space to discuss topology on M v b -metric space and to create an environment for the survival of a unique fixed point. Also, we introduce a notion of a fixed circle and a fixed disc to study the geometry of the set of nonunique fixed points of a discontinuous self-map and establish fixed circle and fixed disc theorems. Further, we verify all these results by illustrative examples to demonstrate the authenticity of the postulates. Towards the end, we solve a fourth order differential equation arising in the bending of an elastic beam.

The Heuristic of Directional Qualitative Semantic: A New Heuristic for Making Decisions about Spinning with Qualitative Reasoning

Multifunctional Robot On Topological Notions (MROTN) is a research program that has as one of its goals to develop qualitative algorithms that make navigation decisions. This article presents new research from MROTN that extends previous results by allowing an agent to carry out qualitative reasoning about direction and spinning. The main result is a new heuristic, the Heuristic of Directional Qualitative Semantic (HDQS), which allows for selecting a spinning action to establish a directional relation between an agent and an object. The HDQS is based on the key idea of encoding directional information into topological relations. The new heuristic is important to the MROTN because it permits the continued development of qualitative navigation methods based on topological notions. We show this by presenting a new version of the Topological Qualitative Architecture of Navigation that uses the HDQS to address situations that require spinning.

Criterion for robustness of global asymptotic stability to external perturbations of linear time-varying systems

ABSTRACT In this paper, we establish a new criterion for robustness of global asymptotic stability of zero solution of LTV system in the presence of external perturbations (disturbances). To prove the result, logarithmic norm will be used under which the stability becomes a topological notion depending on the chosen vector norm in the state-space

Contra-continuous Functions Defined Through Λ1-closed Sets

We introduce some variants of contra-continuity in terms of ΛI -closed sets, namely contra-ΛI - continuous, contra quasi-ΛI -continuous and contra ΛI -irresolute functions. The relationships between these functions are investigated and their respective characterizations are established. Moreover, we study their behavior of several topological notions under the direct and inverse images of these functions.

Clopen objects, connected objects, and normalized topological categories

In this paper we extend the notion of a clopen subset of a topological space to that of a clopen subobject in a topological category. We relate clopen subobjects to the concept of normalized topological category via a theorem which, in turn, shows why certain subsets must be clopen in a topological space. We employ clopen subobjects to extend the topological notion of connectedness to topological categories. Examples of clopen subobjects and connected objects in several topological categories are given.

Interiorist interventions and archival strategies: Contemporary terms for museums and creative practitioners

The contested interior space of the museum and its artifacts are receiving due attention. Museum objectors are altering content, examining unacknowledged assumptions and biases, and developing new ways of seeing and experiencing that will act as new models for critical, creative discourse. Museum objectors range from entire countries (notably Greece, which has been asking the British Museum to return the marble gods of the Parthenon since 1833), to community groups and creative practitioners. A prime example of the latter type of museum objector was Marcel Duchamp. His interiorist interventions of the 1930s and 1940s set the tone for creative practice within the notion of interiority, informed exhibition display and experience, and expanded museum discourse. This paper proposes that Marcel Duchamp extended his creative artistic practice into the field of interior design to subvert critical discourse on authoritarian aesthetics and display. Within this context of critical reflection, the museum as an artistic medium is also fast becoming an accepted methodology of creative practice. To examine this practical, creative work, we first draw on Jacques Derrida’s theoretical discussion about the concept of ‘the archive,’ to acknowledge the language of interiority and to focus on the ‘curated interior’ by juxtaposing the domestic and institutional interior and the notion of topology as a form of creative methodology. As a stepping-off-point, we will explore Marcel Duchamp’s conceptual nuances of the interior, specifically his La Boîte-en-valise, as a masterful blending of the archive (collection), the melding of creative disciplines, and commentary on the museum practices. We will then examine the challenges faced by two American and one Canadian museums’ exhibitions in the 1980s, followed by a review of more recent works highlighting progressive change within these museums. The works by Spring Hurlbut in 2001 and Kent Monkman and Wangechi Mutu in 2019 illustrate evolving discussions about the use of creative practitioners working with museum collections and interiors. As the landscape of the curated interior within the institutional setting (museum) has changed, it is a crucial time to take stock of the different possibilities when addressing the spaces between the memories of the archive: what is remembered, how it was/is remembered? This new methodology of museum as artistic medium is substantially expanding modern critical discourse on the interiority and the archive.

Regularized Generative Adversarial Network

We propose a framework for generating samples from a probability distribution that differs from the probability distribution of the training set. We use an adversarial process that simultaneously trains three networks, a generator and two discriminators. We refer to this new model as regularized generative adversarial network (RegGAN). We evaluate RegGAN on a synthetic dataset composed of gray scale images and we further show that it can be used to learn some pre-specified notions in topology (basic topology properties). The work is motivated by practical problems encountered while using generative methods in the art world.

On contra Λ_I^s -continuous functions and their applications

In this work, we introduce and study the classes of contra ΛIs-continuous, contra quasi-ΛIs-continuous and contra ΛIs-irresolute functions in a topological space endowed with an ideal. We investigate the relationships among these functions and their respective characterizations. Also, we analyze the behavior of certain topological notions under direct and inverse images of these new classes of functions.

Soft $ \alpha $-separation axioms and $ \alpha $-fixed soft points

<abstract> Soft set theory is a theme of interest for many authors working in various areas because of its rich potential for applications in many directions. it received the attention of the topologists who always seeking to generalize and apply the topological notions on different structures. To contribute to this research area, in this paper, we formulate new soft separation axioms, namely $ tt $-soft $ \alpha T_i\; (i = 0, 1, 2, 3, 4) $ and $ tt $-soft $ \alpha $-regular spaces. They are defined using total belong and total non-belong relations with respect to ordinary points. This way of definition helps us to generalize existing comparable properties via general topology and to remove a strict condition of the shape of soft open and closed subsets of soft $ \alpha $-regular spaces. With the help of examples, we show the relationships between them as well as with soft $ \alpha T_i\; (i = 0, 1, 2, 3, 4) $ and soft $ \alpha $-regular spaces. Also, we explore under what conditions they are kept between soft topological space and its parametric topological spaces. We characterize $ tt $-soft $ \alpha T_1 $ and $ tt $-soft $ \alpha $-regular spaces and give some conditions that guarantee the equivalence of $ tt $-soft $ \alpha T_i\; (i = 0, 1, 2) $ and the equivalence of $ tt $-soft $ \alpha T_i\; (i = 1, 2, 3) $. Further, we investigate some interrelations of them and some soft topological notions such as soft compactness, product soft spaces and sum of soft topological spaces. In the end, we study the main properties of of $ \alpha $-fixed soft point theorem. </abstract>

Rough sets theory via new topological notions based on ideals and applications

&lt;abstract&gt;&lt;p&gt;There is a close analogy and similarity between topology and rough set theory. As, the leading idea of this theory is depended on two approximations, namely lower and upper approximations, which correspond to the interior and closure operators in topology, respectively. So, the joined study of this theory and topology becomes fundamental. This theory mainly propose to enlarge the lower approximations by adding new elements to it, which is an equivalent goal for canceling elements from the upper approximations. For this intention, one of the primary motivation of this paper is the desire of improving the accuracy measure and reducing the boundary region. This aim can be achieved easily by utilizing ideal in the construction of the approximations as it plays an important role in removing the vagueness of concept. The emergence of ideal in this theory leads to increase the lower approximations and decrease the upper approximations. Consequently, it minimizes the boundary and makes the accuracy higher than the previous. Therefore, this work expresses the set of approximations by using new topological notions relies on ideals namely $ \mathcal{I} $-$ {\delta_{\beta}}_{J} $-open sets and $ \mathcal{I} $-$ {\bigwedge_{\beta}}_{J} $-sets. Moreover, these notions are also utilized to extend the definitions of the rough membership relations and functions. The essential properties of the suggested approximations, relations and functions are studied. Comparisons between the current and previous studies are presented and turned out to be more precise and general. The brilliant idea of these results is increased in importance by applying it in the chemical field as it is shown in the end of this paper. Additionally, a practical example induced from an information system is introduced to elucidate that the current rough membership functions is better than the former ones in the other studies.&lt;/p&gt;&lt;/abstract&gt;