Point Set Topology Research Articles

Seismic 3D full-horizon tracking based on a knowledge graph to represent the stratigraphic sequence relationship

Seismic 3D full-horizon tracking is a fundamental and crucial step in sequence analysis and reservoir modeling. Existing automatic full-horizon tracking approaches lack effective methods for representing the stratigraphic sequence relationships in seismic data. However, the inability to represent the stratigraphic sequence relationships fully and accurately makes it challenging to address discontinuous areas affected by faults and unconformities. To address this issue, we develop a knowledge graph representing the stratigraphic sequence relationship, which enables the simultaneous extraction of all horizon surfaces once the stratigraphic distribution of the seismic data is obtained. This method first generates horizon patches and calculates the fault attributes, followed by the construction of an initial knowledge graph that characterizes the overall distribution of horizon patches and faults. The initial knowledge graph comprises nodes and edges. The nodes represent horizon patches, and their attributes cover the geographical location information of the patches and faults. Simultaneously, the edges represent the relationship between the horizon patches, including the stratigraphic sequence relationship, and their attributes illustrate the potential for connecting these patches. Furthermore, we introduce a multilayer knowledge graph based on the point-set topology to fuse the nodes. This allows for the continuous merging of the horizon patches to obtain horizon surfaces across discontinuities with the constraints of fault attributes and stratigraphic sequence relationships in 3D space. Synthetic and field examples demonstrate that our approach can effectively represent stratigraphic sequence relationships and accurately track horizons located in discontinuous areas with faults and unconformities.

DEGREE OF SATISFIABILITY IN HEYTING ALGEBRAS

Abstract We investigate degree of satisfiability questions in the context of Heyting algebras and intuitionistic logic. We classify all equations in one free variable with respect to finite satisfiability gap, and determine which common principles of classical logic in multiple free variables have finite satisfiability gap. In particular we prove that, in a finite non-Boolean Heyting algebra, the probability that a randomly chosen element satisfies $x \vee \neg x = \top $ is no larger than $\frac {2}{3}$ . Finally, we generalize our results to infinite Heyting algebras, and present their applications to point-set topology, black-box algebras, and the philosophy of logic.

A Spatial Relation Model of Three-Dimensional Electronic Navigation Charts Based on Point-Set Topology Theory

Spatial relation models are the basis for realising three-dimensional spatial analysis. More researchers are now focusing on models that combine topological relations with distance or directional relations; however, a model that unifies all three relations has not yet been developed. In particular, it is more effective to use different spatial relations between features with different spatial characteristics in three-dimensional electronic navigation charts (3D ENC). Therefore, this paper proposes a 3D ENC feature spatial relation model (3DSRM) based on point-set topology theory, which combines 3D topological relations, distance relations and directional relations, and uses a unified model framework to describe 64 topological relations of 3D ENC features from both horizontal and vertical directions. Through the comparison and derivation of feature topological relations, it is demonstrated that the model can distinguish 3D spatial topological relations more comprehensively, realise the mutual derivation between spatial relations and spatial features, and improve the robustness of spatial relations judgement. The model can be used to judge the topological relations between features, realise 3D topological relation checking and topological creation of complex features, and improve the accuracy and interactivity of 3D ENC.

Formal topological description of three-dimensional topological entities based on k-dimensional pseudo-manifold

This paper clarifies the necessity of studying three-dimensional spatial data models, analyzes the research progress and existing problems of topological spatial relationship description methods, and proposes a complete and formal description framework of three-dimensional topological spatial relations based on point set topological theory and dimension expansion method. On this basis, the topological spatial relations existing in three-dimensional spatial objects are classified, and five fundamental topological spatial relations are defined. The mutual exclusivity and completeness of the minimal set of 3D topological space relations are proved. Based on point set topology and k-dimensional pseudo-manifold, this paper proposes the formal description of three-dimensional topological properties and spatial relations between spatial entities. The simplicial homology is applied to manifolds, the topological properties of three-dimensional space entities are revealed, and the formal description method of three-dimensional space entities is given through the geometric structural relationship between simplex and simplicial k-complex.

On a nation as a topological space

This paper introduces point-set topology into international interactions. Nations are sets of people who interact if there is a well-defined function between them. To do all these, we need to have the structure that describes how such nations interact. This calls for a topology. The kind of topology we construct in this perspective is made up by decision spaces. We first begin by developing a mathematical representation of a decision space, and use such spaces to develop a topology on a nation. Subsequently, we revisit some properties of the interior, closure, limit, and boundary points with respect to this topology and the new concept of ϕ−proximity. Finally, we define and develop ϕ−connectedness of subspaces of a nation.

Subspace spanning graph topological spaces of graphs

A collection of spanning subgraphs TS, of a graph G is said to be a spanning graph topology if it satisfies the three axioms: Nn, K0 ∈ TS where, n = |V (G)|, the collection is closed under any union and finite intersection. Let (X, T) be a topological space in point set topology and Y ⊆ X then, TY = {U ∩ Y : U ∈ T} is a topological space called a subspace topology or relative topology defined by T on Y . In this paper we discusses the subspace spanning graph topology defined by the graph topology TS on a spanning subgraph H of G.

New types of connectedness and intermediate value theorem in ideal topological spaces

The definitions of new type separated subsets are given in ideal topological spaces. By using these definitions, we introduce new types of connectedness. It is shown that these new types of connectedness are more general than some previously defined concepts of connectedness in ideal topological spaces. The new types of connectedness are compared with well-known connectedness in point-set topology. Then, the intermediate value theorem for ideal topological spaces is given. Also, for some special cases, it is shown that the intermediate value theorem in ideal topological spaces and the intermediate value theorem in topological spaces coincide.

Preliminary Concept and Result of Fuzzy Metric Space in Point Set Topology

The innovative concept of fuzzy mathematics has become one of the interesting areas of research since last fifty-five years, which was first introduced by Zadeh in 1965. Many researchers connected the fuzzy concept in different forms of metric spaces. The results about the point sets have discussed, mainly topological properties of sets, connecting with the fuzzy metric space. The propose of this paper is to study the point set topology in fuzzy metric space, especially by introducing the concept of open and closed balls and discuss some of the common properties. Moreover, we introduce the concept of compactness and pre-compactness in fuzzy metric space.

Identification of enclaves and exclaves by computation based on point-set topology

Enclaves and exclaves have special roles in geography and are of particular importance to fields such as (geo)politics and economy. However, enclaves and exclaves have not been defined with sufficient formality for automatic identification yet. To identify enclaves and exclaves more generally by computational means than current definitions existing in the literature, this article proposes expressive and generalized mathematical definitions of enclaves and exclaves based on point-set topology. A novel Boundary Extended 16-Intersection Model is developed in this article to identify enclaves, and 74 possible spatial configurations of enclaves are distinguished according to conditions of intersections for polygons in partitions and enclaves. The classic Dimensionally Extended 9-Intersection Model is employed to identify exclaves, and two possible spatial configurations of exclaves are distinguished according to conditions of intersections for polygons in partitions and exclaves. Applications of the proposed definitions are exemplified by the identification of enclaves and exclaves in prototypes and in the real world.

On the lambda function and a quantification of Torhorst Theorem

For any compact K⊂Cˆ we define a map λK:Cˆ→N∪{∞}, called the lambda function of K. The lambda function of a continuum K has the property that λK(x)=0 for all x∈Cˆ if and only if K is locally connected. We establish several inequalities and a gluing lemma for the lambda functions. These inequalities reflect the relationship between the topology of K and the complexity that may be involved in the boundary of a component of Cˆ∖K. One of them, called the lambda inequality, generalizes and quantifies the Torhorst Theorem. We also find three sufficient conditions under each of which the lambda inequality becomes an equality. These results cover Whyburn's Theorem on the partial converse to the Torhorst Theorem. The gluing lemma for the lambda functions is of research interest in the study of point set topology and of polynomial Julia sets as well.

Research on the mapping grinding of dimple surface with ordered pattern based on topological theory

Since the structured dimple surface is one of the important functional drag-reducing surfaces, the research on its efficient fabrication technology is of great significance for the practical application of this surface. Therefore, in order to grind structured dimple surface, a topological mapping grinding method for structured dimple surface was proposed based on grinding kinematics principle and point set topology. Firstly, the topological spaces of the workpiece and the grinding wheel were established, and the topological features of the structured dimple surface and the structured grinding wheel were extracted based on the analysis of the topography features of the structured surface with ordered pattern. Then, the topology mapping equation of the grinding process was constructed based on the analysis of the generating mechanism of the dimple surface about grinding geometry, and the structured grinding wheel was designed according to the topology mapping equation. Finally, according to the grinding geometry simulation, the influence of grinding parameters on the generating of dimple surface topography was studied, and the grinding experiment was carried out. The results show that the structured grinding wheel designed based on the topological features of the structured dimple surface can achieve the grinding of the structured dimple surface. The ground dimple surface is a topological dimple surface, and its feature parameters can be changed with the change of grinding parameters, but the topological feature attributes remain unchanged under the condition of satisfying the proper grinding speed ratio.

An elementary proof of the Brouwer’s fixed point theorem

We provide an elementary proof of the Brouwer’s fixed point theorem based solely on introductory topological concepts such as compactness and connectedness both covered in a first undergraduate course in point-set topology.

A study of interrelationships between rough set model accuracy and granule cover refinement processes

Rough set model accuracy varies when granules change. Refinement is one way to characterize granule evolutions, but it has not been formally defined in rough set theory so far. Topologically, a granule cover W is called a refinement of another granule cover F if there exists some FW∈F such that W⊆FW for each W∈W. In this article, we show that the concept refinement in topology is too abstract to describe the variability of the rough set model along with the change in granules. By abstracting from some important granule refinements, such as reductions and minimal descriptions, we propose two novel granule cover refinements named base-type refinements and base-preserving refinements, which combine both point-set topology and rough set theory. We analyze the accuracy relationships of some typical approximation operators between granule covers and their base-type or base-preserving refinements. We also show that higher accuracy implies finer granule covers for some approximation operators. These are promising results for rough set model accuracy comparisons. Furthermore, the results hint at how to avoid exponentially increased computation complexity on rough set model accuracy comparisons by checking refinement relationships of granule covers directly instead of computing on them using approximation operators.

Tangible Topology Through the Lens of Limits

ABSTRACT Point-set topology is among the most abstract branches of mathematics in that it lacks tangible notions of distance, length, magnitude, order, and size. There is no shape, no geometry, no algebra, and no direction. Everything we are used to visualizing is gone. In the teaching and learning of mathematics, this can present a conundrum. Yet, this very property makes point set topology perfect for teaching and learning abstract mathematical concepts. It clears our minds of preconceived intuitions and expectations and forces us to think in new and creative ways. In this paper, we present guided investigations into topology through questions and thinking strategies that open up fascinating problems. They are intended for faculty who already teach or are thinking about teaching a class in topology or abstract mathematical reasoning for undergraduates. They can be used to build simple to challenging projects in topology, proofs, honors programs, and research experiences.

On the topological characterization of gestures in a convenient category of spaces

ABSTRACT In this paper, we reconsider the topological characterization of gestures in a convenient category of spaces mentioned by Mazzola in 2009, recovering Arias's 2018 result that the relevant equivalence is a homeomorphism. We also show the topological characterization of gestures extends to an adjunction between the category of gestures and the category of continuous maps whose domain is a one-dimensional CW complex. Our arguments utilize only basic tools from category theory and almost no point-set topology. With little generality lost from our restriction of spaces considered, the ease of conceptualization in this setting provides an advantageous entry point for researchers interested in studying or applying gestures but who may not have advanced knowledge of category theory, algebraic geometry, or point-set topology. Thus one may view this work, in part, as an advertisement for the topological interpretation of gestures developed within.

Research on Authentication of User Information in Cloud Manufacturing

Background: Aiming at the problem of user information security authentication in cloud manufacturing, a user information security authentication model is proposed. The model is introduced in detail. Methods: In the security authentication model, the point set topology group fractal transformation algorithm is combined with the biometric identification technology to acquire the user biometric information to generate the secret key. Combined with the AES (Advanced Encryption Standard) encryption algorithm, the user biometric information is encrypted. Results: It provides a stronger guarantee for the user's security certification. We revised all patents relating to pharmaceutical formulations of applicability in authentication of user information in cloud manufacturing. Conclusion: Experiments show that the research has credibility and practicability, which improves the overall security performance of cloud manufacturing user information authentication and access.

A scalable fuzzy division model of topological space on GIS

It is difficult to describe topological relationship of complicated geographical phenomena and entity with fuzziness and uncertainty in the real world. Aiming at the above problem, this paper has summarized to the topological relationship of between spatial entities with 4-intersection model and 9-intersection model firstly. On the basis of point-set topology, the formal description model of the topological relationship between fuzzy geographical entities is analysed and gives a detailed and accurate formalization-description and strict math formula. Lastly, this paper proposes a method, which improves and expands topological division to between fuzzy geographical entities. It shows that analysing result, the model can analyse and divide the entity fuzzily from simple entity to complicated entity.

Short Note on Topological Fuzzy Spaces of the Behaviour of GS fuzzy Closed Sets

Many fuzzy topologists have very good interest in generalized fuzzy closed sets and in fuzzy point set topology. Here, properties of GS  in fuzzy topological spaces and its relationship with other generalized fuzzy closed sets has been discussed.

Cryptomorphic topological structures: A computational, relation-algebraic approach

In this paper we present an approach to pointless topology that is based on the study of the membership or is-element-of relation within a typed or categorical version of the component-free calculus of relations. This allows us to study several other approaches to topology, including point-set topology, on an abstract and component-free level. In particular, we will show that topologies defined by open sets, closed sets, a family of neighbourhood systems, a topological kernel-mapping, a Kuratowski closure-mapping, or a topological Aumann contact relation are cryptomorphic concepts, i.e., each concept can bijectively be transformed into any other of these concepts. All transformations are specified via relation-algebraic expressions which, in case of set-theoretic relations (that is, in case of point-set topology) can immediately be executed by the specific purpose computer algebra system RelView.

INFINITE COMBINATORICS PLAIN AND SIMPLE

AbstractWe explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. Our main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic.