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In this paper, we present a way of constructing a topology on a vertex set of a graph using monophonic eccentric neighborhoods of the graph G. In this type of construction, we characterize those graphs that induced the indiscrete topology, the discrete topology, and a particular point topology.

In this study, we aim to solve the open problem posed by Zahiri and Borzooei [23] , i.e., given an arbitrary linear topological BL-algebra ( L , U ) , under what condition does its topology U coincides with the new one V induced by a system of filters. We investigate this problem for both finite BL-algebras and infinite BL-algebras. In the finite case, we give a necessary and sufficient condition. In the infinite case, we consider the conditions where the original topology U is zero-dimensional and ( L , V ) is an l–BL-algebra, respectively. In addition, we modify some examples of topological BL-algebras mentioned by Borzooei et al.

In this paper, after discussing the one point compactification of the Khalimsky line (resp. the Khalimsky plane), denoted by ( Z ⁎ , κ ⁎ ) (resp. ( ( Z 2 ) ⁎ , ( κ 2 ) ⁎ ) ), we study various properties of these compactifications associated with the semi- T 1 2 axiom, a non-Alexandroff structure, a non-cut-point space and so forth. We also investigate dense subsets and nowhere dense subsets of ( Z ⁎ , κ ⁎ ) and ( ( Z 2 ) ⁎ , ( κ 2 ) ⁎ ) . Finally, motivated by a particular point topology and an excluded point topology, we develop two kinds of new topologies as quotient topological spaces of ( Z ⁎ , κ ⁎ ) called an excluded two points topology and a cofinite particular point topology.

We generalize Brazas' topology on the fundamental group to the whole universal path space X ˜ , i.e., to the set of homotopy classes of all based paths. We develop basic properties of the new notion and provide a complete comparison of the obtained topology with the established topologies, in particular with the Lasso topology and the CO topology, i.e., the topology that is induced by the compact-open topology. It turns out that the new topology is the finest topology contained in the CO topology, for which the action of the fundamental group on the universal path space is a continuous group action.

We compute for the first time, the number of fuzzy topologies defined on a finite set and having a small number of open sets. Certain cases, where the number of open sets is large, are also considered. Several well known results are obtained as corollaries. The paper is ended by some questions for future investigations.

The aim of this paper is to introduce a single structure which carries the subsets of X as well as the subsets of Y for studying the information about the ordered pair (A, B) of subsets of X and Y. Such a structure is called a binary structure from X to Y. Mathematically a binary structure from X to Y is defined as a set of ordered pairs (A, B) where AX and BY. The purpose of this paper is to introduce a new topology between two sets called a binary topology and investigate its basic properties where a binary topology from X to Y is a binary structure satisfying certain axioms that are analogous to the axioms of topology. MSC 2010: 54A05, 54A99.

In this paper, we introduce and study a metric-like (non-necessarily metric) topology that is weaker than the original topology of a given topological space. The results are used to provide more useful and more general versions of some of the classical fixed point theorems.

To the best of our knowledge, till date there are only two directions along which fuzzy topology (general topology) can be compared, one is the generalization of fuzzy topology (such as fuzzy supra topology, generalized fuzzy topology etc.) and the other is the stronger form of fuzzy topology which is known as Alexandroff fuzzy topology. It means that a given topology is always linear. This paper aims to propose the third direction that is a new parallel form of fuzzy topology (or a non-linear topology) called fuzzy independent topology which is neither a generalization nor a stronger form of the given fuzzy topology though it is a unique natural offshoot of the given fuzzy topology but very rare in existence and that has been shown by defining fuzzy $$\gamma ^{*}$$ -open set in the sense of Dimitrije Andrijevic, by proving that fuzzy $$\gamma ^{*}$$ -open set and fuzzy open set are completely independent of each other though the collection of fuzzy $$\gamma ^{*}$$ -open sets are themselves a fuzzy topology therein. At the same time we claim that it is beyond the scope of general topology with the existing generalized open sets in literature and consequently we move one more step towards learning the difference between topology and fuzzy topology. To this end, we study the fundamental properties of the new structure. Also we study some of the basic properties and characterizations of fuzzy $$\gamma ^{*}$$ -open sets and few of their applications. The investigation enables us to present a new covering property of the given fuzzy topological space and the preservation theory is obtained. Finally to illustrate the advantage of the proposed concept, we compare the obtained results with some already existing ones.

Let CL ( X ) denote the set of all nonempty closed subsets of a topological space X . In this paper, we introduce the definitions of k F -covers and c V -covers of X . Under Fell topology τ F and Vietoris topology τ V on CL ( X ) , we investigate the following selection principles: Rothberger properties, Menger properties and Hurewicz properties, and also characterize tightness-like covering properties: tightness, Set-tightness and T-tightness.

Stephen Hawking, among others, has proposed that the topological stability of a property of spacetime is a necessary condition for it to be physically significant. What counts as stable, however, depends crucially on the choice of topology. Some physicists have thus suggested that one should find a canonical topology, a single “right” topology for every inquiry. While certain such choices might be initially motivated, some little-discussed examples of Geroch and some propositions of my own show that the main candidates—and each possible choice, to some extent—faces the horns of a no-go result. I suggest that instead of trying to decide what the “right” topology is for all problems, one should let the details of particular types of problems guide the choice of an appropriate topology.

The empirical study of visual space (VS) has centered on determining its geometry, whether it is a perspective projection, flat or curved, Euclidean or non-Euclidean, whereas the topology of space consists of those properties that remain invariant under stretching but not tearing. For that reason distance is a property not preserved in topological space whereas the property of spatial order is preserved. Specifically the topological properties of dimensionality, orientability, continuity, and connectivity define “real” space as studied by physics and are the spatial properties that characterize the physical universe as being an integral whole. By contrast the geometrical analysis of VS has taken little cognizance of its topology. Instead such properties have been presupposed a priori rather than being established a posteriori by empirical means, perhaps because these properties are self-evident. Applying the method of coordinative definition expounded by Hans Reichenbach for determining geometrical and topological properties of physical space (PS), it can be shown that VS fulfills the topological criteria of being a “real” space sui generis. Though theorized to be produced by the brain, the topology of VS is not topologically equivalent (homeomorphic) with the structure and activity of the brain because, as will be shown, the topology of VS cannot be formed from the topology of the brain without tearing and/or cutting and pasting.

Some kind of abstract density topology in a topological Baire space is considered. The semiregularization of this type of topology on the real line in many cases is the coarsest topology for which real functions continuous with respect to the abstract density topology are continuous.

Let k be a ring, X be a k -scheme and R be a k -algebra endowed with an arbitrary topology. In this text, we introduce the fine topology on X ( R ) , which is based on Grothendieck's definition of a topology for affine k -schemes. We prove that the fine topology is functorial in both X and R and that it coincides with Grothendieck's topology for affine k -schemes, with the strong topology for k -varieties over topological fields k and with the adelic topology for k -varieties over a global field k . In some concluding remarks, we explain how properties of the topology of R are reflected in geometric properties of the fine topology, and discuss a possible application to higher local fields.

In [9], two new kinds of topologies called the open-point topology and the bi-point-open topology on C(X), the set of all real-valued continuous functions on a Tychonoff space X, have been introduced and their separation and countability properties have been studied. In the present paper, we study the submetrizability and cardinal functions such as extent, cellularity, weight, pseudocharacter, character and tightness of the spaces C(X) equipped with the open-point and bi-point-open topologies.

We consider T0-topologies on n-element set that contain more than 2n elements. We solve a problem of listing and counting of such topologies. For this purpose we introduce the notions of an index of the topology and a vector of the topology. We study their properties and single out all possible classes of the topologies under the consideration. We formulate and prove a theorem related to the number of the topologies in each of the classes.

Mapping is an important task for mobile robots. The assessment of the quality of maps in a simple, efficient and automated way is not trivial and an ongoing research topic. Here, a new approach for the evaluation of 2D grid maps is presented. This structure-based method makes use of a topology graph, i.e., a topological representation that includes abstracted local metric information. It is shown how the topology graph is constructed from a Voronoi diagram that is pruned and simplified such that only high level topological information remains to concentrate on larger, topologically distinctive places. Several methods for computing the similarity of vertices in two topology graphs, i.e., for performing a place-recognition, are presented. Based on the similarities, it is shown how subgraph-isomorphisms can be efficiently computed and two topology graphs can be matched. The match between the graphs is then used to calculate a number of standard map evaluation attributes like coverage, global accuracy, relative accuracy, consistency, and brokenness. Experiments with robot generated maps are used to highlight the capabilities of the proposed approach and to evaluate the performance of the underlying algorithms.

This paper formulates the coordination control for continuous- and discrete-time homogeneous linear multiagent systems under network topology without any structural confinement. This general setting is proved to unify most of the existing models, such as (leaderless and leader-following) synchronization/consensus control and containment control, which are usually considered separately using different analyses, within the same framework. More important, it unravels the most general collective behavior inherent in the network of diffusively coupled agents with identical dynamics: agents outside the closed strong components asymptotically move into the convex space spanned by those agents in the closed strong components. Coordination analysis for such a general framework is successfully performed based on the exploitation of tools from matrix analysis, linear system theory, and algebraic graph theory.

In this paper, we propose an integration of rough sets, matroids and topology to exploit the advantages of three theories. First, we consider topologies induced by binary relations and illustrate that binary relations can be researched by means of topology. Next, similarity of binary relations based on rough set theory and topology is introduced and the fact that every binary relation is solely similar to some preorder relation is proved. Finally, as an application of the similarity, topological structures of matroids are given.

In this paper, we introduce the notions of an α -Λ-sets and an α -V-sets in topological space. We study the fundamental properties of α -Λ-sets and α -V-sets. Also, we investigate the topologies defined by these families of sets.

In the definition of a set-open topology on C ( X ) , the set of all real-valued continuous functions on a Tychonoff space X , we use a certain family of subsets of X and open subsets of R . But instead of using this traditional way to define topologies on C ( X ) , in this paper, we adopt a different approach to define two interesting topologies on C ( X ) . We call them the open-point and the bi-point-open topologies and study the separation and countability properties of these topologies.

Summary In view of the requirement for instantaneity on power system topology analysis, a novel power system topology analysis method is proposed in this paper. For the plant-station topology method, the nodes that represent the circuit breakers are numbered according to the graph theory. On this basis, combining the main wiring characteristics and the switch status information, the plant-station topology is analyzed using the topological base theorem. When the switch status changes, the affected part of topology is modified using the radius search method, so that reforming of the topology is avoided and fast update of topology is guaranteed. When dealing with inter-plant-station topology, the incidence matrix notation method is used to judge the connectivity of the graph and mark the tree branches and link branches in the network, thus conducting static topology analysis of the power network. And then, when the network topology changes, the property of the changing branch is determined with the help of the loop matrix. On this basis, the local network topology and the loop matrix are updated by means of the broken circle method and the radius search method. Calculation examples verify the advantage of the proposed methods over traditional methods in calculation speed. With the proposed topology analysis methods, fast tracking of the switch status can be realized. Copyright © 2015 John Wiley & Sons, Ltd.

In two-dimensions, a fracture network consist of a system of branches and nodes that can be used to define both geometrical features, such as length and orientation, and the relationship between elements of the network – topology. Branch lengths are preferred to trace lengths as they can be uniquely defined, have less censoring and are more clustered around a mean value. Many important properties of networks are more related to topology than geometry. The proportions of isolated (I), abutting (Y) and crossing (X) nodes provide a basis for describing the topology that can be easily applied, even with limited access to the network as a whole. Node counting also provides an unbiased estimate of frequency and can be used in conjunction with fracture intensity to estimate the characteristic length and dimensionless intensity of the fractures. The nodes can be used to classify branches into three types – those with two I-nodes, one I-node and no I-nodes (or two connected nodes). The average number of connections per branch provides a measure of connectivity that is almost completely independent of the topology. We briefly discuss the extension of topological concepts to 3-dimensions.

Let L be a complete lattice. On L we define the so called quasi Scott topology, denoted by qSc. This topology is always larger than or equal to the Scott topology and smaller than or equal to the strong Scott topology. Results concerning the above topology are given. Also, we introduce and investigate the notions of q-continuous and q-algebraic complete lattices. Finally, we give and examine the quasi Lawson topology on a complete lattice. 1. Preliminaries Our reference for complete lattices are (2, 3, 8, 9). We shall frequently denote complete lattices with their underlying sets and write L for (L;D). The top element and the bottom element of a complete lattice L will be denoted by 1L and 0L, respectively.

This is a study of the completeness properties of the space Crc(X) of continuous real-valued functions on a Tychonov space X, where the function space has the C-compact-open topology. Investigate the properties such as completely metrizable, Čech-complete, pseudocomplete and almost Čech-complete. The work was supported by Project 15-16-1-6 comprehensive program of UB RAS.

In this paper, we define and study a new topology constructed from any given topology on a set, using irreducible sets. The manner in which this derived topology is obtained is inspired by how the Scott topology on a poset is constructed from its Alexandroff topology. This derived topology leads us to a weak notion of sobriety called k-bounded sobriety. We investigate the properties of this derived topology and k-bounded sober spaces. A by-product of our theory is a novel type of compactness, which involves crucially the Scott irreducible families of open sets. Some related applications on posets are also given.

In this paper, the spectral topology and quasi-spectral topology of proper prime A-ideals in an MV-module are introduced. We show that the spectral topology of proper ⋅-prime ideals of a PMV-algebra with unity for product, is the same as the spectral topology of proper prime ideals in an MV-algebra. Also we show that the set of all prime A-ideals in an MV-module with spectral topology is not T0 and T1 topological spaces but quasi-spectral topology is T0-space and is not T1-space. Finally, we investigate when the set of all prime A-ideals in an MV-module are Hausdorff and disconnected.

This paper addresses the problem of robust topology design of the multi-UAV formation. We use a digraph to model the topology of the multi-UAV formation, and propose the definition of anti-k-communication-destroying topology, meaning the system can still performs normally even when any arbitrary k communicating links are destroyed. By exploring the property of this kind of topology based on graph theory, we propose the algorithm Uniform-Cost Forest Search, UCFS, which is an extension of the classical search strategy uniform-cost search. The proposed algorithm would establish the anti-k-communication-destroying topology for the multi-UAV formation, with k + 1 minimal-cost edge-independent paths found for each UAV. Proof as well as examples validating the algorithm are provided in the paper. Finally, potential research directions are discussed.

The chapter provides a brief exposition of point set topology. In particular, it aims to make readers from the engineering community feel comfortable with the subject, especially with those topics required in latter chapters. The implicit appearance of topological concepts in the context of continuum mechanics is sketched first. Afterwards, topological concepts like interior and boundary of sets, continuity of mappings, etc., are discussed within metric spaces before the introduction of the concept of topological space.

Let $\Omega$ be a strongly pseudoconvex domain. We introduce the Mabuchi space of strongly plurisubharmonic functions in $\Omega$. We study metric properties of this space using Mabuchi geodesics and establish regularity properties of the latter, especially in the ball. As an application we study the existence of local K\"ahler-Einstein metrics., Comment: 25 pages

In this paper, we have introduce and study the concepts of ω*-open sets and ω*-open open topology for function spaces.

In this paper, we study extensively the compact-GδGδ-open topology on the set C(X)C(X) of all real-valued continuous functions on a Tychonoff space X . In particular, we study the induced map, submetrizability, completeness properties, countability properties, and nuclear and reflexive properties of the compact-GδGδ-open topology on C(X)C(X).

There are exactly three regular planar grids, which are formed by tiling the 2-dimensional Euclidean space with regular triangles, squares, and hexagons. The topology of the square grid is well-understood, but it cannot be said of the remaining two regular sampling schemes. This work deals with the topological properties of digital binary pictures sampled on the triangular grid. Some characterizations of simple pixels and sufficient conditions for topology preserving operators are reported. These results provide the theoretical background to various topological algorithms including thinning, shrinking, generating discrete Voronoi diagrams, and contour smoothing on the triangular grid.

The most common construction of a generalized covering space is that of a topologized (appropriate version of a) path space X ˜ . There have been three suggested topologies on it, each with its advantages and disadvantages. They are called the Whisker topology, the Lasso topology and the quotient of the compact open topology. In this paper we study the relationship between these topologies. The main result consists of an example demonstrating that the Lasso topology is not finer that the compact open topology. Our results also apply to the topology of the fundamental group which appears naturally as a subspace of X ˜ .

The present paper aims to address a long-standing and challenging problem in structural topology optimization: explicit feature control of the optimal topology. The basic idea is to resort to the level set solution framework and impose constraints on the extreme values of the signed distance level set function used for describing the topology of the structure. Numerical examples are also presented and discussed to illustrate the effectiveness of the proposed approach.

For a signature L with at least one constant symbol, an L-structure is called minimal if it has no proper substructures. Let be the set of isomorphism types of minimal L-structures. The elements of can be identified with ultrafilters of the Boolean algebra of quantifier-free L-sentences, and therefore one can define a Stone topology on . This topology on generalizes the topology of the space of n-marked groups. We introduce a natural ultrametric on , and show that the Stone topology on coincides with the topology of the ultrametric space iff the ultrametric space is compact iff L is locally finite (that is, L contains finitely many n-ary symbols for any ). As one of the applications of compactness of the Stone topology on , we prove compactness of certain classes of metric spaces in the Gromov-Hausdorff topology. This slightly refines the known result based on Gromov's ideas that any uniformly totally bounded class of compact metric spaces is precompact.

We characterize the completely regular Lindelof spaces X such that C ( X ) with the finest splitting topology is Frechet, a topological group, or a k -space.

Rough set theory is a powerful tool for dealing with uncertainty, granularity, and incompleteness of knowledge in information systems. In addition, digital topology deals with properties and features of two-dimensional or three-dimensional digital images that correspond to topological properties of objects. So, we try to describe the relationship between rough sets and digital topology. Firstly, we will study the classifications of topologies in rough sets. Secondly, we will use the upper approximation operator to span the digital line, which is the basic building block of the digital spaces.

Let $X$ be the prime spectrum of a ring. In [arXiv:0707.1525] the authors define a topology on $X$ by using ultrafilters and they show that this topology is precisely the constructible topology. In this paper we generalize the construction given in [arXiv:0707.1525] and, starting from a set $X$ and a collection of subsets $\mathcal{F}$ of $X$, we define by using ultrafilters a topology on $X$ in which $\mathcal F$ is a collection of clopen sets. We use this construction for giving a new characterization of spectral spaces and several new examples of spectral spaces., 16 pages. To appear in Communications in Algebra

This paper will discuss about a new topology, obtained from a grill and a filter on the same set. The Characterizations and open base of the new topology are also aim of this paper. The generalized continuity is also a part of this paper.

Mathematical theorems in control theory are only of interest in so far as their assumptions relate to practical situations. The space of systems with transfer functions in , for example, has many advantages mathematically, but includes large classes of non–physical systems, and one must be careful in drawing inferences from results in that setting. Similarly, the graph topology has long been known to be the weakest, or coarsest, topology in which (1) feedback stability is a robust property (i.e. preserved in small neighbourhoods) and (2) the map from open-to-closed-loop transfer functions is continuous. However, it is not known whether continuity is a necessary part of this statement, or only required for the existing proofs. It is entirely possible that the answer depends on the underlying classes of systems used. The class of systems we concern ourselves with here is the set of systems that can be approximated, in the graph topology, by real rational transfer function matrices. That is, lumped parameter...

This paper continues to develop the theory of categorically algebraic (catalg) topology, introduced as a common framework for the majority of the existing many-valued topological settings, to provide convenient means of interaction between different approaches. Motivated by the results of universal topology of H. Herrlich, we show that a concrete category is fibre-small and topological if and only if it is concretely isomorphic to a subcategory of a category of catalg topological structures, which is definable by topological co-axioms.

This paper presents a method for computing topology changes for triangle meshes in an interactive geometric modeling environment. Most triangle meshes in practice do not exhibit desirable geometric properties, so we develop a solution that is independent of standard assumptions and robust to geometric errors. Specifically, we provide the first method for topology change applicable to arbitrary non-solid, non-manifold, non-closed, self-intersecting surfaces. We prove that this new method for topology change produces the expected conventional results when applied to solid (closed, manifold, non-self-intersecting) surfaces---that is, we prove a backwards-compatibility property relative to prior work. Beyond solid surfaces, we present empirical evidence that our method remains tolerant to a variety of surface aberrations through the incorporation of a novel error correction scheme. Finally, we demonstrate how topology change applied to non-solid objects enables wholly new and useful behaviors.

Let me start this chapter with a simple why-question: Why general topology? What is the main problem it wishes to solve? The answer is deceivingly simple: general topology aims at analyzing and describing topological spaces.

The talk tackles the problem of reconstructing the tree-like topological structure of a network of linear dynamical systems. A distance function is defined in order to evaluate the “closeness” of two processes and some useful mathematical properties are derived. Theoretical results to guarantee the correctness of the identification procedure for networked linear systems characterized by a tree topology are provided as well. The talk also suggests the approximation of a complex connected network with a tree in order to detect the most meaningful interconnections.

The paper introduces foundations of the "Flexible Routing Method" that belongs to the topologically-geometric type. It develops the idea to divide the routing problem on two separate successive stages: topological and geometrical. At the first stage it was suggested to use a discrete topological model as Delaunay triangulation or/and Voronoi polygons to describe topology. The explicit and implicit topology models are offered which describe the relative topological nets location without specifying their geometrical characteristics. At the second stage possible is the laying the nets of arbitrary configuration: orthogonal, piecewise linear, curvilinear, under arbitrary angles and arbitrary widths.

We describe an explicit metric that induces the Chabauty topology on the space of closed subsets of a proper metric space M., Comment: 3 pages. This article was initially posted as part of the appendix of arXiv:1606.03360, but has been excised for separate publication. v2 includes corrections suggested by the referee

The interpretation of auxiliary entities as boundary entities in previous non-manifold boundary (NMB) representations may change a model's intended topology and increase the complexity of the corresponding data structure. In this paper, entities appearing in a modeling process are classified into boundary and non-boundary entities. Non-boundary entities are usually embedded into embedding-space entities. These embedding relationships are described by inclusion topology. To support inclusion topology, a new mathematical framework-quasi-cell-complex, as well as a topological data structure-Q-Complex, are proposed. Quasi-cell-complex is an extension of cell-complex with inclusion topology supported. Q-Complex is an NMB representation, in which a new topological entity-embedder is created for inclusion topology and zone/disk is adopted to capture the complete adjacencies around a vertex. Thus, Q-Complex allows full adjacencies, incidence-ordering, and inclusion relationships to be derived, and the efficiency of most basic queries is several times faster than most state-of-the-art NMB representations, without increasing storage. Additionally, the benefits of inclusion topology for shape modeling and feature modeling are explored.