Topological Structure Of Space Research Articles

Primal Topologies on Finite‐Dimensional Vector Spaces Induced by Matrices

Given an matrix A, considered as a linear map A : ℝn⟶ℝn, then A induces a topological space structure on ℝn which differs quite a lot from the usual one (induced by the Euclidean metric). This new topological structure on ℝn has very interesting properties with a nice special geometric flavor, and it is a particular case of the so called “primal space,” In particular, some algebraic information can be shown in a topological fashion and the other way around. If X is a non‐empty set and f : X⟶X is a map, there exists a topology τf induced on X by f, defined by τf = {U ⊂ X : f−1(U) ⊂ U}. The pair (X, τf) is called the primal space induced by f. In this paper, we investigate some characteristics of primal space structure induced on the vector space ℝn by matrices; in particular, we describe geometrical properties of the respective spaces for the case.

3D quantitative characterization and flow simulation of granite residual soil based on CT scanning

<p indent="0mm">This paper aims to reveal the water transport mechanism in soils with macropores. Based on the combination of the X-ray CT scanning and image processing technologies, a three-dimensional (3D) equivalent pore network model (PNM) with macropores (&gt;0.15 mm) is constructed, and the size of the optimal representative volume unit (REV) is determined to be 300×300×300. Then, the quantitative characterization of the topological structure of the macropore space is carried out followed by the PNM single-phase flow and the 3D macropore water-gas two-phase flow simulation. The results show that there are more pores with smaller pore diameters, and more than 80% of the pores are concentrated within <sc>0.15–1.00 mm.</sc> The pore topological space structure is quite different, and the coordination number is mainly less than 20, but some can reach more than 60. The average coordination number of pores is between 3–7, and the average connected pore radius is greater than <sc>2.00 mm.</sc> The single-phase flow simulation find that the complexity of the pore structure and the resolution of X-ray CT scanning resulted in a certain difference between the simulated and measured permeability, but the difference of the main seepage direction is only 3.81%. The two-phase flow simulation find that the front edge of flow is almost linear at the beginning of seepage, then it tend to be fingering, and finally, it become irregular. The residual gas in the pores is mainly distributed in blocks at the edges of pores, dead ends, and sudden changes of pore throats. With the increasing seepage time, the saturation of the water phase shows an upward trend, while the saturation of the gas phase shows a downward trend, then gradually tend to be stable, and finally, the saturation of the water phase and the residual gas phase maintain at 94.92% and 5.08%, respectively. This study can better reveal the meso-seepage mechanism of reconstructed soil, and provide a certain reference for further understanding the law of pore flow in soil.

On the Structure of Topological Spaces

The structure of topological spaces is analysed here through the lenses of fibrous preorders. Each topological space has an associated fibrous preorder and those fibrous preorders which return a topological space are called spatial. A special class of spatial fibrous preorders consisting of an interconnected family of preorders indexed by a unitary magma is called Cartesian and is studied here. Topological spaces that are obtained from those fibrous preorders, with a unitary magma I, are called I-Cartesian and are characterized. The characterization reveals a hidden structure on such spaces. Several other characterizations are obtained, and special attention is drawn to the case of a monoid equipped with a topology. A wide range of examples is provided, as well as general procedures to obtain topologies from other data types such as groups and their actions. Metric spaces and normed spaces are considered as well.

Темная материя как эффект фрактальности топологической структуры пространства

Темная материя как эффект фрактальности топологической структуры пространства

Structure theorems for finite unions of subspaces of special kind

We study the internal structure of topological spaces X which can be represented as the union of a finite collection of subspaces belonging to some nice class of spaces. Several closely related structure theorems are established. In particular, they concern the finite unions of subspaces with the weight ≤τ, the finite unions of subspaces with a point-countable base, and the finite unions of metrizable subspaces. As a corollary, we extend to finite unions the classical Mischenko's Theorem on metrizability of compacta with a point-countable base [11] (see Theorem 11). A few other applications of the structure theorems are given, in particular, to homogeneous spaces (Corollaries 5 and 10).

Modeling and Analyzing Distributed Computation in Monotone Spaces with Structural Map

Distributed computation follows the models of discrete structures in combinatorial forms. In higher-dimensions, the simplex structures of topological spaces as well as homology are employed to model and analyze distributed asynchronous computations. However, the monotone spaces are the general forms of topological spaces and can be effectively employed to analyze distributed computation. This paper proposes an analytical model of distributed computation in monotone spaces. It is illustrated that, the modeling of distributed computation in monotone spaces helps in determining consistent cuts under closure and convergence of computation. Furthermore, a connective mapping between the simplexes and monotone is constructed.

Topology from Neighbourhoods

Summary Using Mizar [9], and the formal topological space structure (FMT_Space_Str) [19], we introduce the three U-FMT conditions (U-FMT filter, U-FMT with point and U-FMT local) similar to those VI , VII , VIII and VIV of the proposition 2 in [10]: If to each element x of a set X there corresponds a set B(x) of subsets of X such that the properties VI , VII , VIII and VIV are satisfied, then there is a unique topological structure on X such that, for each x ∈ X, B(x) is the set of neighborhoods of x in this topology. We present a correspondence between a topological space and a space defined with the formal topological space structure with the three U-FMT conditions called the topology from neighbourhoods. For the formalization, we were inspired by the works of Bourbaki [11] and Claude Wagschal [31].

Some results of calculus for functions having values in an intuitionistic fuzzy n-normed linear space

In this article we study the topological structure of an intuitionistic fuzzy n-normed linear space (IFnNLS). We establish necessary and sufficient conditions for a sequence of functions having values in an IFnNLS to be uniformly convergent. Further, using the vector topological structure of the space, derivative and Riemann integration of a function having values in an IFnNLS are defined and basic results related to these concepts are studied.

On the topological structure of spaces of fuzzy compacta

We prove that the space of fuzzy compacta in a Peano continuum is homeomorphic to the Hilbert space ℓ 2 . As a corollary we find that the spaces of fuzzy compacta in R n and ℓ 2 are also homeomorphic to Hilbert space.

Lower normal topological spaces and lower continuity

In this paper we formulate a new structure of topological spaces which we call it lower normal spaces. This concept of spaces is directly and nature connection with the lower transversal continuous mappings on topological spaces. In this sense, we shall study spaces in which it is possible in the same way to separate two disjoint closed sets by a lower continuous real valued function. Applications in nonlinear functional analysis are considered. The concept of lower normal spaces is closely connected with the concept of normal topological spaces and the results of Alexandroff, Urysohn, Tietze, Lebesgue, Dieudonné, Tychonoff, Lefschetz, and Vietoris.

IDEAS OF GEOMETRIZATION, GEOMETRIC INVARIANTS OF LOW-DIMENSIONAL MANIFOLDS, AND TOPOLOGICAL QUANTUM FIELD THEORIES

IDEAS OF GEOMETRIZATION, GEOMETRIC INVARIANTS OF LOW-DIMENSIONAL MANIFOLDS, AND TOPOLOGICAL QUANTUM FIELD THEORIES

Multirectangular characteristics for Köthe power spaces of the second kind

Isomorphisms of the so-called Köthe power spaces of the second kind are considered. These spaces are determined by a pair of sequences of positive numbers. Counting functions for a pair of sequences (m-rectangular characteristics of the corresponding Köthe space of the second kind) are introduced. They are shown to be invariant under isomorphisms. The proof is based on the construction of special compound invariants suitable for the class of spaces under consideration. New results on the linear topological structure of spaces of analytic functions in multicircular domains are obtained as an application.Bibliography: 29 titles.

Dark matter from a gas of wormholes

The simplistic model of the classical spacetime foam is considered, which consists of static wormholes embedded in Minkowski spacetime. We explicitly demonstrate that such a foam structure leads to a topological bias of point-like sources which can equally be interpreted as the presence of a dark halo around any point source. It is shown that a non-trivial halo appears on scales where the topological structure possesses local inhomogeneity, while the homogeneous structure reduces to a constant renormalization of the intensity of sources. We also show that in general dark halos possess both (positive and negative) signs depending on scales and specific properties of the topological structure of space.

The Shape of Urban Experience: A Reevaluation of Lynch's Five Elements

In this paper I attempt to develop a comprehensive, robust model of urban morphology from a phenomenological and behavioral perspective. I do so by comparing the findings of two extensive empirical studies of users' experiences of urban public space: one primarily examining people's perceptions in relation to the practical task of wayfinding, and the other my own research into people's playful behavior in Melbourne, London, Berlin, and New York. Lynch himself called attention to “our delight … in ambiguity, mystery … surprise and disorder”, but little is known about what role specific spatial conditions might have in framing such experiences, or indeed about the diversity of impractical activities people actually pursue in urban spaces. With this paper I seek to fill these gaps. Three elements common to both studies (paths, nodes, and edges) describe the fundamental topological structure of space in relation to movement and visibility. I focus on the four spatial elements which differ between the two models (landmarks, districts, thresholds, and props). Field observations illustrate ways in which the latter two spatial elements frame particular noninstrumental, ‘playful’ experiences which are characteristic of the urban condition: spontaneous encounters with strangers; unfamiliar and risky bodily experiences; distraction, and interpreting new meanings in the urban fabric.

On open-set lattices and some of their applications in semantics

In this article, we present the theory of Kripke semantics, along with the mathematical framework and applications of Kripke semantics. We take the Kripke-Sato approach to define the knowledge operator in relation to Hintikka's possible worlds model, which is an application of the semantics of intuitionistic logic and modal logic. The applications are interesting from the viewpoint of agent interactives and process interaction. We propose (i) an application of possible worlds semantics, which enables the evaluation of the truth value of a conditional sentence without explicitly defining the operator “→” (implication), through clustering on the space of events (worlds) using the notion of neighborhood; and (ii) a semantical approach to treat discrete dynamic process using Kripke-Beth semantics. Starting from the topological approach, we define the measure-theoretical machinery, in particular, we adopt the methods developed in stochastic process—mainly the martingale—to our semantics; this involves some Boolean algebraic (BA) manipulations. The clustering on the space of events (worlds), using the notion of neighborhood, enables us to define an accessibility relation that is necessary for the evaluation of the conditional sentence. Our approach is by taking the neighborhood as an open set and looking at topological properties using metric space, in particular, the so-called ε-ball; then, we can perform the implication by computing Euclidean distance, whenever we introduce a certain enumerative scheme to transform the semantic objects into mathematical objects. Thus, this method provides an approach to quantify semantic notions. Combining with modal operators Ki operating on E set, it provides a more-computable way to recognize the “indistinguishability” in some applications, e.g., electronic catalogue. Because semantics used in this context is a local matter, we also propose the application of sheaf theory for passing local information to global information. By looking at Kripke interpretation as a function with values in an open-set lattice 𝒪U, which is formed by stepwise verification process, we obtain a topological space structure. Now, using the measure-theoretical approach by taking the Borel set and Borel function in defining measurable functions, this can be extended to treat the dynamical aspect of processes; from the stochastic process, considered as a family of random variables over a measure space (the probability space triple), we draw two strong parallels between Kripke semantics and stochastic process (mainly martingales): first, the strong affinity of Kripke-Beth path semantics and time path of the process; and second, the treatment of time as parametrization to the dynamic process using the technique of filtration, adapted process, and progressive process. The technique provides very effective manipulation of BA in the form of random variables and σ-subalgebra under the cover of measurable functions. This enables us to adopt the computational algorithms obtained for stochastic processes to path semantics. Besides, using the technique of measurable functions, we indeed obtain an intrinsic way to introduce the notion of time sequence. © 2003 Wiley Periodicals, Inc.

Preduals of Spaces of Vector-Valued Holomorphic Functions

For U a balanced open subset of a Frechet space E and F a dual-Banach space we introduce the topology τγ on the space \(H\left( {U,F} \right)\) of holomorphic functions from U into F. This topology allows us to construct a predual for \(\left( {H\left( {U,F} \right),\tau \delta } \right)\) which in turn allows us to investigate the topological structure of spaces of vector-valued holomorphic functions. In particular, we are able to give necessary and sufficient conditions for the equivalence and compatibility of various topologies on spaces of vector-valued holomorphic functions.

Effective representations of the space of linear bounded operators

&lt;p&gt;Representations of topological spaces by infinite sequences of symbols are used in computable analysis to describe computations in topological spaces with the help of Turing machines. From the computer science point of view such representations can be considered as data structures of topological spaces. Formally, a representation of a topological space is a surjective mapping from Cantor space onto the corresponding space. Typically, one is interested in admissible, i.e. topologically well-behaved representations which are continuous and characterized by a certain maximality condition. We discuss a number of representations of the space of linear bounded operators on a Banach space. Since the operator norm topology of the operator space is nonseparable in typical cases, the operator space cannot be represented admissibly with respect to this topology. However, other topologies, like the compact open topology and the Fell topology (on the operator graph) give rise to a number of promising representations of operator spaces which can partially replace the operator norm topology. These representations reflect the information which is included in certain data structures for operators, such as programs or enumerations of graphs. We investigate the sublattice of these representations with respect to continuous and computable reducibility. Certain additional conditions, such as finite dimensionality, let some classes of representations collapse, and thus, change the corresponding graph. Altogether, a precise picture of possible data structures for operator spaces and their mutual relation can be drawn.&lt;/p&gt;

Classifying overlay structures of topological spaces

In 1972, R.H. Fox has generalized the classical classification theorem for covering mappings to a classification theorem for overlay structures over an arbitrary connected metric space. In the present paper we generalize his result to connected topological spaces. To achieve this we use ANR-resolutions of the base space, the fact that overlay mappings are pull-backs of covering mappings over ANRs and the classical classification theorem.

集合住宅の空間構造に関する基礎的研究

In this study, we showed how to understand the characteristics of the spatial structure in multi-story housing. We defined "the spatial unit" with be based on the notion of "control". We examined "the spatial structure in multi-story housing" by "the mutual relation of the spatial units " and "the collective form of the spatial units". Then we defined "the topological structure of space" and "the physical structure of space". We modeled with the spatial structure of multi-story housing and grasped its characteristics of the spatial structure.

Domain theoretic models of topological spaces

A model of a space X is simply a continuous dcpo D and a homeomorphism ∅: X → max D, where max D is given its inherited Scott topology. We show that a space has a coherent model iff it has a Scott domain model and investigate the topological structure of spaces which have Gδ models.