scholarly journals On Product of Smooth Neutrosophic Topological Spaces

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1557
Author(s):  
Kalaivani Chandran ◽  
Swathi Sundari Sundaramoorthy ◽  
Florentin Smarandache ◽  
Saeid Jafari
Keyword(s):  
Topological Spaces ◽  
Product Topology ◽  
Definition Of ◽  
Natural Way

In this paper, we develop the notion of the basis for a smooth neutrosophic topology in a more natural way. As a sequel, we define the notion of symmetric neutrosophic quasi-coincident neighborhood systems and prove some interesting results that fit with the classical ones, to establish the consistency of theory developed. Finally, we define and discuss the concept of product topology, in this context, using the definition of basis.

scholarly journals £-Single Valued Extremally Disconnected Ideal Neutrosophic Topological Spaces

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 53
Author(s):  
Fahad Alsharari
Keyword(s):  
Topological Spaces ◽  
Neutrosophic Sets ◽  
Extremally Disconnected ◽  
Fundamental Properties ◽  
Separation Axioms ◽  
New Concepts ◽  
Definition Of

This paper aims to mark out new concepts of r-single valued neutrosophic sets, called r-single valued neutrosophic £-closed and £-open sets. The definition of £-single valued neutrosophic irresolute mapping is provided and its characteristic properties are discussed. Moreover, the concepts of £-single valued neutrosophic extremally disconnected and £-single valued neutrosophic normal spaces are established. As a result, a useful implication diagram between the r-single valued neutrosophic ideal open sets is obtained. Finally, some kinds of separation axioms, namely r-single valued neutrosophic ideal-Ri (r-SVNIRi, for short), where i={0,1,2,3}, and r-single valued neutrosophic ideal-Tj (r-SVNITj, for short), where j={1,2,212,3,4}, are introduced. Some of their characterizations, fundamental properties, and the relations between these notions have been studied.

scholarly journals Generalized Knaster-Kuratowski-Mazurkiewicz type theorems and applications to minimax inequalities

2017 ◽  
Vol 20 (K2) ◽  
pp. 131-140
Author(s):  
Linh Manh Ha
Keyword(s):  
Variational Inequalities ◽  
Equilibrium Problems ◽  
Applied Mathematics ◽  
Sufficient Conditions ◽  
Topological Spaces ◽  
Minimax Theorems ◽  
Minimax Inequalities ◽  
Special Cases ◽  
Kkm Mappings ◽  
Definition Of

Knaster-Kuratowski-Mazurkiewicz type theorems play an important role in nonlinear analysis, optimization, and applied mathematics. Since the first well-known result, many international efforts have been made to develop sufficient conditions for the existence of points intersection (and their applications) in increasingly general settings: Gconvex spaces [21, 23], L-convex spaces [12], and FCspaces [8, 9]. Applications of Knaster-Kuratowski-Mazurkiewicz type theorems, especially in existence studies for variational inequalities, equilibrium problems and more general settings have been obtained by many authors, see e.g. recent papers [1, 2, 3, 8, 18, 24, 26] and the references therein. In this paper we propose a definition of generalized KnasterKuratowski-Mazurkiewicz mappings to encompass R-KKM mappings [5], L-KKM mappings [11], T-KKM mappings [18, 19], and many recent existing mappings. Knaster-KuratowskiMazurkiewicz type theorems are established in general topological spaces to generalize known results. As applications, we develop in detail general types of minimax theorems. Our results are shown to improve or include as special cases several recent ones in the literature.

scholarly journals Topological characterizations of amenability and congeniality of bases

2020 ◽  
Vol 21 (1) ◽  
pp. 1
Author(s):  
Sergio R. López-Permouth ◽  
Benjamin Stanley
Keyword(s):  
Direct Product ◽  
Topological Spaces ◽  
Countable Dimension ◽  
One To One ◽  
Natural Way

<div>We provide topological interpretations of the recently introduced notions of amenability and congeniality of bases of innite dimensional algebras. In order not to restrict our attention only to the countable dimension case, the uniformity of the topologies involved is analyzed and therefore the pertinent ideas about uniform topological spaces are surveyed.</div><div><p>A basis B over an innite dimensional F-algebra A is called amenable if F<sup>B</sup>, the direct product indexed by B of copies of the eld F, can be made into an A-module in a natural way. (Mutual) congeniality is a relation that serves to identify cases when different amenable bases yield isomorphic A-modules.</p><p>(Not necessarily mutual) congeniality between amenable bases yields an epimorphism of the modules they induce. We prove that this epimorphism is one-to-one only if the congeniality is mutual, thus establishing a precise distinction between the two notions.</p></div>

PAIRWISE α-PARACOMPACT SUBSETS

2021 ◽  
Vol 10 (5) ◽  
pp. 2631-2639
Author(s):  
M. Benbettiche ◽  
H.Z. Hdeib
Keyword(s):  
Topological Spaces ◽  
Definition Of ◽  
Bitopological Spaces

In this paper we define pairwise-$\alpha$-paracompact subsets as a generalisation of $\alpha$-paracompact subsets in topological spaces and study their properties and its relations with other bitopological spaces. Several theorems are obtained concerning pairwise-$\alpha$-paracompact sets. We introduced the definition of $p$-paracompact bitoplogical spaces and study their properties and its relations with other bitopological spaces.

Left Cauchy Integral Bases in Linear Topological Spaces

1970 ◽  
Vol 13 (4) ◽  
pp. 431-439 ◽  
Author(s):  
James A. Dyer
Keyword(s):  
Linear Topological Space ◽  
Integral Representations ◽  
Topological Spaces ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Natural Analogue ◽  
Cauchy Integral ◽  
Integral Basis ◽  
Integral Bases ◽  
Definition Of

The purpose of this paper is to consider a representation for the elements of a linear topological space in the form of a σ-integral over a linearly ordered subset of V; this ordered subset is what will be called an L basis. The formal definition of an L basis is essentially an abstraction from ideas used, often tacitly, in proofs of many of the theorems concerning integral representations for continuous linear functionals on function spaces.The L basis constructed in this paper differs in several basic ways from the integral basis considered by Edwards in [5]. Since the integrals used here are of Hellinger type rather than Radon type one has in the approximating sums for the integral an immediate and natural analogue to the partial sum operators of summation basis theory.

scholarly journals Near-rings of quotients of endomorphism near-rings

1975 ◽  
Vol 19 (4) ◽  
pp. 345-352 ◽  
Author(s):  
Michael Holcombe
Keyword(s):  
Additive Group ◽  
Finite Products ◽  
Definition Of ◽  
Group Object ◽  
Rings Of Quotients ◽  
Natural Way

Let be a category with finite products and a final object and let X be any group object in . The set of -morphisms, (X, X) is, in a natural way, a near-ring which we call the endomorphism near-ring of X in Such nearrings have previously been studied in the case where is the category of pointed sets and mappings, (6). Generally speaking, if Γ is an additive group and S is a semigroup of endomorphisms of Γ then a near-ring can be generated naturally by taking all zero preserving mappings of Γ into itself which commute with S (see 1). This type of near-ring is again an endomorphism near-ring, only the category is the category of S-acts and S-morphisms (see (4) for definition of S-act, etc.).

scholarly journals Boundaries in digital planes

1990 ◽  
Vol 3 (1) ◽  
pp. 27-55 ◽  
Author(s):  
Efim Khalimsky ◽  
Ralph Kopperman ◽  
Paul R. Meyer
Keyword(s):  
Jordan Curve ◽  
Ordered Set ◽  
Continuous Functions ◽  
Jordan Curve Theorem ◽  
Connected Sets ◽  
Digital Plane ◽  
Parameter Interval ◽  
Totally Ordered Set ◽  
Definition Of ◽  
Natural Way

The importance of topological connectedness properties in processing digital pictures is well known. A natural way to begin a theory for this is to give a definition of connectedness for subsets of a digital plane which allows one to prove a Jordan curve theorem. The generally accepted approach to this has been a non-topological Jordan curve theorem which requires two different definitions, 4-connectedness, and 8-connectedness, one for the curve and the other for its complement.In [KKM] we introduced a purely topological context for a digital plane and proved a Jordan curve theorem. The present paper gives a topological proof of the non-topological Jordan curve theorem mentioned above and extends our previous work by considering some questions associated with image processing:How do more complicated curves separate the digital plane into connected sets? Conversely given a partition of the digital plane into connected sets, what are the boundaries like and how can we recover them? Our construction gives a unified answer to these questions.The crucial step in making our approach topological is to utilize a natural connected topology on a finite, totally ordered set; the topologies on the digital spaces are then just the associated product topologies. Furthermore, this permits us to define path, arc, and curve as certain continuous functions on such a parameter interval.

scholarly journals Objets compacts dans les topos

1986 ◽  
Vol 40 (2) ◽  
pp. 203-217 ◽  
Author(s):  
Eduardo J. Dubuc ◽  
Jacques Penon
Keyword(s):  
Differential Geometry ◽  
Topological Space ◽  
Compact Manifolds ◽  
Topological Spaces ◽  
Internal Logic ◽  
Definition Of

AbstractIt is well known that compact topological spaces are those space K for which given any point x0 in any topological space X, and a neighborhood H of the fibre -1 {x0} KXX, then there exists a neighborhood U of x0 such that -1UH. If now is an object in an arbitrary topos, in the internal logic of the topos this property means that, for any A in and B in K, we have (-1AB)=AB. We introduce this formula as a definition of compactness for objects in an arbitrary topos. Then we prove that in the gross topoi of algebraic, analytic, and differential geometry, this property characterizes exactly the complete varieties, the compact (analytic) spaces, and the compact manifolds, respectively.

scholarly journals CONTACT STRUCTURES, σ-CONFOLIATIONS AND CONTAMINATIONS IN 3-MANIFOLDS

2009 ◽  
Vol 11 (02) ◽  
pp. 201-264 ◽  
Author(s):  
ULRICH OERTEL ◽  
JACEK ŚWIATKOWSKI
Keyword(s):  
Contact Structure ◽  
Contact Structures ◽  
Tight Contact ◽  
Branched Surface ◽  
Definition Of ◽  
Branched Surfaces ◽  
Natural Way

We propose in this paper a method for studying contact structures in 3-manifolds by means of branched surfaces. We explain what it means for a contact structure to be carried by a branched surface embedded in a 3-manifold. To make the transition from contact structures to branched surfaces, we first define auxiliary objects called σ-confoliations and pure contaminations, both generalizing contact structures. We study various deformations of these objects and show that the σ-confoliations and pure contaminations obtained by suitably modifying a contact structure remember the contact structure up to isotopy. After defining tightness for all pure contaminations in a natural way, generalizing the definition of tightness for contact structures, we obtain some conditions on (the embedding of) a branched surface in a 3-manifold sufficient to guarantee that any pure contamination carried by the branched surface is tight. We also find conditions sufficient to prove that a branched surface carries only overtwisted (non-tight) contact structures. Our long-term goal in developing these methods is twofold: Not only do we want to study tight contact structures and pure contaminations, but we also wish to use them as tools for studying 3-manifold topology.

scholarly journals Lie groupoids and algebroids applied to the study of uniformity and homogeneity of Cosserat media

2018 ◽  
Vol 15 (08) ◽  
pp. 1830003 ◽  
Author(s):  
Víctor Manuel Jiménez ◽  
Manuel de León ◽  
Marcelo Epstein
Keyword(s):  
Second Order ◽  
Lie Algebroid ◽  
Lie Groupoids ◽  
Lie Groupoid ◽  
Text Structures ◽  
Cosserat Medium ◽  
Homogeneity Property ◽  
Definition Of ◽  
Cosserat Media ◽  
Natural Way

A Lie groupoid, called second-order non-holonomic material Lie groupoid, is associated in a natural way to any Cosserat medium. This groupoid is used to give a new definition of homogeneity which does not depend on a material archetype. The corresponding Lie algebroid, called second-order non-holonomic material Lie algebroid, is used to characterize the homogeneity property of the material. We also relate these results with the previously obtained ones in terms of non-holonomic second-order [Formula: see text]-structures.

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