scholarly journals On topological soft sets

2020 ◽  
Vol 16 (2) ◽  
pp. 5-22
Author(s):  
M. Burç Kandemir ◽  
B. Tanay
Keyword(s):  
Topological Space ◽  
Structural Properties ◽  
Soft Set ◽  
Point Of View ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Soft Sets ◽  
Closed Sets ◽  
Necessary And Sufficient

AbstractIn this paper, we have established topological soft sets over generalized topological spaces and topological spaces, and studied its structural properties. We have derived a topological soft set in any given topological space, and from this point of view, we have given necessary and sufficient condition for homeomorphic Alexandroff spaces using topological soft set technique. At last, we have derived a topological soft set using closed sets in any topological space and we have given necessary and sufficient condition for arbitrary homeomorphic topological spaces using them.

scholarly journals Minimal bases and minimal sub-bases for topological spaces

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 1957-1965
Author(s):  
Yiliang Li ◽  
Jinjin Li ◽  
Jun-e Feng ◽  
Hongkun Wang
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
General Topological Space ◽  
Minimal Base ◽  
Necessary And Sufficient

This paper investigates minimal bases and minimal sub-bases for topological spaces. First, a necessary and sufficient condition is derived for the existence of minimal base for a general topological space. Then the concept of minimal sub-base for a topological space is proposed and its properties are discussed. Finally, for Alexandroff spaces, some special results with respect to minimal bases and minimal sub-bases are illustrated.

On Metrizability of Topological Spaces

1968 ◽  
Vol 20 ◽  
pp. 795-804 ◽  
Author(s):  
Carlos J. R. Borges
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Metrizable Space ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Moore Space ◽  
Necessary And Sufficient ◽  
Preceding Question

Our present work is divided into three sections. In §2 we study the metrizability of spaces with a Gδ-diagonal (see Definition 2.1). In §3 we study the metrization of topological spaces by means of collections of (not necessarily continuous) real-valued functions on a topological space. Our efforts, in §§2 and 3, are directed toward answering the following question: “Is every normal, metacompact (see Definition 2.4) Moore space a metrizable space?” which still remains unsolved. (However, Theorems 2.12 through 2.15 and Theorem 3.1 may be helpful in answering the preceding question.) In §4 we prove an apparently new necessary and sufficient condition for the metrizability of the Stone-Čech compactification of a metrizable space and hence for the compactness of a metric space.

scholarly journals Lorentz Boosts and Wigner Rotations: Self-Adjoint Complexified Quaternions

Physics ◽  
2021 ◽  
Vol 3 (2) ◽  
pp. 352-366
Author(s):  
Thomas Berry ◽  
Matt Visser
Keyword(s):  
Point Of View ◽  
Sufficient Condition ◽  
Use Of Self ◽  
Necessary And Sufficient Condition ◽  
Wigner Rotation ◽  
Inertial Frames ◽  
Explicit Formulae ◽  
Specific Implementation ◽  
Necessary And Sufficient ◽  
Lorentz Boosts

In this paper, Lorentz boosts and Wigner rotations are considered from a (complexified) quaternionic point of view. It is demonstrated that, for a suitably defined self-adjoint complex quaternionic 4-velocity, pure Lorentz boosts can be phrased in terms of the quaternion square root of the relative 4-velocity connecting the two inertial frames. Straightforward computations then lead to quite explicit and relatively simple algebraic formulae for the composition of 4-velocities and the Wigner angle. The Wigner rotation is subsequently related to the generic non-associativity of the composition of three 4-velocities, and a necessary and sufficient condition is developed for the associativity to hold. Finally, the authors relate the composition of 4-velocities to a specific implementation of the Baker–Campbell–Hausdorff theorem. As compared to ordinary 4×4 Lorentz transformations, the use of self-adjoint complexified quaternions leads, from a computational view, to storage savings and more rapid computations, and from a pedagogical view to to relatively simple and explicit formulae.

Some Structural Properties of Systolic Tree Automata

1989 ◽  
Vol 12 (4) ◽  
pp. 571-585
Author(s):  
E. Fachini ◽  
A. Maggiolo Schettini ◽  
G. Resta ◽  
D. Sangiorgi
Keyword(s):  
Structural Properties ◽  
Stability Problem ◽  
Sufficient Condition ◽  
Tree Automata ◽  
Necessary And Sufficient Condition ◽  
The Stability ◽  
Necessary And Sufficient

We prove that the classes of languages accepted by systolic automata over t-ary trees (t-STA) are always either equal or incomparable if one varies t. We introduce systolic tree automata with base (T(b)-STA), a subclass of STA with interesting properties of modularity, and we give a necessary and sufficient condition for the equivalence between a T(b)-STA and a t-STA, for a given base b. Finally, we show that the stability problem for T(b)-ST A is decidible.

scholarly journals ‎INSERTION OF A CONTRA-CONTINUOUS FUNCTION BETWEEN TWO ‎‎COMPARABLE CONTRA-$\alpha-$CONTINUOUS (CONTRA-$C-$CONTINUOUS) FUNCTIONS

2019 ◽  
pp. 13
Author(s):  
Majid Mirmiran ◽  
Binesh Naderi
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

‎A necessary and sufficient condition in terms of lower cut sets ‎are given for the insertion of a contra-continuous function ‎between two comparable real-valued functions on such topological ‎spaces that kernel of sets are open‎. 

Separable Topological Space of Hereditary

NUTA Journal ◽  
2020 ◽  
Vol 7 (1-2) ◽  
pp. 68-70
Author(s):  
Raj Narayan Yadav ◽  
Bed Prasad Regmi ◽  
Surendra Raj Pathak
Keyword(s):  
Topological Space ◽  
Topological Property ◽  
Sufficient Condition ◽  
Topological Properties ◽  
Necessary And Sufficient Condition ◽  
Separable Space ◽  
Separable Topological Space ◽  
Necessary And Sufficient

A property of a topological space is termed hereditary ifand only if every subspace of a space with the property also has the property. The purpose of this article is to prove that the topological property of separable space is hereditary. In this paper we determine some topological properties which are hereditary and investigate necessary and sufficient condition functions for sub-spaces to possess properties of sub-spaces which are not in general hereditary.

scholarly journals Metrisation of Moore spaces and abstract topological manifolds

1997 ◽  
Vol 56 (3) ◽  
pp. 395-401 ◽  
Author(s):  
David L. Fearnley
Keyword(s):  
Recent Work ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Moore Space ◽  
Global Coherence ◽  
Topological Manifolds ◽  
Necessary And Sufficient ◽  
Moore Spaces ◽  
Significant Class

The problem of metrising abstract topological spaces constitutes one of the major themes of topology. Since, for each new significant class of topological spaces this question arises, the problem is always current. One of the famous metrisation problems is the Normal Moore Space Conjecture. It is known from relatively recent work that one must add special conditions in order to be able to get affirmative results for this problem. In this paper we establish such special conditions. Since these conditions are characterised by local simplicity and global coherence they are referred to in this paper generically as “abstract topological manifolds.” In particular we establish a generalisation of a classical development of Bing, giving a proof which is complete in itself, not depending on the result or arguments of Bing. In addition we show that the spaces recently developed by Collins designated as “W satisfying open G(N)” are metrisable if they are locally separable and locally connected and regular. Finally, we establish a new necessary and sufficient condition for spaces to be metrisable.

Construction of sheaves by tolerance relations

2016 ◽  
Vol 09 (02) ◽  
pp. 1650035 ◽  
Author(s):  
M. P. K. Kishore ◽  
R. V. G. Ravi Kumar ◽  
P. Vamsi Sagar
Keyword(s):  
Topological Space ◽  
Universal Algebra ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

In this paper a method to construct a global sheaf space over a topological space for an arbitrary set using tolerance relations is proposed. It is observed that in general, the sheaf constructed by this method is different from the sheaf constructed by the method discussed in [U. M. Swamy, Representation of Universal algebra by sheaves, Proc. Amer. Math. Soc. 45 (1974) 55–58]. A necessary and sufficient condition for embedding a non-empty set into the set of all global sections of a sheaf over given topological space is established, and further an application over graphs is studied.

On the Modulii of Analytic Functions

1970 ◽  
Vol 13 (3) ◽  
pp. 325-327 ◽  
Author(s):  
Malcolm J. Sherman
Keyword(s):  
Analytic Functions ◽  
Point Of View ◽  
Sufficient Condition ◽  
Two Dimensional ◽  
Necessary And Sufficient Condition ◽  
Concrete Form ◽  
Norm Equality ◽  
Image Position ◽  
Necessary And Sufficient

The problem to be considered in this note, in its most concrete form, is the determination of all quartets f1, f2, g1, g2 of functions analytic on some domain and satisfying*where p > 0. When p = 2 the question can be reformulated in terms of finding a necessary and sufficient condition for (two-dimensional) Hilbert space valued analytic functions to have equal pointwise norms, and the answer (Theorem 1) justifies this point of view. If p ≠ 2, the problem is solved by reducing to the case p = 2, and the reformulation in terms of the norm equality of lp valued analytic functions gives no clue to the answer.

Anselm's Account of Freedom

2001 ◽  
Vol 31 (2) ◽  
pp. 221-244 ◽  
Author(s):  
Thomas Williams ◽  
Sandra Visser
Keyword(s):  
Moral Responsibility ◽  
Point Of View ◽  
Alternative Possibilities ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Freedom Of Choice ◽  
Necessary And Sufficient

According to Anselm's official definition, freedom of choice is ‘the power to preserve rectitude of will for the sake of that rectitude itself.’ From the point of view of contemporary metaphysics, this is one of the most unhelpful definitions imaginable. Does such freedom require alternative possibilities, for example? Is it compatible with causal determination? Is the exercise of such freedom a necessary and sufficient condition for moral responsibility? The definition sheds no light on these questions.And so we need to move on from Anselm's definition to Anselm's account of freedom. Here, though, we encounter the opposite problem. Where Anselm's definition seems not to answer these questions at all, Anselm's account seems to answer all these questions, sometimes with a yes and sometimes with a no. Consider the question about alternative possibilities. In De libertate arbitrii, Anselm seems clearly to deny that freedom involves alternative possibilities. God, the good angels, and the blessed dead cannot do otherwise than preserve rectitude, but they are still free- freer, in fact, than those who are capable of abandoning rectitude.

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