scholarly journals Soft Open Bases and a Novel Construction of Soft Topologies from Bases for Topologies

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 672 ◽  
Author(s):  
José Carlos R. Alcantud
Keyword(s):  
Topological Space ◽  
General Construction ◽  
Topological Spaces ◽  
Soft Sets ◽  
Extensive Discussion ◽  
Soft Topology

Soft topology studies a structure on the collection of all soft sets on a given set of alternatives (the relevant attributes being fixed). It is directly inspired by the axioms of a topological space. This paper contributes to the theoretical bases of soft topology in various ways. We extend a general construction of soft topologies from topologies on the set of alternatives in two different directions. An extensive discussion with criteria about what a soft counterpart of “topological separability” should satisfy is also given. The interactions of the properties that arise with separability, and of second-countability and its soft counterpart, are studied under the general mechanisms that generate soft topological spaces. The first non-trivial examples of soft second-countable soft topological spaces are produced as a consequence.

α -CONNECTEDNESS BETWEEN SOFT SETS

2019 ◽  
Vol 7 (4) ◽  
pp. 09-14
Author(s):  
Alpa Singh Rajput ◽  
S. S. Thakur
Keyword(s):  
Image Processing ◽  
Image Segmentation ◽  
Topological Space ◽  
Topological Spaces ◽  
Soft Sets ◽  
Soft Topology ◽  
Feasible Path ◽  
Soft Topological Space

Purpose of the study: In the present paper the concept of soft α -connectedness between soft sets in soft topological spaces has been introduced and studied. The notion of connectedness captures the idea of hanging-togetherness of image elements in an object by given a firmness of connectedness to every feasible path between every possible pair of image elements. It is an important tool for the designing of algorithms for image segmentation. The purpose of this paper is to extend the concept of α –connectedness between sets in soft topology. Main Findings: If a soft topological space (X, τ, E) is soft α -connected between a pair of its soft sets, then it is not necessarily that it is soft α -connected between each pair of its soft sets and so it is not necessarily soft α -connected. Applications of this study: Image Processing. Novelty/Originality of this study: Extend of α -connectedness between soft sets in soft topology.

scholarly journals Soft Semi Local Functions in Soft Ideal Topological Spaces

2019 ◽  
Vol 12 (3) ◽  
pp. 857-869
Author(s):  
Fatouh Gharib ◽  
Alaa Mohamed Abd El-latif
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Local Function ◽  
Soft Sets ◽  
Soft Topology ◽  
Local Functions ◽  
Equivalent Conditions

In this paper, we define a soft semi local function (F, E) ∗s ( ˜I, τ ) by using semi open soft sets in a soft ideal topological space (X, τ, E, ˜I). This concept is discussed with a view to find new soft topologies from the original one, called ∗s-soft topology. Some properties and characterizations of soft semi local function are explored. Finally, the notion of soft semi compatibility of soft ideals with soft topologies is introduced and some equivalent conditions concerning this topic are established here.

scholarly journals On Two Classes of Soft Sets in Soft Topological Spaces

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 265 ◽  
Author(s):  
Samer Al Ghour ◽  
Worood Hamed
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Logical Connective ◽  
Soft Sets ◽  
Natural Properties ◽  
Soft Topological Space

In this paper, we define soft ω -open sets and strongly soft ω -open sets as two new classes of soft sets. We study the natural properties of these types of soft sets and we study the validity of the exact versions of some known results in ordinary topological spaces regarding ω -open sets in soft topological spaces. Also, we study the relationships between the ω -open sets of a given indexed family of topological spaces and the soft ω -open sets (resp. strongly soft ω -open sets) of their generated soft topological space. These relationships form a biconditional logical connective which is a symmetry. As an application of strongly soft ω -open sets, we characterize soft Lindelof (resp. soft weakly Lindelof) soft topological spaces.

scholarly journals New Soft Structure: Infra Soft Topological Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Tareq M. Al-shami
Keyword(s):  
Topological Space ◽  
Soft Set ◽  
Topological Spaces ◽  
Closure Operators ◽  
Relative Topology ◽  
Closed Sets ◽  
Soft Topology ◽  
Soft Limit ◽  
Basic Concepts ◽  
Soft Topological Space

It is always convenient to find the weakest conditions that preserve some topologically inspired properties. To this end, we introduce the concept of an infra soft topology which is a collection of subsets that extend the concept of soft topology by dispensing with the postulate that the collection is closed under arbitrary unions. We study the basic concepts of infra soft topological spaces such as infra soft open and infra soft closed sets, infra soft interior and infra soft closure operators, and infra soft limit and infra soft boundary points of a soft set. We reveal the main properties of these concepts with the help of some elucidative examples. Then, we present some methods to generate infra soft topologies such as infra soft neighbourhood systems, basis of infra soft topology, and infra soft relative topology. We also investigate how we initiate an infra soft topology from crisp infra topologies. In the end, we explore the concept of continuity between infra soft topological spaces and determine the conditions under which the continuity is preserved between infra soft topological space and its parametric infra topological spaces.

scholarly journals Caliber and Chain Conditions in Soft Topologies

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2349
Author(s):  
José Carlos R. Alcantud ◽  
Tareq M. Al-shami ◽  
A. A. Azzam
Keyword(s):  
Future Research ◽  
Topological Spaces ◽  
Soft Sets ◽  
Theoretical Underpinning ◽  
Chain Conditions ◽  
Soft Topology ◽  
Uncertain Knowledge ◽  
Point Set ◽  
Fruitful Field

In this paper, we contribute to the growing literature on soft topology. Its theoretical underpinning merges point-set or classical topology with the characteristics of soft sets (a model for the representation of uncertain knowledge initiated in 1999). We introduce two types of axioms that generalize suitable concepts of soft separability. They are respectively concerned with calibers and chain conditions. We investigate explicit procedures for the construction of non-trivial soft topological spaces that satisfy these new axioms. Then we explore the role of cardinality in their study, and the relationships among these and other properties. Our results bring to light a fruitful field for future research in soft topology.

scholarly journals On Soft Metric Spaces

2016 ◽  
Vol 12 (3) ◽  
pp. 6103-6110
Author(s):  
Radwa mohamed Hassan
Keyword(s):  
Mathematical Theory ◽  
General Framework ◽  
Metric Spaces ◽  
Topological Spaces ◽  
Topological Properties ◽  
Soft Sets ◽  
Soft Topology ◽  
Practical Life ◽  
Definition Of ◽  
Metric Function

After the famous article of Moldotsove [10] in 1999 which initiate the theory of soft sets as a mathematical theory to deal  with the uncertainty problems, many research works in the softbmathematics and its applications in various fields are appeared.   In [17], the authers introduced a new definition of the soft metric function using the soft elements. By this definition each soft metric in view of Das and Samanta [6] is also a soft metric in our concept but the converse is not true. In the present paper, some soft topological properties are given in details, namely (soft compactness, soft sequentially compactness, continuity and uniformly continues of soft functions between soft topological spaces).  We hope that the findings in thispaper will help researcher enhance and promote the further study on soft topology to carry out a general framework for theirapplications in practical life.

scholarly journals A Review of Fuzzy Soft Topological Spaces, Intuitionistic Fuzzy Soft Topological Spaces and Neutrosophic Soft Topological Spaces

2020 ◽  
pp. 96-104
Author(s):  
admin admin ◽  
◽  
◽  
◽  
M M.Karthika ◽  
...  
Keyword(s):  
Topological Space ◽  
Fuzzy Sets ◽  
Topological Structure ◽  
Intuitionistic Fuzzy Set ◽  
Soft Set ◽  
Topological Spaces ◽  
Soft Sets ◽  
Structure Space ◽  
Intuitionistic Fuzzy ◽  
Soft Topological Space

The notion of fuzzy sets initiated to overcome the uncertainty of an object. Fuzzy topological space, in- tuitionistic fuzzy sets in topological structure space, vagueness in topological structure space, rough sets in topological space, theory of hesitancy and neutrosophic topological space, etc. are the extension of fuzzy sets. Soft set is a family of parameters which is also a set. Fuzzy soft topological space, intuitionistic fuzzy soft and neutrosophic soft topological space are obtained by incorporating soft sets with various topological structures. This motivates to write a review and study on various soft set concepts. This paper shows the detailed review of soft topological spaces in various sets like fuzzy, Intuitionistic fuzzy set and neutrosophy. Eventually, we compared some of the existing tools in the literature for easy understanding and exhibited their advantages and limitations.

γ-Operation and Some Types of Soft Sets and Soft Continuity of (Supra) Soft Topological Spaces

2016 ◽  
pp. 127-171
Author(s):  
Ali Kandil ◽  
Osama A. El-Tantawy ◽  
Sobhy A. El-Sheikh ◽  
A. M. Abd El-latif
Keyword(s):  
Topological Space ◽  
Soft Set ◽  
Topological Spaces ◽  
Topological Properties ◽  
Soft Sets ◽  
Soft Mapping ◽  
Separation Axioms ◽  
Different Types

The main purpose of this chapter is to introduce the notions of ?-operation, pre-open soft set a-open sets, semi open soft set and ß-open soft sets to soft topological spaces. We study the relations between these different types of subsets of soft topological spaces. We introduce new soft separation axioms based on the semi open soft sets which are more general than of the open soft sets. We show that the properties of soft semi Ti-spaces (i=1,2) are soft topological properties under the bijection and irresolute open soft mapping. Also, we introduce the notion of supra soft topological spaces. Moreover, we introduce the concept of supra generalized closed soft sets (supra g-closed soft for short) in a supra topological space (X,µ,E) and study their properties in detail.

scholarly journals Bioperators on soft topological spaces

2021 ◽  
Vol 6 (11) ◽  
pp. 12471-12490
Author(s):  
Baravan A. Asaad ◽  
◽  
Tareq M. Al-shami ◽  
Abdelwaheb Mhemdi ◽  
◽  
...  
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Closed Sets ◽  
Fundamental Properties ◽  
Soft Topology ◽  
Open Set ◽  
Soft Topological Space

<abstract><p>To contribute to soft topology, we originate the notion of soft bioperators $ \tilde{\gamma} $ and $ {\tilde{\gamma}}^{'} $. Then, we apply them to analyze soft $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-open sets and study main properties. We also prove that every soft $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-open set is soft open; however, the converse is true only when the soft topological space is soft $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-regular. After that, we define and study two classes of soft closures namely $ Cl_{(\tilde{\gamma}, {\tilde{\gamma}}^{'})} $ and $ \tilde{\tau}_{(\tilde{\gamma}, {\tilde{\gamma}}^{'})} $-$ Cl $ operators, and two classes of soft interior namely $ Int_{(\tilde{\gamma}, {\tilde{\gamma}}^{'})} $ and $ \tilde{\tau}_{(\tilde{\gamma}, {\tilde{\gamma}}^{'})} $-$ Int $ operators. Moreover, we introduce the notions of soft $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-$ g $.closed sets and soft $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-$ T_{\frac{1}{2}} $ spaces, and explore their fundamental properties. In general, we explain the relationships between these notions, and give some counterexamples.</p></abstract>

scholarly journals On Soft Semi Weakly Generalized Closed Set in Soft TopologicalSpaces

2019 ◽  
Vol 8 (12) ◽  
pp. 1252-1256 ◽  
Keyword(s):  
Social Science ◽  
Topological Space ◽  
Life Science ◽  
Soft Sets ◽  
Science And Engineering ◽  
Closed Set ◽  
Soft Topology ◽  
Soft Topological Space

Soft sets has helped the development of soft topological space and it was also applied in the field of life science, Social science and Engineering. Many researchers developed various ideas based on the properties of soft topology. The article deals with study of properties in soft topological space based on soft semi weakly generalized closed set

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