General Topology-I

Here, we employ partial belong and total non-belong relations to construct a study consisting of two parts: one of them is related to soft topologies, and the other one is concerned with real-life applications. In the first part, we define a new class of soft separation axioms, namely e-soft $$T_i$$-spaces $$(i=0, 1, 2, 3, 4)$$. We formulate these spaces with respect to distinct ordinary points using partial belong and total non-belong relations. The merits of using these two relations are that they help to generate a wider class of soft spaces and open up the way for more real-life applications. With the help of examples, we show the relationships between them and investigate the interrelations between them and their parametric topological spaces. Also, we study under what condition the concepts of soft $$T_i$$, p-soft $$T_i$$ and e-soft $$T_i$$ are equivalent. Furthermore, we scrutinize their behaviours in terms of soft subspaces, soft topological properties and finite soft product spaces. In the second part, we introduce an algorithm using partial belong relations in a decision-making problem in order to bring out the optimal choices.

Partial belong relation on soft separation axioms and decision-making problem, two birds with one stone

This study introduces a new family of soft separation axioms and a real-life application utilizing partial belong and natural non-belong relations. First, we initiate the concepts of w-soft 
$$T_i$$

-spaces 
$$(i=0, 1, 2, 3, 4)$$

 with respect to distinct ordinary points. These concepts generate a wider family of soft spaces compared with soft 
$$T_i$$

-spaces, p-soft 
$$T_i$$

-spaces and e-soft 
$$T_i$$

-spaces. We illustrate the relationships between w-soft 
$$T_i$$

-spaces with the help of examples and discuss some sufficient conditions of soft topological spaces to be w-soft 
$$T_i$$

-spaces. Additionally, we point out that stable or soft regular spaces are sufficient conditions for the equivalence among the concepts of soft 
$$T_i$$

, p-soft 
$$T_i$$

 and w-soft 
$$T_i$$

. We highlight on explaining the links between w-soft 
$$T_i$$

-spaces and their parametric topological spaces and studying the role of enriched spaces in these links. Furthermore, we prove that w-soft 
$$T_i$$

-spaces are hereditary and topological properties, and they are preserved under finite product soft spaces. Finally, we propose an algorithm to bring out the optimal choices. This algorithm is based on dividing the whole parameters set into parameter sets and then apply a partial belong relation in the favorite soft sets. This application is supported with an interesting example to show how to implement this algorithm.

https://link.springer.com/content/pdf/10.1007/s40314-020-01161-3.pdf

Applications of partial belong and total non-belong relations on soft separation axioms and decision-making problem

An improvement of rough sets’ accuracy measure using containment neighborhoods with a medical application

The desire of generalizing some set-theoretic properties to the soft set theory motivated many researchers to define various types of soft operators. For example, they redefined the complement of a soft set, and soft union and intersection between two soft sets in a way that satisfies De Morgan's laws. In this paper, we introduce and study the concepts of $T$-soft subset and $T$-soft equality relations. Then, we utilize them to define the concepts of $T$-soft union and $T$-soft intersection for arbitrary family of soft sets. By $T$-soft union, we successfully keep some classical properties via soft set theory. We conclude this work by giving and investigating new types of soft linear equations with respect to some soft equality relations. Illustrative examples are provided to elucidate main obtained results.

https://journals.tubitak.gov.tr/math/issues/mat-20-44-4/mat-44-4-25-2005-117.pdf

$T$-soft equality relation

The main aim of the present paper is to define new soft separation axioms which lead us, first, to generalize existing comparable properties via general topology, second, to eliminate restrictions on the shape of soft open sets on soft regular spaces which given in [22], and third, to obtain a relationship between soft Hausdorff and new soft regular spaces similar to those exists via general topology. To this end, we define partial belong and total non belong relations, and investigate many properties related to these two relations. We then introduce new soft separation axioms, namely p-soft Ti-spaces (i = 0, 1, 2, 3, 4), depending on a total non belong relation, and study their features in detail. With the help of examples, we illustrate the relationships among these soft separation axioms and point out that p-soft Ti-spaces are stronger than soft Ti-spaces, for i = 0, 1, 4. Also, we define a p-soft regular space, which is weaker than a soft regular space and verify that a p-soft regular condition is sufficient for the equivalent among p-soft Ti-spaces, for i = 0, 1, 2. Furthermore, we prove the equivalent among finite p-soft Ti-spaces, for i = 1, 2, 3 and derive that a finite product of p-soft Ti-spaces is p-soft Ti, for i = 0, 1, 2, 3, 4. In the last section, we show the relationships which associate some p-soft Ti-spaces with soft compactness, and in particular, we conclude under what conditions a soft subset of a p-soft T2-space is soft compact and prove that every soft compact p-soft T2-space is soft T3-space. Finally, we illuminate that some findings obtained in general topology are not true concerning soft topological spaces which among of them a finite soft topological space need not be soft compact.

/pdf/partial-soft-separation-axioms-and-soft-compact-spaces-1gp9t3h17c.pdf

Tareq M. Al-shami

Papers

Partial soft separation axioms and soft compact spaces

Partial belong relation on soft separation axioms and decision-making problem, two birds with one stone

Applications of partial belong and total non-belong relations on soft separation axioms and decision-making problem

An improvement of rough sets’ accuracy measure using containment neighborhoods with a medical application

$T$-soft equality relation