Partial soft separation axioms and soft compact spaces

In the current study the researchers have been introduced a modern kind of soft separation axioms which is named i-soft separation axioms by using the concept of soft i-open sets [17] in soft topological spaces, the relations among i-soft separation axioms and many examples about it are investigated. Further, they found that soft separation axioms imply i-soft separation axioms, but, the converses may not be true. Also, many theorems have been proved which are clarified the properties of i-soft separation axioms

The paper deals with a soft topological space which is dened over an initial universe set U with a xed set of parameters E . The main goal is to point out that any soft topological space is homeomorphic to a topological space $( E \times U, \tau )$ where $\tau$ is an arbitrary topology on the product $E \times U$ , consequently many soft topological notions and results can be derived from general topology. Furthermore, in many papers some notions are introduced by dierent ways and it would be good to give a unied approach for a transfer of topological notions to a soft set theory and to create a bridge between general topology and soft set theory.

This paper defines the base of soft topology, studies the properties of base. We give the definition of the product over the soft topological space, and discuss the relative properties of soft topological product space, and finally generate these results to the soft rough topological space.

Realizing the importance of separation axioms in classifications of topological spaces and studying certain properties of fixed points, we formulate new soft separation axioms, namely tt-soft and tt-soft b-regular spaces. Their definitions depend on three factors: soft b-open sets, total belong and total non-belong relations. In fact, they are genuine generalizations of p-soft -spaces in the cases of i = 0, 1, 2. With the help of examples, we study the relationships between them as well as with soft and soft b-regular spaces. Some interesting properties of them are obtained under the conditions of soft hyperconnected and extended soft topological spaces. Also, we show that they are preserved under finite product soft spaces and soft -homeomorphism mappings. Finally, we introduce a concept of b-fixed soft points and investigate its main properties.

By applying a new approach to concepts of both soft belonging which is called soft element and two distinct soft elements, we success to introduce and study the concepts of ${T_0}\left( {X,\tilde \tau ,A} \right)$, ${T_1}\left( {X,\tilde \tau ,A} \right)$, ${T_2}\left( {X,\tilde \tau ,A} \right)$ (soft hausdorff), ${T_3}\left( {X,\tilde \tau ,A} \right)$, (soft regularity), ${T_4}\left( {X,\tilde \tau ,A} \right)$ (soft normality) soft separation axioms in fuzzifying soft topology, and give some of their equivalents as well as the relation with each other of the five axioms.

Soft ω-local indiscreetness as a weaker form of both soft local countability and soft local indiscreetness is introduced. Then soft ω-regularity as a weaker form of both soft regularity and soft ω-local indiscreetness is defined and investigated. Additionally, soft ω-T2 as a new soft topological property that lies strictly between soft T2 and soft T1 is defined and investigated. It is proved that soft anti-local countability is a sufficient condition for equivalence between soft ω-locally indiscreetness (resp. soft ω-regularity) and soft locally indiscreetness (resp. soft ω-regularity). Additionally, it is proved that the induced topological spaces of a soft ω-locally indiscrete (resp. soft ω-regular, soft ω-T2) soft topological space are (resp. ω-regular, ω-T2) topological spaces. Additionally, it is proved that the generated soft topological space of a family of ω-locally indiscrete (resp. ω-regular, ω-T2) topological spaces is soft ω-locally indiscrete and vice versa. In addition to these, soft product theorems regarding soft ω-regular and soft ω-T2 soft topological spaces are obtained. Moreover, it is proved that soft ω-regular and soft ω-T2 are hereditarily under soft subspaces.

<abstract><p>This paper begins with an introduction to some soft topological operators that will be used to characterize several soft separation axioms followed by their main properties. Then, we define a new soft separation axiom called "soft $ T_D $-space" and analyze its main properties. We also show that this space precisely lies between soft $ T_0 $ and soft $ T_1 $-spaces. Finally, we characterize soft $ T_i $-spaces, for $ i = 0, 1, D $, in terms of the stated operators.</p></abstract>

In this paper, we introduce the concept of soft g*closed sets and soft g* open sets in soft topological space which are defined over an initial universe set with a fixed set of parameters together with its corresponding soft g*closure and soft g*interior operators. Also, we introduce the concept of soft and regular spaces and investigate the relationship between them.

" In this paper, we use the concept of the soft turing point and join it with separation axioms in soft topological space and investigate the relationship between them and study the most important properties and results of it.

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.

In this paper, some new generalized R-KKM type theorems for generalized R-KKM mappings with finitely closed values and with finitely open values are established in noncompact topological spaces without any convexity structure under much weaker assumptions. As applications, some new minimax inequalities, saddle point theorem and equilibrium existence theorem for equilibrium problems with lower and upper bounds are established in general noncompact topological spaces. These theorems unify and generalize many known results in the literature.

The present paper comes across the main steps that were laid from Zadeh’s fuzziness and Atanassov’s intuitionistic fuzzy sets to Smarandache’s indeterminacy and to Molodstov’s soft sets. Two hybrid methods for assessment and decision making, respectively, under fuzzy conditions are also presented using suitable examples that use soft sets and real intervals as tools. The decision making method improves on an earlier method of Maji et al. Further, it is described how the concept of topological space, the most general category of mathematical spaces, can be extended to fuzzy structures and how to generalize the fundamental mathematical concepts of limit, continuity compactness and Hausdorff space within such kinds of structures. In particular, fuzzy and soft topological spaces are defined and examples are given to illustrate these generalizations.

Emerging trends in soft set theory and related topics.

Near soft sets are considered as mathematical tools for dealing with ambiguities. In this study, we describe the near soft connectedness in near soft topological spaces, and introduce its concerned properties. Near soft sets are considered as mathematical tools for dealing with ambiguities. In this study, we describe the near soft connectedness in near soft topological spaces, and introduce its concerned properties. Near soft sets are considered as mathematical tools for dealing with ambiguities. In this study, we describe the near soft connectedness in near soft topological spaces, and introduce its concerned properties.

In the present paper, soft -open and soft -closed sets in soft topological spaces are defined over an initial universe with a set of parameters. A necessary and sufficient condition for a soft set to be soft -open set in soft topological space is stated and proved. A detailed study is carried out on properties of soft -interier and soft -closure of soft sets.