The journal of mathematics and computer science, Oct 9, 2023
The aim of this study is to introduce and investigate two new classes of separation axioms called... more The aim of this study is to introduce and investigate two new classes of separation axioms called supra soft R 0 and supra soft R 1 . They are defined in the spaces of supra soft topologies by using the notions of supra soft open sets and supra soft closure operator. We discuss the basic properties and characterizations of them. We also study the relationships between these classes and some other supra soft separation axioms with many results and explanative examples. Moreover, the connections between the properties of these classes and those in some generated soft topologies are presented. Finally, we show that these classes are preserved under subspaces, which means they are supra soft topological properties.
Journal of mathematics and computer science, Apr 18, 2024
In this article, we study the concept of finite open sets in the spaces of supra soft-topologies.... more In this article, we study the concept of finite open sets in the spaces of supra soft-topologies. We aim by defining this concept to furnish the researchers with a tool they generate new forms of topological concepts via supra soft-topologies such as continuity, compactness, separation axioms, etc. We discuss the main properties of this concept and prove that this class of soft sets is neither infra soft-topology nor supra soft-topology. Also, we reveal its connections with other celebrated generalizations of supra soft-open sets and conclude the conditions they guarantee the equivalence between them. After that, we display some soft operators, like interior and closure, inspired by the classes of supra finite soft-open and supra finite soft-closed sets. Finally, we set forth some types of continuity defined by these classes and describe their main characterizations and behaviors in different cases such as decomposition theorems and transition between the realms of of supra soft-topologies and their parametric supra topologies. The obtained results and implementations are articulated with the aid of some counterexamples.
A useful tool for expressing fuzziness and uncertainty is the recently developed n,m-rung orthopa... more A useful tool for expressing fuzziness and uncertainty is the recently developed n,m-rung orthopair fuzzy set (n,m-ROFS). Due to their superior ability to manage uncertain situations compared to theories of q-rung orthopair fuzzy sets, the n,m-rung orthopair fuzzy sets have variety of applications in decision-making in daily life. To deal with ambiguity and unreliability in multi-attribute group decision-making, this study introduces a novel tool called k n m -rung picture fuzzy set (k n m -RPFS). The suggested k n m -RPFS incorporates all of the benefits of n,m-ROFS and represents both the quantitative and qualitative analyses of the decision-makers. The presented model is a fruitful advancement of the q-rung picture fuzzy set (q-RPFS). Furthermore, numerous of its key notions, including as complement, intersection, and union are explained and illustrated with instances. In many decision-making situations, the main benefit of k n m -rung picture fuzzy sets is the ability to express more uncertainty than q-rung picture fuzzy sets. Then, along with their numerous features, we discover the basic set of operations for the k n m -rung picture fuzzy sets. Importantly, we present a novel operator, k n m -rung picture fuzzy weighted power average (k n m -RPFWPA) over k n m -rung picture fuzzy sets, and use it to multi-attribute decision-making issues for evaluating alternatives with k n m -rung picture fuzzy information. Additionally, we use this operator to pinpoint the countries with the highest expat living standards and demonstrate how to choose the best option by comparing aggregate findings and applying score values. Finally, we compare the outcomes of the q-RPFEWA, SFWG, PFDWA, SFDWA, and SFWA operators to those of the k n m -RPFWPA operator. m -Rung picture fuzzy set • k n m -Rung picture fuzzy weighted power average operator • Multi-attribute decision-making problem
In this paper, we hybridize the bipolar fuzzy set (BFS) theory with T-rough sets (T-RSs) and init... more In this paper, we hybridize the bipolar fuzzy set (BFS) theory with T-rough sets (T-RSs) and initiate the novel idea of T-rough BFSs (T-RBFSs). The concept presented in this article has never been discussed earlier. Moreover, we investigate the axiomatic systems of T-RBFSs in detail. Meanwhile, we address a decision-making (DM) problem having data endowed with fuzziness and bipolarity in the framework of the T-RBFSs. We also propose an algorithm for this application. This algorithm facilitates tackling the case when there is a team of decision-makers instead of a single decision-maker and when the objects of one set need to be approximated by grading the objects of some other set. Moreover, a practical application of T-RBFSs in DM problems is given, accompanied by a practical example, which provides the optimal and the worst decision between some objects. Finally, a comparative analysis of the recommended study with several prevailing approaches is given to endorse the advantages of the suggested research.
A topological space X is countably paracompact if and only if X satisfies the condition (A): For ... more A topological space X is countably paracompact if and only if X satisfies the condition (A): For any decreasing sequence {Fi} of non-empty closed sets with $${\bigcap_{i=1}^{\infty} F_{i} = \emptyset}$$⋂i=1∞Fi=∅ there exists a sequence {Gi} of open sets such that $${\bigcap_{i=1}^{\infty}\overline{G_{i}}=\emptyset}$$⋂i=1∞Gi¯=∅ and $${F_{i} \subset G_{i}}$$Fi⊂Gi for every i. We will show, by an example, that this is not true in generalized topological spaces. In fact there is a $${\mu}$$μ-normal generalized topological space satisfying the analogue of A which is not even countably $${\mu}$$μ-metacompact. Then we study the relationships between countably $${\mu}$$μ-paracompactness, countably $${\mu}$$μ-metacompactness and the condition corresponding to condition A in generalized topological spaces.
In topological spaces every normal space with a countable base is metacompact. We show that this ... more In topological spaces every normal space with a countable base is metacompact. We show that this is not necessarily true in generalized topological spaces; more exactly we give an example of a μ-normal space with a countable μ-base which has a μ-open cover with no μ-open point-finite refinement.
In this paper, we first explore more properties of soft somewhere dense continuous functions. The... more In this paper, we first explore more properties of soft somewhere dense continuous functions. Then, we discuss the preservation of soft Baire property and soft Baire category. We give some concrete examples to illustrate how our findings extend some conclusions and connections made in the literature.
This work introduces weakly soft β-open subsets, a new family of soft-open sets. By this family, ... more This work introduces weakly soft β-open subsets, a new family of soft-open sets. By this family, we expand a soft topology to a soft structure which is neither supra-soft topology nor infra-soft topology. The connections between this class of soft sets and other celebrated classes via soft topology are examined with some elucidative examples. Also, it is established some relationships under conditions of extended and hyperconnected soft topologies. Furthermore, the interior and closure operators are structured along with weakly soft β-open and weakly soft β-closed sets. Finally, the class of weakly soft β-continuous functions is introduced and its main characterizations are studied. It is investigated the systematic relationships and findings that are lost for this kind of soft continuity as well as it is shown the conditions required to maintain some of these relationships such as full, extended and hyperconnected soft topologies.
Fuzzy set (FS) theory and rough sets (RSs) are constructed to accommodate the data uncertainty. I... more Fuzzy set (FS) theory and rough sets (RSs) are constructed to accommodate the data uncertainty. In contrast, the bipolar FS (BFS) theory can tackle the uncertainty and the bipolarity of the data in different circumstances. This article aims to introduce the idea of rough bipolar fuzzy ideals in semigroup (SG), which is a generalization of the concept of rough BFSs (RBFSs) in an SG. We also investigate the roughness in the bipolar fuzzy subsemigroup (BF-SSG) with the help of congruence relation (cng-R) defined on the SG and studied some relevant structural properties. Moreover, the idea is extended to the rough bipolar fuzzy left ideal, rough bipolar fuzzy right ideal, rough bipolar fuzzy two-sided ideal, rough bipolar fuzzy interior ideal, and rough bipolar fuzzy bi-ideal in SGs. Further, it is seen that cng-Rs and complete cng-Rs play vital roles in the construction of rough approximations of bipolar fuzzy ideals. Consequently, their associated properties are explored by using cng-...
Our target in the present work, is presenting the idea of weakly soft preopen (ws-preopen) subset... more Our target in the present work, is presenting the idea of weakly soft preopen (ws-preopen) subsets and studying some of its characterizations. With the assistance of some elucidative examples, the interrelationships between ws-preopen sets and some extensions of soft open sets are studied. Under some conditions such as extended and hyperconnected soft topologies, several motivating results and relationships are acquired. The interior and closure operators that built through ws-preopen and ws-preclosed subsets are introduced. Their main features that construe the relations among them are established. Soft continuity with respect to theses classes of soft subsets are studied and their substantial characteristics are investigated. Generally, the systematic relations and outcomes that are lost through the scope of this study are discussed. The proposed line in the current study will present new ways to discover novel concepts in the field of soft topology.
In this paper, the concept of neutrosophic topological spaces is introduced. We dene and study&l... more In this paper, the concept of neutrosophic topological spaces is introduced. We dene and study<br> the properties of neutrosophic open sets, closed sets, interior and closure. The set of all generalize<br> neutrosophic pre-closed sets GNPC( ) and the set of all neutrosophic -open sets in a neutrosophic topological<br> space (X; ) can be considered as examples of generalized neutrosophic topological spaces. The concept<br> of neutrosophic continuity is dened and we studied their properties. We dene and study the properties<br> of neutrosophic compact, -Lindelof and -countably compact spaces. We prove that for a countable<br> neutrosophic -space X: -countably compactness and -compactness are equivalent. We give an example of<br> a neutrosophic -space X which has a neutrosophic countable -base but it is not neutrosophic -countably<br> compact .
In this paper, the concept of neutrosophic μ−topological spaces is introduced. We define and stud... more In this paper, the concept of neutrosophic μ−topological spaces is introduced. We define and study the properties of neutrosophic μ−open sets, μ−closed sets, μ−interior and μ−closure. The set of all generalize neutrosophic pre-closed sets GNPC(τ) and the set of all neutrosophic α-open sets in a neutrosophic topological space (X, τ) can be considered as examples of generalized neutrosophic μ−topological spaces. The concept of neutrosophic μ − continuity is defined and we studied their properties. We define and study the properties of neutrosophic μ − compact, μ-Lindelöf and μ-countably compact spaces. We prove that for a countable neutrosophic μ-space X: μ-countably compactness and μ-compactness are equivalent. We give an example of a neutrosophic μ-space X which has a neutrosophic countable μ-base but it is not neutrosophic μ-countably compact . neutrosophic μ−topological spaces;μ−open; μ−closed; μ−interior; μ−closure; generalize neutrosophic pre-closed sets; neutrosophic α-open set...
In this paper, the concept of neutrosophic μ−topological spaces is introduced. We define and stud... more In this paper, the concept of neutrosophic μ−topological spaces is introduced. We define and study the properties of neutrosophic μ−open sets, μ−closed sets, μ−interior and μ−closure. The set of all generalize neutrosophic pre-closed sets GNPC(τ) and the set of all neutrosophic α-open sets in a neutrosophic topological space (X, τ) can be considered as examples of generalized neutrosophic μ−topological spaces. The concept of neutrosophic μ − continuity is defined and we studied their properties. We define and study the properties of neutrosophic μ − compact, μ-Lindelöf and μ-countably compact spaces. We prove that for a countable neutrosophic μ-space X: μ-countably compactness and μ-compactness are equivalent. We give an example of a neutrosophic μ-space X which has a neutrosophic countable μ-base but it is not neutrosophic μ-countably compact . neutrosophic μ−topological spaces;μ−open; μ−closed; μ−interior; μ−closure; generalize neutrosophic pre-closed sets; neutrosophic α-open set...
We answer the following question: Are neutrosophic µ-compactness and neutrosophic µ-countably com... more We answer the following question: Are neutrosophic µ-compactness and neutrosophic µ-countably compactness equivalent? which posted in [10]. Since every neutrosophic topology is neutrosophic µ-topology, we answer the question for neutrosophic topological spaces, more precisely, we give an example of neutrosophic topology which is neutrosophic countably compact but not neutrosophic compact
We answer the following question: Are neutrosophic μ-compactness and neutrosophic μ-countably com... more We answer the following question: Are neutrosophic μ-compactness and neutrosophic μ-countably compactness equivalent? which posted in [10]. Since every neutrosophic topology is neutrosophic μ-topology, we answer the question for neutrosophic topological spaces, more precisely, we give an example of neutrosophic topology which is neutrosophic countably compact but not neutrosophic compact
The journal of mathematics and computer science, Oct 9, 2023
The aim of this study is to introduce and investigate two new classes of separation axioms called... more The aim of this study is to introduce and investigate two new classes of separation axioms called supra soft R 0 and supra soft R 1 . They are defined in the spaces of supra soft topologies by using the notions of supra soft open sets and supra soft closure operator. We discuss the basic properties and characterizations of them. We also study the relationships between these classes and some other supra soft separation axioms with many results and explanative examples. Moreover, the connections between the properties of these classes and those in some generated soft topologies are presented. Finally, we show that these classes are preserved under subspaces, which means they are supra soft topological properties.
Journal of mathematics and computer science, Apr 18, 2024
In this article, we study the concept of finite open sets in the spaces of supra soft-topologies.... more In this article, we study the concept of finite open sets in the spaces of supra soft-topologies. We aim by defining this concept to furnish the researchers with a tool they generate new forms of topological concepts via supra soft-topologies such as continuity, compactness, separation axioms, etc. We discuss the main properties of this concept and prove that this class of soft sets is neither infra soft-topology nor supra soft-topology. Also, we reveal its connections with other celebrated generalizations of supra soft-open sets and conclude the conditions they guarantee the equivalence between them. After that, we display some soft operators, like interior and closure, inspired by the classes of supra finite soft-open and supra finite soft-closed sets. Finally, we set forth some types of continuity defined by these classes and describe their main characterizations and behaviors in different cases such as decomposition theorems and transition between the realms of of supra soft-topologies and their parametric supra topologies. The obtained results and implementations are articulated with the aid of some counterexamples.
A useful tool for expressing fuzziness and uncertainty is the recently developed n,m-rung orthopa... more A useful tool for expressing fuzziness and uncertainty is the recently developed n,m-rung orthopair fuzzy set (n,m-ROFS). Due to their superior ability to manage uncertain situations compared to theories of q-rung orthopair fuzzy sets, the n,m-rung orthopair fuzzy sets have variety of applications in decision-making in daily life. To deal with ambiguity and unreliability in multi-attribute group decision-making, this study introduces a novel tool called k n m -rung picture fuzzy set (k n m -RPFS). The suggested k n m -RPFS incorporates all of the benefits of n,m-ROFS and represents both the quantitative and qualitative analyses of the decision-makers. The presented model is a fruitful advancement of the q-rung picture fuzzy set (q-RPFS). Furthermore, numerous of its key notions, including as complement, intersection, and union are explained and illustrated with instances. In many decision-making situations, the main benefit of k n m -rung picture fuzzy sets is the ability to express more uncertainty than q-rung picture fuzzy sets. Then, along with their numerous features, we discover the basic set of operations for the k n m -rung picture fuzzy sets. Importantly, we present a novel operator, k n m -rung picture fuzzy weighted power average (k n m -RPFWPA) over k n m -rung picture fuzzy sets, and use it to multi-attribute decision-making issues for evaluating alternatives with k n m -rung picture fuzzy information. Additionally, we use this operator to pinpoint the countries with the highest expat living standards and demonstrate how to choose the best option by comparing aggregate findings and applying score values. Finally, we compare the outcomes of the q-RPFEWA, SFWG, PFDWA, SFDWA, and SFWA operators to those of the k n m -RPFWPA operator. m -Rung picture fuzzy set • k n m -Rung picture fuzzy weighted power average operator • Multi-attribute decision-making problem
In this paper, we hybridize the bipolar fuzzy set (BFS) theory with T-rough sets (T-RSs) and init... more In this paper, we hybridize the bipolar fuzzy set (BFS) theory with T-rough sets (T-RSs) and initiate the novel idea of T-rough BFSs (T-RBFSs). The concept presented in this article has never been discussed earlier. Moreover, we investigate the axiomatic systems of T-RBFSs in detail. Meanwhile, we address a decision-making (DM) problem having data endowed with fuzziness and bipolarity in the framework of the T-RBFSs. We also propose an algorithm for this application. This algorithm facilitates tackling the case when there is a team of decision-makers instead of a single decision-maker and when the objects of one set need to be approximated by grading the objects of some other set. Moreover, a practical application of T-RBFSs in DM problems is given, accompanied by a practical example, which provides the optimal and the worst decision between some objects. Finally, a comparative analysis of the recommended study with several prevailing approaches is given to endorse the advantages of the suggested research.
A topological space X is countably paracompact if and only if X satisfies the condition (A): For ... more A topological space X is countably paracompact if and only if X satisfies the condition (A): For any decreasing sequence {Fi} of non-empty closed sets with $${\bigcap_{i=1}^{\infty} F_{i} = \emptyset}$$⋂i=1∞Fi=∅ there exists a sequence {Gi} of open sets such that $${\bigcap_{i=1}^{\infty}\overline{G_{i}}=\emptyset}$$⋂i=1∞Gi¯=∅ and $${F_{i} \subset G_{i}}$$Fi⊂Gi for every i. We will show, by an example, that this is not true in generalized topological spaces. In fact there is a $${\mu}$$μ-normal generalized topological space satisfying the analogue of A which is not even countably $${\mu}$$μ-metacompact. Then we study the relationships between countably $${\mu}$$μ-paracompactness, countably $${\mu}$$μ-metacompactness and the condition corresponding to condition A in generalized topological spaces.
In topological spaces every normal space with a countable base is metacompact. We show that this ... more In topological spaces every normal space with a countable base is metacompact. We show that this is not necessarily true in generalized topological spaces; more exactly we give an example of a μ-normal space with a countable μ-base which has a μ-open cover with no μ-open point-finite refinement.
In this paper, we first explore more properties of soft somewhere dense continuous functions. The... more In this paper, we first explore more properties of soft somewhere dense continuous functions. Then, we discuss the preservation of soft Baire property and soft Baire category. We give some concrete examples to illustrate how our findings extend some conclusions and connections made in the literature.
This work introduces weakly soft β-open subsets, a new family of soft-open sets. By this family, ... more This work introduces weakly soft β-open subsets, a new family of soft-open sets. By this family, we expand a soft topology to a soft structure which is neither supra-soft topology nor infra-soft topology. The connections between this class of soft sets and other celebrated classes via soft topology are examined with some elucidative examples. Also, it is established some relationships under conditions of extended and hyperconnected soft topologies. Furthermore, the interior and closure operators are structured along with weakly soft β-open and weakly soft β-closed sets. Finally, the class of weakly soft β-continuous functions is introduced and its main characterizations are studied. It is investigated the systematic relationships and findings that are lost for this kind of soft continuity as well as it is shown the conditions required to maintain some of these relationships such as full, extended and hyperconnected soft topologies.
Fuzzy set (FS) theory and rough sets (RSs) are constructed to accommodate the data uncertainty. I... more Fuzzy set (FS) theory and rough sets (RSs) are constructed to accommodate the data uncertainty. In contrast, the bipolar FS (BFS) theory can tackle the uncertainty and the bipolarity of the data in different circumstances. This article aims to introduce the idea of rough bipolar fuzzy ideals in semigroup (SG), which is a generalization of the concept of rough BFSs (RBFSs) in an SG. We also investigate the roughness in the bipolar fuzzy subsemigroup (BF-SSG) with the help of congruence relation (cng-R) defined on the SG and studied some relevant structural properties. Moreover, the idea is extended to the rough bipolar fuzzy left ideal, rough bipolar fuzzy right ideal, rough bipolar fuzzy two-sided ideal, rough bipolar fuzzy interior ideal, and rough bipolar fuzzy bi-ideal in SGs. Further, it is seen that cng-Rs and complete cng-Rs play vital roles in the construction of rough approximations of bipolar fuzzy ideals. Consequently, their associated properties are explored by using cng-...
Our target in the present work, is presenting the idea of weakly soft preopen (ws-preopen) subset... more Our target in the present work, is presenting the idea of weakly soft preopen (ws-preopen) subsets and studying some of its characterizations. With the assistance of some elucidative examples, the interrelationships between ws-preopen sets and some extensions of soft open sets are studied. Under some conditions such as extended and hyperconnected soft topologies, several motivating results and relationships are acquired. The interior and closure operators that built through ws-preopen and ws-preclosed subsets are introduced. Their main features that construe the relations among them are established. Soft continuity with respect to theses classes of soft subsets are studied and their substantial characteristics are investigated. Generally, the systematic relations and outcomes that are lost through the scope of this study are discussed. The proposed line in the current study will present new ways to discover novel concepts in the field of soft topology.
In this paper, the concept of neutrosophic topological spaces is introduced. We dene and study&l... more In this paper, the concept of neutrosophic topological spaces is introduced. We dene and study<br> the properties of neutrosophic open sets, closed sets, interior and closure. The set of all generalize<br> neutrosophic pre-closed sets GNPC( ) and the set of all neutrosophic -open sets in a neutrosophic topological<br> space (X; ) can be considered as examples of generalized neutrosophic topological spaces. The concept<br> of neutrosophic continuity is dened and we studied their properties. We dene and study the properties<br> of neutrosophic compact, -Lindelof and -countably compact spaces. We prove that for a countable<br> neutrosophic -space X: -countably compactness and -compactness are equivalent. We give an example of<br> a neutrosophic -space X which has a neutrosophic countable -base but it is not neutrosophic -countably<br> compact .
In this paper, the concept of neutrosophic μ−topological spaces is introduced. We define and stud... more In this paper, the concept of neutrosophic μ−topological spaces is introduced. We define and study the properties of neutrosophic μ−open sets, μ−closed sets, μ−interior and μ−closure. The set of all generalize neutrosophic pre-closed sets GNPC(τ) and the set of all neutrosophic α-open sets in a neutrosophic topological space (X, τ) can be considered as examples of generalized neutrosophic μ−topological spaces. The concept of neutrosophic μ − continuity is defined and we studied their properties. We define and study the properties of neutrosophic μ − compact, μ-Lindelöf and μ-countably compact spaces. We prove that for a countable neutrosophic μ-space X: μ-countably compactness and μ-compactness are equivalent. We give an example of a neutrosophic μ-space X which has a neutrosophic countable μ-base but it is not neutrosophic μ-countably compact . neutrosophic μ−topological spaces;μ−open; μ−closed; μ−interior; μ−closure; generalize neutrosophic pre-closed sets; neutrosophic α-open set...
In this paper, the concept of neutrosophic μ−topological spaces is introduced. We define and stud... more In this paper, the concept of neutrosophic μ−topological spaces is introduced. We define and study the properties of neutrosophic μ−open sets, μ−closed sets, μ−interior and μ−closure. The set of all generalize neutrosophic pre-closed sets GNPC(τ) and the set of all neutrosophic α-open sets in a neutrosophic topological space (X, τ) can be considered as examples of generalized neutrosophic μ−topological spaces. The concept of neutrosophic μ − continuity is defined and we studied their properties. We define and study the properties of neutrosophic μ − compact, μ-Lindelöf and μ-countably compact spaces. We prove that for a countable neutrosophic μ-space X: μ-countably compactness and μ-compactness are equivalent. We give an example of a neutrosophic μ-space X which has a neutrosophic countable μ-base but it is not neutrosophic μ-countably compact . neutrosophic μ−topological spaces;μ−open; μ−closed; μ−interior; μ−closure; generalize neutrosophic pre-closed sets; neutrosophic α-open set...
We answer the following question: Are neutrosophic µ-compactness and neutrosophic µ-countably com... more We answer the following question: Are neutrosophic µ-compactness and neutrosophic µ-countably compactness equivalent? which posted in [10]. Since every neutrosophic topology is neutrosophic µ-topology, we answer the question for neutrosophic topological spaces, more precisely, we give an example of neutrosophic topology which is neutrosophic countably compact but not neutrosophic compact
We answer the following question: Are neutrosophic μ-compactness and neutrosophic μ-countably com... more We answer the following question: Are neutrosophic μ-compactness and neutrosophic μ-countably compactness equivalent? which posted in [10]. Since every neutrosophic topology is neutrosophic μ-topology, we answer the question for neutrosophic topological spaces, more precisely, we give an example of neutrosophic topology which is neutrosophic countably compact but not neutrosophic compact
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