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Keywords = soft induced topological spaces

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11 pages, 271 KiB  
Article
A New Approach to Soft Continuity
by Sandeep Kaur, Tareq M. Al-shami, Alkan Özkan and M. Hosny
Mathematics 2023, 11(14), 3164; https://doi.org/10.3390/math11143164 - 19 Jul 2023
Cited by 9 | Viewed by 1301
Abstract
The concept of continuity in topological spaces has a very important place. For this reason, a great deal of work has been done on continuity, and many generalizations of continuity have been obtained. In this work, we seek to find a new approach [...] Read more.
The concept of continuity in topological spaces has a very important place. For this reason, a great deal of work has been done on continuity, and many generalizations of continuity have been obtained. In this work, we seek to find a new approach to the study of soft continuity in soft topological spaces in connection with an induced mapping based on soft sets. By defining the *-image of a soft set, we define an induced soft mapping and present its related properties. To elaborate on the obtained results and relationships, we furnish a number of illustrative examples. Full article
(This article belongs to the Special Issue Recent Advances on Fuzzy Topology)
18 pages, 358 KiB  
Article
Two New Families of Supra-Soft Topological Spaces Defined by Separation Axioms
by Tareq M. Al-shami, José Carlos R. Alcantud and A. A. Azzam
Mathematics 2022, 10(23), 4488; https://doi.org/10.3390/math10234488 - 28 Nov 2022
Cited by 16 | Viewed by 1763
Abstract
This paper contributes to the field of supra-soft topology. We introduce and investigate supra pp-soft Tj and supra pt-soft Tj-spaces (j=0,1,2,3,4). These are [...] Read more.
This paper contributes to the field of supra-soft topology. We introduce and investigate supra pp-soft Tj and supra pt-soft Tj-spaces (j=0,1,2,3,4). These are defined in terms of different ordinary points; they rely on partial belong and partial non-belong relations in the first type, and partial belong and total non-belong relations in the second type. With the assistance of examples, we reveal the relationships among them as well as their relationships with classes of supra-soft topological spaces such as supra tp-soft Tj and supra tt-soft Tj-spaces (j=0,1,2,3,4). This work also investigates both the connections among these spaces and their relationships with the supra topological spaces that they induce. Some connections are shown with the aid of examples. In this regard, we prove that for i=0,1, possessing the Ti property by a parametric supra-topological space implies possessing the pp-soft Ti property by its supra-soft topological space. This relationship is invalid for the other types of soft spaces introduced in previous literature. We derive some results of pp-soft Ti-spaces from the cardinality numbers of the universal set and a set of parameters. We also demonstrate how these spaces behave as compared to their counterparts studied in soft topology and its generalizations (such as infra-soft topologies and weak soft topologies). Moreover, we investigated whether subspaces, finite product spaces, and soft S Full article
(This article belongs to the Section D2: Operations Research and Fuzzy Decision Making)
13 pages, 802 KiB  
Article
Soft Regular Generalized ω-Closed Sets and Soft ω-T1/2 Spaces
by Samer Al Ghour
Axioms 2022, 11(10), 529; https://doi.org/10.3390/axioms11100529 - 3 Oct 2022
Cited by 2 | Viewed by 1787
Abstract
Soft rgω-closed sets are introduced as a new class of soft sets that strictly contain the classes of soft rg-closed sets and soft gω-closed sets. Furthermore, the behavior of soft rgω-closed sets with [...] Read more.
Soft rgω-closed sets are introduced as a new class of soft sets that strictly contain the classes of soft rg-closed sets and soft gω-closed sets. Furthermore, the behavior of soft rgω-closed sets with respect to soft unions, soft intersections, and soft subspaces, as well as induced soft topologies are investigated. Moreover, soft ω-T1/2 spaces which is a weaker form soft T1/2 spaces is defined and investigated. In addition to these, the characterizations of soft rg-T1/2 spaces and soft rgω-T1/2 spaces are discussed. The work also looks at the relationship between our novel notions in soft topological spaces and their analogs in topological spaces. Full article
(This article belongs to the Special Issue Differential Geometry and Its Application)
12 pages, 290 KiB  
Article
On Soft Generalized ω-Closed Sets and Soft T1/2 Spaces in Soft Topological Spaces
by Samer Al Ghour
Axioms 2022, 11(5), 194; https://doi.org/10.3390/axioms11050194 - 21 Apr 2022
Cited by 17 | Viewed by 2546
Abstract
In this paper, we define a soft generalized ω-closed set, which is a generalization of both the soft ω-closed set and the soft generalized closed set. We show that the classes of generalized closed sets and generalized ω-closed sets coincide [...] Read more.
In this paper, we define a soft generalized ω-closed set, which is a generalization of both the soft ω-closed set and the soft generalized closed set. We show that the classes of generalized closed sets and generalized ω-closed sets coincide in soft anti-locally countable soft topological spaces. Additionally, in soft locally countable soft topological spaces, we show that every soft set is a soft generalized ω-closed set. Furthermore, we prove that the classes of soft generalized closed sets and soft generalized ω-closed sets coincide in the soft topological space (X,τω,A). In addition to these, we determine the behavior of soft generalized ω-closed sets relative to soft unions, soft intersections, soft subspaces, and generated soft topologies. Furthermore, we investigate soft images and soft inverse images of soft generalized closed sets and soft generalized ω-closed sets under soft continuous, soft closed soft transformations. Finally, we continue the study of soft T1/2 spaces, in which we obtain two characterizations of these soft spaces, and investigate their behavior with respect to soft subspaces, soft transformations, and generated soft topologies. Full article
(This article belongs to the Special Issue Advances in General Topology and Its Application)
13 pages, 289 KiB  
Article
Soft -Open Sets and the Soft Topology of Soft δω-Open Sets
by Samer Al Ghour
Axioms 2022, 11(4), 177; https://doi.org/10.3390/axioms11040177 - 15 Apr 2022
Cited by 9 | Viewed by 2456
Abstract
The author devotes this paper to defining a new class of soft open sets, namely soft Rω-open sets, and investigating their main features. With the help of examples, we show that the class of soft Rω-open sets lies strictly [...] Read more.
The author devotes this paper to defining a new class of soft open sets, namely soft Rω-open sets, and investigating their main features. With the help of examples, we show that the class of soft Rω-open sets lies strictly between the classes of soft regular open sets and soft open sets. We show that soft Rω-open subsets of a soft locally countable soft topological space coincide with the soft open sets. Moreover, we show that soft Rω-open subsets of a soft anti-locally countable coincide with the soft regular open sets. Moreover, we show that the class of soft Rω-open sets is closed under finite soft intersection, and as a conclusion, we show that this class forms a soft base for some soft topology. In addition, we define the soft δω-closure operator as a new operator in soft topological spaces. Moreover, via the soft δω-closure operator, we introduce soft δω-open sets as a new class of soft open sets which form a soft topology. Moreover, we study the correspondence between soft δω-open in soft topological spaces and δω-open in topological spaces. Full article
(This article belongs to the Special Issue Differential Geometry and Its Application)
14 pages, 295 KiB  
Article
Between the Classes of Soft Open Sets and Soft Omega Open Sets
by Samer Al Ghour
Mathematics 2022, 10(5), 719; https://doi.org/10.3390/math10050719 - 24 Feb 2022
Cited by 18 | Viewed by 2078
Abstract
In this paper, we define the class of soft ω0-open sets. We show that this class forms a soft topology that is strictly between the classes of soft open sets and soft ω-open sets, and we provide some sufficient conditions [...] Read more.
In this paper, we define the class of soft ω0-open sets. We show that this class forms a soft topology that is strictly between the classes of soft open sets and soft ω-open sets, and we provide some sufficient conditions for the equality of the three classes. In addition, we show that soft closed soft ω-open sets are soft ω0-open sets in soft Lindelof soft topological spaces. Moreover, we study the correspondence between soft ω0-open sets in soft topological spaces and ω0-open sets in topological spaces. Furthermore, we investigate the relationships between the soft α-open sets (respectively, soft regular open sets, soft β-open sets) of a given soft anti-locally countable soft topological space and the soft α-open sets (respectively, soft regular open sets, soft β-open sets) of the soft topological space of soft ω0-open sets generated by it. Finally, we introduce ω0-regularity in topological spaces via ω0-open sets, which is strictly between regularity and ω-regularity, and we also introduce soft ω0-regularity in soft topological spaces via soft ω0-open sets, which is strictly between soft regularity and soft ω-regularity. We investigate relationships regarding ω0-regularity and soft ω0-regularity. Moreover, we study the correspondence between soft ω0-regularity in soft topological spaces and ω0-regularity in topological spaces. Full article
(This article belongs to the Special Issue Computing Mathematics with Fuzzy Sets)
12 pages, 766 KiB  
Article
Soft Semi ω-Open Sets
by Samer Al Ghour
Mathematics 2021, 9(24), 3168; https://doi.org/10.3390/math9243168 - 9 Dec 2021
Cited by 5 | Viewed by 2695
Abstract
In this paper, we introduce the class of soft semi ω-open sets of a soft topological space (X,τ,A), using soft ω-open sets. We show that the class of soft semi ω-open sets contains [...] Read more.
In this paper, we introduce the class of soft semi ω-open sets of a soft topological space (X,τ,A), using soft ω-open sets. We show that the class of soft semi ω-open sets contains both the soft topology τω and the class of soft semi-open sets. Additionally, we define soft semi ω-closed sets as the class of soft complements of soft semi ω-open sets. We present here a study of the properties of soft semi ω-open sets, especially in (X,τ,A) and (X,τω,A). In particular, we prove that the class of soft semi ω-open sets is closed under arbitrary soft union but not closed under finite soft intersections; we also study the correspondence between the soft topology of soft semi ω-open sets of a soft topological space and their generated topological spaces and vice versa. In addition to these, we introduce the soft semi ω-interior and soft semi ω-closure operators via soft semi ω-open and soft semi ω-closed sets. We prove several equations regarding these two new soft operators. In particular, we prove that these operators can be calculated using other usual soft operators in both of (X,τ,A) and (X,τω,A), and some equations focus on soft anti-locally countable soft topological spaces. Full article
(This article belongs to the Special Issue Fuzzy Topology)
11 pages, 277 KiB  
Article
Soft ωp-Open Sets and Soft ωp-Continuity in Soft Topological Spaces
by Samer Al Ghour
Mathematics 2021, 9(20), 2632; https://doi.org/10.3390/math9202632 - 19 Oct 2021
Cited by 18 | Viewed by 2251
Abstract
We define soft ωp-openness as a strong form of soft pre-openness. We prove that the class of soft ωp-open sets is closed under soft union and do not form a soft topology, in general. We prove that soft [...] Read more.
We define soft ωp-openness as a strong form of soft pre-openness. We prove that the class of soft ωp-open sets is closed under soft union and do not form a soft topology, in general. We prove that soft ωp-open sets which are countable are soft open sets, and we prove that soft pre-open sets which are soft ω-open sets are soft ωp-open sets. In addition, we give a decomposition of soft ωp-open sets in terms of soft open sets and soft ω-dense sets. Moreover, we study the correspondence between the soft topology soft ωp-open sets in a soft topological space and its generated topological spaces, and vice versa. In addition to these, we define soft ωp-continuous functions as a new class of soft mappings which lies strictly between the classes of soft continuous functions and soft pre-continuous functions. We introduce several characterizations for soft pre-continuity and soft ωp-continuity. Finally, we study several relationships related to soft ωp-continuity. Full article
(This article belongs to the Special Issue Fuzzy Topology)
13 pages, 291 KiB  
Article
Weaker Forms of Soft Regular and Soft T2 Soft Topological Spaces
by Samer Al Ghour
Mathematics 2021, 9(17), 2153; https://doi.org/10.3390/math9172153 - 3 Sep 2021
Cited by 21 | Viewed by 2567
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Computing Mathematics with Fuzzy Sets)
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