Soft $ \alpha $-separation axioms and $ \alpha $-fixed soft points

2021

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.

Section: Introductionmentioning

confidence: 99%

Weaker Forms of Soft Regular and Soft T2 Soft Topological Spaces

2021

“…Then, many topological concepts were modified to include soft topology. The concepts of soft topology and their applications is still a hot area of research (see for example [1,2,[5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]).…”

Section: Introductionmentioning

confidence: 99%

Soft ωp-Open Sets and Soft ωp-Continuity in Soft Topological Spaces

2021

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.

“…For a STS (U, τ, E), the members τ are called soft open sets. Soft topological concepts and their applications are still a hot area of research [1,2,[5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. The concept of ω-open sets in TSs is defined in [23] as follows: let (U, ) be a TS and V ⊆ U, then V is ω-open set in (U, ) if for each x ∈ V, there is W ∈ such that x ∈ W and W − V is countable, or equivalently, V is ω-open set in (U, ) if and only if for each x ∈ V, there is W ∈ and a countable set C ⊆ U such that x ∈ W −C ⊆ V. Denote the family of all ω-open sets in the TS (U, ) by ω .…”

Section: Introductionmentioning

confidence: 99%

Some Modifications of Pairwise Soft Sets and Some of Their Related Concepts

2021

In this paper, we first define soft u-open sets and soft s-open as two new classes of soft sets on soft bitopological spaces. We show that the class of soft p-open sets lies strictly between these classes, and we give several sufficient conditions for the equivalence between soft p-open sets and each of the soft u-open sets and soft s-open sets, respectively. In addition to these, we introduce the soft u-ω-open, soft p-ω-open, and soft s-ω-open sets as three new classes of soft sets in soft bitopological spaces, which contain soft u-open sets, soft p-open sets, and soft s-open sets, respectively. Via soft u-open sets, we define two notions of Lindelöfeness in SBTSs. We discuss the relationship between these two notions, and we characterize them via other types of soft sets. We define several types of soft local countability in soft bitopological spaces. We discuss relationships between them, and via some of them, we give two results related to the discrete soft topological space. According to our new concepts, the study deals with the correspondence between soft bitopological spaces and their generated bitopological spaces.