Weaker Forms of Soft Regular and Soft T2 Soft Topological Spaces

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-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.

Between the Classes of Soft Open Sets and Soft Omega Open Sets

“…(Y, µ) is ω 0 -regular but not regular. Thus, by Corollary 10 and Corollary 4 of [37], (Y, τ(µ), B) is soft ω 0 -regular but not soft regular. Now we give an example to show that the converse of Theorem 35 need not be true, in general: Example 12.…”

“…In this paper, we follow the notions and terminologies as they appear in [23,34,37]. Throughout this paper, ST and STS will denote topological space and soft topological space, respectively.…”

“…Soft ω-open sets were defined and investigated by Al Ghour and Hamed [35], who used them to characterize soft Lindelofness and soft weakly Lindelofness. Then several research papers related to soft ω-open sets were introduced [36,37]. The authors of [38] defined and investigated ω 0 -open sets as a generalization of open sets, which are a strong form of ω-open sets.…”

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

“…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]).…”

“…(b) [2] soft anti-locally countable for every F ∈ τ − {0 A }, F is not a countable soft set. (c) [5] soft ω-regular whenever S is soft closed and a x ∈1 A − S, then we find K ∈ τ and N ∈ τ ω such that a x ∈K, S ⊆N, and K ∩N = 0 A . Theorem 1 ([4]).…”