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Strong Form of Soft Semi-Open Sets in Soft Topological Spaces
International Journal of Fuzzy Logic and Intelligent Systems 2021;21(2):159-168
Published online June 25, 2021
© 2021 Korean Institute of Intelligent Systems.

Samer Al Ghour

Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan
Correspondence to: Samer Al Ghour (algore@just.edu.jo)
Received April 19, 2021; Revised June 3, 2021; Accepted June 8, 2021.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Soft ωs-open sets as a class of soft sets that lies strictly between soft open sets and soft semi-open sets is introduced. The natural properties of soft ωs-open sets are described. Using soft ωs-open sets, soft ωs-closure and soft ωs-interior as new soft operators are defined and investigated. Furthermore, the relationships regarding generated soft topological spaces and generated topological spaces are studied.
Keywords : Soft ω-open sets, Soft semi-open sets, Soft ωs-open, Generated soft topology
1. Introduction

Throughout this paper, we follow the notions and terminologies as used in [1] and [2], and for simplicity, STS stands for soft topological space. For the purpose of dealing with uncertain objects, Molodtsov [3] introduced soft sets in 1999. Let Y be a universal set and B be a set of parameters. A soft set over Y relative to B is a function H : B(Y). The family of all soft sets over Y relative to B is denoted by SS (Y, B). Let HSS (Y, B), then H is a countable soft set if H (b) is a countable set for all bB. The family of all members of SS (Y, B) that are countable is denoted by CSS (Y, B). Furthermore, the null soft set and the absolute soft set are denoted by 0B and 1B, respectively. The authors in [4] define the notion of STSs as follows: The triplet (Y, τ, B), where τSS (Y, B) is a STS if 0B and 1Bτ, τ is closed under finite soft intersection, and τ is closed under arbitrary soft union. If (Y, τ, B) is a STS, then the members of τ are called soft open sets and their soft complements are called soft closed sets. Topologists recently have applied various topological concepts to soft topological spaces ([1,2,525]).

Chen [26] introduced the concept of soft semi-open sets, a weaker form of soft open sets in STSs. This paper was followed by many more papers about soft semi-open sets and their modifications. In this paper, we introduce soft ωs-open sets as a class of soft sets that lies strictly between soft open sets and soft semi-open sets. We present the natural properties of soft ωs-open sets. Furthermore, using ωs-open sets, we define and investigate soft ωs-closure and ωs-interior as new soft topological operators. We also compare our proposed concepts and their corresponding previous analogous concepts in Theorems 2.2, 2.17, 3.3, and 3.19, with the comparisons supported by examples. We present several sufficient conditions for the equivalence between the proposed concepts and their corresponding previous analogous concepts. We next discuss relationships between relative topological spaces and generated soft topological spaces. Finally, we raise an open question related to soft ωs-open.

In [27,28], the authors showed that soft sets are a class of special information systems. This insight constitutes our motivation to study the structures of soft sets for information systems. Therefore, this paper does not only form the theoretical basis for further applications of soft topology, such as soft ωs-mappings and soft ωs-compactness, but also comments on the future development of information systems.

The following definitions and results will be used in the sequel:

Definition 1.1

Let (X, ℑ) be a topological space and let DX. The D is called

(a) [29] semi-open if there is V ∈ ℑ such that VDCl(V). SO(X, ℑ) will denote the family of all semi-open sets in (X, ℑ).

(b) [30] ωs-open if there is V ∈ ℑ such that VDClω (V). ωs(X, ℑ) will denote the family of all ωs-open sets in (X, ℑ).

Definition 1.2 [26]

Let (X, τ, A) be a STS and let KSS(X, A). Then

(a)K is called a soft semi-open set in (X, τ, A) if there exists Fτ such that F ⊆̃ K ⊆̃ Clτ (F). SO(X, τ, A) will denote the family of all soft semi-open sets in (X, τ, A).

(b)K is called a soft semi-closed set in (X, τ, A) if 1AKSO(X, τ, A).

Definition 1.3 [26]

Let (X, τ, A) be a STS and HSS(X, A).

(a) The soft semi-closure of H in (X, τ, A) is denoted by S-Clτ (H) and defined by S-Clτ (H) = ∩̃ {M : M is soft semi-closed in (X, τ, A) and H ⊆̃ M}.

(b) The soft semi-interior of H in (X, τ, A) is denoted by S-intτ (H) and defined by S-intτ (H) = ∪̃ {K : K is soft semi-open in (X, τ, A) and K ⊆̃ H}.

As defined in [2], a STS (X, τ, A) is called soft anti-locally countable if for every Fτ – {0A}, FCSS(X, A).

Proposition 1.4 [2]

Let (X, τ, A) be soft anti-locally countable. Then for all Gτω, Clτ (G) = Clτω (G). As defined in [2], a STS (X, τ, A) is called soft locally countable if for axSP(X, A) there exists GCSS(X, A) ∩ τ such that ax ∊̃ G.

Proposition 1.5 [2]

A STS (X, τ, A) is soft locally countable if and only if (X, τω, A) is a discrete STS.

Proposition 1.6 [2]

For any STS (X, τ, A) we have τω = (τω)ω.

Proposition 1.7 [26]

Let (X, τ, A) be a STS and HSS(X, A). Then

(a) S-Clτ (H) is the smallest soft semi-closed in (X, τ, A) containing H.

(b) H is soft semi-closed in (X, τ, A) if and only if H = S-Clτ (H).

2. Soft ωs-Open sets

Definition 2.1

Let (X, τ, A) be a STS and let KSS(X, A). Then K is called a soft ωs-open set in (X, τ, A) if there exists Fτ such that F ⊆̃ K ⊆̃ Clτω (F). ωs(X, τ, A) will denote the family of all soft ωs-open sets in (X, τ, A).

Theorem 2.2

Let (X, τ, A) be a STS. Then τωs(X, τ, A) ⊆ SO(X, τ, A).

Proof

Let Gτ, choose F = G. Then Fτ and F ⊆̃ G ⊆̃ Clτω (F) which implies that Gωs(X, τ, A). This shows that τωs(X, τ, A). Let Kωs(X, τ, A), then by definition we find Fτ such that F ⊆̃ K ⊆̃ Clτω (F) ⊆̃ Clτ (F) which implies that KSO(X, τ, A). It follows that ωs(X, τ, A) ⊆ SO(X, τ, A).

The following three examples show that none of the two inclusions in Theorem 2.2 is equality in general:

Example 2.3

Let X = ℝ and A = [0, 1]. Let F, GSS(X, A) such that for every aA, F(a) = ℕ and G(a) = ℚc. Let τ = {0A, 1A, F, G, F ∪̃ G}. One can easily check that Clτω (F) = F, Clτ (F) = 1AG and Clτω (G) = 1AF. Hence 1AGSO(X, τ, A) – ωs(X, τ, A) and 1AFωs(X, τ, A) – τ.

Example 2.4

Let X = ℝ and A = ℕ. Let τ = {FSS(X, A) : F(a) belongs to the usual topology on ℝ}.

Let GSS(X, A) such that for every aA, G(a) = [0, 1). Then Gωs(X, τ, A) – τ.

Example 2.5

Let X = ℚ and A = ℝ. Let F, GSS(X, A) such that for every aA, F(a) = {1} and G(a) = {1, 2}. Let τ = {0A, 1A, F}. Then GSO(X, τ, A) – ωs(X, τ, A).

Theorem 2.6

If (X, τ, A) is a soft anti-locally countable STS, then ωs(X, τ, A) = SO(X, τ, A).

Proof

By Theorem 2.2, we only need to see that SO(X, τ, A) ⊆ ωs(X, τ, A). Let KSO(X, τ, A), then there is Fτ such that F ⊆̃ K ⊆̃ Clτ (F). By Proposition 1.4, Clτ (F) = Clτω (F). Therefore, Kωs(X, τ, A). This shows that SO(X, τ, A) ⊆ ωs(X, τ, A).

Theorem 2.7

For any STS (X, τ, A), SO(X, τω, A) = ωs(X, τω, A).

Proof

Let (X, τ, A) be STS. By Theorem 2.2, we only need to show that SO(X, τω, A) ⊆̃ ωs(X, τω, A). Let KSO(X, τω, A), then there is Fτω such that F ⊆̃ K ⊆̃ Cl(τω)ω (F). As by Proposition 1.6, (τω)ω = τω, then Cl(τω)ω (F) = Clτω (F). Hence, Kωs(X, τω, A). This ends the proof that SO(X, τω, A) ⊆ ωs(X, τω, A).

Theorem 2.8

If (X, τ, A) is a soft locally countable STS, then τ = ωs(X, τ, A).

Proof

By Theorem 2.2, we only need to see that ωs(X, τ, A) ⊆ τ. Let Kωs(X, τ, A). Then there is Fτ such that F ⊆̃ K ⊆̃ Clτω (F). By Proposition 1.5, Clτω (F) = F which implies that K = F. Thus, Kτ. This shows that τ = ωs(X, τ, A).

Soft ω-open sets and soft ωs-open sets are independent of each other:

Example 2.9

Let X = ℝ and A = ℕ. Let F, G, KSS(X, A) defined by F(a) = {a} ∪ (2, 3), G(a) = [0, ∞) and K(a) = (2, 3) for all aA. Let τ = {0A, 1A, F}. As (X, τ, A) is soft anti-locally countable, then by Proposition 1.5, Clτω (F) = Clτ (F) = 1A. So Gωs(X, τ, A) – τω and Kτωωs(X, τ, A).

Theorem 2.10

Let (X, τ, A) be a STS and let KSS(X, A). Then K is soft ωs-open in (X, τ, A) if and only if K ⊆̃ Clτω (intτ (K)).

Proof

Necessity

Suppose that K is soft ωs-open set in (X, τ, A). Then we find Fτ such that F ⊆̃ K ⊆̃ Clτω (F). As F ⊆̃ K, then F = intτ (F) ⊆̃ intτ (K), which implies that Clτω(F) ⊆̃ Clτω (intτ (K)). It follows that K ⊆̃ Clτω (intτ (K)).

Sufficiency

Suppose that K ⊆̃ Clτω (intτ (K)). Put F = intτ (K). Then Fτ with F ⊆̃ K ⊆̃ Clτω (intτ (K)). Hence, K is soft ωs-open set in (X, τ, A).

Theorem 2.11

Let (X, τ, A) be a STS. If {Kα : α ∈ Δ} ⊆ ωs(X, τ, A), then α Δ ˜ K α ω s ( X , τ , A ).

Proof

For every α ∈ Δ, we find Fατ such that Fα ⊆̃ Kα ⊆̃ Clτω (Fα). Then α Δ ˜ F α τ and α Δ ˜ F α ˜ α Δ ˜ K α ˜ α Δ ˜ C l τ ω ( K α ) ˜ C l τ ω ( α Δ ˜ K α ). Therefore, α Δ ˜ K α ω s ( X , τ , A ).

Soft intersection of two soft ωs-open sets is not a soft ωs open in general:

Example 2.12

Let X = ℝ and A = ℕ. Let F, GSS(X, A) defined by F(a) = (0, ∞) and G(a) = (–∞, 0) for all aA. Let τ = {0A, 1A, F, G, F ∪̃ G}. As (X, τ, A) is soft anti-locally countable, then by Proposition 1.4, Clτω (F) = Clτ (F) = 1AG and Clτω (G) = Clτ (G) = 1AF. Then 1AF, 1AGωs(X, τ, A) but ((1AF) ∩̃ (1AG))(a) = {0} for all aA and hence (1AF) ∩̃ (1AG) ∉ ωs(X, τ, A).

Theorem 2.13

Let (X, τ, A) be a STS. If Gτ and Kωs(X, τ, A), then G ∩̃ Kωs(X, τ, A).

Proof

Let Gτ and Kωs(X, τ, A). As Kωs(X, τ, A), then there is Fτ such that F ⊆̃ K ⊆̃ Clτω (F). Then G ∩̃ Fτ and G ∩̃ F ⊆̃ G ∩̃ K ⊆̃ G ∩̃ Clτω (F) ⊆̃ Clτω (G ∩̃ F). This ends the proof that G ∩̃ Kωs(X, τ, A).

Theorem 2.14

Let (X, τ, A) be a STS. If Kωs(X, τ, A) and K ⊆̃ G ⊆̃ Clτω (K), then Gωs(X, τ, A).

Proof

Suppose that Kωs(X, τ, A) and K ⊆̃ G ⊆̃ Clτω (K). As Kωs(X, τ, A), then there is Fτ such that F ⊆̃ K ⊆̃ Clτω (F). As K ⊆̃ Clτω (F), then Clτω (K) ⊆̃ Clτω (F). As G ⊆̃ Clτω (K), then G ⊆̃ Clτω (F). Thus, we have Fτ and F ⊆̃ K ⊆̃ G ⊆̃ Clτω (K) ⊆̃ Clτω (F).

Therefore, Gωs(X, τ, A).

Theorem 2.15

For any STS (X, τ, A),

τ = { i n t τ ( K ) : K ω s ( X , τ , A ) } .

Proof

As intτ (K) ∈ τ for every Kωs(X, τ, A), then {intτ (K) : Kωs(X, τ, A)} ⊆ τ. Conversely, let Fτ, then F = intτ (F). On the other hand, by Theorem 2.2, Fωs(X, τ, A). It follows that τ ⊆ {intτ (K) : Kωs(X, τ, A)}.

Definition 2.16

Let (X, τ, A) be a STS and let MSS(X, A). Then M is called soft ωs-closed set in (X, τ, A) if 1AM is soft ωs-open set in (X, τ, A).

Theorem 2.17

Let (X, τ, A) be a STS. Then

(a) Every soft closed set in (X, τ, A) is soft ωs-closed set in (X, τ, A).

(b) Every soft ωs-closed set in (X, τ, A) is soft semi-closed set in (X, τ, A).

(c) 0A and 1A are soft ωs-closed sets in (X, τ, A).

Proof

(a) Let M be a soft closed set in (X, τ, A), then 1AMτ and by Theorem 2.2 1AMωs(X, τ, A). Thus, M is a soft ωs-closed set in (X, τ, A).

(b) Let M be a soft ωs-closed set in (X, τ, A), then 1AMωs-(X, τ, A) and by Theorem 2.2 1AMSO(X, τ, A). Thus, M is soft semi-closed set in (X, τ, A).

(c) Follows by (a).

One can use Example 2.3 to show that none of the implications in Theorem 2.15 (a), (b) is reversible in general.

Theorem 2.18

Let (X, τ, A) be a STS. If Mα is soft ωs-closed in (X, τ, A) for all α ∈ Δ, then α Δ ˜ M α is soft ωs-closed in (X, τ, A).

Proof

Suppose that Mα is soft ωs-closed in (X, τ, A) for all α ∈ Δ. Then for every α ∈ Δ, 1AMαωs(X, τ, A). Thus, by Theorem 2.9, we have 1 A - ( α Δ ˜ M α ) = α Δ ˜ ( 1 A - M α ) ω s ( X , τ , A ). It follows that α Δ ˜ M α is soft ωs-closed in (X, τ, A).

Soft ωs-closedness is not closed under finite soft union:

Example 2.19

Consider (X, τ, A), F, and G as in Example 2.12. As 1AF, 1AGωs(X, τ, A), then F and G are soft ωs-closed in (X, τ, A). As 1A – (F ∪̃ G) = (1AF) ∩̃ (1AG) ∉ ωs(X, τ, A), then F ∪̃ G is not soft ωs-closed in (X, τ, A).

Theorem 2.20

Let (X, τ, A) be a STS. If M is soft closed in (X, τ, A) and N is soft ωs-closed in (X, τ, A), then M ∪̃ N is soft ωs-closed in (X, τ, A).

Proof

Suppose that M is soft closed in (X, τ, A) and N is soft ωs-closed in (X, τ, A). Then 1AMτ and 1ANωs(X, τ, A). So by Theorem 2.13, 1A – (M ∪̃ N) = (1AM) ∩̃ (1AN) ∈ ωs(X, τ, A). It follows that M ∪̃ N is soft ωs-closed in (X, τ, A).

Theorem 2.21

Let (X, τ, A) be a STS and let MSS(X, A). Then M is soft ωs-closed in (X, τ, A) if and only if

i n t τ ω ( C l τ ( M ) ) ˜ M .

Proof

Necessity

Assume that M is soft ωs-closed in (X, τ, A). Then 1AM is soft ωs-open in (X, τ, A). Theorem Theorem 2.10 implies that 1AM ⊆̃ Clτω (intτ (1AM)), and so

i n t τ ω ( C l τ ( M ) ) = E x t τ ω ( 1 A - C l τ ( M ) ) = E x t τ ω ( E x t τ ( M ) ) = 1 A - C l τ ω ( E x t τ ( M ) ) = 1 A - C l τ ω ( i n t τ ( 1 A - M ) ) ˜ M .

Sufficiency

Suppose that intτω (Clτ (M)) ⊆̃ M. Then

1 A - M 1 A - i n t τ ω ( C l τ ( M ) ) = 1 A - E x t τ ω ( 1 A - C l τ ( M ) ) = 1 A - E x t τ ω ( E x t τ ( M ) ) = C l τ ω ( E x t τ ( M ) ) = C l τ ω ( i n t τ ( 1 A - M ) ) .

Therefore, by Theorem 2.10, 1AM is soft ωs-open in (X, τ, A) and so M is soft ωs-closed in (X, τ, A).

Theorem 2.22

If fpu : (X, τ, A) → (Y, σ, B) is a soft open mapping such that fpu : (X, τω, A) → (Y, σω, B) is soft continuous, then for every Kωs(X, τ, A), we have fpu(K) ∈ ωs(Y, σ, B).

Proof

Let Kωs(X, τ, A). Then there exists Fτ such that F ⊆̃ K ⊆̃ Clτω (F). Thus we have fpu (F) ⊆̃ fpu(K) ⊆̃ fpu (Clτω (F)). As fpu : (X, τ, A) → (Y, σ, B) is soft open mapping, then fpu (F) ∈ σ. As fpu : (X, τω, A) → (Y, σω, B) is soft continuous, then fpu (Clτω (F)) ⊆̃ Clτω (fpu (F)). Therefore, fpu(K) ∈ ωs(Y, σ, B).

In Theorem 2.22, we cannot drop the condition ‘soft open mapping’:

Example 2.23

Let X = ℝ, A = ℕ, F, GSS(X, A) with F(a) = ℝ and G(a) = {0} for all aA, τ = SS(X, A), and σ = {0B, 1B, F}. Define p : XX and u : AA by p (x) = 0 for all xX and u(a) = a for all aA. Then clearly that fpu : (X, τω, A) → (X, τω, A) is soft continuous, but Gωs (X, τω, A) while fpu (G) = Gωs (X, τω, A).

3. Soft ωs-Closure and Soft ωs-Interior

Definition 3.1

Let (X, τ, A) be a STS and HSS(X, A). The soft ωs-closure of H in (X, τ, A) is denoted by ωs-Clτ (H) and defined by ωs-Clτ (H) = ∩̃ {M : M is soft ωs-closed in (X, τ, A) and H ⊆̃ }.

Theorem 3.2

Let (X, τ, A) be a STS and HSS(X, A). Then

(a) ωs-Clτ (H) is the smallest soft ωs-closed in (X, τ, A) containing H.

(b) H is soft ωs-closed in (X, τ, A) if and only if H = ωs-Clτ (H).

Proof

(a) Follows from Definition 3.1 and Theorem 2.18.

(b) Follows immediately by (a).

Theorem 3.3

Let (X, τ, A) be a STS and HSS(X, A). Then H ⊆̃ S-Clτ (H) ⊆̃ ωs-Clτ (H) ⊆̃ Clτ (H).

Proof

Follows from the definitions and parts (a) and (b) of Theorem 2.17.

In Theorem 3.3 S-Clτ (H) ≠ ωs-Clτ (H) and ωs-Clτ (H) ≠ Clτ (H), in general:

Example 3.4

Consider Example 2.3. We proved that 1AGSO(X, τ, A) – ωs(X, τ, A) and 1AFωs(X, τ, A) – τ. So, we have F as an ωs-closed set in (X, τ, A) that is not closed set in (X, τ, A), and G as a semi-closed set in (X, τ, A) that is not ωs-closed set in (X, τ, A). Thus, by Proposition 3.9 (2) of [26] and Theorem 3.2 (b), S-Clτ (G) = Gωs-Clτ (H) and ωs-Clτ (F) = FClτ (F).

Theorem 3.5

Let (X, τ, A) be a STS and HSS(X, A). Then

(a) If H is soft closed in (X, τ, A), then H = S-Clτ (H) = ωs-Clτ (H) = Clτ (H).

(b) If H is soft ωs-closed in (X, τ, A), then H = S-Clτ (H) = ωs-Clτ (H).

Proof

(a) Suppose that soft H is closed in (X, τ, A). Then H = Clτ (H) and by Theorem 3.3 we have H = S-Clτ (H) = ωs-Clτ (H) = Clτ (H).

(b) Suppose that H is soft ωs-closed in (X, τ, A). Then by Theorem 3.2 (b) we have H = ωs-Clτ (H), and by Theorem 3.3 we conclude that H = S-Clτ (H) = ωs-Clτ (H).

Theorem 3.6

If (X, τ, A) is a soft anti-locally countable STS then for every HSS(X, A), S-Clτ (H) = ωs-Clτ (H).

Proof

As (X, τ, A) is soft anti-locally countable, then by Theorem 2.6 we have ωs(X, τ, A) = SO(X, τ, A) and hence S-Clτ (H) = ωs-Clτ (H).

Theorem 3.7

If (X, τ, A) is a soft locally countable STS then for every HSS(X, A), ωs-Clτ (H) = Clτ (H).

Proof

As (X, τ, A) is soft locally countable, then by Theorem 2.8 we have τ = ωs(X, τ, A) and hence ωs-Clτ (H) = Clτ (H).

Theorem 3.8

If (X, τ, A) is a STS and HSS(X, A), then S-Clτω (H) = ωs-Clτω (H).

Proof

By Theorem 2.7 we have SO(X, τω, A) = ωs(X, τω, A) and hence S-Clτω (H) = ωs-Clτω (H).

Theorem 3.9

Let (X, τ, A) be a STS and H, KSS(X, A). Then

(a) If H ⊆̃ K, then ωs-Clτ (H) ⊆̃ ωs-Clτ (K).

(b) ωs-Clτ (H) ∪̃ ωs-Clτ (K) ⊆̃ ωs-Clτ (H ∪̃ K).

(c) ωs-Clτ (H ∩̃ K) ⊆̃ ωs-Clτ (H) ∩̃ ωs-Clτ (K).

Proof

(a) Suppose that H ⊆̃ K. As K ⊆̃ ωs-Clτ (K), then H ⊆̃ ωs-Clτ (K). By Theorem 3.2 (a), ωs-Clτ (K) is soft ωsclosed in (X, τ, A). Again by Theorem 3.2 (a), ωs-Clτ (H) ⊆̃ ωs-Clτ (K).

(b) As H ⊆̃ H ∪̃ K and K ⊆̃ H ∪̃ K, then by (a), ωs-Clτ (H) ⊆̃ ωs-Clτ (H ∪̃ K) and ωs-Clτ (K) ⊆̃ ωs-Clτ (H ∪̃ K). Hence ωs-Clτ (H) ∪̃ ωs-Clτ (K) ⊆̃ ωs-Clτ (H ∪̃ K).

(c) As H ∩̃ K ⊆̃ H and H ∩̃ K ⊆̃ K, then by (a), ωs-Clτ (H ∩̃ K) ⊆̃ ωs-Clτ (H) and ωs-Clτ (H ∩̃ K) ⊆̃ ωs-Clτ (K). Hence ωs-Clτ (H ∩̃ K) ⊆̃ ωs-Clτ (H) ∩̃ ωs-Clτ (K).

The soft inclusion in Theorem 3.9 (b) cannot be replaced by soft equality in general:

Example 3.10

Consider (X, τ, A) as in Example 2.19. We proved that F and G are soft ωs-closed in (X, τ, A) while F ∪̃ G is not soft ωs-closed in (X, τ, A). Therefore, by Theorem 3.2 (b), ωs-Clτ (F) ∪̃ ωs-Clτ (G) = F ∪̃ G while ωs-Clτ (F ∪̃ G) ≠ F ∪̃ G.

Theorem 3.11

If (X, τ, A) is a soft locally countable STS, then for any H, KSS(X, A) we have ωs-Clτ (H) ∪̃ ωs-Clτ (K) = ωs-Clτ (H ∪̃ K).

Proof

As (X, τ, A) is soft locally countable, then by Theorem 3.7,

ω s - C l τ ( H ˜ K ) = C l τ ( H ˜ K ) = C l τ ( H ) ˜ C l τ ( K ) = ω s - C l τ ( H ) ˜ ω s - C l τ ( K ) .

Theorem 3.12

Let (X, τ, A) be a STS and let H, KSS(X, A). If H is soft closed in (X, τ, A), then ωs-Clτ (H) ∪̃ ωs-Clτ (K) = ωs-Clτ (H ∪̃ K).

Proof

By Theorem 3.9 (b) we only need to show that ωs-Clτ (H ∪̃ K) ⊆̃ ωs-Clτ (H) ∪̃ ωs-Clτ (K). As H is soft closed in (X, τ, A), then by Theorem 3.5 (a), H = ωs-Clτ (H). So by Theorem 2.20, ωs-Clτ (H) ∪̃ ωs-Clτ (K) is ωs-closed in (X, τ, A). As H ∪̃ K ⊆̃ ωs-Clτ (H) ∪̃ ωs-Clτ (K), then by Theorem 3.2 (a) we have ωs-Clτ (H ∪̃ K) ⊆̃ ωs-Clτ (H) ∪̃ ωs-Clτ (K).

The soft inclusion in Theorem 3.9 (c) cannot be replaced by soft equality in general:

Example 3.13

Let X = ℤ × ℤ, A = ℝ and τ = {FSS(X, A) : F(a) ∈ τcof for every aA}. Let H, KSS(X, A) defined by H(a) = ℤ × {0} and K(a) = {0} × ℤ for all aA. Then (H ∩̃ K) (a) = {(0, 0)} for all aA. Note that MSS(X, A) is soft closed in (X, τ, A) if and only if for all aA, either M(a) is finite or M (a) = ℤ × ℤ. Then Clτ (H) = Clτ (K) = 1A and Clτ (H ∩̃ K) = H ∩̃ K. As (X, τ, A) is soft locally countable, then by Theorem 3.7, ωs-Clτ (H) = Clτ (H) = 1A, ωs-Clτ (K) = Clτ (K) = 1A, and ωs-Clτ (H ∩̃ K) = H ∩̃ K. Thus, ωs-Clτ (H) ∩̃ ωs-Clτ (K) = 1A while ωs-Clτ (H ∩̃ K) ωs-Clτ (H ∩̃ K) ≠ 1A.

Theorem 3.14

Let (X, τ, A) be a STS and let HSS(X, A). Then S-Clτ (Clτ (H)) = Clτ (S-Clτ (H)) = ωs-Clτ (Clτ (H)) = Clτ (ωs-Clτ (H)) = Clτ (H).

Proof

As Clτ (H) is soft closed in (X, τ, A), then, by Theorem 3.5 (a) we have Clτ (H) = S-Clτ (Clτ (H)) = ωs-Clτ (Clτ (H)).

By Theorem 3.3, H ⊆̃ S-Clτ (H) ⊆̃ ωs-Clτ (H) ⊆̃ Clτ (H). So, Clτ (H) ⊆̃ Clτ (S-Clτ (H)) ⊆̃ Clτ (ωs-Clτ (H)) ⊆̃ Clτ (Clτ (H)) = Clτ (H). Hence, we have Clτ (H) = Clτ (S-Clτ (H)) = Clτ (ωs-Clτ (H)).

Theorem 3.15

Let (X, τ, A) be a STS and let HSS(X, A). Then ωs-Clτ (S-Clτ (H)) = S-Clτ (ωs-Clτ (H)) = ωs-Clτ (H).

Proof

As ωs-Clτ (H) is soft ωs-closed in (X, τ, A), then by Theorem 3.5 (b) we have ωs-Clτ (H) = S-Clτ (ωs-Clτ (H)).

By Theorem 3.3, H ⊆̃ S-Clτ (H) ⊆̃ ωs-Clτ (H). So by Theorem 3.9 (a), ωs-Clτ (H) ⊆̃ ωs-Clτ (S-Clτ (H)) ⊆̃ ωs-Clτ (ωs-Clτ (H)) = ωs-Clτ (H). Therefore,

ω s - C l τ ( S - C l τ ( H ) ) = ω s - C l τ ( H ) .

Theorem 3.16

Let (X, τ, A) be a STS and let HSS(X, A). Then ax ∊̃ ωs-Clτ (H) if and only if for all Kωs(X, τ, A) with ax ∊̃ K we have K ∩̃ H ≠ 0A.

Proof

Necessity

Suppose that ax ∊̃ ωs-Clτ (H) and suppose to the contrary that there is Kωs(X, τ, A) such that ax ∊̃ K and K ∩̃ H = 0A. So we have H ⊆̃ 1AK with 1AK is soft ωs-closed set in (X, τ, A), and hence ωs-Clτ (H) ⊆̃ 1AK. As ax ∊̃ ωs-Clτ (H), then ax ∊̃ 1AK, a contradiction.

Sufficiency

By contradiction. Then we have ax ∊̃ (1Aωs-Clτ (H)) ∈ ωs(X, τ, A). Then by assumption (1Aωs-Clτ (H)) ∩̃ H = 0A, a contradiction.

Definition 3.17

Let (X, τ, A) be a STS and HSS(X, A). The soft ωs-interior of H in (X, τ, A) is denoted by ωs-intτ (H) and defined by ωs-intτ (H) = ∪̃ {K : K is soft ωs-open in (X, τ, A) and K ⊆̃ H}.

Theorem 3.18

Let (X, τ, A) be a STS and HSS(X, A). Then

(a) ωs-intτ (H) is the largest soft ωs-open in (X, τ, A) contained in H.

(b) H is soft ωs-open in (X, τ, A) if and only if H = ωs intτ (H).

Proof

(a) Follows from Definition 3.17 and Theorem 2.11.

(b) Follows immediately by (a).

Theorem 3.19

Let (X, τ, A) be a STS and HSS(X, A). Then intτ (H) ⊆̃ ωs-intτ (H) ⊆̃ S-intτ (H) ⊆̃ H.

Proof

Follows from the definitions and Theorem 2.2.

In Theorem 3.19 intτ (H) ≠ ωs-intτ (H) and ωs-intτ (H) ≠ S-intτ (H), in general:

Example 3.20

Consider Example 2.3. We proved that 1AGSO(X, τ, A) – ωs(X, τ, A) and 1AFωs(X, τ, A) – τ. Thus, by Proposition 3.9 (2) and Theorem 3.2 (b), S-intτ (1AG) = 1AGωs-intτ (1AG) and ωs-intτ (1AF) = 1AFintτ (1AF).

Theorem 3.21

Let (X, τ, A) be a STS and HSS(X, A). Then

(a) If H is soft open in (X, τ, A), then H = S-intτ (H) = ωs-intτ (H) = intτ (H).

(b) If H is soft ωs-open in (X, τ, A), then H = S-intτ (H) = ωs-intτ (H).

Proof

(a) Suppose that H is soft open in (X, τ, A). Then H = intτ (H) and by Theorem 3.19 we have H = S-intτ (H) = ωs-intτ (H) = intτ (H).

(b) Suppose that H is soft ωs-open in (X, τ, A). Then by Theorem 3.2 (b), H = ωs-intτ (H). Thus, by Theorem 3.3 we ωs-intτ (H) ⊆̃ S-intτ (H) ⊆̃ ωs-intτ (H). Therefore, H = S-intτ (H) = ωs-intτ (H).

Theorem 3.22

If (X, τ, A) is a soft anti-locally countable STS then for every HSS(X, A), S-intτ (H) = ωs intτ (H).

Proof

As (X, τ, A) is soft anti-locally countable, then by Theorem 2.6 we have ωs(X, τ, A) = SO(X, τ, A) and hence S-intτ (H) = ωs-intτ (H).

Theorem 3.23

If (X, τ, A) is a soft locally countable STS then for every HSS(X, A), ωs-intτ (H) = intτ (H).

Proof

As (X, τ, A) is soft locally countable, then by Theorem 2.8 we have τ = ωs(X, τ, A) and hence ωs-intτ (H) = intτ (H).

Theorem 3.24

If (X, τ, A) is a STS and HSS(X, A), then S-intτω (H) = ωs-intτω (H).

Proof

By Theorem 2.7 we have SO(X, τω, A) = ωs(X, τω, A) and hence S-intτω (H) = ωs-intτω (H).

Theorem 3.25

Let (X, τ, A) be a STS and H, KSS(X, A). Then

(a) If H ⊆̃ K, then ωs-intτ (H) ⊆̃ ωs-intτ (K).

(b) ωs-intτ (H) ∪̃ ωs-intτ (K) ⊆̃ ωs-intτ (H ∪̃ K).

(c) ωs-intτ (H ∩̃ K) ⊆̃ ωs-intτ (H) ∩̃ ωs-intτ (K).

Proof

(a) Suppose that H ⊆̃ K. Then we have ωs-intτ (H) ⊆̃ H ⊆̃ K. By Theorem 3.18 (a), ωs-intτ (H) is soft ωs-open in (X, τ, A). Again by Theorem 3.18 (a), ωs-intτ (H) ⊆̃ ωs-intτ (K).

(b) As H ⊆̃ H ∪̃ K and K ⊆̃ H ∪̃ K, then by (a), ωs-intτ (H) ⊆̃ ωs-intτ (H ∪̃ K) and ωs-intτ (K) ⊆̃ ωs-intτ (H ∪̃ K). Hence ωs-intτ (H) ∪̃ ωs-intτ (K) ⊆̃ ωs-intτ (H ∪̃ K).

(c) As H ∩̃ K ⊆̃ H and H ∩̃ K ⊆̃ K, then by (a), ωs-intτ (H ∩̃ K) ⊆̃ ωs-intτ (H) and ωs-intτ (H ∩̃ K) ⊆̃ ωs-intτ (K). Hence ωs-intτ (H ∩̃ K) ⊆̃ ωs-intτ (H) ∩̃ ωs-intτ (K).

The soft inclusion in Theorem 3.26 (c) cannot be replaced by soft equality in general:

Example 3.27

Consider (X, τ, A) as in Example 2.12. We proved that 1AF and 1AG are soft ωs-open in (X, τ, A) while (1AF) ∩̃ (1AG) is not soft ωs-open in (X, τ, A). Therefore, by Theorem 3.18 (b),

ω s - i n t τ ( 1 A - F ) ˜ ω s - i n t τ ( 1 A - G ) = ( 1 A - F ) ˜ ( 1 A - G ) ,

while

ω s - i n t τ ( ( 1 A - F ) ˜ ( 1 A - G ) ( 1 A - F ) ˜ ( 1 A - G ) .

Theorem 3.28

If (X, τ, A) is a soft locally countable STS, then for any H, KSS(X, A) we have ωs-intτ (H) ∩̃ ωs-intτ (K) = ωs-intτ (H ∩̃ K).

Proof

As (X, τ, A) is soft locally countable, then by Theorem 3.23,

ω s - i n t τ ( H ˜ K ) = i n t τ ( H ˜ K ) = C l τ ( H ) ˜ C l τ ( K ) = ω s - C l τ ( H ) ˜ ω s - C l τ ( K ) .

Theorem 3.29

Let (X, τ, A) be a STS and let H, KSS(X, A). If H is soft open in (X, τ, A), then ωs-intτ (H) ∩̃ ωs-intτ (K) = ωs-intτ (H ∩̃ K).

Proof

By Theorem 3.25 (c) we only need to show that ωs-intτ (H) ∩̃ ωs-intτ (K) ⊆̃ ωs-intτ (H ∩̃ K). As H is soft open in (X, τ, A), then by Theorem 3.18 (b), H = ωs-intτ (H). So by Theorem 2.13, ωs-intτ (H) ∩̃ ωs-intτ (K) is soft ωs-open in (X, τ, A). As ωs-intτ (H) ∩̃ ωs-intτ (K) ⊆̃ H ∩̃ K, then by Theorem 3.18 (a) we have ωs-intτ (H) ∩̃ ωs-intτ (K) ⊆̃ ωs-intτ (H ∩̃ K).

The soft inclusion in Theorem 3.25 (b) cannot be replaced by soft equality in general:

Example 3.30

Consider (X, τ, A) in Example 3.13 and let B = {–n : n ∈ ℕ}. Let H, KSS(X, A) defined by H(a) = ℤ × ℕ and K(a) = ({0} ∪ B) × ℤ for all aA. As (X, τ, A) is soft locally countable, then by Theorem 3.23, ωs-intτ (H) = intτ (H) = 0A, ωs-intτ (K) = intτ (K) = 0A, and ωs-Clτ (H ∪̃ K) = intτ (H ∪̃ K) = intτ (1A) = 1A. Thus, ωs-intτ (H) ∪̃ ωs-intτ (K) = 0A while ωs-Clτ (H ∪̃ K) ≠ 0A.

Theorem 3.31

Let (X, τ, A) be a STS and let HSS(X, A). Then intτ (H) = ωs-intτ (intτ (H)) = S-intτ (intτ (H)) = intτ (ωs-intτ (H)) = intτ (S-intτ (H)).

Proof

As intτ (H) is soft open in (X, τ, A), then by Theorem 3.21 (a) we have intτ (H) = S-intτ (intτ (H)) = ωs-intτ (intτ (H)).

By Theorem 3.19 we have intτ (H) ⊆̃ ωs-intτ (H) ⊆̃ S-intτ (H) ⊆̃ H. So, intτ (H) = intτ (intτ (H)) ⊆̃ intτ (ωs-intτ (H)) ⊆̃ intτ (S-intτ (H)) ⊆̃ intτ (H).

Hence,

i n t τ ( H ) = i n t τ ( ω s - i n t τ ( H ) ) = i n t τ ( S - i n t τ ( H ) ) .

Theorem 3.32

Let (X, τ, A) be a STS and let HSS(X, A). Then ωs-intτ (S-intτ (H)) = S-intτ (ωs-intτ (H)) = ωs-intτ (H).

Proof

As ωs-intτ (H) is soft ωs-open in (X, τ, A), then by Theorem 3.21 (b) we have ωs-intτ (H) = S-intτ (ωs-intτ (H))

By Theorem 3.19 we have

ω s - i n t τ ( H ) ˜ S - i n t τ ( H ) ˜ H .

So by Theorem 3.25 (a) we have ωs-intτ (H) = ωs-intτ (ωs-intτ (H)) ⊆̃ ωs-intτ (S-intτ (H)) ⊆̃ ωs-intτ (H). Therefore, ωs-intτ (H) = ωs-intτ (S-intτ (H)).

Theorem 3.33

Let (X, τ, A) be a STS and let HSS(X, A). Then

(a) ωs-intτ (1AH) ∩̃ ωs-Clτ (H) = 0A.

(b) 1A = ωs-intτ (1AH) ∪̃ ωs-Clτ (H).

(c) 1Aωs-Clτ (H) = ωs-intτ (1AH) and 1Aωs-intτ (1AH) = ωs-Clτ (H).

Proof

(a) Suppose to the contrary that there exists ax ∊̃ (ωs-intτ (1AH) ∩̃ ωs-Clτ (H)). Then we have ax ∊̃ ωs-intτ (1AH) ∈ ωs(X, τ, A) and ax ∊̃ ωs-Clτ (H). Thus by Theorem 3.16 ωs-intτ (1AH) ∩̃ H ≠ 0A. But ωs-intτ (1AH) ∩̃ H ⊆̃ (1AH) ∩̃ H = 0A, a contradiction.

(b) It is sufficient to show that

1 A - ω s - C l τ ( H ) ˜ ω s - i n t τ ( 1 A - H ) .

Let ax ∊̃ (1Aωs-Clτ (H)), then by Theorem 3.16 there is Kωs(X, τ, A) such that ax ∊̃ K and K ∩̃ H = 0A. So, we have ax ∊̃ K ⊆̃ 1AH and hence axωs-intτ (1AH).

(c) Follows from (a) and (b).

4. Relative Topological Spaces and Generated Soft Topological Spaces

At this stage, we believe that the following four questions are natural:

Question 4.1

Let (X, τ, A) and let Kωs (X, τ, A). Is it true that K(a) ∈ ωs (X, τa) for all aA?

Question 4.2

Let (X, τ, A) and let KSS(X, A) such that K(a) ∈ ωs (X, τa) for all aA. Is it true that Kωs (X, τ, A)?

Question 4.3

Let (X, τ, A) and let KSO (X, τ, A). Is it true that K(a) ∈ SO (X, τa) for all aA?

Question 4.4

Let (X, τ, A) and let KSS(X, A) such that K(a) ∈ SO (X, τa) for all aA. Is it true that KSO (X, τ, A)?

We will leave Question 4.3 as an open question. However, in the following example we will give negative answers for Questions 4.1, 4.2, and 4.4:

Example 4.5

Let X = ℝ and A = {a, b}. Let F, G, H, KSS(X, A) defined by

F ( a ) = c , F ( b ) = { 1 } , G ( a ) = { 2 } , G ( b ) = c , H ( a ) = c , H ( b ) = , K ( a ) = { 2 } , K ( b ) = { 1 } .

Let τ = {0A, 1A, F, G, F ∪̃ G}. As clearly that (X, τ, A) is soft anti-locally countable, then by Theorem 2.6 ωs(X, τ, A) = SO(X, τ, A). It is not difficult to check that HSO(X, τ, A) = ωs (X, τ, A) while H (b) ∉ ωs (X, τb), which gives a negative answer for Questions 4.1. Also it is not difficult to check that K(a) ∈ ωs (X, τa) ⊆ SO(X, τa) and K (b) ∈ ωs(X, τb) ⊆ SO(X, τb) while Kωs(X, τ, A) = SO(X, τ, A), which gives negative answers for Questions 4.2 and 4.4.

If we add the condition ‘(X, τ, A) is soft locally countable’, then Question 4.1 will have a positive answer:

Theorem 4.6

Let (X, τ, A) be a soft locally countable and let Kωs (X, τ, A). Then K(a) ∈ ωs (X, τa) for every aA.

Proof

Let aA. As (X, τ, A) is soft locally countable and Kωs (X, τ, A), then by Theorem 2.8, Kτ and so K(a) ∈ τaωs (X, τa).

Lemma 4.7

Let {(X, ℑa) : aA} be an indexed family of topological spaces and let τ = a A a. Let HSS(X, A), then for every aA, Cl(τa)ω (H (a)) = (Clτω (H)) (a).

Proof

Let aA. By Proposition 7 of [4], Cl(τa)ω (H(a)) ⊆ (Clτω (H)) (a). To see that (Clτω (H)) (a) ⊆ Cl(τa)ω (H (a)), let x ∈ (Clτω (H)) (a) and let U ∈ (τa)ω such that xU. By Theorem 7 of [2], (τa)ω = (τω)a. Then aUτω. As axaUτω and axClτω (H), then by Lemma 5 of [2], H ∩̃ aU ≠ 0A. Thus, we must have (H ∩̃ aU) (a) = H (a) ∩ U ≠ ∅︀. It follows that xCl(τω)a (H(a)).

If we add the condition ‘τ is a generated soft topology’, then Questions 4.1 and 4.2 will have positive answers:

Theorem 4.8

Let {(X, ℑa) : aA} be an indexed family of topological spaces and let τ = a A a. Let HSS(X, A). Then Hωs(X, τ, A) if and only if H(a) ∈ ωs(X, τa) for every aA.

Proof

Necessity

Suppose that Hωs(X, τ, A) and let aA. Choose Fτ such that F ⊆̃ H ⊆̃ Clτω (F). So, F(a) ⊆ H(a) ⊆ (Clτω (F)) (a). As F(a) ∈ τa and by Lemma 4.7 we have (Clτω (F)) (a) = Cl(τa)ω (F(a)), then H(a) ∈ ωs(X, τa).

Sufficiency

Suppose that H(a) ∈ ωs(X, τa) for every aA. Then for every aA, there exists Uaτa = ℑa such that UaH(a) ⊆ Cl(τa)ω (Ua). Let KSS(X, A) with K(a) = Ua ∈ ℑa for every aA. Then K a A a = τ

and by Lemma 4.7, (Clτω (K)) (a) = Cl(τa)ω (K(a)) = Cl(τa)ω (Ua) for all aA. As for every aA, K(a) = UaH(a) ⊆ Cl(τa)ω (Ua) = (Clτω (K)) (a), then K ⊆̃ H ⊆̃ Clτω (K). It follows that Hωs(X, τ, A).

Lemma 4.9

Let {(X, ℑa) : aA} be an indexed family of topological spaces and let τ = a A a. Let HSS(X, A), then for every aA, Clτa (H(a)) = (Clτ (H)) (a).

Proof

Let aA. By Proposition 7 of [4], Clτa (H(a)) ⊆ (Clτ (H)) (a). To see that (Clτ (H)) (a) ⊆ Clτa (H(a)), let x ∈ (Clτ (H)) (a) and let Uτa = ℑa such that xU. Then we have ax ∊̃ aUτ and ax ∊̃ Clτ (H), and by Lemma 5 of [2], H ∩̃ aU ≠ 0A. Thus, we must have (H ∩̃ aU) (a) = H(a) ∩ U ≠ ∅︀. It follows that xClτa (H (a)).

If we add the condition ‘τ is a generated soft topology’, then Questions 4.3 and 4.4 will have positive answers:

Theorem 4.10

Let {(X, ℑa) : aA} be an indexed family of topological spaces and let τ = a A a. Let HSS(X, A). Then HSO(X, τ, A) if and only if H(a) ∈ SO(X, τa) for every aA.

Proof

Necessity

Suppose that HSO(X, τ, A). Then there exists Fτ such that F ⊆̃ H ⊆̃ Clτ (F). So, F(a) ⊆ H(a) ⊆ (Clτ (F)) (a). As F(a) ∈ τa and by Lemma 4.9 we have (Clτ (F)) (a) = Clτa (F (a)), then H(a) ∈ SO(X, τa).

Sufficiency

Suppose that H(a) ∈ SO(X, τa) for every aA. Then for every aA, there exists Uaτa = ℑa such that UaH(a) ⊆ Clτa (Ua). Let KSS(X, A) with K(a) = Ua ∈ ℑa for every aA. Then K a A a = τ and by Lemma 4.9, (Clτ (K)) (a) = Clτa (K (a)) = Clτa (Ua) for all aA. As for every aA, K(a) = UaH(a) ⊆ Clτa (Ua) = (Clτ (K)) (a), then K ⊆̃ H ⊆̃ Clτ (K). It follows that HSO(X, τ, A).

5. Conclusion


We introduced soft ωs-open sets as a new class of soft sets. We then proved that the class of soft ωs-open sets lies strictly between the classes of soft open sets and soft semi-open sets. Furthermore, we studied two new soft operators, soft ωs-closure and soft ωs-interior, as well as relationships regarding relative topological spaces and generated soft topological. Finally, we end by raising an open question related to soft ωs-open. We believe the proposed concept can open up doors to research areas such as soft ωs-mappings and soft ωs-compactness.

Conflict of Interest

No potential conflict of interest relevant to this article was reported.

Conflict of Interest

No potential conflict of interest relevant to this article was reported.


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Biography

Samer Al Ghour received the Ph.D. in Mathematics from University of Jordan, Jordan in 1999. Currently, he is currently a professor at the Department of Mathematics and Statistics, Jordan University of Science and Technology, Jordan. His research interests is include general topology, fuzzy topology, and soft set theory.

E-mail: algore@just.edu.jo