International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 89-99

**Published online** March 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.1.89

© The 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 (Samer Al Ghour)

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 noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

The soft *θ*_{ω}-closure operator is defined as a new soft operator that lies strictly between the usual soft closure and the soft *θ*-closure. Sufficient conditions are provided for equivalence between the soft *θ*_{ω}-closure and usual soft closure operators, and between the soft *θ*_{ω}-closure and soft *θ*-closure operators. Via the soft *θ*_{ω}-closure operator, the soft *θ*_{ω}-open sets are defined as a new class of soft sets that lies strictly between the class of soft open sets and the class of soft *θ*-open sets. It is proven that the class of soft *θ*_{ω}-open sets form a new soft topology. The soft ω-regularity is characterized via both the soft *θ*_{ω}-closure operator and soft *θ*_{ω}-open sets. The soft product theorem and several soft mapping theorems are introduced. The correspondence between the soft topology of the soft *θ*_{ω}-open sets of soft topological space and their generated topological spaces, and vice versa, are studied. In addition to these, soft *θ*_{ω}-continuity as a strong form of soft *θ*-continuity is introduced and investigated.

**Keywords**: Soft *θ*-closure, Soft *θ*-open, Soft ω-regular, Soft product, Soft *θ*-continuity, Soft generated soft topological space.

This paper follows the concepts and terminology that appear in [1–3]. In this paper, TS and STS denote the topological space and soft topological space, respectively. The concept of soft sets, which was introduced by Molodtsov [4] in 1999, is a general mathematical tool for dealing with uncertainty. Let _{A}_{A}_{A}_{A}_{A}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}

The following definitions and results are used throughout this work:

Let

(a) [1] _{Y}

(b) [1] _{Y}

(c) [46] _{x}

The set of all soft points in

Let _{x}_{x}_{x}_{x}_{x}

Let (_{a}

Let (

defines a soft topology on

Let _{a}

Then,

Let

for each (

Let (

Recall that a soft set

An STS (

(a) [50] soft locally indiscrete if every soft open set is soft closed;

(b) [2] soft locally countable if for each _{x}_{x}

(c) [2] soft anti-locally countable if for every _{A}

(d) [6] soft

(e) [6] soft _{x}_{A}_{ω}_{x}_{A}

In this section, we introduce the _{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}

Let (

(a) A soft point _{x}_{x}_{θ}_{τ}_{A}_{x}

(b) _{θ}

(c) _{A}

(d) The family of all soft _{θ}

Let (

(a) (_{θ}

(b) _{θ}_{θ}

The following is the main definition of this work.

Let (

(a) A soft point _{x}_{ω}_{x}_{θ}_{ω} (_{τ}_{ω} (_{A}_{x}

(b) _{ω}_{θ}_{ω} (

(c) _{ω}_{A}_{ω}

(d) The family of all soft _{ω}_{θ}_{ω}.

Let (

(a) _{τ}_{θ}_{ω} (_{θ}

(b) If _{ω}

(c) If _{ω}

(a) To demonstrate that _{τ}_{θ}_{ω} (_{x}_{τ}_{x}_{x}_{τ}_{A}_{τ}_{ω} (_{τ}_{ω} (_{A}_{x}_{θ}_{ω} (_{θ}_{ω} (_{θ}_{x}_{θ}_{ω} (_{x}_{x}_{θ}_{ω} (_{τ}_{ω} (_{A}_{τ}_{ω} (_{τ}_{τ}_{A}_{x}_{θ}

(b) Suppose that _{θ}_{θ}_{ω} (_{ω}

(c) Suppose that _{ω}_{θ}_{ω} (_{τ}

Let (

(a) _{τ}_{θ}_{ω} (F).

(b) If _{ω}

(a) From Theorem 2.4 (a), _{τ}_{θ}_{ω} (_{θ}_{ω} (_{τ}_{x}_{θ}_{ω} (_{x}_{τ}_{ω} (_{A}_{τ}_{ω} (_{A}_{x}_{τ}

(b) Suppose that _{τ}_{θ}_{ω} (_{ω}

Let (

(a) _{τ}_{θ}_{ω} (

(b) If _{ω}

The proof follows from Theorem 2.5 and Theorem 7 of [6].

Let (

(a) _{τ}_{θ}_{ω} (

(b) If _{ω}

The proof follows from Theorem 2.5 and Theorem 6 of [6].

Let (

(a) _{θ}_{θ}_{ω} (

(b) If _{ω}

(a) Using Theorem 2.4(a), _{θ}_{ω} (_{θ}_{θ}_{θ}_{ω} (_{x}_{θ}_{x}_{τ}_{A}_{τ}_{ω} (_{τ}_{τ}_{ω} (_{A}_{x}_{θ}_{ω} (

(b) Suppose that _{ω}_{θ}_{ω} (_{θ}

For any STS (_{θ}_{θ}_{ω} ⊆

Let _{θ}_{A}_{A}_{ω}_{θ}_{ω}. Therefore, _{θ}_{θ}_{ω}. To demonstrate that _{θ}_{ω} ⊆ _{θ}_{ω}; then, 1_{A}_{ω}_{A}

Let (_{θ}_{τ}

Let _{τ}_{θ}_{θ}_{τ}_{x}_{θ}_{x}_{τ}_{A}_{y}_{τ}_{A}_{τ}_{θ}

Let (

(a) If _{A}_{θ}_{ω} (_{θ}_{ω} (

(b) For each _{θ}_{ω} (_{θ}_{ω} (_{θ}_{ω} (

(c) For each _{θ}_{ω} (

(d) For each _{ω}_{θ}_{ω} (_{τ}

(e) For each _{θ}_{θ}_{ω} (_{τ}

(a) Let _{x}_{θ}_{ω} (_{x}_{x}_{θ}_{ω} (_{τ}_{ω} (_{A}_{τ}_{ω} (_{A}_{x}_{θ}_{ω} (

(b) As _{θ}_{ω} (_{θ}_{ω} (_{θ}_{ω} (_{x}_{A}_{θ}_{ω} (_{θ}_{ω} (_{x}_{τ}_{ω} (_{A}_{τ}_{ω} (_{A}_{x}

Hence, _{x}_{A}_{θ}_{ω} (

(c) Let _{A}_{θ}_{ω} (_{x}_{A}_{θ}_{ω} (_{x}_{τ}_{ω} (_{A}_{θ}_{ω} (_{A}_{A}_{θ}_{ω} (

(d) Let _{ω}_{τ}_{θ}_{ω} (_{θ}_{ω} (_{τ}_{x}_{θ}_{ω} (_{A}_{τ}_{x}_{θ}_{ω}(_{x}_{A}_{τ}_{τ}_{ω} (1_{A}_{τ}_{A}

This is a contradiction.

(e) follows from (d) and Lemma 2.10.

Let (

(a) 0_{A}_{A}_{ω}

(b) The finite soft union of soft _{ω}_{ω}

(c) The arbitrary soft intersection of soft _{ω}_{ω}

(a) The proof follows from Theorems 2.2(a) and 2.4(b).

(b) We demonstrate that the soft union of two soft _{ω}_{ω}_{ω}_{θ}_{ω} (_{θ}_{ω} (

Hence, _{ω}

(c) Let _{λ}_{ω}_{λ}_{θ}_{ω} (_{λ}_{θ}_{ω} ( ∩? {_{λ}_{λ}_{x}_{θ}_{ω} ( ∩? {_{λ}_{x}_{τ}_{ω} (_{λ}_{A}_{τ}_{ω} (_{λ}_{A}_{x}_{θ}_{ω} (_{λ}_{λ}_{x}_{λ}

For any STS (_{θ}_{ω}

(1) According to Theorem 2.12(a), 0_{A}_{A}_{ω}_{A}_{A}_{θ}_{ω}.

(2) Let _{θ}_{ω}; then, 1_{A}_{A}_{ω}

Then, using Theorem 2.12(b), 1_{A}_{ω}_{θ}_{ω}.

(3) Let _{λ}_{θ}_{ω} for every _{A}_{λ}_{ω}

is soft _{ω}_{λ}_{θ}_{ω}.

Let (_{θ}_{ω} if and only if, for each _{x}_{x}_{τ}_{ω} (

Let _{θ}_{ω}, and let _{x}_{A}_{ω}_{x}_{A}_{A}_{A}_{ω}_{θ}_{ω} (1_{A}_{A}_{x}_{A}_{θ}_{ω} (1_{A}_{x}_{τ}_{ω} (_{A}_{A}_{x}_{τ}_{ω} (

Suppose that for each _{x}_{x}_{τ}_{ω} (_{θ}_{ω}. Then, _{θ}_{ω}(1_{A}_{A}_{x}_{θ}_{ω} (1_{A}_{A}_{x}_{τ}_{ω} (_{x}_{τ}_{ω} (_{A}_{A}_{x}_{A}_{θ}_{ω} (1_{A}

Every soft open, soft _{ω}

Let (_{x}_{τ}_{ω} (_{x}_{τ}_{ω} (_{ω}

Every countable soft open set in an STS is soft _{ω}

Using Theorem 2(d) of [2], countable soft sets in an STS are soft

For any STS (

(a) (

(b) _{θ}_{ω}.

(c) For every _{θ}_{ω} (_{τ}

(a) ⇒ (b): Suppose that (_{θ}_{ω}, let _{x}_{x}_{τ}_{ω}(_{θ}_{ω}. However, from Theorem 2.9, we obtain _{θ}_{ω} ⊆

(b) ⇒ (c): Suppose that _{θ}_{ω} and let _{θ}_{ω} (_{τ}_{x}_{A}_{τ}_{A}_{τ}_{A}_{τ}_{θ}_{ω}. Thus, according to Theorem 2.14, there exists _{x}_{τ}_{ω} (_{A}_{τ}_{x}_{τ}_{ω} (_{τ}_{ω} (_{τ}_{A}_{x}_{A}_{θ}_{ω} (_{τ}_{θ}_{ω} (

(c) ⇒ (a): Suppose that _{θ}_{ω} (_{τ}_{x}_{A}_{θ}_{ω} (1_{A}_{τ}_{A}_{A}_{x}_{A}_{θ}_{ω} (1_{A}_{x}_{τ}_{ω} (_{A}_{A}_{x}_{τ}_{ω} (

The inclusions in Theorem 2.9 are not equalities in general.

Let

Then,

and

Let (_{τ}_{a} (_{τ}

Let (_{θ}_{a}_{θ}

Suppose that _{θ}_{x}_{τ}_{a}_{τ}_{a} (_{τ}_{a}_{θ}

Let _{a}_{a}_{θ}

Suppose that

for all

Suppose that _{a}_{θ}_{x}_{a}_{θ}_{a}_{ℑ}_{a} (_{x}_{V}_{Cl}_{ℑa (}_{V}_{ )} ⊆?

Let (_{θ}_{θ}

For each _{a}

Let (_{θ}_{ω}, _{a}_{θ}_{ω} for all

Suppose that _{θ}_{ω} and let _{x}_{θ}_{ω}, there exists _{x}_{τ}_{ω} (_{a}_{(}_{τ}_{a}_{)}_{ω,} (_{(}_{τ}_{ω}_{)}_{a} (_{τ}_{ω} (_{a}_{θ}_{ω}.

Let _{a}_{a}_{θ}_{ω} for all

Suppose that

Suppose that _{a}_{θ}_{ω} for all _{x}_{a}_{θ}_{ω}. Thus, there exists _{a}_{(ℑ}_{a}_{)}_{ω} (_{x}_{V}_{Cl}_{(ℑa)ω}_{(}_{V}_{)} ⊆?

Let (_{θ}_{ω} if and only if _{θ}_{ω} for all

For each _{a}

Let (_{θ}_{ω}, _{θ}_{ω} and _{θ}_{ω}.

Let _{x}_{y}_{(}_{x,y}_{)} ?? _{θ}_{ω}, there exists _{(}_{x,y}_{)} ?? _{(}_{τ}_{*}_{σ}_{)}_{ω} (_{(}_{x,y}_{)} ?? _{(}_{x,y}_{)} ?? _{τ}_{ω} (_{σ}_{ω} (_{(}_{τ}_{*}_{σ}_{)}_{ω} (_{(}_{τ}_{*}_{σ}_{)}_{ω} (

Thus, we obtain _{x}_{τ}_{ω} (_{y}_{τ}_{ω} (_{θ}_{ω} and _{θ}_{ω}.

Let (_{θ}_{ω} and _{θ}_{ω}. Is it true that _{θ}_{ω} ?

If _{pu}_{pu}_{ω}

Let

Define

For every soft closed set _{pu}_{pu}_{ω}_{A}_{pu}_{A}_{pu}

Let _{pu}_{pu}_{pu}_{ω}_{pu}_{θ}_{θ}_{ω}

Let _{θ}_{y}_{pu}_{x}_{y}_{pu}_{x}_{x}_{θ}_{x}_{τ}_{y}_{pu}_{x}_{pu}_{pu}_{τ}_{pu}_{pu}_{pu}_{pu}_{ω}_{pu}_{τ}_{τ}_{ω} (_{pu}_{pu}_{τ}_{pu}_{θ}_{ω}. It follows that _{pu}_{θ}_{θ}_{ω}

Let _{pu}_{pu}_{pu}_{ω}_{ω}_{pu}_{θ}_{ω}_{θ}_{ω}

Let _{θ}_{ω} and let _{y}_{pu}_{x}_{y}_{pu}_{x}_{x}_{θ}_{ω}, there exists _{x}_{τ}_{ω} (_{y}_{pu}_{x}_{pu}_{pu}_{τ}_{ω} (_{pu}_{pu}_{pu}_{ω}_{ω}_{pu}_{τ}_{ω} (_{τ}_{ω} (_{pu}_{pu}_{τ}_{ω} (_{pu}_{pu}_{θ}_{ω}.

Let _{pu}_{pu}_{pu}_{ω}_{ω}_{pu}_{θ}_{ω}_{θ}_{ω}

Let _{θ}_{ω} and let _{pu}_{x}_{pu}_{x}_{τ}_{ω} (_{pu}_{pu}_{ω}_{ω}_{pu}_{θ}_{ω}_{θ}_{ω}

In this section, we introduce and investigate soft _{ω}

A soft function _{pu}_{x}_{pu}_{x}_{x}_{pu}_{τ}_{σ}

Let _{pu}

Suppose that _{pu}_{pu}_{x}_{p}_{(}_{x}_{)} ?? _{V}_{pu}_{x}_{pu}_{τ}_{(}_{ℑ}_{)}(_{τ}_{(ℵ)}(_{V}_{τ}_{(}_{ℑ}_{)}(_{pu}_{τ}_{(}_{ℑ}_{)}(_{τ}_{(ℵ)}(_{V}_{ℑ}(_{(}_{τ}_{(}_{ℑ}_{))}_{a} (_{τ}_{(}_{ℑ}_{)}(_{τ}_{(ℵ)}(_{V}_{Cl}_{ℵ}_{(}_{V}_{)}(_{ℵ}(_{ℑ}(_{ℵ}(

Suppose that _{x}_{pu}_{x}_{p}_{(}_{x}_{)} ?? _{ℑ}(_{ℵ}(_{ℵ}(_{(}_{τ}_{(ℵ))}_{u (a)} (_{ℵ}(_{τ}_{(}_{ℑ}_{)} (_{S}_{Cl}_{ℑ}_{ (}_{S}_{)}, and thus, _{pu}_{τ}_{(}_{ℑ}_{)}(_{S}_{pu}_{Cl}_{ℑ}_{ (}_{S}_{)}) = _{p}_{(}_{Cl}_{ℑ}_{ (}_{S}_{))}. Therefore, we obtain _{x}_{S}_{pu}_{τ}_{(}_{ℑ}_{)}(_{S}_{τ}_{(ℵ)}(_{pu}

A soft function _{pu}_{ω}_{x}_{pu}_{x}_{x}_{pu}_{τ}_{σ}_{ω} (

Let _{pu}_{ω}_{ω}

Suppose that _{pu}_{ω}_{pu}_{x}_{p}_{(}_{x}_{)} ?? _{V}_{pu}_{ω}_{x}_{pu}_{τ}_{(}_{ℑ}_{)}(_{(}_{τ}_{(ℵ))}_{ω} (_{V}_{τ}_{(}_{ℑ}_{)}(_{pu}_{τ}_{(}_{ℑ}_{)}(_{τ}_{(ℵ)}(_{V}_{ℑ}(_{(}_{τ}_{(}_{ℑ}_{))}_{a} (_{τ}_{(}_{ℑ}_{)}(_{(}_{τ}_{(ℵ))}_{ω} (_{V}_{Cl}_{ℵω }_{(}_{V}_{)}(_{ℵ}_{ω} (_{ℑ}(_{ℵ}_{ω} (_{ω}

Suppose that _{ω}_{x}_{pu}_{x}_{p}_{(}_{x}_{)} ?? _{ω}_{ℑ}(_{ℵ}_{ω} (_{ℵ}_{ω} (_{(}_{τ}_{(ℵ}_{ω}_{))}_{u(a)} (_{ℵ}_{ω} (_{τ}_{(}_{ℑ}_{)}(_{S}_{Cl}_{ℑ}_{ (}_{S}_{)}, and thus, _{pu}_{τ}_{(}_{ℑ}_{)}(_{S}_{pu}_{Cl}_{ℑ}_{ (}_{S}_{)}) = _{p}_{(}_{Cl}_{ℑ}_{ (}_{S}_{))}. Therefore, we obtain _{x}_{S}_{pu}_{τ}_{(}_{ℑ}_{)}(_{S}_{(}_{τ}_{(ℵ))}_{ω} (_{pu}_{ω}

Every soft _{ω}

Let _{pu}_{ω}_{x}_{pu}_{x}_{ω}_{pu}_{x}_{pu}_{τ}_{σ}_{ω} (_{σ}_{ω} (_{σ}_{pu}_{τ}_{σ}_{pu}

The converse of Theorem 3.5 is not true in general, as clarified by the following example:

Let _{ω}_{pu}_{ω}

If _{pu}_{pu}_{ω}

Suppose that _{pu}_{x}_{pu}_{x}_{pu}_{x}_{pu}_{τ}_{σ}_{σ}_{σ}_{ω} (_{pu}_{τ}_{σ}_{ω} (_{pu}_{ω}

Every soft continuous function is soft

The following two examples demonstrate that soft continuity and soft _{ω}

Let _{ω}_{pu}_{ω}

Let _{ω}_{pu}_{ω}

The following result provides a sufficient condition for a soft _{ω}

If _{pu}_{ω}_{pu}

Suppose that _{pu}_{ω}_{x}_{pu}_{x}_{pu}_{x}_{σ}_{ω} (_{pu}_{ω}_{x}_{pu}_{τ}_{σ}_{ω} (

Hence, _{pu}

Let _{pu}_{x}_{pu}_{x}_{θ}_{x}_{pu}_{σ}_{ω}(_{pu}_{ω}

Let _{x}_{pu}_{x}_{θ}_{x}_{pu}_{σ}_{ω} (_{x}_{θ}_{x}_{τ}_{x}_{pu}_{τ}_{pu}_{σ}_{ω}(_{pu}_{ω}

Let _{pu}_{pu}_{ω}

Using Theorem 3.12, it is sufficient to prove that for every _{x}_{pu}_{x}_{θ}_{x}_{pu}_{σ}_{ω} (_{x}_{pu}_{x}_{θ}_{pu}_{x}

If _{pu}_{ω}_{pu}_{θ}_{θ}_{ω} (_{pu}

Suppose that _{pu}_{ω}_{y}_{pu}_{θ}_{y}_{θ}_{ω} (_{pu}_{y}_{y}_{pu}_{θ}_{x}_{θ}_{y}_{pu}_{x}_{pu}_{ω}_{x}_{pu}_{τ}_{σ}_{ω} (_{x}_{x}_{θ}_{τ}_{A}_{pu}_{τ}_{pu}_{A}_{B}_{pu}_{τ}_{pu}_{τ}_{pu}_{σ}_{ω} (_{pu}_{σ}_{ω}(_{pu}_{B}_{y}_{θ}_{ω} (_{pu}

We have defined and investigated the _{ω}_{ω}_{ω}_{ω}

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

- Al Ghour, S, and Bin-Saadon, A (2019). On some generated soft topological spaces and soft homogeneity. Heliyon.
*5*. article no. e02061 - Al Ghour, S, and Hamed, W (2020). On two classes of soft sets in soft topological spaces. Symmetry.
*12*. article no. 265 - Al Ghour, S, and Irshedat, B (2017). The topology of θω-open sets. Filomat.
*31*, 5369-5377. https://doi.org/10.2298/FIL1716369A - Molodtsov, D (1999). Soft set theory?first results. Computers & Mathematics with Applications.
*37*, 19-31. https://doi.org/10.1016/S0898-1221(99)00056-5 - Shabir, M, and Naz, M (2011). On soft topological spaces. Computers & Mathematics with Applications.
*61*, 1786-1799. https://doi.org/10.1016/j.camwa.2011.02.006 - Al Ghour, S (2021). Weaker forms of soft regular and soft T2 soft topological spaces. Mathematics.
*9*. article no. 2153 - Al Ghour, S (2021). Soft minimal soft sets and soft prehomogeneity in soft topological spaces. International Journal of Fuzzy Logic and Intelligent Systems.
*21*, 269-279. https://doi.org/10.5391/IJFIS.2021.21.3.269 - Kocinac, LD, Al-shami, TM, and Cetkin, V (2021). Selection principles in the context of soft sets: Menger spaces. Soft Computing.
*25*, 12693-12702. https://doi.org/10.1007/s00500-021-06069-6 - Al-shami, TM (2021). New soft structure: infra soft topological spaces. Mathematical Problems in Engineering.
*2021*. article no. 3361604 - Al-shami, TM (2021). Infra soft compact spaces and application to fixed point theorem. Journal of Function Spaces.
*2021*. article no. 3417096 - Asaad, BA, Al-shami, TM, and Mhemdi, A (2021). Bioperators on soft topological spaces. AIMS Mathematics.
*6*, 12471-12490. https://doi.org/10.3934/math.2021720 - Al-shami, TM, and Mhemdi, A (2021). Belong and nonbelong relations on double-framed soft sets and their applications. Journal of Mathematics.
*2021*. article no. 9940301 - Alkhazaleh, S, and Marei, EA (2021). New soft rough set approximations. International Journal of Fuzzy Logic and Intelligent Systems.
*21*, 123-134. https://doi.org/10.5391/IJFIS.2021.21.2.123 - Al Ghour, S (2021). Strong form of soft semi-open sets in soft topological spaces. International Journal of Fuzzy Logic and Intelligent Systems.
*21*, 159-168. https://doi.org/10.5391/IJFIS.2021.21.2.159 - Al Ghour, S (2021). Soft ω-paracompactness in soft topological spaces. International Journal of Fuzzy Logic and Intelligent Systems.
*21*, 57-65. https://doi.org/10.5391/IJFIS.2021.21.1.57 - Al-shami, TM, and Abo-Tabl, ESA (2021). Connectedness and local connectedness on infra soft topological spaces. Mathematics.
*9*. article no. 1759 - Mousarezaei, R, and Davvaz, B (2021). On soft topological polygroups and their examples. International Journal of Fuzzy Logic and Intelligent Systems.
*21*, 29-37. https://doi.org/10.5391/IJFIS.2021.21.1.29 - Oztunc, S, Aslan, S, and Dutta, H (2021). Categorical structures of soft groups. Soft Computing.
*25*, 3059-3064. https://doi.org/10.1007/s00500-020-05362-0 - Al-Shami, TM, and Abo-Tabl, EA (2021). Soft α-separation axioms and α-fixed soft points. AIMS Mathematics.
*6*, 5675-5694. https://doi.org/10.3934/math.2021335 - Al-Shami, TM (2021). Bipolar soft sets: relations between them and ordinary points and their applications. Complexity.
*2021*. article no. 6621854 - Al-Shami, TM, Alshammari, I, and Asaad, BA (2020). Soft maps via soft somewhere dense sets. Filomat.
*34*, 3429-3440. https://doi.org/10.2298/FIL2010429A - Oguz, G (2020). Soft topological transformation groups. Mathematics.
*8*. article no. 1545 - Min, WK (2020). On soft ω-structures defined by soft sets. International Journal of Fuzzy Logic and Intelligent Systems.
*20*, 119-123. https://doi.org/10.5391/IJFIS.2020.20.2.119 - Cetkin, V, Guner, E, and Aygun, H (2020). On 2S-metric spaces. Soft Computing.
*24*, 12731-12742. https://doi.org/10.1007/s00500-020-05134-w - El-Shafei, ME, and Al-Shami, TM (2020). Applications of partial belong and total non-belong relations on soft separation axioms and decision-making problem. Computational and Applied Mathematics.
*39*. article no. 138 - Alcantud, JCR (2020). Soft open bases and a novel construction of soft topologies from bases for topologies. Mathematics.
*8*. article no. 672 - Bahredar, AA, and Kouhestani, N (2020). On ?-soft topological semigroups. Soft Computing.
*24*, 7035-7046. https://doi.org/10.1007/s00500-020-04826-7 - Al-shami, TM, and El-Shafei, ME (2020). Partial belong relation on soft separation axioms and decision-making problem, two birds with one stone. Soft Computing.
*24*, 5377-5387. https://doi.org/10.1007/s00500-019-04295-7 - Al-shami, TM, Kocinac, LD, and Asaad, BA (2020). Sum of soft topological spaces. Mathematics.
*8*. article no. 990 - Al-Saadi, HS, and Min, WK (2017). On soft generalized closed sets in a soft topological space with a soft weak structure. International Journal of Fuzzy Logic and Intelligent Systems.
*17*, 323-328. https://doi.org/10.5391/IJFIS.2017.17.4.323 - Velichko, NV (1966). H-closed topological spaces. Mat Sb.
*70*, 98-112. - Hassan, JA, and Labendia, MA (2022). θs-open sets and θscontinuity of maps in the product space. Journal of Mathematics and Computer Science.
*25*, 182-190. https://doi.org/10.22436/jmcs.025.02.07 - Osipov, AV (2021). On the cardinality of S(n)-spaces. Quaestiones Mathematicae.
*44*, 121-128. https://doi.org/10.2989/16073606.2019.1672112 - Periyasamy, P, and Ramesh, PR (2020). Local δ-closure functions in ideal topological spaces. Advances in Mathematics: Scientific Journal.
*9*, 2427-2436. https://doi.org/10.37418/amsj.9.5.6 - Ramirez-Paramo, A (2019). A generalization of some cardinal function inequalities. Topology and its Applications.
*256*, 228-234. https://doi.org/10.1016/j.topol.2019.02.009 - Babinkostova, L, Pansera, BA, and Scheepers, M (2019). Selective versions of θ-density. Topology and its Applications.
*258*, 268-281. https://doi.org/10.1016/j.topol.2019.02.061 - Modak, S, and Noiri, T (2019). Some generalizations of locally closed sets. Iranian Journal of Mathematical Sciences and Informatics.
*14*, 159-165. https://doi.org/10.7508/ijmsi.2019.01.014 - Reyes, JD, and Morales, AR (2018). The weak Urysohn number and upper bounds for cardinality of Hausdorff spaces. Houston Journal of Mathematics.
*44*, 1389-1398. - Al Ghour, S, and Al-Zoubi, S (2021). A new class between theta open sets and theta omega open sets. Heliyon.
*7*. article no. e05996 - Al Ghour, S, and Irshidat, B (2020). On θω continuity. Heliyon.
*6*. article no. e03349 - Butanas, LLL, and Labendia, MA (2020). θω-connected space and θω-continuity in the product space. Poincare Journal of Analysis and Applications.
*7*, 79-88. https://doi.org/10.46753/pjaa.2020.v07i01.008 - Latif, RM (2020). Theta-ω-mappings in topological spaces. WSEAS Transactions on Mathematics.
*19*, 186-207. https://doi.org/10.37394/23206.2020.19.18 - Al Ghour, S, and El-Issa, S (2019). θω-Connectedness and ω-R1 properties. Proyecciones (Journal of Mathematics).
*38*, 921-942. https://doi.org/10.22199/issn.0717-6279-2019-05-0059 - Georgiou, DN, Megaritis, AC, and Petropoulos, VI (2013). On soft topological spaces. Applied Mathematics & Information Sciences.
*7*, 1889-1901. https://doi.org/10.12785/amis/070527 - Al-shami, TM, and Kocinac, LD (2019). The equivalence between the enriched and extended soft topologies. Applied and Computational Mathematics.
*18*, 149-162. - Das, S, and Samanta, S (2013). Soft metric. Annals of Fuzzy Mathematics and Informatics.
*6*, 77-94. - Nazmul, S, and Samanta, SK (2013). Neighbourhood properties of soft topological spaces. Annals of Fuzzy Mathematics and Informatics.
*6*, 1-15. - Babitha, KV, and Sunil, J (2010). Soft set relations and functions. Computers & Mathematics with Applications.
*60*, 1840-1849. https://doi.org/10.1016/j.camwa.2010.07.014 - Aygunoglu, A, and Aygun, H (2012). Some notes on soft topological spaces. Neural Computing and Applications.
*21*, 113-119. https://doi.org/10.1007/s00521-011-0722-3 - Sayed, ME, and El-Bably, MK (2017). Soft simply open sets in soft topological space. Journal of Computational and Theoretical Nanoscience.
*14*, 4100-4103. https://doi.org/10.1166/jctn.2017.6792 - Fomin, S (1943). Extensions of topological spaces. Annals of Mathematics.
*43*, 471-480. https://doi.org/10.2307/1968976

E-mail: algore@just.edu.jo

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 89-99

**Published online** March 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.1.89

Copyright © The 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 (Samer Al Ghour)

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 noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

The soft *θ*_{ω}-closure operator is defined as a new soft operator that lies strictly between the usual soft closure and the soft *θ*-closure. Sufficient conditions are provided for equivalence between the soft *θ*_{ω}-closure and usual soft closure operators, and between the soft *θ*_{ω}-closure and soft *θ*-closure operators. Via the soft *θ*_{ω}-closure operator, the soft *θ*_{ω}-open sets are defined as a new class of soft sets that lies strictly between the class of soft open sets and the class of soft *θ*-open sets. It is proven that the class of soft *θ*_{ω}-open sets form a new soft topology. The soft ω-regularity is characterized via both the soft *θ*_{ω}-closure operator and soft *θ*_{ω}-open sets. The soft product theorem and several soft mapping theorems are introduced. The correspondence between the soft topology of the soft *θ*_{ω}-open sets of soft topological space and their generated topological spaces, and vice versa, are studied. In addition to these, soft *θ*_{ω}-continuity as a strong form of soft *θ*-continuity is introduced and investigated.

**Keywords**: Soft *&theta,*-closure, Soft *&theta,*-open, Soft &omega,-regular, Soft product, Soft *&theta,*-continuity, Soft generated soft topological space.

This paper follows the concepts and terminology that appear in [1–3]. In this paper, TS and STS denote the topological space and soft topological space, respectively. The concept of soft sets, which was introduced by Molodtsov [4] in 1999, is a general mathematical tool for dealing with uncertainty. Let _{A}_{A}_{A}_{A}_{A}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}

The following definitions and results are used throughout this work:

Let

(a) [1] _{Y}

(b) [1] _{Y}

(c) [46] _{x}

The set of all soft points in

Let _{x}_{x}_{x}_{x}_{x}

Let (_{a}

Let (

defines a soft topology on

Let _{a}

Then,

Let

for each (

Let (

Recall that a soft set

An STS (

(a) [50] soft locally indiscrete if every soft open set is soft closed;

(b) [2] soft locally countable if for each _{x}_{x}

(c) [2] soft anti-locally countable if for every _{A}

(d) [6] soft

(e) [6] soft _{x}_{A}_{ω}_{x}_{A}

In this section, we introduce the _{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}_{ω}

Let (

(a) A soft point _{x}_{x}_{θ}_{τ}_{A}_{x}

(b) _{θ}

(c) _{A}

(d) The family of all soft _{θ}

Let (

(a) (_{θ}

(b) _{θ}_{θ}

The following is the main definition of this work.

Let (

(a) A soft point _{x}_{ω}_{x}_{θ}_{ω} (_{τ}_{ω} (_{A}_{x}

(b) _{ω}_{θ}_{ω} (

(c) _{ω}_{A}_{ω}

(d) The family of all soft _{ω}_{θ}_{ω}.

Let (

(a) _{τ}_{θ}_{ω} (_{θ}

(b) If _{ω}

(c) If _{ω}

(a) To demonstrate that _{τ}_{θ}_{ω} (_{x}_{τ}_{x}_{x}_{τ}_{A}_{τ}_{ω} (_{τ}_{ω} (_{A}_{x}_{θ}_{ω} (_{θ}_{ω} (_{θ}_{x}_{θ}_{ω} (_{x}_{x}_{θ}_{ω} (_{τ}_{ω} (_{A}_{τ}_{ω} (_{τ}_{τ}_{A}_{x}_{θ}

(b) Suppose that _{θ}_{θ}_{ω} (_{ω}

(c) Suppose that _{ω}_{θ}_{ω} (_{τ}

Let (

(a) _{τ}_{θ}_{ω} (F).

(b) If _{ω}

(a) From Theorem 2.4 (a), _{τ}_{θ}_{ω} (_{θ}_{ω} (_{τ}_{x}_{θ}_{ω} (_{x}_{τ}_{ω} (_{A}_{τ}_{ω} (_{A}_{x}_{τ}

(b) Suppose that _{τ}_{θ}_{ω} (_{ω}

Let (

(a) _{τ}_{θ}_{ω} (

(b) If _{ω}

The proof follows from Theorem 2.5 and Theorem 7 of [6].

Let (

(a) _{τ}_{θ}_{ω} (

(b) If _{ω}

The proof follows from Theorem 2.5 and Theorem 6 of [6].

Let (

(a) _{θ}_{θ}_{ω} (

(b) If _{ω}

(a) Using Theorem 2.4(a), _{θ}_{ω} (_{θ}_{θ}_{θ}_{ω} (_{x}_{θ}_{x}_{τ}_{A}_{τ}_{ω} (_{τ}_{τ}_{ω} (_{A}_{x}_{θ}_{ω} (

(b) Suppose that _{ω}_{θ}_{ω} (_{θ}

For any STS (_{θ}_{θ}_{ω} ⊆

Let _{θ}_{A}_{A}_{ω}_{θ}_{ω}. Therefore, _{θ}_{θ}_{ω}. To demonstrate that _{θ}_{ω} ⊆ _{θ}_{ω}; then, 1_{A}_{ω}_{A}

Let (_{θ}_{τ}

Let _{τ}_{θ}_{θ}_{τ}_{x}_{θ}_{x}_{τ}_{A}_{y}_{τ}_{A}_{τ}_{θ}

Let (

(a) If _{A}_{θ}_{ω} (_{θ}_{ω} (

(b) For each _{θ}_{ω} (_{θ}_{ω} (_{θ}_{ω} (

(c) For each _{θ}_{ω} (

(d) For each _{ω}_{θ}_{ω} (_{τ}

(e) For each _{θ}_{θ}_{ω} (_{τ}

(a) Let _{x}_{θ}_{ω} (_{x}_{x}_{θ}_{ω} (_{τ}_{ω} (_{A}_{τ}_{ω} (_{A}_{x}_{θ}_{ω} (

(b) As _{θ}_{ω} (_{θ}_{ω} (_{θ}_{ω} (_{x}_{A}_{θ}_{ω} (_{θ}_{ω} (_{x}_{τ}_{ω} (_{A}_{τ}_{ω} (_{A}_{x}

Hence, _{x}_{A}_{θ}_{ω} (

(c) Let _{A}_{θ}_{ω} (_{x}_{A}_{θ}_{ω} (_{x}_{τ}_{ω} (_{A}_{θ}_{ω} (_{A}_{A}_{θ}_{ω} (

(d) Let _{ω}_{τ}_{θ}_{ω} (_{θ}_{ω} (_{τ}_{x}_{θ}_{ω} (_{A}_{τ}_{x}_{θ}_{ω}(_{x}_{A}_{τ}_{τ}_{ω} (1_{A}_{τ}_{A}

This is a contradiction.

(e) follows from (d) and Lemma 2.10.

Let (

(a) 0_{A}_{A}_{ω}

(b) The finite soft union of soft _{ω}_{ω}

(c) The arbitrary soft intersection of soft _{ω}_{ω}

(a) The proof follows from Theorems 2.2(a) and 2.4(b).

(b) We demonstrate that the soft union of two soft _{ω}_{ω}_{ω}_{θ}_{ω} (_{θ}_{ω} (

Hence, _{ω}

(c) Let _{λ}_{ω}_{λ}_{θ}_{ω} (_{λ}_{θ}_{ω} ( ∩? {_{λ}_{λ}_{x}_{θ}_{ω} ( ∩? {_{λ}_{x}_{τ}_{ω} (_{λ}_{A}_{τ}_{ω} (_{λ}_{A}_{x}_{θ}_{ω} (_{λ}_{λ}_{x}_{λ}

For any STS (_{θ}_{ω}

(1) According to Theorem 2.12(a), 0_{A}_{A}_{ω}_{A}_{A}_{θ}_{ω}.

(2) Let _{θ}_{ω}; then, 1_{A}_{A}_{ω}

Then, using Theorem 2.12(b), 1_{A}_{ω}_{θ}_{ω}.

(3) Let _{λ}_{θ}_{ω} for every _{A}_{λ}_{ω}

is soft _{ω}_{λ}_{θ}_{ω}.

Let (_{θ}_{ω} if and only if, for each _{x}_{x}_{τ}_{ω} (

Let _{θ}_{ω}, and let _{x}_{A}_{ω}_{x}_{A}_{A}_{A}_{ω}_{θ}_{ω} (1_{A}_{A}_{x}_{A}_{θ}_{ω} (1_{A}_{x}_{τ}_{ω} (_{A}_{A}_{x}_{τ}_{ω} (

Suppose that for each _{x}_{x}_{τ}_{ω} (_{θ}_{ω}. Then, _{θ}_{ω}(1_{A}_{A}_{x}_{θ}_{ω} (1_{A}_{A}_{x}_{τ}_{ω} (_{x}_{τ}_{ω} (_{A}_{A}_{x}_{A}_{θ}_{ω} (1_{A}

Every soft open, soft _{ω}

Let (_{x}_{τ}_{ω} (_{x}_{τ}_{ω} (_{ω}

Every countable soft open set in an STS is soft _{ω}

Using Theorem 2(d) of [2], countable soft sets in an STS are soft

For any STS (

(a) (

(b) _{θ}_{ω}.

(c) For every _{θ}_{ω} (_{τ}

(a) ⇒ (b): Suppose that (_{θ}_{ω}, let _{x}_{x}_{τ}_{ω}(_{θ}_{ω}. However, from Theorem 2.9, we obtain _{θ}_{ω} ⊆

(b) ⇒ (c): Suppose that _{θ}_{ω} and let _{θ}_{ω} (_{τ}_{x}_{A}_{τ}_{A}_{τ}_{A}_{τ}_{θ}_{ω}. Thus, according to Theorem 2.14, there exists _{x}_{τ}_{ω} (_{A}_{τ}_{x}_{τ}_{ω} (_{τ}_{ω} (_{τ}_{A}_{x}_{A}_{θ}_{ω} (_{τ}_{θ}_{ω} (

(c) ⇒ (a): Suppose that _{θ}_{ω} (_{τ}_{x}_{A}_{θ}_{ω} (1_{A}_{τ}_{A}_{A}_{x}_{A}_{θ}_{ω} (1_{A}_{x}_{τ}_{ω} (_{A}_{A}_{x}_{τ}_{ω} (

The inclusions in Theorem 2.9 are not equalities in general.

Let

Then,

and

Let (_{τ}_{a} (_{τ}

Let (_{θ}_{a}_{θ}

Suppose that _{θ}_{x}_{τ}_{a}_{τ}_{a} (_{τ}_{a}_{θ}

Let _{a}_{a}_{θ}

Suppose that

for all

Suppose that _{a}_{θ}_{x}_{a}_{θ}_{a}_{ℑ}_{a} (_{x}_{V}_{Cl}_{ℑa (}_{V}_{ )} ⊆?

Let (_{θ}_{θ}

For each _{a}

Let (_{θ}_{ω}, _{a}_{θ}_{ω} for all

Suppose that _{θ}_{ω} and let _{x}_{θ}_{ω}, there exists _{x}_{τ}_{ω} (_{a}_{(}_{τ}_{a}_{)}_{ω,} (_{(}_{τ}_{ω}_{)}_{a} (_{τ}_{ω} (_{a}_{θ}_{ω}.

Let _{a}_{a}_{θ}_{ω} for all

Suppose that

Suppose that _{a}_{θ}_{ω} for all _{x}_{a}_{θ}_{ω}. Thus, there exists _{a}_{(ℑ}_{a}_{)}_{ω} (_{x}_{V}_{Cl}_{(ℑa)ω}_{(}_{V}_{)} ⊆?

Let (_{θ}_{ω} if and only if _{θ}_{ω} for all

For each _{a}

Let (_{θ}_{ω}, _{θ}_{ω} and _{θ}_{ω}.

Let _{x}_{y}_{(}_{x,y}_{)} ?? _{θ}_{ω}, there exists _{(}_{x,y}_{)} ?? _{(}_{τ}_{*}_{σ}_{)}_{ω} (_{(}_{x,y}_{)} ?? _{(}_{x,y}_{)} ?? _{τ}_{ω} (_{σ}_{ω} (_{(}_{τ}_{*}_{σ}_{)}_{ω} (_{(}_{τ}_{*}_{σ}_{)}_{ω} (

Thus, we obtain _{x}_{τ}_{ω} (_{y}_{τ}_{ω} (_{θ}_{ω} and _{θ}_{ω}.

Let (_{θ}_{ω} and _{θ}_{ω}. Is it true that _{θ}_{ω} ?

If _{pu}_{pu}_{ω}

Let

Define

For every soft closed set _{pu}_{pu}_{ω}_{A}_{pu}_{A}_{pu}

Let _{pu}_{pu}_{pu}_{ω}_{pu}_{θ}_{θ}_{ω}

Let _{θ}_{y}_{pu}_{x}_{y}_{pu}_{x}_{x}_{θ}_{x}_{τ}_{y}_{pu}_{x}_{pu}_{pu}_{τ}_{pu}_{pu}_{pu}_{pu}_{ω}_{pu}_{τ}_{τ}_{ω} (_{pu}_{pu}_{τ}_{pu}_{θ}_{ω}. It follows that _{pu}_{θ}_{θ}_{ω}

Let _{pu}_{pu}_{pu}_{ω}_{ω}_{pu}_{θ}_{ω}_{θ}_{ω}

Let _{θ}_{ω} and let _{y}_{pu}_{x}_{y}_{pu}_{x}_{x}_{θ}_{ω}, there exists _{x}_{τ}_{ω} (_{y}_{pu}_{x}_{pu}_{pu}_{τ}_{ω} (_{pu}_{pu}_{pu}_{ω}_{ω}_{pu}_{τ}_{ω} (_{τ}_{ω} (_{pu}_{pu}_{τ}_{ω} (_{pu}_{pu}_{θ}_{ω}.

Let _{pu}_{pu}_{pu}_{ω}_{ω}_{pu}_{θ}_{ω}_{θ}_{ω}

Let _{θ}_{ω} and let _{pu}_{x}_{pu}_{x}_{τ}_{ω} (_{pu}_{pu}_{ω}_{ω}_{pu}_{θ}_{ω}_{θ}_{ω}

In this section, we introduce and investigate soft _{ω}

A soft function _{pu}_{x}_{pu}_{x}_{x}_{pu}_{τ}_{σ}

Let _{pu}

Suppose that _{pu}_{pu}_{x}_{p}_{(}_{x}_{)} ?? _{V}_{pu}_{x}_{pu}_{τ}_{(}_{ℑ}_{)}(_{τ}_{(ℵ)}(_{V}_{τ}_{(}_{ℑ}_{)}(_{pu}_{τ}_{(}_{ℑ}_{)}(_{τ}_{(ℵ)}(_{V}_{ℑ}(_{(}_{τ}_{(}_{ℑ}_{))}_{a} (_{τ}_{(}_{ℑ}_{)}(_{τ}_{(ℵ)}(_{V}_{Cl}_{ℵ}_{(}_{V}_{)}(_{ℵ}(_{ℑ}(_{ℵ}(

Suppose that _{x}_{pu}_{x}_{p}_{(}_{x}_{)} ?? _{ℑ}(_{ℵ}(_{ℵ}(_{(}_{τ}_{(ℵ))}_{u (a)} (_{ℵ}(_{τ}_{(}_{ℑ}_{)} (_{S}_{Cl}_{ℑ}_{ (}_{S}_{)}, and thus, _{pu}_{τ}_{(}_{ℑ}_{)}(_{S}_{pu}_{Cl}_{ℑ}_{ (}_{S}_{)}) = _{p}_{(}_{Cl}_{ℑ}_{ (}_{S}_{))}. Therefore, we obtain _{x}_{S}_{pu}_{τ}_{(}_{ℑ}_{)}(_{S}_{τ}_{(ℵ)}(_{pu}

A soft function _{pu}_{ω}_{x}_{pu}_{x}_{x}_{pu}_{τ}_{σ}_{ω} (

Let _{pu}_{ω}_{ω}

Suppose that _{pu}_{ω}_{pu}_{x}_{p}_{(}_{x}_{)} ?? _{V}_{pu}_{ω}_{x}_{pu}_{τ}_{(}_{ℑ}_{)}(_{(}_{τ}_{(ℵ))}_{ω} (_{V}_{τ}_{(}_{ℑ}_{)}(_{pu}_{τ}_{(}_{ℑ}_{)}(_{τ}_{(ℵ)}(_{V}_{ℑ}(_{(}_{τ}_{(}_{ℑ}_{))}_{a} (_{τ}_{(}_{ℑ}_{)}(_{(}_{τ}_{(ℵ))}_{ω} (_{V}_{Cl}_{ℵω }_{(}_{V}_{)}(_{ℵ}_{ω} (_{ℑ}(_{ℵ}_{ω} (_{ω}

Suppose that _{ω}_{x}_{pu}_{x}_{p}_{(}_{x}_{)} ?? _{ω}_{ℑ}(_{ℵ}_{ω} (_{ℵ}_{ω} (_{(}_{τ}_{(ℵ}_{ω}_{))}_{u(a)} (_{ℵ}_{ω} (_{τ}_{(}_{ℑ}_{)}(_{S}_{Cl}_{ℑ}_{ (}_{S}_{)}, and thus, _{pu}_{τ}_{(}_{ℑ}_{)}(_{S}_{pu}_{Cl}_{ℑ}_{ (}_{S}_{)}) = _{p}_{(}_{Cl}_{ℑ}_{ (}_{S}_{))}. Therefore, we obtain _{x}_{S}_{pu}_{τ}_{(}_{ℑ}_{)}(_{S}_{(}_{τ}_{(ℵ))}_{ω} (_{pu}_{ω}

Every soft _{ω}

Let _{pu}_{ω}_{x}_{pu}_{x}_{ω}_{pu}_{x}_{pu}_{τ}_{σ}_{ω} (_{σ}_{ω} (_{σ}_{pu}_{τ}_{σ}_{pu}

The converse of Theorem 3.5 is not true in general, as clarified by the following example:

Let _{ω}_{pu}_{ω}

If _{pu}_{pu}_{ω}

Suppose that _{pu}_{x}_{pu}_{x}_{pu}_{x}_{pu}_{τ}_{σ}_{σ}_{σ}_{ω} (_{pu}_{τ}_{σ}_{ω} (_{pu}_{ω}

Every soft continuous function is soft

The following two examples demonstrate that soft continuity and soft _{ω}

Let _{ω}_{pu}_{ω}

Let _{ω}_{pu}_{ω}

The following result provides a sufficient condition for a soft _{ω}

If _{pu}_{ω}_{pu}

Suppose that _{pu}_{ω}_{x}_{pu}_{x}_{pu}_{x}_{σ}_{ω} (_{pu}_{ω}_{x}_{pu}_{τ}_{σ}_{ω} (

Hence, _{pu}

Let _{pu}_{x}_{pu}_{x}_{θ}_{x}_{pu}_{σ}_{ω}(_{pu}_{ω}

Let _{x}_{pu}_{x}_{θ}_{x}_{pu}_{σ}_{ω} (_{x}_{θ}_{x}_{τ}_{x}_{pu}_{τ}_{pu}_{σ}_{ω}(_{pu}_{ω}

Let _{pu}_{pu}_{ω}

Using Theorem 3.12, it is sufficient to prove that for every _{x}_{pu}_{x}_{θ}_{x}_{pu}_{σ}_{ω} (_{x}_{pu}_{x}_{θ}_{pu}_{x}

If _{pu}_{ω}_{pu}_{θ}_{θ}_{ω} (_{pu}

Suppose that _{pu}_{ω}_{y}_{pu}_{θ}_{y}_{θ}_{ω} (_{pu}_{y}_{y}_{pu}_{θ}_{x}_{θ}_{y}_{pu}_{x}_{pu}_{ω}_{x}_{pu}_{τ}_{σ}_{ω} (_{x}_{x}_{θ}_{τ}_{A}_{pu}_{τ}_{pu}_{A}_{B}_{pu}_{τ}_{pu}_{τ}_{pu}_{σ}_{ω} (_{pu}_{σ}_{ω}(_{pu}_{B}_{y}_{θ}_{ω} (_{pu}

We have defined and investigated the _{ω}_{ω}_{ω}_{ω}

- Al Ghour, S, and Bin-Saadon, A (2019). On some generated soft topological spaces and soft homogeneity. Heliyon.
*5*. article no. e02061 - Al Ghour, S, and Hamed, W (2020). On two classes of soft sets in soft topological spaces. Symmetry.
*12*. article no. 265 - Al Ghour, S, and Irshedat, B (2017). The topology of θω-open sets. Filomat.
*31*, 5369-5377. https://doi.org/10.2298/FIL1716369A - Molodtsov, D (1999). Soft set theory?first results. Computers & Mathematics with Applications.
*37*, 19-31. https://doi.org/10.1016/S0898-1221(99)00056-5 - Shabir, M, and Naz, M (2011). On soft topological spaces. Computers & Mathematics with Applications.
*61*, 1786-1799. https://doi.org/10.1016/j.camwa.2011.02.006 - Al Ghour, S (2021). Weaker forms of soft regular and soft T2 soft topological spaces. Mathematics.
*9*. article no. 2153 - Al Ghour, S (2021). Soft minimal soft sets and soft prehomogeneity in soft topological spaces. International Journal of Fuzzy Logic and Intelligent Systems.
*21*, 269-279. https://doi.org/10.5391/IJFIS.2021.21.3.269 - Kocinac, LD, Al-shami, TM, and Cetkin, V (2021). Selection principles in the context of soft sets: Menger spaces. Soft Computing.
*25*, 12693-12702. https://doi.org/10.1007/s00500-021-06069-6 - Al-shami, TM (2021). New soft structure: infra soft topological spaces. Mathematical Problems in Engineering.
*2021*. article no. 3361604 - Al-shami, TM (2021). Infra soft compact spaces and application to fixed point theorem. Journal of Function Spaces.
*2021*. article no. 3417096 - Asaad, BA, Al-shami, TM, and Mhemdi, A (2021). Bioperators on soft topological spaces. AIMS Mathematics.
*6*, 12471-12490. https://doi.org/10.3934/math.2021720 - Al-shami, TM, and Mhemdi, A (2021). Belong and nonbelong relations on double-framed soft sets and their applications. Journal of Mathematics.
*2021*. article no. 9940301 - Alkhazaleh, S, and Marei, EA (2021). New soft rough set approximations. International Journal of Fuzzy Logic and Intelligent Systems.
*21*, 123-134. https://doi.org/10.5391/IJFIS.2021.21.2.123 - Al Ghour, S (2021). Strong form of soft semi-open sets in soft topological spaces. International Journal of Fuzzy Logic and Intelligent Systems.
*21*, 159-168. https://doi.org/10.5391/IJFIS.2021.21.2.159 - Al Ghour, S (2021). Soft ω-paracompactness in soft topological spaces. International Journal of Fuzzy Logic and Intelligent Systems.
*21*, 57-65. https://doi.org/10.5391/IJFIS.2021.21.1.57 - Al-shami, TM, and Abo-Tabl, ESA (2021). Connectedness and local connectedness on infra soft topological spaces. Mathematics.
*9*. article no. 1759 - Mousarezaei, R, and Davvaz, B (2021). On soft topological polygroups and their examples. International Journal of Fuzzy Logic and Intelligent Systems.
*21*, 29-37. https://doi.org/10.5391/IJFIS.2021.21.1.29 - Oztunc, S, Aslan, S, and Dutta, H (2021). Categorical structures of soft groups. Soft Computing.
*25*, 3059-3064. https://doi.org/10.1007/s00500-020-05362-0 - Al-Shami, TM, and Abo-Tabl, EA (2021). Soft α-separation axioms and α-fixed soft points. AIMS Mathematics.
*6*, 5675-5694. https://doi.org/10.3934/math.2021335 - Al-Shami, TM (2021). Bipolar soft sets: relations between them and ordinary points and their applications. Complexity.
*2021*. article no. 6621854 - Al-Shami, TM, Alshammari, I, and Asaad, BA (2020). Soft maps via soft somewhere dense sets. Filomat.
*34*, 3429-3440. https://doi.org/10.2298/FIL2010429A - Oguz, G (2020). Soft topological transformation groups. Mathematics.
*8*. article no. 1545 - Min, WK (2020). On soft ω-structures defined by soft sets. International Journal of Fuzzy Logic and Intelligent Systems.
*20*, 119-123. https://doi.org/10.5391/IJFIS.2020.20.2.119 - Cetkin, V, Guner, E, and Aygun, H (2020). On 2S-metric spaces. Soft Computing.
*24*, 12731-12742. https://doi.org/10.1007/s00500-020-05134-w - El-Shafei, ME, and Al-Shami, TM (2020). Applications of partial belong and total non-belong relations on soft separation axioms and decision-making problem. Computational and Applied Mathematics.
*39*. article no. 138 - Alcantud, JCR (2020). Soft open bases and a novel construction of soft topologies from bases for topologies. Mathematics.
*8*. article no. 672 - Bahredar, AA, and Kouhestani, N (2020). On ?-soft topological semigroups. Soft Computing.
*24*, 7035-7046. https://doi.org/10.1007/s00500-020-04826-7 - Al-shami, TM, and El-Shafei, ME (2020). Partial belong relation on soft separation axioms and decision-making problem, two birds with one stone. Soft Computing.
*24*, 5377-5387. https://doi.org/10.1007/s00500-019-04295-7 - Al-shami, TM, Kocinac, LD, and Asaad, BA (2020). Sum of soft topological spaces. Mathematics.
*8*. article no. 990 - Al-Saadi, HS, and Min, WK (2017). On soft generalized closed sets in a soft topological space with a soft weak structure. International Journal of Fuzzy Logic and Intelligent Systems.
*17*, 323-328. https://doi.org/10.5391/IJFIS.2017.17.4.323 - Velichko, NV (1966). H-closed topological spaces. Mat Sb.
*70*, 98-112. - Hassan, JA, and Labendia, MA (2022). θs-open sets and θscontinuity of maps in the product space. Journal of Mathematics and Computer Science.
*25*, 182-190. https://doi.org/10.22436/jmcs.025.02.07 - Osipov, AV (2021). On the cardinality of S(n)-spaces. Quaestiones Mathematicae.
*44*, 121-128. https://doi.org/10.2989/16073606.2019.1672112 - Periyasamy, P, and Ramesh, PR (2020). Local δ-closure functions in ideal topological spaces. Advances in Mathematics: Scientific Journal.
*9*, 2427-2436. https://doi.org/10.37418/amsj.9.5.6 - Ramirez-Paramo, A (2019). A generalization of some cardinal function inequalities. Topology and its Applications.
*256*, 228-234. https://doi.org/10.1016/j.topol.2019.02.009 - Babinkostova, L, Pansera, BA, and Scheepers, M (2019). Selective versions of θ-density. Topology and its Applications.
*258*, 268-281. https://doi.org/10.1016/j.topol.2019.02.061 - Modak, S, and Noiri, T (2019). Some generalizations of locally closed sets. Iranian Journal of Mathematical Sciences and Informatics.
*14*, 159-165. https://doi.org/10.7508/ijmsi.2019.01.014 - Reyes, JD, and Morales, AR (2018). The weak Urysohn number and upper bounds for cardinality of Hausdorff spaces. Houston Journal of Mathematics.
*44*, 1389-1398. - Al Ghour, S, and Al-Zoubi, S (2021). A new class between theta open sets and theta omega open sets. Heliyon.
*7*. article no. e05996 - Al Ghour, S, and Irshidat, B (2020). On θω continuity. Heliyon.
*6*. article no. e03349 - Butanas, LLL, and Labendia, MA (2020). θω-connected space and θω-continuity in the product space. Poincare Journal of Analysis and Applications.
*7*, 79-88. https://doi.org/10.46753/pjaa.2020.v07i01.008 - Latif, RM (2020). Theta-ω-mappings in topological spaces. WSEAS Transactions on Mathematics.
*19*, 186-207. https://doi.org/10.37394/23206.2020.19.18 - Al Ghour, S, and El-Issa, S (2019). θω-Connectedness and ω-R1 properties. Proyecciones (Journal of Mathematics).
*38*, 921-942. https://doi.org/10.22199/issn.0717-6279-2019-05-0059 - Georgiou, DN, Megaritis, AC, and Petropoulos, VI (2013). On soft topological spaces. Applied Mathematics & Information Sciences.
*7*, 1889-1901. https://doi.org/10.12785/amis/070527 - Al-shami, TM, and Kocinac, LD (2019). The equivalence between the enriched and extended soft topologies. Applied and Computational Mathematics.
*18*, 149-162. - Das, S, and Samanta, S (2013). Soft metric. Annals of Fuzzy Mathematics and Informatics.
*6*, 77-94. - Nazmul, S, and Samanta, SK (2013). Neighbourhood properties of soft topological spaces. Annals of Fuzzy Mathematics and Informatics.
*6*, 1-15. - Babitha, KV, and Sunil, J (2010). Soft set relations and functions. Computers & Mathematics with Applications.
*60*, 1840-1849. https://doi.org/10.1016/j.camwa.2010.07.014 - Aygunoglu, A, and Aygun, H (2012). Some notes on soft topological spaces. Neural Computing and Applications.
*21*, 113-119. https://doi.org/10.1007/s00521-011-0722-3 - Sayed, ME, and El-Bably, MK (2017). Soft simply open sets in soft topological space. Journal of Computational and Theoretical Nanoscience.
*14*, 4100-4103. https://doi.org/10.1166/jctn.2017.6792 - Fomin, S (1943). Extensions of topological spaces. Annals of Mathematics.
*43*, 471-480. https://doi.org/10.2307/1968976

International Journal of Fuzzy Logic and Intelligent Systems 2022;22:89~99 https://doi.org/10.5391/IJFIS.2022.22.1.89

© IJFIS

Copyright © The Korean Institute of Intelligent Systems. / Powered by INFOrang Co., Ltd