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Research article

Simulation of tumor density evolution upon chemotherapy alone or combined with a treatment to reduce lactate levels

  • Received: 30 October 2023 Revised: 12 January 2024 Accepted: 19 January 2024 Published: 26 January 2024
  • MSC : 35Q92, 35Q93, 92C50, 65M60, 35K51, 35K58

  • In this study, we introduced a mathematical model mimicking as much as possible the evolutions and interactions between glioma and lactate in the brain, in order to test different therapies and administration protocols. We simulated both glioma cell density evolution and lactate concentration, and considered two therapies: chemotherapy and a treatment targeting lactate production. Three different protocols for administrating the therapies were tested. We compared the efficiency of the combined therapies, depending on the administration protocols and the dosage of the drugs, in order to evaluate the importance of controlling lactate production. Results show that the use of an agent to reduce lactate concentration permits one to significantly reduce the dose of the chemotherapeutic drug.

    Citation: Hussein Raad, Cyrille Allery, Laurence Cherfils, Carole Guillevin, Alain Miranville, Thomas Sookiew, Luc Pellerin, Rémy Guillevin. Simulation of tumor density evolution upon chemotherapy alone or combined with a treatment to reduce lactate levels[J]. AIMS Mathematics, 2024, 9(3): 5250-5268. doi: 10.3934/math.2024254

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  • In this study, we introduced a mathematical model mimicking as much as possible the evolutions and interactions between glioma and lactate in the brain, in order to test different therapies and administration protocols. We simulated both glioma cell density evolution and lactate concentration, and considered two therapies: chemotherapy and a treatment targeting lactate production. Three different protocols for administrating the therapies were tested. We compared the efficiency of the combined therapies, depending on the administration protocols and the dosage of the drugs, in order to evaluate the importance of controlling lactate production. Results show that the use of an agent to reduce lactate concentration permits one to significantly reduce the dose of the chemotherapeutic drug.



    To deal with vagueness and uncertainty existing in practical situations, e.g., social science, economics, engineering, and medical science, it has been proposed various mathematical instruments. The recent one, introduced by Molodtsov [33], is a soft set whose parameterizations are adequate to process uncertainties and are free from the inherent limitations of the previous instruments. The advantages of soft sets and their applications to different scopes were elaborated in the pioneering work of Molodtsov [33]. In 2002, Maji et al. [30] proposed a technique to handle decision-making problems using soft sets, which was then developed by many researchers. The basic ideas of soft set theory (operators and operations) were initiated by [31]. Then, the authors of [5] showed the awkwardness of these ideas and updated to be consistent with their analogs in the crisp set theory. Also, it was taken advantage of the parameterization families of soft sets to establish several types of operators and operations between soft sets as illustrated in [14,35]. To raise the efficiency of soft set theory to cope with uncertain and complicated issues, it has been hybridized with other vague instruments like fuzzy and rough sets as debated in [13,40,41].

    The year 2011 is the birth of topology induced from soft sets. It was defined by Shabir and Naz [42] and Çaǧman et al. [21] at the same time. Their approaches differ in the way of choosing the set of parameters as variable or constant. Herein, we follow the line of Shabir and Naz, which stipulate the necessity of a constant set of parameters for each element of the soft topology. Afterward, the field of soft topology attracted many researchers and intellectuals who studied the ideas of classical topology via soft topology. Min [32] described the shape of soft open and closed subsets of soft regular spaces and proved the systematic relation between soft T2 and soft T3-spaces. El-Shafei et al. [24] founded a strong family of soft Ti-spaces that preserves more features of classical separation axioms. El-Shafei and Al-shami [23] established another type of soft separation axioms and evinced their relationships with the previous types. Important adjustments for the previous studies of soft separation axioms have been conducted by some authors [8,43]. Recently, Al-shami [11] has investigated how soft separation axioms are applied to select the optimal alternatives for tourism programs.

    Aygünoǧlu and Aygün [20] familiarized the concepts of compact and Lindelöf spaces. Hida [27] presented and described other kinds of compactness. The concepts of covering properties have been popularized with respect to soft regular closed [6], soft somewhat open sets [15] and soft somewhere dense set [17]. An interesting application of soft compactness to information systems was provided by Al-shami [9]. The correspondence between enriched and extended soft topologies was proved by Al-shami and Kočinac [18]. They also pointed out that many topological properties are transposable between this type of soft topology and their parametric topologies.

    The notion of functions between soft topological spaces was defined by Kharal and Ahmad [28], which it refined using the crisp functions and soft points by Al-shami [10]. Specific sorts of soft functions such as soft continuous, open, and closed functions were discussed in [44]. The concepts of Menger spaces [29], maximal topologies [4] and expandable spaces [36] were integrated via soft settings as well. The authors of [26,37,38,39] studied topological structures inspired by the hybridizations of soft sets with fuzzy sets and neutrosophic sets. To expand the topological ideas and relax topological conditions, generalizations of soft open sets have been probed. The main contributions to this topic were done by Chen [22], Akdag and Ozkan [1], Al-shami [7] and Al-Ghour [3] who respectively put forward the concepts of soft semi-open, α-open, somewhere dense and Q-sets. Al-shami et al. [16] defined the concept of weakly soft semi-open sets and debated its main characterizations. Al-shami [12] exploited soft somewhat open sets to discover to what extent the nutrition followed by individuals is convenient for the needs of their bodies.

    The motivations for writing this article are, first, to suggest a new approach to generalizing soft topology inspired by its classical topologies. Second, to offer a new framework to produce soft topological concepts such as soft operators and continuity, which are achieved in this work. Of course, the researchers can explore other notions like soft covering properties and separation axioms via the proposed class of weakly soft α-open sets. Finally, to enhance the importance of the soft topological environment to create various analogs for each classical topological concept.

    The content of this study is regulated as follows. In Section 2, we gather the most essential definitions and properties that are necessary to make this manuscript self-contained. Then, Section 3 provides a novel approach to introduce a new generalization of soft open sets, namely, weakly soft α-sets. It constructs an illustrative example to describe its main characteristics. Section 4 applies this generalization to establish the concepts of weakly soft α-interior, weakly soft α-closure, weakly soft α-boundary, and weakly soft α-limit soft points and explores the relationships between them. Section 5 discusses the idea of weakly soft α-continuity and elucidates that the equivalent conditions of soft continuity are invalid for this type of continuity. We close this work with some conclusions and proposed future work in Section 6.

    In this segment, we recapitulate the fundamentals that are required for the readers to become conscious of the manuscript's context.

    Definition 2.1. [33] A soft set over a nonempty (crisp) set Σ is a set-valued function F from nonempty set of parameters Δ to the power set 2Σ of Σ; it is denoted by the ordered pair (F,Δ).

    That is, a soft set (F,Δ) over Σ provides a parameterized collection of subsets of Σ; so it may represented as follows

    (F,Δ)={(δ,F(δ)):δΔandF(δ)2Σ},

    where each F(δ) is termed a δ-component of (F,Δ). We denote the family of all soft sets over Σ with a set of parameters Δ by 2ΣΔ.

    Through this manuscript, (F,Δ),(G,Δ) denote soft sets over Σ.

    Definition 2.2. [31,34] A soft set (F,Δ) is called:

    (i) Absolute, symbolized by ˜Σ, if F(δ)=Σ for all δΔ.

    (ii) Null, symbolized by ϕ, if F(δ)= for all δΔ.

    (iii) A soft point if there are δΔ and σΣ with F(δ)={σ} and F(a)= for all aΔ{δ}. A soft point is symbolized by σδ. We write σδ(F,Δ) if σF(δ).

    (iv) Pseudo constant if F(δ)=Σ or for all δΔ.

    Definition 2.3. [25] We call (F,Δ) a soft subset of (G,Δ) (or (G,Δ) a soft superset of (G,Δ)), symbolized by (F,Δ)˜(G,Δ) if F(δ)G(δ) for each δΔ.

    Definition 2.4. [5] If G(δ)=ΣF(δ) for all δΔ, then we call (G,Δ) a complement of (F,Δ). The complement of (F,Δ) is symbolized by (F,Δ)c=(Fc,Δ).

    Definition 2.5. [14] Let (F,Δ) and (G,Δ) be soft sets. Then:

    (i) (F,Δ)˜(G,Δ)=(H,Δ), where H(δ)=F(δ)G(δ) for all δΔ.

    (ii) (F,Δ)˜(G,Δ)=(H,Δ), where H(δ)=F(δ)G(δ) for all δΔ.

    (iii) (F,Δ)(G,Δ)=(H,Δ), where H(δ)=F(δ)G(δ) for all δΔ.

    (iv) (F,Δ)×(G,Δ)=(H,Δ), where H(δ1,δ2)=F(δ1)×G(δ2) for all (δ1,δ2)Δ×Δ.

    The adjusted version of the definition of soft functions is given in the following.

    Definition 2.6. [10] Let M: ΣΥ and P: ΔΩ be crisp functions. A soft function MP of 2ΣΔ into 2ΥΩ is a relation such that each σδ2ΣΔ is related to one and only one ϵω2ΥΩ such that

    MP(σδ)=M(σ)P(δ) for all σδ2ΣΔ.

    In addition, M1P(ϵω)=˜σM1(ϵ)δP1(ω)σδ for each ϵω2ΥΩ.

    That is, the image of (F,Δ) and pre-image of (G,Ω) under a soft function MP: 2ΣΔ2ΥΩ are respectively given by:

    MP(F,Δ)=˜σδ(F,Δ)MP(σδ,),

    and

    M1P(G,Ω)=˜ϵω(G,Ω)M1P(ϵω).

    A soft function is described as surjective (resp., injective, bijective) if its two crisp functions satisfy this description.

    Proposition 2.7. [28] Let MP: 2ΣΔ2ΥΩ be a soft function and let (F,Δ) and (G,Δ) be soft subsets of ˜Σ and ˜Υ, respectively. Then

    (i) (F,Δ)˜M1P(MP(F,Δ)).

    (ii) If MP is injective, then (F,Δ)=M1P(MP(F,Δ)).

    (iii) MP(M1P(G,Ω))˜(G,Ω).

    (iv) If MP is surjective, then MP(M1P(G,Ω))=(G,Ω).

    Definition 2.8. [33] A subfamily T of 2ΣΔ is said to be a soft topology if the following terms are satisfied:

    (i) ˜Σ and ϕ are elements of T.

    (ii) T is closed under the arbitrary unions.

    (iii) T is closed under the finite intersections.

    We will call the triplet (Σ,T,Δ) a soft topological space (briefly, softTS). Each element in T is called soft open and its complement is called soft closed.

    Definition 2.9. [33] For a soft subset (F,Δ) of a softTS (Σ,T,Δ), the soft interior and soft closure of (F,Δ), denoted respectively by int(F,Δ) and cl(F,Δ), are defined as follows:

    (i) int(F,Δ)=˜{(G,Δ)T:(G,Δ)˜(F,Δ)}.

    (ii) cl(F,Δ)=˜{(H,Δ):(F,Δ)˜(H,Δ) and (Hc,Δ)T}.

    Definition 2.10. [15] A softTS (Σ,T,Δ) is called full if every non-null soft open set has no empty component.

    Proposition 2.11. [33] Let (Σ,T,Δ) be a softTS. Then

    Tδ={F(δ):(F,Δ)T}

    is a topology on Σ for every δΔ. We will call this topology a parametric topology.

    Definition 2.12 [33] Let (F,Δ) be a soft subset of a softTS (Σ,T,Δ). Then (int(F),Δ) and (cl(F),Δ) are respectively defined by

    int(F)(δ)=int(F(δ)),

    and

    cl(F)(δ)=cl(F(δ)),

    where int(F(δ)) and cl(F(δ)) are respectively the interior and closure of F(δ) in (Σ,Tδ).

    Definition 2.13. [20,34] Let (Σ,T,Δ) be a softTS.

    (i) If all pseudo constant soft sets are elements of T, then T is called an enriched soft topology.

    (ii) T with the property "(F,Δ)T iff F(δ)Tδ for each δΔ" is called an extended soft topology.

    A comprehensive investigation of the extended and enriched soft topologies was comported on [18]. The corresponding between these kinds of soft topologies was one of the valuable results attained in [18]. Henceforth, this sort of soft topology will be called an extended soft topology. Under this soft topology, it was elucidated several consequences that associated soft topology with its parametric topologies. De facto, the next result will be a key point in the proof of many results.

    Theorem 2.14. [18] A softTS (Σ,T,Δ) is extended iff (int(F),Δ)=int(F,Δ) and (cl(F),Δ)=cl(F,Δ) for any soft subset (F,Δ).

    Definition 2.15. A soft subset (F,Δ) of (Σ,T,Δ) is said to be:

    (i) soft α-open [1] if (F,Δ)˜int(cl(int(F,Δ))).

    (ii) soft semi-open [22] if (F,Δ)˜cl(int(F,Δ)).

    (iii) soft β-open [2] if (F,Δ)˜cl(int(cl(F,Δ))).

    (iv) soft sw-open [19] if (F,Δ)=ϕ or int(F,Δ)ϕ.

    Definition 2.16. [44] A soft function MP: (Σ,TΣ,Δ)(Υ,TΥ,Δ) is said to be soft continuous if M1P(F,Δ) is a soft open set where (F,Δ) is soft open.

    Theorem 2.17. [18] If MP: (Σ,T,Δ)(Υ,S,Ω) is soft continuous, then h: (Σ,Tδ)(Υ,SP(δ)) is continuous for each δΔ.

    We introduce the main idea of this manuscript called "weakly soft α-open sets" in this section. We show that this class of soft subset is a novel extension of soft open subsets and it lies between soft α-open and soft sw-open subsets of extended soft topology. Additionally, we construct some counterexamples to point out some divergences between this class and other extensions such as this class is not closed under soft unions. Among other obtained results, we investigate how this class behaves with respect to topological properties and the product of soft spaces.

    Definition 3.1. A soft subset (F,Δ) of (Σ,T,Δ) is called weakly soft α-open if it is a null soft set or there is a component of it which is a nonempty α-open set. That is, F(δ)= for all δΔ or

    F(δ)int(cl(int(F(δ))))

    for some δΔ.

    We call (F,Δ) a weakly soft α-closed set if its complement is weakly soft α-open.

    Proposition 3.2. A subset (F,Δ) of (Σ,T,Δ) is weakly soft α-closed iff

    (F,Δ)=˜Σ

    or

    cl(int(cl(F(δ))))F(δ)Σ

    for some δΔ.

    Proof. "⇒": Let (F,Δ) be a weakly soft α-closed set. Then,

    (Fc,Δ)=ϕ,

    or

    Fc(δ)int(cl(int(Fc(δ)))),

    for some δΔ. This means that

    (F,Δ)=˜Σ,

    or

    cl(int(cl(F(δ))))F(δ)Σ,

    for some δΔ, as required.

    "⇐": Let (F,Δ) be a soft set such that

    (F,Δ)=˜Σ,

    or

    cl(int(cl(F(δ))))F(δ)Σ,

    for some δΔ. Then,

    (Fc,Δ)=ϕ,

    or

    Fc(δ)int(cl(int(Fc(δ)))),

    for some δΔ. This implies that (Fc,Δ) is weakly soft α-open. Hence, (F,Δ) is weakly soft α-closed, as required.

    The following example clarifies that the family of weakly soft α-open (weakly soft α-closed) subsets is not closed under soft union or soft intersection.

    Example 3.3. Let R be the set of real numbers and Δ={δ1,δ2} be a set of parameters. Let T be the soft topology on R generated by

    {(δi,F(δi)):F(δi)=(ai,bi);ai,biR; ai bi and i=1,2}.

    Set

    (F,Δ)={(δ1,(0,1)),(δ2,[0,1])},

    and

    (G,Δ)={(δ1,[0,1]),(δ2,(0,1))},

    over R. It is obvious that (F,Δ) and (G,Δ) are both weakly soft α-open and weakly soft α-closed. On the other hand, their soft union is not weakly soft α-open and their soft intersection is not weakly soft α-closed. Also,

    (H,Δ)={(δ1,(1,6)),(δ2,[2,3])}

    and

    (K,Δ)={(δ1,[2,3]),(δ2,(1,6))}

    are both weakly soft α-open and weakly soft α-closed sets over R. But their soft intersection is not weakly soft α-open and their soft union is not weakly soft α-closed.

    Proposition 3.4. Let (Σ,T,Δ) be a full softTS with the property of soft hyperconnected. Then the soft intersection of soft α-open and weakly soft α-open subsets is weakly soft α-open.

    Proof. Assume that (F,Δ) and (G,Δ) are respectively soft α-open and weakly soft α-open sets. Then there exist a non-null soft open set (U,Δ) and δΔ such that (U,Δ)˜(F,Δ) and G(δ) is a nonempty α-open subset of (Σ,Tδ). So there exists a nonempty open subset Vδ of G(δ). This means that T contains a non-null soft open set (V,Δ) with V(δ)=Vδ. Since T is soft hyperconnected and full, we get VδU(δ). Therefore, G(δ) and U(δ) have a nonempty intersection. It follows from general topology that G(δ)U(δ) is a nonempty α-open subset of (Σ,Tδ). Hence, (F,Δ)˜(G,Δ) is a weakly soft α-open set.

    Corollary 3.5. Let (Σ,T,Δ) be a full softTS with the property of soft hyperconnected. Then the soft intersection of soft open and weakly soft α-open subsets is weakly soft α-open.

    Remark 3.6. Every pseudo constant soft subset (F,Δ) is a weakly soft α-subset because F(δ)= for all δΔ or int(F(δ))=Σ for some δΔ.

    The next propositions are obvious.

    Proposition 3.7. Every soft open set is weakly soft α-open.

    Proposition 3.8. Any soft subset (F,Δ) of (Σ,T,Δ) with F(δ)=Σ (resp., F(δ)=) is weakly soft α-open (resp., weakly soft α-closed).

    In the next result we provide a condition that guarantees the relation between weakly soft α-open sets and some generalizations of soft open sets.

    Proposition 3.9. If (Σ,T,Δ) is extended, then every soft α-open set is weakly soft α-open.

    Proof. Let (F,Δ) be a non-null soft α-open set. Then

    (F,Δ)˜int(cl(int(F,Δ))).

    Since T is an extended soft topology, we get

    F(δ)int(cl(int(F(δ))))

    for each δΔ. This implies that there is a component of (F,Δ) which is a nonempty α-open subset. Hence, (F,Δ) is weakly soft α-open.

    Following similar arguments one can prove the other cases.

    Proposition 3.10. If (Σ,T,Δ) is extended, then every weakly soft α-open set is soft sw-open.

    Proof. Let (F,Δ) be a non-null weakly soft α-open set. Then there is a component of (F,Δ) which is a nonempty α-open set. So int(F(δ)) for some δΔ. Since T is extended, we get

    int(F,Δ)=(int(F),Δ)ϕ.

    This completes the proof.

    The next example elaborates that a condition of "extended soft topology" furnished in Propositions 3.9 and 3.10 is indispensable.

    Example 3.11. Let Σ={σ1,σ2,σ3} be unverse and Δ={δ1,δ2} be a parameters set. Take the family T consisting of ϕ, ˜Σ and the following soft subsets over Σ with Δ

    (F1,Δ)={(δ1,{σ1}),(δ2,)},
    (F2,Δ)={(δ1,),(δ2,{σ1})},
    (F3,Δ)={(δ1,Σ),(δ2,{σ1})},
    (F4,Δ)={(δ1,{σ1}),(δ2,Σ)},
    (F5,Δ)={(δ1,{σ1}),(δ2,{σ1})},
    (F6,Δ)={(δ1,{σ1}),(δ2,{σ2,σ3})},

    and

    (F7,Δ)={(δ1,{σ2,σ3}),(δ2,{σ1})}.

    Then, (Σ,T,Δ) is a softTS. Remark that a soft set

    (H,Δ)={(δ1,{σ1,σ2}),(δ2,{σ1,σ2})}

    is soft α-open because

    int(cl(int(H,Δ)))=˜Σ.

    But it is not a weakly soft α-open set because

    int(cl(int(H(δ1))))=int(cl(int(H(δ2))))={σ1}H(δ1)=H(δ2).

    Also, a soft set

    (G,Δ)={(δ1,{σ2,σ3}),(δ2,{σ3})}

    is a weakly soft α-open set because

    int(cl(int(G(δ1))))=G(δ1).

    But it is not a soft sw-open set because int(G,Δ)=ϕ.

    To demonstrate that the converse of Propositions 3.9 and 3.10 fail, the following example is shown.

    Example 3.12. Let Σ={σ1,σ2,σ3} be unverse and Δ={δ1,δ2} be a parameters set. Take the family T consisting of ϕ, ˜Σ and the following soft subsets over Σ with Δ

    (F1,Δ)={(δ1,{σ1}),(δ2,{σ2,σ3})},
    (F2,Δ)={(δ1,{σ2,σ3}),(δ2,{σ1})},
    (F3,Δ)={(δ1,{σ1}),(δ2,)},
    (F4,Δ)={(δ1,),(δ2,{σ1})},
    (F5,Δ)={(δ1,{σ2,σ3}),(δ2,)},
    (F6,Δ)={(δ1,),(δ2,{σ2,σ3})},
    (F7,Δ)={(δ1,Σ),(δ2,{σ1})},
    (F8,Δ)={(δ1,{σ1}),(δ2,Σ)},
    (F9,Δ)={(δ1,{σ2,σ3}),(δ2,Σ)},
    (F10,Δ)={(δ1,Σ),(δ2,{σ2,σ3})},
    (F11,Δ)={(δ1,{σ1}),(δ2,{σ1})},
    (F12,Δ)={(δ1,{σ2,σ3}),(δ2,{σ2,σ3})},
    (F13,Δ)={(δ1,Σ),(δ2,)},

    and

    (F14,Δ)={(δ1,),(δ2,Σ)}.

    Then, (Σ,T,Δ) is an extended softTS. Remark that a soft set

    (H,Δ)={(δ1,{σ2,σ3}),(δ2,{σ3})}

    is weakly soft α-open. But it is not a soft α-open set because

    int(cl(int(H,Δ)))={(δ1,{σ2,σ3}),(δ2,)}(H,Δ).

    Also, a soft set

    (G,Δ)={(δ1,{σ1,σ2}),(δ2,)}

    is soft sw-open because

    int(G,Δ)={(δ1,{σ1}),(δ2,)}ϕ.

    But it is not a weakly soft α-open set because

    int(cl(int(G(δ1))))={σ1}G(δ1)

    and G(δ2) is empty.

    Proposition 3.13. The image and pre-image of weakly soft α-open set under a soft bi-continuous function (soft open and continuous) is weakly soft α-open.

    Proof. To show the case of image, let MP: (Σ,T,Δ)(Υ,S,Ω) be a soft bi-continuous function and let (F,Δ) be a weakly soft α-subset of (Σ,T,Δ). Suppose that there exists δΔ such that F(δ) is a nonempty α-open subset and let P(δ)=ω. According to Theorem 2.17, it follows from the soft bicontinuity of MP that M: (Σ,Tδ)(Υ,SP(δ)=ω) is a bicontinuous function.

    It is well known that a continuity of M implies that M(cl(V))cl(M(V)), and an openness of M implies that M(int(V))int(M(V)) for each subset V of Σ. This implies that

    M(F(δ))˜M(int(cl(int(F(δ))))˜int(cl(int(M(F(δ))))).

    According to Definition 2.16, we find that M(F(δ)) is a nonempty α-open subset of MP(F,Δ); hence, MP(F,Δ) is a weakly soft α-open subset of (Σ,T,Δ).

    Corollary 3.14. The property of being a weakly soft α-open set is a topological property.

    Proposition 3.15. The product of two weakly soft α-open sets is weakly soft α-open.

    Proof. Suppose that (F,Δ) and (G,Δ) are weakly soft α-open subsets and let

    (H,Δ×Δ)=(F,Δ)×(G,Δ).

    Then there are δ1,δ2Δ such that F(δ1) and G(δ2) are nonempty α-open subsets. Now, (δ1,δ2)Δ×Δ such that

    H(δ1,δ2)=F(δ1)×G(δ2).

    As we know from the classical topology the product of two nonempty α-open subsets is still a nonempty α-open subset; therefore, H(δ1,δ2) is a nonempty α-open subset. Hence, (H,Δ×Δ) is a weakly soft α-open subset.

    As an expected line of this type of study, we build the operators of interior, closure, boundary, and limit inspired by the class of weakly soft α-open and weakly soft α-closed sets. We elucidate their master properties and scrutinize the relationships among them. By some counterexamples, we illustrate that the weakly α-interior (resp., weakly α-closure) of soft subset need not be weakly α-open (resp., weakly α-closed) sets, in general.

    Definition 4.1. The weakly α-interior points of a subset (F,Δ) of (Σ,T,Δ), denoted by intwα(F,Δ), is defined as the union of all weakly soft α-open sets contained in (F,Δ).

    By Example 3.3 we remark that the weakly α-interior points of a subset need not be a weakly α-open set. That is,

    intwα(F,Δ)=(F,Δ)

    does not imply that (F,Δ) is a weakly α-open set.

    One can easily prove the next propositions.

    Proposition 4.2. Let (F,Δ) be a subset of (Σ,T,Δ) and σδ˜Σ. Then σδintwα(F,Δ) iff there is a weakly soft α-open set (G,Δ) contains σδ such that (G,Δ)˜(F,Δ).

    Proposition 4.3. Let (F,Δ), (G,Δ) be soft subsets of (Σ,T,Δ). Then

    (i) intwα(F,Δ)˜(F,Δ).

    (ii) if (F,Δ)˜(G,Δ), then intwα(F,Δ)˜intwα(G,Δ).

    Corollary 4.4. For any two subsets (F,Δ), (G,Δ) of (Σ,T,Δ), we have the following results:

    (i) intwα[(F,Δ)˜(G,Δ)]˜intwα(F,Δ)˜intwα(G,Δ).

    (ii) intwα(F,Δ)˜intwα(G,Δ)˜intwα[(F,Δ)˜(G,Δ)].

    Proof. It automatically comes from the following:

    (i) (F,Δ)˜(G,Δ)˜(F,Δ) and (F,Δ)˜(G,Δ)˜(G,Δ).

    (ii) (F,Δ)˜[(F,Δ)˜(G,Δ)] and (G,Δ)˜[(F,Δ)˜(G,Δ)]. Let

    (E,Δ)={(δ1,{5}),(δ2,{6,7})},
    (F,Δ)={(δ1,(1,2)),(δ2,[1,2])},
    (G,Δ)={(δ1,(2,3)),(δ2,[2,3])},
    (H,Δ)={(δ1,),(δ2,(0,1])},

    and

    (J,Δ)={(δ1,(0,1]),(δ2,)}

    be soft subsets of a softTS given in Example 3.3. We remark the following properties:

    (i) (E,Δ)˜intwα(E,Δ)=ϕ.

    (ii) intwα(E,Δ)˜intwα(H,Δ), whereas (E,Δ)˜(H,Δ).

    (iii) intwα[(F,Δ)˜(G,Δ)]=ϕ, whereas intwα(F,Δ)˜intwα(G,Δ)={(δ1,),(δ2,{2})}.

    (iv)

    intwα(H,Δ)˜intwα(J,Δ)={(δ1,(0,1)),(δ2,(0,1))},

    whereas

    intwα[(H,Δ)˜(J,Δ)]={(δ1,(0,1]),(δ2,(0,1])}.

    Hence, the inclusion relations of Proposition 4.3 and Corollary 4.4 are proper.

    Definition 4.5. The weakly α-closure points of a subset (F,Δ) of (Σ,T,Δ), denoted by clwα(F,Δ), is defined as the intersection of all weakly soft α-closed sets containing (F,Δ).

    By Example 3.3 we remark that the weakly α-closure points of a subset need not be a weakly α-closed set. That is, clwα(F,Δ)=(F,Δ) does not imply that (F,Δ) is a weakly α-closed set.

    Proposition 4.6. Let (F,Δ) be a subset of (Σ,T,Δ) and σδ˜Σ. Then σδclwα(F,Δ) iff (G,Δ)˜(F,Δ)ϕ for each weakly soft α-open set (G,Δ) contains σδ.

    Proof. [] Let σδclwα(F,Δ). Suppose that there is weakly soft α-open set (G,Δ) containing σδ with

    (G,Δ)˜(F,Δ)=ϕ.

    Then

    (F,Δ)˜(Gc,Δ).

    Therefore,

    clwα(F,Δ)˜(Gc,Δ).

    Thus

    σδclwα(F,Δ).

    This is a contradiction, which means that

    (G,Δ)˜(F,Δ)ϕ,

    as required.

    [] Let

    (G,Δ)˜(F,Δ)ϕ

    for each weakly soft α-open set (G,Δ) contains σδ. Suppose that

    σδclwα(F,Δ).

    Then there is a weakly soft α-closed set (H,Δ) containing (F,Δ) with σδ(H,Δ). So

    σδ(Hc,Δ),

    and

    (Hc,Δ)˜(F,Δ)=ϕ.

    This is a contradiction. Hence, we obtain the desired result.

    Corollary 4.7. If

    (F,Δ)˜(G,Δ)=ϕ

    such that (F,Δ) is a weakly soft α-open set and (G,Δ) is a soft set in (Σ,T,Δ), then

    (F,Δ)˜clwα(G,Δ)=ϕ.

    Proof. Obvious.

    Proposition 4.8. The following properties hold for a subset (F,Δ) of (Σ,T,Δ).

    (i) [intwα(F,Δ)]c=clwα(Fc,Δ).

    (ii) [clwα(F,Δ)]c=intwα(Fc,Δ).

    Proof. (i) If

    σδ[intwα(F,Δ)]c,

    then there is a weakly soft α-open set (G,Δ) with

    σδ(G,Δ)˜(F,Δ).

    Therefore,

    (Fc,Δ)˜(G,Δ)=ϕ,

    and hence,

    σδclwα(Fc,Δ).

    Conversely, if σδclwα(Fc,Δ) we can follow the previous steps to verify σδ[intwα(F,Δ)]c.

    (ii) Following similar approach given in (i).

    The next proposition is easy, so we omit its proof.

    Proposition 4.9. Let (F,Δ), (G,Δ) be soft subsets of (Σ,T,Δ). Then

    (i) (F,Δ)˜clwα(F,Δ).

    (ii) if (F,Δ)˜(G,Δ), then clwα(F,Δ)˜clwα(G,Δ).

    Corollary 4.10. The following results hold for any subsets (F,Δ), (G,Δ) of (Σ,T,Δ).

    (i) clwα[(F,Δ)˜(G,Δ)]˜clwα(F,Δ)˜clwα(G,Δ).

    (ii) clwα(F,Δ)˜clwα(G,Δ)˜clwα[(F,Δ)˜(G,Δ)].

    Proof. It automatically comes from the following:

    (i) (F,Δ)˜(G,Δ)˜(F,Δ) and (F,Δ)˜(G,Δ)˜(G,Δ).

    (ii) (F,Δ)˜[(F,Δ)˜(G,Δ)] and (G,Δ)˜[(F,Δ)˜(G,Δ)].

    Let

    (E,Δ)={(δ1,R),(δ2,[0,1))},
    (F,Δ)={(δ1,(1,2)),(δ2,[1,2])},
    (G,Δ)={(δ1,(2,3)),(δ2,[2,3])},

    and

    (H,Δ)={(δ1,R),(δ2,(0,1])}

    be soft subsets of a softTS given in Example 3.3. We remark the following properties:

    (i) clwα(E,Δ)={(δ1,R),(δ2,[0,1])}˜(E,Δ).

    (ii) clwα(E,Δ)˜clwα(H,Δ), whereas (E,Δ)˜(H,Δ).

    (iii) clwα[(F,Δ)˜(G,Δ)]={(δ1,),(δ2,{2})}, whereas clwα(F,Δ)˜clwα(G,Δ)={(δ1,{2}),(δ2,{2})}.

    Hence, the inclusion relations of Proposition 4.9 and Corollary 4.10 are proper.

    Definition 4.11. A soft point σδ is said to be a weakly α-boundary point of a subset (F,Δ) of (Σ,T,Δ) if σδ belongs to the complement of intwα(F,Δ)˜intwα(Fc,Δ).

    All α-boundary points of (F,Δ) is called a weakly α-boundary set, denoted by bwα(F,Δ).

    Proposition 4.12.

    bwα(F,Δ)=clwα(F,Δ)˜clwα(Fc,Δ)

    for every subset (F,Δ) of (Σ,T,Δ).

    Proof.

    bwα(F,Δ)=[intwα(F,Δ)˜intwα(Fc,Δ)]c=[intwα(F,Δ)]c˜[intwα(Fc,Δ)]c(De  Morgans  law)=clwα(Fc,Δ)˜clwα(F,Δ)(Proposition  4.8(ii)).

    Corollary 4.13. For every subset (F,Δ) of (Σ,T,Δ), the following properties hold.

    (i) bwα(F,Δ)=bwα(Fc,Δ).

    (ii) bwα(F,Δ)=clwα(F,Δ)intwα(F,Δ).

    (iii) clwα(F,Δ)=intwα(F,Δ)˜bwα(F,Δ).

    (iv) intwα(F,Δ)=(F,Δ)bwα(F,Δ).

    Proof. (i) Obvious.

    (ii) bwα(F,Δ)=clwα(F,Δ)˜clwα(Fc,Δ)=clwα(F,Δ)[clwα(Fc,Δ)]c. By (ii) of Proposition 4.8 we obtain the required relation.

    (iii) intwα(F,Δ)˜bwα(F,Δ)=intwα(F,Δ)˜[clwα(F,Δ)intwα(F,Δ)]=clwα(F,Δ).

    (iv)

    (F,Δ)bwα(F,Δ)=(F,Δ)[clwα(F,Δ)intwα(F,Δ)]=(F,Δ)˜[clwα(F,Δ)˜(intwα(F,Δ))c]c=(F,Δ)˜[(clwα(F,Δ))c˜intwα(F,Δ)]=[(F,Δ)˜(clwα(F,Δ))c]˜[(F,Δ)˜intwα(F,Δ)]=intwα(F,Δ).

    Proposition 4.14. Let (F,Δ),(G,Δ) be subsets of (Σ,T,Δ), the following properties hold.

    (i) bwα(intwα(F,Δ))˜bwα(F,Δ).

    (ii) bwα(clwα(F,Δ))˜bwα(F,Δ).

    Proof. By substituting in the formula No. (iii) of Corollary 4.13, the proof follows.

    Proposition 4.15. Let (F,Δ) be a subset of (Σ,T,Δ). Then

    (i) (F,Δ)=intwα(F,Δ) iff bwα(F,Δ)˜(F,Δ)=ϕ.

    (ii) (F,Δ)=clwα(F,Δ) iff bwα(F,Δ)˜(F,Δ).

    Proof. (i) Suppose that

    (F,Δ)=intwα(F,Δ).

    Then by (iv) of Corollary 4.13,

    (F,Δ)=intwα(F,Δ)=(F,Δ)bwα(F,Δ),

    and hence,

    bwα(F,Δ)˜(F,Δ)=ϕ.

    Conversely, let σδ(F,Δ). Since σδbwα(F,Δ) and σδclwα(F,Δ), by (iii) of Corollary 4.13, σδintwα(F,Δ). Therefore,

    intwα(F,Δ)=(F,Δ),

    as required.

    (ii) Assume that

    (F,Δ)=clwα(F,Δ).

    Then

    bwα(F,Δ)=clwα(F,Δ)˜clwα(Fc,Δ)˜clwα(F,Δ)=(F,Δ),

    as required.

    Conversely, if bwα(F,Δ)˜(F,Δ), then by (iii) of Corollary 4.13,

    clwα(F,Δ)˜intwα(F,Δ)˜(F,Δ)=(F,Δ),

    and hence,

    clwα(F,Δ)=(F,Δ),

    as required.

    Corollary 4.16. Let (F,Δ) be a subset of (Σ,T,Δ). Then

    intwα(F,Δ)=(F,Δ)=clwα(F,Δ),

    iff

    bwα(F,Δ)=ϕ.

    Definition 4.17. A soft point σδ is said to be a weakly α-limit point of a subset (F,Δ) of (Σ,T,Δ) if

    [(G,Δ)σδ](F,Δ)ϕ,

    for each weakly soft α-open set (G,Δ) containing σδ.

    All weakly α-limit points of (F,Δ) is called a weakly α-derived set and denoted by lwα(F,Δ).

    Proposition 4.18. Let (F,Δ) and (G,Δ) be subsets of (Σ,T,Δ). If (F,Δ)˜(G,Δ), then lwα(F,Δ)˜lwα(G,Δ).

    Proof. Straightforward by Definition 4.17.

    Corollary 4.19. Consider (F,Δ) and (G,Δ) are subsets of (Σ,T,Δ). Then:

    (i) lwα[(F,Δ)˜(G,Δ)]˜lwα(F,Δ)˜lwα(G,Δ).

    (ii) lwα(F,Δ)˜lwα(G,Δ)˜lwα[(F,Δ)˜(G,Δ)].

    Theorem 4.20. Let (F,Δ) be a subset of (Σ,T,Δ), then

    clwα(F,Δ)=(F,Δ)˜lwα(F,Δ).

    Proof. The side

    (F,Δ)˜lwα(F,Δ)˜clwα(F,Δ)

    is obvious. To prove the other side let

    σδ[(F,Δ)˜lwα(F,Δ)].

    Then σδ(F,Δ) and σδlwα(F,Δ). Therefore, there is weakly soft α-open (G,Δ) containing σδ with

    (G,Δ)˜(F,Δ)=ϕ.

    Thus, σδclwα(F,Δ). Hence, we find that

    clwα(F,Δ)=(F,Δ)˜lwα(F,Δ).

    Corollary 4.21. Let (F,Δ) be a weakly soft α-closed subset of (Σ,T,Δ), then lwα(F,Δ)˜(F,Δ).

    This is the last main section we dedicate to tackling the concept of soft continuity via weakly soft α-open sets. We establish its main characterizations and show that loss of the property says that "weakly α-interior of the soft subset is weakly soft α-open" leads to evaporating some descriptions of this type of soft continuity. An elucidative counterexample is supplied.

    Definition 5.1. A soft function MP: (Σ,TΣ,Δ)(Υ,TΥ,Δ) is said to be weakly soft α-continuous if the inverse image of each soft open set is weakly soft α-open.

    It is straightforward to prove the next result, so omit its proof.

    Proposition 5.2. If MP: (Σ,TΣ,Δ)(Υ,TΥ,Δ) is a weakly soft α-continuous function and NK: (Υ,TΥ,Δ)(Γ,TΓ,Δ) is a soft continuous function, then NKMP is weakly soft α-continuous.

    Proposition 5.3. Every soft continuous function is weakly soft α-continuous.

    Proof. It follows from Proposition 3.7.

    Proposition 5.4. Let MP:(Σ,TΣ,Δ)(Υ,TΥ,Δ) be a soft function such that TΣ is extended. Then

    (i) If MP is soft α-continuous, then MP is weakly soft α-continuous.

    (ii) If MP is weakly soft α-continuous, then MP is soft sw-continuous.

    Proof. It respectively follows from Propositions 3.9 and 3.10.

    Proposition 5.5. A soft function MP: (Σ,TΣ,Δ)(Υ,TΥ,Δ) is weakly soft α-continuous iff the inverse image of every soft closed subset is weakly soft α-closed.

    Proof. Necessity: suppose that (F,Δ) is a soft closed subset of (Υ,TΥ,Δ). Then (Fc,Δ) is soft open. Therefore,

    M1P(Fc,Δ)=˜ΣM1P(F,Δ)

    is weakly soft α-open. Thus, M1P(F,Δ) is a weakly soft α-closed set.

    Following similar argument one can prove the sufficient part.

    Theorem 5.6. If MP: (Σ,TΣ,Δ)(Υ,TΥ,Δ) is weakly soft α-continuous, then the next properties are equivalent.

    (i) For each soft open subset (F,Δ) of (Υ,TΥ,Δ), we have M1P(F,Δ)=intwα(M1P(F,Δ)).

    (ii) For each soft closed subset (F,Δ) of (Υ,TΥ,Δ), we have M1P(F,Δ)=clwα(M1P(F,Δ)).

    (iii) clwα(M1P(F,Δ))˜M1P(cl(F,Δ)) for each (F,Δ)˜˜Υ.

    (iv) MP(clwα(G,Δ))˜cl(MP(G,Δ)) for each (G,Δ)˜˜Σ.

    (v) M1P(int(F,Δ))˜intwα(M1P(F,Δ)) for each (F,Δ)˜˜Υ.

    Proof. (i)(ii): Suppose that (F,Δ) is a soft closed subset of (Υ,TΥ,Δ). Then (Fc,Δ) is soft open. Therefore,

    M1P(Fc,Δ)=intwα(M1P(Fc,Δ)).

    According to Proposition 4.8, we obtain

    M1P(F,Δ)=clwα(M1P(F,Δ)).

    (ii)(iii): For any soft set (F,Δ)˜˜Υ, we have

    M1P(cl(F,Δ))=clwα(M1P(cl(F,Δ))).

    Then

    clwα(M1P(F,Δ))˜clwα(M1P(cl(F,Δ)))=M1P(cl(F,Δ)).

    (iii)(iv): It is obvious that

    clwα(G,Δ)˜clwα(M1P(MP(G,Δ)))

    for each (G,Δ)˜˜Σ. By (iii), we get

    clwα(M1P(MP(G,Δ)))˜M1P(cl(MP(G,Δ))).

    Therefore,

    MP(clwα(G,Δ)˜MP(M1P(cl(MP(G,Δ))))˜cl(MP(G,Δ)).

    (iv)(v): Let (F,Δ) be an arbitrary soft set in (Υ,TΥ,Δ). Then

    MP(clwα(M1P(Fc,Δ))˜cl(MP(M1P(Fc,Δ)))˜cl(Fc,Δ)

    So that,

    clwα((M1P(F,Δ))c)˜M1P((int(F,Δ))c.

    Hence,

    M1P(int(F,Δ))˜intwα(M1P(F,Δ)).

    (v)(i): Suppose that (F,Δ) is a soft open subset in (Υ,TΥ,Δ). By (v), we obtain

    M1P(F,Δ)=M1P(int(F,Δ))˜intwα(M1P(F,Δ)).

    But

    intwα(M1P(F,Δ))˜M1P(F,Δ),

    so

    M1P(F,Δ)=intwα(M1P(F,Δ)),

    as required.

    The converse of the above theorem fails. To demonstrate that the next example is furnished.

    Example 5.7. Let Σ={σ1,σ2,σ3} and Υ={υ1,υ2} with Δ={δ1,δ2}. Let

    TΣ={ϕ,˜Σ,(F,Δ),(G,Δ)}

    and

    TΥ={ϕ,˜Υ,(H,Δ)}

    be two soft topologies defined on Σ and Υ, respectively, with the same set of parameters Δ, where

    (F,Δ)={(δ1,{σ1}),(δ2,{σ1})},
    (G,Δ)={(δ1,{σ2,σ3}),(δ2,{σ2,σ3})},

    and

    (H,Δ)={(δ1,{υ1}),(δ2,{υ1})}.

    Consider MP: (Σ,TΣ,Δ)(Υ,TΥ,Δ) is a soft function, where M: ΣΥ is defined as follows

    M(σ1)=M(σ2)=υ1 and  M(σ3)=υ2,

    and P:ΔΔ is the identity function.

    Now,

    M1P(H,Δ)={(δ1,{σ1,σ2}),(δ2,{σ1,σ2})},

    which is not a weakly soft α-open subset because

    int(cl(int({σ1,σ2})))={σ1}.

    Then MP is not weakly soft α-continuous. On the other hand,

    M1P(ϕ)=intwα(M1P(ϕ)),
    M1P(˜Υ)=intwα(M1P(˜Υ)),

    and

    M1P(H,Δ)=intwα(M1P(H,Δ)),

    which means that the all properties given in Theorem 5.6 hold true.

    Now, we introduce the concepts of weakly soft α-open, weakly soft α-closed and weakly soft α-homeomorphism functions.

    Definition 5.8. A soft function MP: (Σ,TΣ,Δ)(Υ,TΥ,Δ) is called:

    (i) weakly soft α-open provided that the image of each soft open set is weakly soft α-open.

    (ii) weakly soft α-closed provided that the image of each soft closed set is weakly soft α-closed.

    Theorem 5.9. Let MP: (Σ,TΣ,Δ)(Υ,TΥ,Δ) be a soft function and (F,Δ) be any soft subset of ˜Σ. Then

    (i) If MP is weakly soft α-open, then MP(int(F,Δ))˜intwα(MP(F,Δ)).

    (ii) If MP is weakly soft α-closed, then clwα(MP(F,Δ)) ˜MP(cl(F,Δ)).

    Proof. (i) Let (F,Δ) be a soft subset of ˜Σ. Then MP(int(F,Δ)) is a weakly soft α-open subset of (Υ,TΥ,Δ) and so

    MP(int(F,Δ))=intwα(MP(int(F,Δ)))˜intwα(MP(F,Δ)).

    (ii) The proof is similar to that of (i).

    Proposition 5.10. A bijective soft function MP: (Σ,TΣ,Δ)(Υ,TΥ,Δ) is weakly soft α-open iff it is weakly soft α-closed.

    Proof. Necessity: let (F,Δ) be a weakly soft α-closed subset of (Σ,TΣ,Δ). Since MP is weakly soft α-open, MP(Fc,Δ) is weakly soft α-open. By bijectiveness of MP, we obtain

    MP(Fc,Δ)=(MP(F,Δ))c.

    So that, MP(F,Δ) is a weakly soft α-closed set. Hence, MP is weakly soft α-closed. To prove the sufficient, we follow similar approach.

    Proposition 5.11. Let MP: (Σ,TΣ,Δ)(Υ,TΥ,Δ) be a weakly soft α-closed function and ˜Γ be a soft closed subset of ˜Σ. Then MPΓ: (Γ,TΓ,Δ)(Υ,TΥ,Δ) is weakly soft α-closed.

    Proof. Suppose that (F,Δ) is a soft closed subset of (Γ,TΓ,Δ). Then there is a soft closed subset (G,Δ) of (Σ,TΣ,Δ) with

    (F,Δ)=(G,Δ)˜˜Γ.

    Since ˜Γ is a soft closed subset of (Σ,TΣ,Δ), then (F,Δ) is also a soft closed subset of (Σ,TΣ,Δ). Since,

    MPΓ(F,Δ)=MP(F,Δ),

    then MPΓ(F,Δ) is a weakly soft α-closed set. Thus, MPΓ is a weakly soft α-closed.

    Proposition 5.12. The next three statements hold for soft functions MP: (Σ,TΣ,Δ)(Υ,TΥ,Δ) and NK: (Υ,TΥ,Δ)(Γ,TΓ,Δ).

    (i) If MP is soft open and NK is soft α-open such that TΓ is extended, then NKMP is weakly soft α-open.

    (ii) If NKMP is weakly soft α-open and MP is surjective soft continuous, then NK is weakly soft α-open.

    (iii) If NKMP is soft open and NK is injective weakly soft α-continuous, then MP is weakly soft α-open.

    Proof. (i) Take

    (F,Δ)ϕ

    as a soft open subset of ˜Σ. So

    MP(F,Δ)ϕ

    is a soft open subset of ˜Υ. Thus, NK(MP(F,Δ)) is a soft α-open subset. According to Proposition 3.9, NK(MP(F,Δ)) is a weakly soft α-open subset. Hence, NKMP is weakly soft α-open.

    (ii) Suppose that (F,Δ)ϕ is a soft open subset of ˜Υ. Then

    M1P(F,Δ)ϕ

    is a soft open subset of ˜Σ. Therefore, (NKMP)(M1P(F,Δ)) is a weakly soft α-open subset of ˜Γ. Since MP is surjective, then

    (NKMP)(M1P(F,Δ))=NK(MP(M1P(F,Δ)))=NK(F,Δ).

    Thus, NK is weakly soft α-open.

    (iii) Let (F,Δ)ϕ be a soft open subset of ˜Σ. Then

    (NKMP)(F,Δ)ϕ

    is a soft open subset of ˜Γ. Therefore, N1K(NKMP(F,Δ)) is a weakly soft α-open subset of ˜Υ. Since NK is injective,

    N1K(NKMP(F,Δ))=(N1KNK)(MP(F,Δ))=MP(F,Δ).

    Thus, MP is weakly soft α-open.

    We cancel the proof of the next finding because it can be obtained following similar approach of the above proposition.

    Proposition 5.13. The next three statements hold for soft functions MP: (Σ,TΣ,Δ)(Υ,TΥ,Δ) and NK: (Υ,TΥ,Δ)(Γ,TΓ,Δ).

    (i) If MP is soft closed and NK is soft α-closed, then NKMP is weakly soft α-closed.

    (ii) If NKMP is weakly soft α-closed and MP is surjective soft continuous, then NK is weakly soft α-closed.

    (iii) If NKMP is soft closed and NK is injective weakly soft α-continuous, then MP is weakly soft α-closed.

    Definition 5.14. A bijective soft function MP in which is weakly soft α-continuous and weakly soft α-open is called a weakly soft α-homeomorphism.

    It is well known that soft topology is defined as a family of soft sets fulfilling the basic axioms of general topology and creates a family of general topologies. it is crucial to examine the connections between these classical topologies and the soft topology that they generate. In this paper, we have benefited from the fruitful variety existing via soft topology to introduce a novel class of generalizations of soft open subsets called "weakly soft α-open sets", which we have constructed using its corresponding notion via parametric topologies.

    First, we have studied the basic properties of this class and showed that this class lost the property of closing under arbitrary soft unions, which is satisfied by the previous famous generalizations. With respect to its relationship with the previous generalizations, we have demonstrated that it lies between soft α-open and soft sw-open subsets of extended (hyperconnected) soft topology. Then, we have introduced the concepts of interior, closure, boundary, and limit soft points via weakly soft α-open and weakly soft α-closed sets. We have scrutinized main their characterizations and inferred the formulas that connected each other. In the end, we have discussed the concepts of soft continuity, openness and closeness defined by weakly soft α-open and weakly soft α-closed sets.

    Among the unique properties obtained in this study is that most descriptions of soft continuity have been evaporated for this type of continuity, which is due to the loss of the properties report that "a soft subset (F,Δ) is weakly soft α-open iff intwα(F,Δ)=(F,Δ)" and "a soft subset (F,Δ) is weakly soft α-closed iff clwα(F,Δ)=(F,Δ)". To illustrate these divergences between this class and other generalizations, we have provided some counterexamples. To avoid this irregular behavior, we plan to produce another type of soft continuity inspired by weakly soft α-open sets. Moreover, we intend to discuss other topological ideas that can be formulated using this class of soft sets, e.g., covering property and separation axioms.

    This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2023/R/1444).

    The authors declare that they have no competing interests.



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