Research article

On the significance of membrane unfolding in mechanosensitive cell spreading: Its individual and synergistic effects


  • Mechanosensitivity of cell spread area to substrate stiffness has been established both through experiments and different types of mathematical models of varying complexity including both the mechanics and biochemical reactions in the cell. What has not been addressed in previous mathematical models is the role of cell membrane dynamics on cell spreading, and an investigation of this issue is the goal of this work. We start with a simple mechanical model of cell spreading on a deformable substrate and progressively layer mechanisms to account for the traction dependent growth of focal adhesions, focal adhesion induced actin polymerization, membrane unfolding/exocytosis and contractility. This layering approach is intended to progressively help in understanding the role each mechanism plays in reproducing experimentally observed cell spread areas. To model membrane unfolding we introduce a novel approach based on defining an active rate of membrane deformation that is dependent on membrane tension. Our modeling approach allows us to show that tension-dependent membrane unfolding plays a critical role in achieving the large cell spread areas experimentally observed on stiff substrates. We also demonstrate that coupling between membrane unfolding and focal adhesion induced polymerization works synergistically to further enhance cell spread area sensitivity to substrate stiffness. This enhancement has to do with the fact that the peripheral velocity of spreading cells is associated with contributions from the different mechanisms by either enhancing the polymerization velocity at the leading edge or slowing down of the retrograde flow of actin within the cell. The temporal evolution of this balance in the model corresponds to the three-phase behavior observed experimentally during spreading. In the initial phase membrane unfolding is found to be particularly important.

    Citation: Magdalena A. Stolarska, Aravind R. Rammohan. On the significance of membrane unfolding in mechanosensitive cell spreading: Its individual and synergistic effects[J]. Mathematical Biosciences and Engineering, 2023, 20(2): 2408-2438. doi: 10.3934/mbe.2023113

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  • Mechanosensitivity of cell spread area to substrate stiffness has been established both through experiments and different types of mathematical models of varying complexity including both the mechanics and biochemical reactions in the cell. What has not been addressed in previous mathematical models is the role of cell membrane dynamics on cell spreading, and an investigation of this issue is the goal of this work. We start with a simple mechanical model of cell spreading on a deformable substrate and progressively layer mechanisms to account for the traction dependent growth of focal adhesions, focal adhesion induced actin polymerization, membrane unfolding/exocytosis and contractility. This layering approach is intended to progressively help in understanding the role each mechanism plays in reproducing experimentally observed cell spread areas. To model membrane unfolding we introduce a novel approach based on defining an active rate of membrane deformation that is dependent on membrane tension. Our modeling approach allows us to show that tension-dependent membrane unfolding plays a critical role in achieving the large cell spread areas experimentally observed on stiff substrates. We also demonstrate that coupling between membrane unfolding and focal adhesion induced polymerization works synergistically to further enhance cell spread area sensitivity to substrate stiffness. This enhancement has to do with the fact that the peripheral velocity of spreading cells is associated with contributions from the different mechanisms by either enhancing the polymerization velocity at the leading edge or slowing down of the retrograde flow of actin within the cell. The temporal evolution of this balance in the model corresponds to the three-phase behavior observed experimentally during spreading. In the initial phase membrane unfolding is found to be particularly important.



    Most real-world problems in engineering, medical science, economics, the environment, and other fields are full of uncertainty. The soft set theory was proposed by Molodtsov [25], in 1999, as a mathematical model for dealing with uncertainty. This is free of the obstacles associated with previous theories including fuzzy set theory, rough set theory, and so on. The nature of parameter sets related to soft sets, in particular, provides a uniform framework for modeling uncertain data. This results in the rapid development of soft set theory in a short period of time, as well as diverse applications of soft sets in real life.

    Influenced by the standard postulates of traditional topological space, Shabir and Naz [29], and Çağman et al. [18], separately, established another branch of topology known as "soft topology", which is a mixture of soft set theory and topology. This work was essential in building the subject of soft topology. Despite the fact that many studies followed their directions and many ideas appeared in soft contexts such as those discussed in [2,3,12,13,14]. However, significant contributions can indeed be made.

    The separation axioms are just axioms in the sense that you could add these conditions as extra axioms to the definition of topological space to achieve a more restricted definition of what a topological space is. These axioms have a great role in developing (classical) topology. Correspondingly, soft separation axioms are a significant aspect in the later development of soft topology; see for example [4,6,7,8,19,24,29]. A specific type of separation axioms was defined by Aull and Thron [15]. This axiom performs as an important part in the development other disciplines like Locale Theory [28], Logic and Information Theory [16] and Philosophy [27]. First, motivating the role of "TD-spaces", we generalize this separation axiom in the language of soft set theory under the name of "soft TD-spaces", and study their primary properties. Second, most of the given soft separation axioms were characterized by soft open, soft closed, or soft closure, we want to describe them differently. As a result, this work is demonstrated. Finally, the desire of describing some soft Ti-spaces using new soft operators motivates us to present the operators of "soft kernel" and "soft shell".

    The body of the paper is structured as follows: In Section 2, we present an overview of the literature on soft set theory and soft topology. Section 3 focuses on the concepts of soft topological operators and their main properties for characterization of soft separation axioms. Section 4 introduces a new soft separation axiom called a soft "TD-space". The relationships of soft TD-spaces with known soft separation axioms are determined. Furthermore, we characterize soft TD-spaces via soft operators proposed in Section 3. In Section 5, we offer characterizations of soft T0-spaces and soft T1-spaces through the given operators. We end our paper with a brief summary and conclusions (Section 6).

    Let X be a domain set and E be a set of parameters. A pair (F,E)={(e,F(e)):eE} is said to be a soft set [25] over X, where F:E2X is a set-valued mapping. The set of all soft sets on X parameterized by E is identified by SE(X). We call a soft set (F,E) over X a soft element [29], denoted by ({x},E), if F(e)={x} for each eE, where xX. It is said that a soft element ({x},E) is in (F,E) (briefly, x(F,E)) if xF(e) for each eE. On the other hand, x(F,E) if xF(e) for some eE. This implies that if ({x},E)˜(F,E)=Φ, then x(F,E). We call a soft set (F,E) over X a soft point [10,26], denoted by xe, if F(e)={x} and x(e)= for each eE with ee, where eE and xX. An argument xe(F,E) means that xF(e). The set of all soft points over X is identified by PE(X). A soft set (X,E)(F,E) (or simply (F,E)c) is the complement of (F,E), where Fc:E2X is given by Fc(e)=XF(e) for each eE. If (F,E)SE(X), it is denoted by Φ if F(e)= for each eE and is denoted by ˜X if F(e)=X for each eE. Evidently, ˜Xc=Φ and Φc=˜X. A soft set (F,E) is called degenerate if (F,E)={xe} or (F,E)=Φ. It is said that (A,E1) is a soft subset of (B,E2) (written by (A,E1)˜(B,E2), [22]) if E1E2 and A(e)B(e) for each eE1, and (A,E1)=(B,E2) if (A,E1)˜(B,E2) and (B,E2)˜(A,E1). The union of soft sets (A,E),(B,E) is represented by (F,E)=(A,E)˜(B,E), where F(e)=A(e)B(e) for each eE, and intersection of soft sets (A,E),(B,E) is given by (F,E)=(A,E)˜(B,E), where F(e)=A(e)B(e) for each eE, (see [9]).

    Definition 2.1. [29] A collection T of SE(X) is said to be a soft topology on X if it satisfies the following axioms:

    (T.1) Φ,˜XT.

    (T.2) If (F1,E),(F2,E)T, then (F1,E)˜(F2,E)T.

    (T.3) If {(Fi,E):iI}˜T, then ˜iI(Fi,E)T.

    Terminologically, we call (X,T,E) a soft topological space on X. The elements of T are called soft open sets. The complement of every soft open or elements of Tc are called soft closed sets. The lattice of all soft topologies on X is referred to TE(X), (see [1]).

    Definition 2.2. [11] Let F˜SE(X). The intersection of all soft topologies on X containing F is called a soft topology generated by F and is referred to T(F).

    Definition 2.3. [29] Let (B,E)SE(X) and TTE(X).

    (1) The soft closure of (B,E) is cl(B,E):=˜{(F,E):(B,E)˜(F,E),(F,E)Tc}.

    (2) The soft interior of (B,E) is int(B,E):=˜{(F,E):(F,E)˜(B,E),(F,E)T}.

    Definition 2.4. [18] Let (B,E)SE(X) and TTE(X). A point xePE(X) is called a soft limit point of (B,E) if (G,E)˜(B,E){xe}Φ for all (G,E)T with xe(G,E). The set of all soft limit points is symbolized by der(B,E). Then cl(F,E)=(F,E)˜der(F,E) (see Theorem 5 in [18]).

    Definition 2.5. [21] Let TTE(X). A set (A,E)SE(X) is called soft locally closed if there exist (G,E)T and (F,E)Tc such that (A,E)=(G,E)˜(F,E). The family of all soft locally closed sets in X is referred to LC(X).

    Definition 2.6. [20] Let TTE(X) and let (A,E)SE(X). A point xe(A,E) is called soft isolated if there exists (G,E)T such that (G,E)˜(A,E)={xe}. It is called soft weakly isolated if there exists (G,E)T with xe(G,E) such that (G,E)˜(A,E)˜cl(xe). Let I(A,E), WI(A,E) respectively denote the set of all soft isolated and soft weakly isolated points of (A,E).

    Definition 2.7. [17] A soft space (X,E,T) (or simply soft topology TTE(X)) is called

    (1) Soft T0 if for every xe,yePE(X) with xeye, there exist (U,E),(V,E)T such that xe(U,E), ye(U,E) or ye(V,E), xe(V,E).

    (2) Soft T1 if for every xe,yePE(X) with xeye, there exist (U,E),(V,E)T such that xe(U,E), ye(U,E) and ye(V,E), xe(V,E).

    The above soft separation axioms have been defined by Sabir and Naz [29] with respect to soft elements.

    Lemma 2.8. [17,Theorem 4.1] Let TTE(X). Then T is soft T1 iff cl(xe)={xe} for every xePE(X).

    In this section, we define "soft kernel" and "soft shell" as two topological operators. Then the connections between these operators and soft closure and soft derived set operators are obtained. The presented results will be used to characterize several soft separation axioms.

    Definition 3.1. Let (F,E)SE(X) and let TTE(X). The soft kernel of (F,E) is defined by:

    ker(F,E):=˜{(G,E):(G,E)T,(F,E)˜(G,E)}.

    Lemma 3.2. Let (F,E),(G,E)SE(X) and TTE(X). The following properties are valid:

    (1) (F,E)˜ker(F,E).

    (2) ker(F,E)˜ker(ker(F,E)).

    (3) (F,E)˜(G,E)ker(F,E)˜ker(G,E).

    (4) ker[(F,E)˜(G,E)]˜ker(F,E)˜ker(G,E).

    (5) ker[(F,E)˜(G,E)]=ker(F,E)˜ker(G,E).

    Proof. Standard.

    From Definitions 2.3 and 3.1, it is obtained that

    Definition 3.3. Let xePE(X) and TTE(X). Then

    (1) ker({xe}):=˜{(G,E):(G,E)T,xe(G,E)}.

    (2) cl({xe}):=˜{(F,E):(F,E)Tc,xe(F,E)}.

    In the sequel, we interchangeably use xe or {xe} for the one point soft set containing xe.

    Lemma 3.4. For (F,E)SE(X) and TTE(X), we have

    ker(F,E)={xePE(X):cl(xe)˜(F,E)Φ}.

    Proof. Let xeker(F,E). If cl(xe)˜(F,E)=Φ, then (F,E)˜˜Xcl(xe). Therefore, ˜Xcl(xe)T such that it contains (F,E) but not xe, a contradiction.

    Conversely, if xeker(F,E) and cl(xe)˜(F,E)Φ, then there is (G,E)T such that (F,E)˜(G,E) but xe(G,E) and yecl(xe)˜(F,E). Therefore, ˜X(G,E)Tc including xe but not ye. This contradicts to yecl(xe)˜(F,E). Thus, xeker(F,E).

    Definition 3.5. Let (F,E),(G,E)SE(X) and TTE(X). It is said that (F,E) is separated in a weak sense from (G,E) (symbolized by (F,E)(G,E)) if there exists (H,E)T with (F,E)˜(H,E) such that (H,E)˜(G,E)=Φ.

    We have the following observation in light of Lemma 3.4 and Definition 3.5.

    Remark 3.6. For xe,yePE(X) and TTE(X), we have

    (1) cl(xe)={ye:ye  xe}.

    (2) ker(xe)={ye:xe  ye}.

    Definition 3.7. For xePE(X) and TTE(X), we define:

    (1) The soft derived set of xe as der(xe)=cl(xe){xe}.

    (2) The soft shell of xe as shel(xe)=ker(xe){xe}.

    (3) The soft set xe=cl(xe)˜ker(xe).

    We have the following remark in view of Definition 3.7 and Remark 3.6.

    Remark 3.8. For xe,yePE(X) and TTE(X), we have

    (1) der(xe)={ye:yexe,ye  xe}.

    (2) shel(xe)={ye:yexe,xe  ye}.

    Example 3.9. Let X={0,1,2} and let E={e1,e2} be a set of parameters. Consider the following soft topology on X:

    T={Φ,(F,E),G,E),(H,E),˜X},

    where, (F,E)={(e1,{0}),(e2,)}, (G,E)={(e1,{0,1}),(e2,)}, and (H,E)={(e1,{0,2}),(e2,X)}. By an easy computation, one can conclude the following:

    ker({1e1})=(G,E)ker({1e2})=(H,E)shel({1e1})=(F,E)shel({1e2})={(e1,{0,2}),(e2,{0,2})}cl({1e1})={1e1}cl({1e2})={(e1,{2}),(e2,X)}der({1e1})=Φder({1e2})={(e1,{2}),(e2,{0,2})}.

    Lemma 3.10. The following properties are valid for every xe,yePE(X) and TTE(X):

    (1) yeker(xe)xecl(ye).

    (2) yeshel(xe)xeder(ye).

    (3) yecl(xe)cl(ye)˜cl(xe).

    (4) yeker(xe)ker(ye)˜ker(xe).

    Proof. (1) and (2) follow, respectively, from Remarks 3.6 and 3.8.

    (3) Straightforward.

    (4) Let zeker(ye). By (1), yecl(ze) and so cl(ye)˜cl(ze) (by (3)). By hypothesis, yeker(xe) and so xecl(ye). Therefore, cl(xe)˜cl(ye). Finally, we get cl(xe)˜cl(ze) and then xecl(ze). By (1), zeker(xe). Thus, ker(ye)˜ker(xe).

    Lemma 3.11. Let TTE(X) and let xePE(X). Then

    (1) shel(xe) is degenerate iff for every yePE(X) with yexe, der(xe)˜der(ye)=Φ.

    (2) der(xe) is degenerate iff for every yePE(X) with yexe, shel(xe)˜shel(ye)=Φ.

    Proof. (1) If der(xe)˜der(ye)Φ, then there exists zePE(X) such that zeder(xe), zeder(ye). Therefore, zeyexe for which zecl(xe) and zecl(ye). By Lemma 3.10 (1), xe,yeker(ze). Thus, xe,yeker(ze)ze=shel(ze). This proves that shel(xe) is not degenerate.

    Conversely, if xe,yeshel(ze), then xeye, xeze and so xeker(ze), yeker(ze). Therefore, zecl(xe)˜cl(ye) and thus zeder(xe)˜der(ye). But this is impossible, hence der(xe)˜der(ye)=Φ.

    Lemma 3.12. Let TTE(X) and let xe,yePE(X). Then

    (1) If yexe, then ye=xe.

    (2) Either ye=xe or ye˜xe=Φ.

    Proof. (1) If yexe, then yecl(xe) and yeker(xe). When yecl(xe), by Lemma 3.10 (1), xeker(ye). By Lemma 3.10 (3) and (4), cl(ye)˜cl(xe) and ker(xe)˜ker(ye). When yeker(xe), by Lemma 3.10 (2), xecl(ye). By Lemma 3.10 (3) and (4), cl(xe)˜cl(ye) and ker(ye)˜ker(xe). Summing up all these together, we get cl(xe)=cl(ye) and ker(xe)=ker(ye). Thus, ye=xe.

    (2) It can be deduced from (1).

    Lemma 3.13. Let TTE(X) and let xe,yePE(X). Then ker(xe)ker(ye) iff cl(xe)cl(ye).

    Proof. If ker(xe)ker(ye), then one can find zeker(xe) but zeker(ye). From zeker(xe), we get xecl(ze) and then cl(xe)˜cl(ze). Since zeker(ye), by Lemma 3.10 (1), cl(ze)˜ye=Φ. Therefore, cl(ze)˜ye=Φ implies yecl(xe). Hence, cl(ye)cl(xe).

    The converse can be proved in a similar manner to the first part.

    Definition 4.1. Let TTE(X). We call T a soft TD-space if der(xe) is a soft closed set for every xePE(X).

    Theorem 4.2. Let TTE(X). Then

    (1) If T is soft T1, then it is soft TD.

    (2) If T is soft TD, then it is soft T0.

    Proof. (1) If T is soft T1, by Lemma 2.8, for every xePE(X), cl(xe)={xe}, so der(xe)=ΦTc. Thus, T is soft TD.

    (2) Let xe,yePE(X) with xeye. If yeder(xe), then [der(xe)]c is a soft open set that includes xe but not ye. If yeder(xe) and since xeye, then ye[cl(xe)]c and [cl(xe)]cT with xe[cl(xe)]c. Consequently, T is soft T0.

    The reverse of the above implications may not be true, as illustrated by the examples below.

    Example 4.3. Let X be an infinite and let E be a set of parameters. For a fixed pePE(X), the soft topology T on X is given by T={(F,E)SE(X):pe(F,E)or(F,E)=˜X}. We first need to check T is soft TD. Indeed, take xePE(X), if xe=pe, then der(xe)=Φ. If xepe, then der(xe)={pe}. Therefore, in either cases, der(xe) is soft closed. On the other hand, for any xepe, cl(xe)={xe,pe}{xe}, which means {xe} is not a closed set. Hence T is not soft T1.

    Example 4.4. Let E={e1,e2} be a set of parameters and let T be a soft topology on the set of real numbers R generated by

    {{(e1,B(e1)),(e2,B(e2))}:B(e1)=(a,b),B(e2)=(c,);a,b,cR;a<b}.

    Let xe1,ye2PE(X) with xe1ye2. W.l.o.g, we assume x<y. Take (G,E)={(e1,),(e2,(x,))}. Then (G,E) is a soft open set containing ye2 but not xe1 and hence T is soft T0. But then der(ye2)={(e1,),(e2,(,y))} is not soft closed, and consequently T is not soft TD.

    Proposition 4.5. Let TTE(X). Then T is soft TD iff {xe}LC(X) for every xePE(X).

    Proof. Let xePE(X). We need to prove that {xe} can be written as an intersection of a soft open set with a soft closed set. Set (G,E)=[der(xe)]c and (F,E)=cl(xe). Then (G,E)T and (F,E)Tc such that {xe}=(G,E)˜(F,E).

    Conversely, (w.l.o.g) we set {xe}=(G,E)˜cl(xe). Now, der(xe)=cl(xe){xe}=cl(xe)[(G,E)˜cl(xe)]= cl(xe)˜(G,E)c. Since finite intersections of soft closed sets are soft closed, so der(xe) is soft closed.

    Proposition 4.6. Let TTE(X). Then T is soft TD iff for every xePE(X), there exists (G,E)T including xe such that (G,E){xe}T.

    Proof. Take a point xePE(X). If we set (G,E)=[der(xe)]c, then (G,E)T containing xe. Now,

    (G,E){xe}=˜Xder(xe)˜{xe}c=˜X˜(der(xe))c˜{xe}c=˜X˜[der(xe)˜{xe}]c=˜Xcl(xe).

    Thus, (G,E){xe}T.

    Conversely, suppose for every xePE(X), there exists xe(G,E)T such that (G,E){xe}T. Therefore, {xe}=(G,E)˜[(G,E){xe}]c. By Proposition 4.5, T is soft TD.

    Proposition 4.7. For a soft topology TTE(X), the following properties are equivalent:

    (1) T is soft TD.

    (2) der(der(A,E))˜der(A,E) for every (A,E)SE(X).

    (3) der(A,E)Tc for every (A,E)SE(X).

    Proof. (1)(2) Let xeder(der(A,E)). Then every (G,E)T with xe(G,E) includes some points of der(A,E). Since T is soft TD,

    (H,E){xe}˜der(A,E)Φ,

    where (H,E)=[der(xe)]c˜(G,E). Suppose yeder(A,E) with yexe. Then ye(H,E)˜(G,E). Since yeder(A,E), then (H,E)T contains a point ze of (A,E) except ye. Indeed, zexe and then every (G,E) with xe(G,E) contains some points of (A,E) except xe. Hence, xeder(A,E).

    (2)(3) Since cl(der(A,E))=der(der(A,E))˜der(A,E)˜der(A,E), so der(A,E)Tc.

    (3)(1) It is evident.

    Proposition 4.8. Let (A,E)SE(X), TTE(X) and (F,E)Tc. The following properties are equivalent:

    (1) T is soft TD.

    (2) For every xePE(X), [cl(xe)]c˜{xe}T.

    (3) Every xeWI(A,E)xeI(A,E).

    (4) Every xeWI(F,E)xeI(F,E).

    Proof. (1)(2) Given xePE(X), by Proposition 4.6, there is (G,E)T such that xe(G,E) and (G,E){xe}T. Therefore, (G,E){xe}=(G,E)cl(xe). Since T is soft TD, so (G,E)der(xe)=(G,E)cl(xe)˜{xe}T. But, for every xe[cl(xe)]c˜{xe}, we have

    xe(G,E)=(G,E)der(xe)˜{xe}˜[cl(xe)]c˜{xe}.

    Thus, [cl(xe)]c˜{xe}T.

    (2)(3) Suppose xeWI(A,E). Then there is (G,E)T such that

    xe(G,E)˜(A,E)˜cl(xe).

    By (2), [cl(xe)]c˜{xe}T. But,

    (G,E)˜(A,E)˜[[cl(xe)]c˜{xe}]={xe}.

    Hence, xeI(A,E).

    (3)(4) Clear.

    (4)(1) Given xePE(X), we can easily conclude from the definition that xeWI(cl(xe)). By (4), xeI(cl(xe)), and so there exists (G,E)T such that (G,E)˜cl(xe)={xe}. Therefore, (G,E){xe}=(G,E)cl(xe)T. By Proposition 4.6, T is soft TD.

    Summing up all the above findings yields the following characterization:

    Theorem 4.9. For a soft topology TTE(X), the following properties are equivalent:

    (1) T is soft TD.

    (2) {xe}LC(X) for every xePE(X).

    (3) der(A,E)Tc for every (A,E)SE(X).

    (4) der(der(A,E))˜der(A,E) for every (A,E)SE(X).

    (5) xePE(X), there exists (G,E)T with xe(G,E) such that (G,E){xe}T.

    (6) xePE(X), [cl(xe)]c˜{xe}T.

    (7) xeWI(A,E)xeI(A,E), where (A,E)SE(X).

    (8) xeWI(F,E)xeI(F,E), where (F,E)Tc.

    The properties of soft topological operators derived in Section 2 are used to develop new characterizations of soft Ti-spaces for i=0,1.

    Proposition 5.1. For a soft topology TTE(X), the following properties are equivalent:

    (1) T is soft T0.

    (2) For every xe,yePE(X) with xeye, either xeye or yexe.

    (3) yecl(xe)xecl(ye).

    (4) For every xe,yePE(X) with xeye, cl(xe)cl(ye).

    Proof. (1)(2) It is just a reword of the definition.

    (2)(3) Let yecl(xe). For every (G,E)T that contains ye, (G,E)˜{xe}Φ and so yexe. If xe=ye, then there is nothing to prove. Otherwise, by (2), xeye. Therefore, there exists (H,E)T such that xe(H,E) and (H,E)˜{ye}=Φ. Hence, xecl(ye).

    (3)(4) Suppose the negative of (4) holds. Then cl(xe)˜cl(ye) and cl(ye)˜cl(xe). Since yecl(ye), then it implies that cl(ye)cl(xe) and so yecl(xe). By (3), xecl(ye) implies xecl(xe) which is impossible.

    (4)(1) Suppose xe,yePE(X) with xeye, cl(xe)cl(ye). This means that there is zePE(X) for which zecl(xe) but zecl(ye). We claim that xecl(ye). Otherwise, we will have {xe}cl(ye) and so cl(xe)cl(ye). This implies that zecl(ye), a contradiction to the selection of ze. Set (G,E)=[cl(ye)]c. Therefore, (G,E)T such that xe(G,E) and ye(G,E). Hence, T is soft T0.

    Proposition 5.2. For a soft topology TTE(X), the following properties are equivalent:

    (1) T is soft T0.

    (2) For every xe,yePE(X), yeker(xe)xeker(ye).

    (3) For every xe,yePE(X) with xeye, ker(xe)ker(ye).

    Proof. Follows from Lemma 3.10 (1) and Proposition 5.1.

    Proposition 5.3. A soft topology TTE(X) is soft T0 iff yeder(xe) implies cl(ye)˜der(xe) for every xe,yePE(X).

    Proof. Given xe,yePE(X). If yeder(xe), then yexe and xecl(ye) (as T is soft T0), then cl(ye)˜der(xe).

    Conversely, let xe,yePE(X) be such that xeye. If yeder(xe), then cl(ye)˜der(xe). This means that yecl(xe) and xecl(ye). From Proposition 5.1, T is soft T0.

    Proposition 5.4. A soft topology TTE(X) is soft T0 iff yeshel(xe) implies ker(ye)˜shel(xe) for every xe,yePE(X).

    Proof. By Proposition 5.3 and Lemma 3.10, we can obtain the proof.

    Proposition 5.5. A soft topology TTE(X) is soft T0 iff [cl(xe)˜{ye}]˜[{xe}˜cl(ye)] is degenerate for every xe,yePE(X).

    Proof. Assume xe,yePE(X) and T is soft T0. By Proposition 5.1, for every xe,yePE(X), if yecl(xe), then xecl(ye). Therefore, [cl(xe)˜{ye}]˜[{xe}˜cl(ye)]={ye} is a degenerated soft set. Otherwise, [cl(xe)˜{ye}]˜[{xe}˜cl(ye)]={xe} which is also degenerate.

    Conversely, if the given condition is satisfied, then the result is either Φ,{xe}, or {ye}. For the case of Φ, the conclusion is obvious. If [cl(xe)˜{ye}]˜[{xe}˜cl(ye)]={xe} implies xecl(ye) and cl(xe)˜{ye}=Φ. Therefore, yecl(xe). The case of {ye} is similar to the latter one. Hence, T is soft T0.

    Proposition 5.6. A soft topology TTE(X) is soft T0 iff der(xe)˜shel(xe)=Φ for every xePE(X).

    Proof. If der(xe)˜shel(xe)Φ, then there is xePE(X) such that zeder(xe) and zeshel(xe). Indeed, zexe and so zecl(xe) and zeker(xe). By Remark 3.6, zexe and xeze implies that T cannot be soft T0, a contradiction.

    Conversely, if der(xe)˜shel(xe)=Φ, then for each zexe, either zecl(xe) or zeker(xe). Therefore, either zecl(xe) or xecl(ze). By Proposition 5.1 (3), T is soft T0.

    Proposition 5.7. A soft topology TTE(X) is soft T0 iff xe={xe} for every xePE(X).

    Proof. It is a consequence of Definition 3.7 and Proposition 5.6.

    Proposition 5.8. A soft topology TTE(X) is soft T0 iff der(xe) is a union of soft closed sets for every xePE(X).

    Proof. Since, for every xePE(X), der(xe)Tc, then for every zeder(xe) we must have (G,E)T such that xe(G,E) and ze(G,E). Therefore, (F,E)=(G,E)cTc with with ze(F,E) but xe(F,E). This means that zeder(xe), we have

    ze(F,E)˜cl(xe)˜der(xe).

    Since (F,E)˜cl(xe)Tc, so der(xe) is a union of soft closed sets.

    Conversely, let der(xe)=˜iI(Fi,E), where (Fi,E)Tc. If zeder(xe), then ze(Fi,E) for some i but xe(Fi,E). Therefore, (Fi,E)cT such that xe(Fi,E)c but ze(Fi,E)c. If zeder(xe) and zexe, then ze[cl(xe)]c and [cl(xe)]cT for which xe[cl(xe)]c. This proves that T is soft T0.

    Summing up all the above propositions yields the following characterization:

    Theorem 5.9. For a soft topology TTE(X), the following properties are equivalent:

    (1) T is soft T0.

    (2) For every xe,yePE(X) with xeye, either xeye or yexe.

    (3) For every xe,yePE(X), yecl(xe)xecl(ye).

    (4) For every xe,yePE(X), yeder(xe)cl(ye)˜der(xe)

    (5) For every xe,yePE(X) with xeye, cl(xe)cl(ye).

    (6) For every xe,yePE(X), yeker(xe)xeker(ye).

    (7) For every xe,yePE(X), yeshel(xe)ker(ye)˜shel(xe)

    (8) For every xe,yePE(X) with xeye, ker(xe)ker(ye).

    (9) For every xe,yePE(X) [cl(xe)˜{ye}]˜[{xe}˜cl(ye)] is degenerate.

    (10) For every xePE(X), der(xe)˜shel(xe)=Φ.

    (11) For every xePE(X), der(xe) is a union of soft closed sets.

    (12) For every xePE(X), xe={xe}.

    Theorem 5.10. For a soft topology TTE(X), the following properties are equivalent:

    (1) T is soft T1.

    (2) For every xe,yePE(X) with xeye, xeye.

    (3) For every xePE(X), cl(xe)={xe}.

    (4) For every xePE(X), der(xe)=Φ.

    (5) For every xePE(X), ker(xe)={xe}.

    (6) For every xePE(X), shel(xe)=Φ.

    (7) For every xe,yePE(X) with xeye, cl(xe)˜cl(ye)=Φ.

    (8) For every xe,yePE(X) with xeye, ker(xe)˜ker(ye)=Φ.

    Proof. One can easily notice that all the statements are rephrases of (1) with the help of Lemmas in Section 3. Last statement means xeker(ye) and yeker(xe). Equivalently, yecl(xe) and xecl(ye). This guarantees the existence of two sets (G,E),(H,E)T such that xe(G,E),ye(G,E) and ye(H,E),xe(H,E). Thus, T is soft T1.

    We close this investigation with the following remark:

    Remark 5.11. Section 2 recalls soft points and soft elements, which are two distinct types of soft point theory. We have employed the concept of soft points throughout this paper, although most of the (obtained) results are invalid for soft elements. The reasons can be found in [30], Examples 3.14–3.21. The divergences between axioms via classical and soft settings were studied in detail in [5].

    Soft separation axioms are a collection of conditions for classifying a system of soft topological spaces according to particular soft topological properties. These axioms are usually described in terms of soft open or soft closed sets in a topological space.

    In this work, we have proposed soft topological operators that will be used to characterize certain soft separation axioms and named them "soft kernel" and "soft shell". The interrelations between the latter soft operators and soft closure or soft derived set operators have been discussed. Moreover, we have introduced soft TD-spaces as a new soft separation axiom that is weaker than soft T1 but stronger than soft T0-spaces. It should be noted that TD-spaces have applications in other (applied) disciplines. Some examples have been provided, illustrating that soft TD-spaces are at least different from soft T1 and soft T0-spaces. The soft topological operators mentioned above are used to obtain new characterizations of soft Ti-spaces for i=0,1, and D. Ultimately, we have analyzed the validity of our findings in relation to two different theories of soft points.

    In the upcoming work, we shall define the axioms given herein and examine their properties via other soft structures like infra soft topologies and supra soft topologies. We will also conduct a comparative study between these axioms and their counterparts introduced with respect to different types of belonging and non-belonging relations. Moreover, we will generalize the concept of functionally separation axioms [23] to soft settings and investigate its relationships with the other types of soft separation axioms.

    The authors declare that they have no competing interests.



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