To explore the soft ultrasound marker (USM) combined with non-invasive prenatal testing (NIPT) in diagnosing fetal chromosomal abnormalities based on machine learning and data mining techniques.
To analyze the data of ultrasonic examination from 856 cases with high-risk single pregnancy during early and middle pregnancy stage. NIPT was applied in 642 patients. All 856 patients accepted amniocentesis and chromosome karyotype analysis to determine the efficacy of USM, Down's syndrome screening, and NIPT in detecting fetal chromosomal abnormalities.
Among the 856 fetuses, 129 fetuses (15.07%) with single positive USM and 36 fetuses (4.21%) with two or more positive USM. There were 81 fetuses (9.46%) with chromosomal abnormalities. In the group with multiple USM, chromosomal abnormalities were found in 36.11% of them. It was higher than the group without USM, which was 6.22% (P < 0.01), and the group with just a single USM (19.38%, P < 0.05). The sensitivity, specificity and accuracy were 96.72%, 98.45% and 98.29% when the combination of USM, Down's syndrome screening and NIPT was used to diagnose fetal chromosomal abnormalities further evaluating the accuracy and effectiveness of the above diagnostic criteria and methods with mainstream Classifiers based evaluation indicators of accuracy, f1 score, AUC.
The combination of USM, Down's syndrome screening and NIPT is valuable for the diagnosis of fetal chromosomal abnormalities.
Citation: Xianfeng Xu, Liping Wang, Xiaohong Cheng, Weilin Ke, Shenqiu Jie, Shen Lin, Manlin Lai, Linlin Zhang, Zhenzhou Li. Machine learning-based evaluation of application value of the USM combined with NIPT in the diagnosis of fetal chromosomal abnormalities[J]. Mathematical Biosciences and Engineering, 2022, 19(4): 4260-4276. doi: 10.3934/mbe.2022197
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To explore the soft ultrasound marker (USM) combined with non-invasive prenatal testing (NIPT) in diagnosing fetal chromosomal abnormalities based on machine learning and data mining techniques.
To analyze the data of ultrasonic examination from 856 cases with high-risk single pregnancy during early and middle pregnancy stage. NIPT was applied in 642 patients. All 856 patients accepted amniocentesis and chromosome karyotype analysis to determine the efficacy of USM, Down's syndrome screening, and NIPT in detecting fetal chromosomal abnormalities.
Among the 856 fetuses, 129 fetuses (15.07%) with single positive USM and 36 fetuses (4.21%) with two or more positive USM. There were 81 fetuses (9.46%) with chromosomal abnormalities. In the group with multiple USM, chromosomal abnormalities were found in 36.11% of them. It was higher than the group without USM, which was 6.22% (P < 0.01), and the group with just a single USM (19.38%, P < 0.05). The sensitivity, specificity and accuracy were 96.72%, 98.45% and 98.29% when the combination of USM, Down's syndrome screening and NIPT was used to diagnose fetal chromosomal abnormalities further evaluating the accuracy and effectiveness of the above diagnostic criteria and methods with mainstream Classifiers based evaluation indicators of accuracy, f1 score, AUC.
The combination of USM, Down's syndrome screening and NIPT is valuable for the diagnosis of fetal chromosomal abnormalities.
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)):e∈E} is said to be a soft set [25] over X, where F:E→2X 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 e∈E, where x∈X. It is said that a soft element ({x},E) is in (F,E) (briefly, x∈(F,E)) if x∈F(e) for each e∈E. On the other hand, x∉(F,E) if x∉F(e) for some e∈E. 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 e′∈E with e′≠e, where e∈E and x∈X. An argument xe∈(F,E) means that x∈F(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:E→2X is given by Fc(e)=X−F(e) for each e∈E. If (F,E)∈SE(X), it is denoted by Φ if F(e)=∅ for each e∈E and is denoted by ˜X if F(e)=X for each e∈E. 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 E1⊆E2 and A(e)⊆B(e) for each e∈E1, 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 e∈E, 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 e∈E, (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,˜X∈T.
(T.2) If (F1,E),(F2,E)∈T, then (F1,E)˜∩(F2,E)∈T.
(T.3) If {(Fi,E):i∈I}˜⊆T, then ˜∪i∈I(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 T∈TE(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 T∈TE(X). A point xe∈PE(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 T∈TE(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 T∈TE(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 T∈TE(X)) is called
(1) Soft T0 if for every xe,ye′∈PE(X) with xe≠ye′, 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,ye′∈PE(X) with xe≠ye′, 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 T∈TE(X). Then T is soft T1 iff cl(xe)={xe} for every xe∈PE(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 T∈TE(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 T∈TE(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 xe∈PE(X) and T∈TE(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 T∈TE(X), we have
ker(F,E)={xe∈PE(X):cl(xe)˜∩(F,E)≠Φ}. |
Proof. Let xe∈ker(F,E). If cl(xe)˜∩(F,E)=Φ, then (F,E)˜⊆˜X−cl(xe). Therefore, ˜X−cl(xe)∈T such that it contains (F,E) but not xe, a contradiction.
Conversely, if xe∉ker(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 ye′∈cl(xe)˜∩(F,E). Therefore, ˜X−(G,E)∈Tc including xe but not ye′. This contradicts to ye′∈cl(xe)˜∩(F,E). Thus, xe∈ker(F,E).
Definition 3.5. Let (F,E),(G,E)∈SE(X) and T∈TE(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,ye′∈PE(X) and T∈TE(X), we have
(1) cl(xe)={ye′:ye′ ⊬ xe}.
(2) ker(xe)={ye′:xe ⊬ ye′}.
Definition 3.7. For xe∈PE(X) and T∈TE(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,ye′∈PE(X) and T∈TE(X), we have
(1) der(xe)={ye′:ye′≠xe,ye′ ⊬ xe}.
(2) shel(xe)={ye′:ye′≠xe,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,ye′∈PE(X) and T∈TE(X):
(1) ye′∈ker(xe)⟺xe∈cl(ye′).
(2) ye′∈shel(xe)⟺xe∈der(ye′).
(3) ye′∈cl(xe)⟹cl(ye′)˜⊆cl(xe).
(4) ye′∈ker(xe)⟹ker(ye′)˜⊆ker(xe).
Proof. (1) and (2) follow, respectively, from Remarks 3.6 and 3.8.
(3) Straightforward.
(4) Let ze∗∈ker(ye′). By (1), ye′∈cl(ze∗) and so cl(ye′)˜⊆cl(ze∗) (by (3)). By hypothesis, ye′∈ker(xe) and so xe∈cl(ye′). Therefore, cl(xe)˜⊆cl(ye′). Finally, we get cl(xe)˜⊆cl(ze∗) and then xe∈cl(ze∗). By (1), ze∗∈ker(xe). Thus, ker(ye′)˜⊆ker(xe).
Lemma 3.11. Let T∈TE(X) and let xe∈PE(X). Then
(1) shel(xe) is degenerate iff for every ye′∈PE(X) with ye′≠xe, der(xe)˜∩der(ye′)=Φ.
(2) der(xe) is degenerate iff for every ye′∈PE(X) with ye′≠xe, shel(xe)˜∩shel(ye′)=Φ.
Proof. (1) If der(xe)˜∩der(ye′)≠Φ, then there exists ze∗∈PE(X) such that ze∗∈der(xe), ze∗∈der(ye′). Therefore, ze∗≠ye′≠xe for which ze∗∈cl(xe) and ze∗∈cl(ye′). By Lemma 3.10 (1), xe,ye′∈ker(ze∗). Thus, xe,ye′∈ker(ze∗)−ze∗=shel(ze∗). This proves that shel(xe) is not degenerate.
Conversely, if xe,ye′∈shel(ze∗), then xe≠ye′, xe≠ze∗ and so xe∈ker(ze∗), ye′∈ker(ze∗). Therefore, ze∗∈cl(xe)˜∩cl(ye′) and thus ze∗∈der(xe)˜∩der(ye′). But this is impossible, hence der(xe)˜∩der(ye′)=Φ.
Lemma 3.12. Let T∈TE(X) and let xe,ye′∈PE(X). Then
(1) If ye′∈⟨xe⟩, then ⟨ye′⟩=⟨xe⟩.
(2) Either ⟨ye′⟩=⟨xe⟩ or ⟨ye′⟩˜∩⟨xe⟩=Φ.
Proof. (1) If ye′∈⟨xe⟩, then ye′∈cl(xe) and ye′∈ker(xe). When ye′∈cl(xe), by Lemma 3.10 (1), xe∈ker(ye′). By Lemma 3.10 (3) and (4), cl(ye′)˜⊆cl(xe) and ker(xe)˜⊆ker(ye′). When ye′∈ker(xe), by Lemma 3.10 (2), xe∈cl(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 T∈TE(X) and let xe,ye′∈PE(X). Then ker(xe)≠ker(ye′) iff cl(xe)≠cl(ye′).
Proof. If ker(xe)≠ker(ye′), then one can find ze∗∈ker(xe) but ze∗∉ker(ye′). From ze∗∈ker(xe), we get xe∈cl(ze∗) and then cl(xe)˜⊆cl(ze∗). Since ze∗∉ker(ye′), by Lemma 3.10 (1), cl(ze∗)˜∩ye′=Φ. Therefore, cl(ze∗)˜∩ye′=Φ implies ye′∉cl(xe). Hence, cl(ye′)≠cl(xe).
The converse can be proved in a similar manner to the first part.
Definition 4.1. Let T∈TE(X). We call T a soft TD-space if der(xe) is a soft closed set for every xe∈PE(X).
Theorem 4.2. Let T∈TE(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 xe∈PE(X), cl(xe)={xe}, so der(xe)=Φ∈Tc. Thus, T is soft TD.
(2) Let xe,ye′∈PE(X) with xe≠ye′. If ye′∈der(xe), then [der(xe)]c is a soft open set that includes xe but not ye′. If ye′∉der(xe) and since xe≠ye′, then ye′∈[cl(xe)]c and [cl(xe)]c∈T 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 pe∈PE(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 xe∈PE(X), if xe=pe, then der(xe)=Φ. If xe≠pe, then der(xe)={pe}. Therefore, in either cases, der(xe) is soft closed. On the other hand, for any xe≠pe, 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,c∈R;a<b}. |
Let xe1,ye2∈PE(X) with xe1≠ye2. 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 T∈TE(X). Then T is soft TD iff {xe}∈LC(X) for every xe∈PE(X).
Proof. Let xe∈PE(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 T∈TE(X). Then T is soft TD iff for every xe∈PE(X), there exists (G,E)∈T including xe such that (G,E)−{xe}∈T.
Proof. Take a point xe∈PE(X). If we set (G,E)=[der(xe)]c, then (G,E)∈T containing xe. Now,
(G,E)−{xe}=˜X−der(xe)˜⋂{xe}c=˜X˜⋂(der(xe))c˜⋂{xe}c=˜X˜⋂[der(xe)˜⋃{xe}]c=˜X−cl(xe). |
Thus, (G,E)−{xe}∈T.
Conversely, suppose for every xe∈PE(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 T∈TE(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 xe∈der(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 ye′∈der(A,E) with ye′≠xe. Then ye′∈(H,E)˜⊆(G,E). Since ye′∈der(A,E), then (H,E)∈T contains a point ze∗ of (A,E) except ye′. Indeed, ze∗≠xe and then every (G,E) with xe∈(G,E) contains some points of (A,E) except xe. Hence, xe∈der(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), T∈TE(X) and (F,E)∈Tc. The following properties are equivalent:
(1) T is soft TD.
(2) For every xe∈PE(X), [cl(xe)]c˜∪{xe}∈T.
(3) Every xe∈WI(A,E)⟹xe∈I(A,E).
(4) Every xe∈WI(F,E)⟹xe∈I(F,E).
Proof. (1)⟹(2) Given xe∈PE(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 xe∈WI(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, xe∈I(A,E).
(3)⟹(4) Clear.
(4)⟹(1) Given xe∈PE(X), we can easily conclude from the definition that xe∈WI(cl(xe)). By (4), xe∈I(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 T∈TE(X), the following properties are equivalent:
(1) T is soft TD.
(2) {xe}∈LC(X) for every xe∈PE(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) ∀xe∈PE(X), there exists (G,E)∈T with xe∈(G,E) such that (G,E)−{xe}∈T.
(6) ∀xe∈PE(X), [cl(xe)]c˜∪{xe}∈T.
(7) ∀xe∈WI(A,E)⟹xe∈I(A,E), where (A,E)∈SE(X).
(8) ∀xe∈WI(F,E)⟹xe∈I(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 T∈TE(X), the following properties are equivalent:
(1) T is soft T0.
(2) For every xe,ye′∈PE(X) with xe≠ye′, either xe⊢ye′ or ye′⊢xe.
(3) ye′∈cl(xe)⟹xe∉cl(ye′).
(4) For every xe,ye′∈PE(X) with xe≠ye′, cl(xe)≠cl(ye′).
Proof. (1)⟹(2) It is just a reword of the definition.
(2)⟹(3) Let ye′∈cl(xe). For every (G,E)∈T that contains ye′, (G,E)˜∩{xe}≠Φ and so ye′⊬xe. If xe=ye′, then there is nothing to prove. Otherwise, by (2), xe⊢ye′. Therefore, there exists (H,E)∈T such that xe∈(H,E) and (H,E)˜∩{ye′}=Φ. Hence, xe∉cl(ye′).
(3)⟹(4) Suppose the negative of (4) holds. Then cl(xe)˜⊆cl(ye′) and cl(ye′)˜⊆cl(xe). Since ye′∈cl(ye′), then it implies that cl(ye′)∈cl(xe) and so ye′∈cl(xe). By (3), xe∉cl(ye′) implies xe∉cl(xe) which is impossible.
(4)⟹(1) Suppose xe,ye′∈PE(X) with xe≠ye′, cl(xe)≠cl(ye′). This means that there is ze∗∈PE(X) for which ze∗∈cl(xe) but ze∗∉cl(ye′). We claim that xe∉cl(ye′). Otherwise, we will have {xe}∉cl(ye′) and so cl(xe)∉cl(ye′). This implies that ze∗∈cl(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 T∈TE(X), the following properties are equivalent:
(1) T is soft T0.
(2) For every xe,ye′∈PE(X), ye′∈ker(xe)⟹xe∉ker(ye′).
(3) For every xe,ye′∈PE(X) with xe≠ye′, ker(xe)≠ker(ye′).
Proof. Follows from Lemma 3.10 (1) and Proposition 5.1.
Proposition 5.3. A soft topology T∈TE(X) is soft T0 iff ye′∈der(xe) implies cl(ye′)˜⊆der(xe) for every xe,ye′∈PE(X).
Proof. Given xe,ye′∈PE(X). If ye′∈der(xe), then ye′≠xe and xe∉cl(ye′) (as T is soft T0), then cl(ye′)˜⊆der(xe).
Conversely, let xe,ye′∈PE(X) be such that xe≠ye′. If ye′∈der(xe), then cl(ye′)˜⊆der(xe). This means that ye′∈cl(xe) and xe∉cl(ye′). From Proposition 5.1, T is soft T0.
Proposition 5.4. A soft topology T∈TE(X) is soft T0 iff ye′∈shel(xe) implies ker(ye′)˜⊆shel(xe) for every xe,ye′∈PE(X).
Proof. By Proposition 5.3 and Lemma 3.10, we can obtain the proof.
Proposition 5.5. A soft topology T∈TE(X) is soft T0 iff [cl(xe)˜∩{ye′}]˜⋃[{xe}˜∩cl(ye′)] is degenerate for every xe,ye′∈PE(X).
Proof. Assume xe,ye′∈PE(X) and T is soft T0. By Proposition 5.1, for every xe,ye′∈PE(X), if ye′∈cl(xe), then xe∉cl(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 xe∈cl(ye′) and cl(xe)˜∩{ye′}=Φ. Therefore, ye′∉cl(xe). The case of {ye′} is similar to the latter one. Hence, T is soft T0.
Proposition 5.6. A soft topology T∈TE(X) is soft T0 iff der(xe)˜∩shel(xe)=Φ for every xe∈PE(X).
Proof. If der(xe)˜∩shel(xe)≠Φ, then there is xe∈PE(X) such that ze∗∈der(xe) and ze∗∈shel(xe). Indeed, ze∗≠xe and so ze∗∈cl(xe) and ze∗∈ker(xe). By Remark 3.6, ze∗⊬xe and xe⊬ze∗ implies that T cannot be soft T0, a contradiction.
Conversely, if der(xe)˜∩shel(xe)=Φ, then for each ze∗≠xe, either ze∗∈cl(xe) or ze∗∈ker(xe). Therefore, either ze∗∈cl(xe) or xe∈cl(ze∗). By Proposition 5.1 (3), T is soft T0.
Proposition 5.7. A soft topology T∈TE(X) is soft T0 iff ⟨xe⟩={xe} for every xe∈PE(X).
Proof. It is a consequence of Definition 3.7 and Proposition 5.6.
Proposition 5.8. A soft topology T∈TE(X) is soft T0 iff der(xe) is a union of soft closed sets for every xe∈PE(X).
Proof. Since, for every xe∈PE(X), der(xe)∈Tc, then for every ze∗∈der(xe) we must have (G,E)∈T such that xe∈(G,E) and ze∗∉(G,E). Therefore, (F,E)=(G,E)c∈Tc with with ze∗∈(F,E) but xe∉(F,E). This means that ∀ze∗∈der(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)=˜⋃i∈I(Fi,E), where (Fi,E)∈Tc. If ze∗∈der(xe), then ze∗∈(Fi,E) for some i but xe∉(Fi,E). Therefore, (Fi,E)c∈T such that xe∈(Fi,E)c but ze∗∉(Fi,E)c. If ze∗∉der(xe) and ze∗≠xe, then ze∗∈[cl(xe)]c and [cl(xe)]c∈T 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 T∈TE(X), the following properties are equivalent:
(1) T is soft T0.
(2) For every xe,ye′∈PE(X) with xe≠ye′, either xe⊢ye′ or ye′⊢xe.
(3) For every xe,ye′∈PE(X), ye′∈cl(xe)⟹xe∉cl(ye′).
(4) For every xe,ye′∈PE(X), ye′∈der(xe)⟹cl(ye′)˜⊆der(xe)
(5) For every xe,ye′∈PE(X) with xe≠ye′, cl(xe)≠cl(ye′).
(6) For every xe,ye′∈PE(X), ye′∈ker(xe)⟹xe∉ker(ye′).
(7) For every xe,ye′∈PE(X), ye′∈shel(xe)⟹ker(ye′)˜⊆shel(xe)
(8) For every xe,ye′∈PE(X) with xe≠ye′, ker(xe)≠ker(ye′).
(9) For every xe,ye′∈PE(X) [cl(xe)˜∩{ye′}]˜⋃[{xe}˜∩cl(ye′)] is degenerate.
(10) For every xe∈PE(X), der(xe)˜∩shel(xe)=Φ.
(11) For every xe∈PE(X), der(xe) is a union of soft closed sets.
(12) For every xe∈PE(X), ⟨xe⟩={xe}.
Theorem 5.10. For a soft topology T∈TE(X), the following properties are equivalent:
(1) T is soft T1.
(2) For every xe,ye′∈PE(X) with xe≠ye′, xe⊢ye′.
(3) For every xe∈PE(X), cl(xe)={xe}.
(4) For every xe∈PE(X), der(xe)=Φ.
(5) For every xe∈PE(X), ker(xe)={xe}.
(6) For every xe∈PE(X), shel(xe)=Φ.
(7) For every xe,ye′∈PE(X) with xe≠ye′, cl(xe)˜∩cl(ye′)=Φ.
(8) For every xe,ye′∈PE(X) with xe≠ye′, 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 xe∉ker(ye′) and ye′∉ker(xe). Equivalently, ye′∉cl(xe) and xe∉cl(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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