Soft separation axioms and their properties are popular topic in the research of soft topological spaces. Two types of separation axioms Ti-I and Ti-II (i = 0, 1, ⋯ , 4) which take single point soft sets and soft points as separated objects have been given in [18] and [30] respectively. In this paper we show that a soft T0-II(T1-II, T2-II, and T4-II respectively) space is a soft T0-I(T1-I, T2-I, and T4-I respectively) space, if the initial universe set X and the parameter set E are sets of two elements. Some examples are given to explain that a soft Ti-I may not to be a soft Ti-II space (i = 0, 1, ⋯ , 4).
In order to solve the problem of data uncertainty, mathematicians have put forward research schemes such as fuzzy set theory, interval mathematics theory and rough set theory. However, these theories also have their own defects. In the year 1999, Molodtsov initiated soft set theory [1] which is a mathematical tool for parameterizing research objects of sets. At present this theory has been widely combined with the theories of algebra [4–10], fuzzy set [11–13], decision-making [14–16], and topology [17–22]. Many rich results have been obtained.
Soft separation axioms which are draw lessons from general topology theory are very important and popular topic in the research of soft topological spaces. They are established by different separated objects with new types of belong and non-belong relations [18, 25–28]. In the study of these fields, soft separation axioms defined in [18] and [27] take single point soft sets and soft points as separated objects respectively. They are usually distinguished by symbols and (see [37, 38]). Studies about the relationships among soft separation axioms are investigated in [25, 37]. Singh and Noorie [37] demonstrated that and are independent of each other if i = 0, 1. Al-shami [38] has corrected some obtained results in previous papers and illustrated the relationships among soft -spaces by two figures clearly. Furthermore Al-shami et al. initiated the notions of ordered soft separation axioms, namely p-soft Ti-ordered spaces are introduced and the relationship among them have been illustrated in [32, 33]. Some applications of soft separation axioms in real life have also been launched in [36, 40].
Then He [30] put forward a new type of soft separation axioms. We can find that the separated objects in [30] are the same as those in [27]. The difference is that any pair of different soft points can be separated if the separation axioms defined in [30] hold. However, this is not the case in [27]. The main purpose of this paper is to study the relationship between soft separation axioms defined in [18] and [30]. For the convenience of expression, we add Roman letters to distinguish these soft separations axioms discussed here.
The rest of this paper is organized as follows. The second section briefly reviews some basic notions on soft theory. The third section lists two types of separation axioms in soft topological spaces, and then compares the differences among them by examples. Some properties on the relationship of separation axioms are also shown.
Preliminaries
In this section, first, we present some basic definitions and results of soft set theory. Let X be an initial universe set and E be a set of parameters which usually are initial attributes, characteristics, or properties of objects in X.
Definition 2.1.([1]) Let X be an initial universe set and E be a set of parameters. Let denote the power set of X and A ⊆ E. A pair (F, A) is called a soft set over X, where F is a mapping given by .
In other words, a soft set over X is a parameterized family of subsets of the universe X. For ɛ ∈ A, F (ɛ) may be considered as the set of ɛ-approximate elements of the soft set (F, A). The family of all soft sets over X with a parameters set E is denoted by SSE (X).
Definition 2.2.([18]) Let Y be a non-empty subset of X, then denotes the soft set (Y, E) over X for which Y (e) = Y, for all e ∈ E. If Y is a single point set, such as Y = {x} and x ∈ X, then is called a single point soft set.
Definition 2.3.([10]) For two soft sets (F, A) and (G, B) over a common universe X, (F, A) is a soft subset of (G, B), denoted by (F, A) ⊆ (G, B), if
(i) A ⊆ B, and
(ii) ∀ɛ ∈ A, F (ɛ) is a subset of G (ɛ).
If (F, A) ⊆ (G, B) and (G, B) ⊆ (F, A), we write (F, A) = (G, B) and say (F, A) is equal to (G, B).
Definition 2.4.([18]) Let (F, E) be a soft sets over a common universe X and x ∈ X. We say that read as x belongs to the soft set (F, E) whenever x ∈ F (e) for all e ∈ E.
Definition 2.5.([2]) The union of two soft sets (F, A) and (G, B) over a common universe X is the soft set (H, C), where C = A ∪ B, and ∀e ∈ C,
We write (F, A) ∪ (G, B) = (H, C).
For the union of a family of soft sets {(Fi, Ai) : i ∈ I},
where F (e) = ⋃ {Fi (e) : i ∈ Ie} and Ie = {i ∈ I : e ∈ Ai} (see [9, 12]).
Definition 2.6.([3]) The intersection of two soft sets (F, A) and (G, B) over a common universe X such that A∩ B ≠ ∅ is the soft set (H, C), where C = A ∩ B, and ∀e ∈ C, H (e) = F (e) ∩ G (e) . We write (F, A) ∩ (G, B) = (H, C).
Definition 2.7.([18]) Let be the collection of soft sets over X, then is said to be a soft topology on X if
(1) belong to
(2) the union of any number of soft sets in belongs to
(3) the intersection of any two soft sets in belongs to .
The triplet is called a soft topological space over X. The members of are said to be soft open sets in X.
The definition of soft topological spaces means the family of soft sets in a soft topology is closed with respect to the operations of union and finite intersection.
Example 2.1. Let X = {x, y}, E = {e1, e2} and , where F1 (e1) = {x} , F1 (e2) = {y} ;
F2 (e1) = {y} , F2 (e2) = {x}.
Clearly, is a soft topological space over X.
In the following we give two examples of non-soft topological spaces. They have been mistakenly studied as examples of topological spaces in [30] and [41].
Example 2.2. 2 Let X = {x, y}, E = {e1, e2} and , where
F1 (e1) = {x} , F1 (e2) = {y},
F2 (e1) = {y} , F2 (e2) =∅,
F3 (e1) = ∅ , F3 (e2) = {x},
F4 (e1) = {y} , F4 (e2) = {y},
F5 (e1) = {x} , F5 (e2) =∅,
F6 (e1) = X, F6 (e2) = {y},
F7 (e1) = {x} , F7 (e2) = X,
F8 (e1) = {y} , F8 (e2) = {x},
F9 (e1) = X, F9 (e2) =∅,
F10 (e1) = {y} , F10 (e2) = X,
F11 (e1) = {x} , F11 (e2) = {x} .
The triple given in [30] is not a soft topological space over X, since . Assume that three mappings F12, F13, F14 are defined as:
F12 (e1) = X, F12 (e2) = {x},
F13 (e1) = ∅ , F13 (e2) = X,
F14 (e1) = ∅ , F14 (e2) = {y} .
Let . Then is a soft topological space.
Example 2.3. Let X = {x1, x2, x3, x4}, E ={ e1, e2 } and , where
The triple is given in [41]. It is not a soft topological space over X, since the following soft sets (F3, E) ∪ (F4, E) , (F2, E) ∪ (F4, E) , (F4, E) ∪ (F8, E) , (F4, E) ∪ (F12, E) , (F8, E) ∪ (F12, E) , (F11, E) ∪ (F13, E) are not in .
Suppose that the mappings F23, F24, F25, F26, F27 are defined as follows:
Definition 2.8.([3]) The relative complement of a soft set (F, A) is denoted by (F, A) ′ and is defined by (F, A) ′ = (F′, A) where is a mapping given by F′ (e) = X - F (e) for all e ∈ E.
Definition 2.9. ([18]) Let be a soft topological space over X. If , then (F, E) is called a soft closed set.
Definition 2.10. ([24]) A soft set (P, E) is called a soft point, denoted by , if there exist e0 ∈ E and x0 ∈ X such that P (e0) = {x0} and P (e) =∅ for all e ∈ E - {e0}. The family of all soft points in SSE (X) is denoted by SP (SSE (X)).
In [24] a soft point is called be a soft element. Here we call it a soft point with commonly used name. In fact a soft point in [24] is a special case of a soft point given in [23].
For a soft point and a soft set (G, E), we write if (see [23]). It is clearly that if and only if x0 ∈ G (e0). For another symbol , there exists a similar relationship that if and only if . Obviously we have if . But the converse is generally not true. Essentially the relationships between soft sets are all inclusion relations ⊆.
By introducing soft points and single point soft sets, then the soft separation axioms appear in soft topological spaces naturally.
Separation axioms in soft topological spaces
Separation axioms are important topological properties. It is of practical significance to discuss some relevant separation axioms in soft set theory [34–39]. In a sense it indicates the information represented by soft sets is incompatible, if the intersection of two soft sets is a empty set. The separation axioms of soft topological spaces further indicate that, for any two different basic elements (such as soft points), incompatible soft sets with more information can be found.
At present, there are different ways to take “points” in the separation of soft topological spaces. References [18, 27] took single point soft sets or soft points generated by elements in the initial set as separation objects respectively. In this paper in order to distinguish the definitions of soft separations in [18] and [30], these same names are marked differently. The following sections are treated similarly.
Soft T0 - I space and soft T0 - II space
Definition 3.1.([18]) Let be a soft topological space over X. For each x, y ∈ X such that x ≠ y, if there exists a soft open set (F, E) or (G, E) such that and or and , then is called a soft T0-I space.
Definition 3.2.([30]) Let be a soft topological space over X. and are any two different soft points in SP (SSE (X)). If there exists a soft open set (F, E) or (G, E) such that and or and , then is called a soft T0-II space.
Obviously separability of soft topological spaces means objects such as and can be separated by soft open sets. For two finite sets X and E, the cardinality of SP (SSE (X)) is generally bigger than that of the set X. For example, let |X| = m and |E| = n, then |SP (SSE (X)) | = m × n. Therefore, it is a bit more complicated to verify the separation axioms T0-II than the separation T0-I.
A soft topological space does not necessarily have the above separation axioms. It is shown in the following examples.
Example 3.1. Let be a soft topological space over X as shown in Example 2.1. For all x ∈ X, if , then . While . It indicates that x and y cannot be separated by soft open sets. So is not a soft T0-I space.
Take . We have . The soft set (F2, E) is the only one nontrivial soft open set such that or . Thus is also not a soft T0-II space.
Remark 3.1.It should be pointed out that there exist differences between the separation axioms and T0-II defined in [27] and [30] respectively. Obviously, a soft T0-II topological space is soft . While the opposite is generally not true by Example 3.1. Because is a soft- space, but it is not a soft T0-II topological space. Similar relations exist between the soft separation axioms Ti-II and .
Example 3.2.Let X = {x, y}, E = {e1, e2} and , where
Then is a soft topological space.
There are only a couple of different elements x and y in the set X. For x and y, we have and . So is a soft T0-I space.
Take . We have . While (F, E) is the only one nontrivial soft open set such that or . Thus is not a soft T0-II space.
Example 3.2 shows that a soft T0-I space may not to be a soft T0-II space.
Proposition 3.1. Let X = {x, y}, E = {e1, e2}. If a soft topology over X and E contains any of the following six soft sets (Fi, E) (i = 1, 2, ⋯ , 6), then is a soft T0-I space, where
F1 (e1) = {x} , F1 (e2) = {x},
F2 (e1) = {y} , F2 (e2) = {y},
F3 (e1) = {x} , F3 (e2) = X,
F4 (e1) = {y} , F4 (e2) = X,
F5 (e1) = X, F5 (e2) = {x},
F6 (e1) = X, F6 (e2) = {y}.
Proof. For all i ∈ I, we have and or and . Then, by Definition 3.1, we obtain the conclusion straightforwardly.
Proposition 3.2.Let X = {x, y}, E = {e1, e2}. If is a soft T0-II space, then is a soft T0-I space.
Proof. (Proof by contradiction) Assume that is not a soft T0-I space. We denote , where Fi is defined as given in Proposition 3.1. Then .
We have . These four soft sets form six pairs of soft points as follows:
Because is a soft T0-II space and , the soft topology contains at least one of the soft sets and (F8, E), where
F7 (e1) = { x } , F7 (e2) = { y } ,
F8 (e1) = { y } , F8 (e2) = { x } .
For the group (iii), we have . Then contains at least one of the soft sets and (F10, E), where
F9 (e1) = X, F9 (e2) = ∅ ,
F10 (e1) = ∅ , F10 (e2) = X .
For the group (iv), we have . Then contains at least one of the soft sets and (F10, E). For the group (vi), we have . Then contains at least one of the soft sets and (F8, E).
Furthermore, soft sets for separating these soft points are further determined below according to the closure of on the union operation. If the soft set (F9, E) separating the group (iv) and any soft set separating the group (vi), or (F10, E) separating the group (iv) and any soft set separating the group (i) belong to a same topology, there must have a soft set (Fi, E) in the set . For example, if (F9, E) and belong to , then . It is contradictory to . So or is the only soft sets separating the group (iv).
If , then the soft set separating the group (vi) is or (F8, E). If , then . That is to say, we always have . But it is clearly that (F8, E) and any soft set separating the group (iii) cannot exist in a same topology which is not a soft T0-I topology. Thus the group (iii) cannot be separated.
If , then or (F10, E) is the soft set separating the group (iii). If , then . Thus we always have . While the soft set (F10, E) and any soft set separating the group (i) cannot exist in a same topology which is not a soft T0-I topology. Then the group (i) cannot be separated.
So the assumption is not true in any case. That is to say, is a soft T0-I space.
Soft T1-I space and soft T1-II space
Definition 3.3.([18]) Let be a soft topological space over X. For each x, y ∈ X such that x ≠ y, if there exist two soft open sets (F, E) and (G, E) such that and , and , then is called a soft T1-I space.
Definition 3.4.([30]) Let be a soft topological space over X, and be any two different soft points in SP (SSE (X)). If there exist soft open sets (F, E) and (G, E) such that and , and , then is called a soft T1-II space.
Firstly, a soft topological space satisfying separation axioms T1-I and T1-II is constructed.
Example 3.3. Let X = {x}, E = {e1, e2} and . The triple is a soft topological space over X. Since X is a single point set, is a soft T1-I topological space.
Here . We have , and . Thus and can be separated by themselves. Then is a soft T1-II space.
Example 3.4. Let X = {x, y}, E = {e1, e2} and , where
For x and y, we have , and , . Then is a soft T1-I topological space.
Let be a nontrivial soft set. If , then x ∈ H (e1). We have (H, E) = (F, E). Thus . That is to say, if , then . The converse is also true. So is not a soft T1-II topological space.
Proposition 3.3. Let X = {x1, x2, ⋯ , xm}, E = {e1, e2, ⋯ , en}. If is a soft T1-II space, then (F, E) belongs to , where
Proof. First, we show that . Since is a soft T1-II space and for all i = 1, 2, ⋯ , n and j ≠ 1, there exist soft open sets (H1j, E) such that x1 ∈ H1j (e1) , xi ∉ H1j (ej).
For and (i ≠ 1), there exist soft open sets (H2i, E) such that x1 ∈ H2i (e1) , xi ∉ H2i (e1). We have
Since is closed on the operation intersection, we have .
Similarly we can show that for all j = 2, 3, ⋯ , n. Since is closed on the operation union, we have
Remark 3.2. Example 3.4 shows that a soft T1-I space may not to be a soft T1-II space. Meanwhile it indicates that the converse of Proposition 3.3 may not be true.
By Proposition 3.3 and Definition 3.3, we can obtain the following conclusion immediately.
Proposition 3.4.If the initial universe set X and the parameter set E are sets of two elements, then a soft T1-II space over X is a soft T1-I space.
Soft T2-I space and soft T2-II space
Definition 3.5. ([18]) Let be a soft topological space over X. For any x, y ∈ X such that x ≠ y, if there exist soft open sets (F, E) and (G, E) such that , and , then is called a soft T2-I space.
Definition 3.6. ([30]) Let be a soft topological space over X, and be any two different soft points in SP (SSE (X)). If there exist soft open sets (F, E) and (G, E) such that , and , then is called a soft T2-II space.
By definitions defined above, every soft T2-I space is a soft T1-I space, and every soft T1-I space is a soft T0-I space. Every soft T2-II space is a soft T1-II space, and every soft T1-II space is a soft T0-II space (see [18, 27]).
Example 3.5. As shown in Example 3.4, we known that is not a soft T2-II topological space.
For x and y, there exist two soft sets (F, E) and (G, E) such that , and . Then is a soft T2-I space. This example shows that a soft T2-I space may not to be a soft T2-II space.
Proposition 3.5.Let X = {x, y}, E = {e1, e2}. If is a soft T1-II space over X, then it is a soft T2-I space.
Proof. By Proposition 3.3, we have (F, E) and (G, E) defined in Proposition 3.3 belong to . Thus , and . Then is a soft T2-I space.
By Proposition 3.5, we have the following conclusion.
Proposition 3.6.If the initial universe set X and the parameter set E are sets of two elements, then a soft T2-II space over X is a soft T2-I space.
Remark 3.3. Proposition 3.2, Proposition 3.4 and Proposition 3.6 only proved the case of two elements. In general, if X and E are finite sets or arbitrary sets, is this conclusion correct that a soft T2-II(T1-II and T0-II respectively) space over X is a soft T2-I (T1-I and T0-I respectively) space?
Soft T3-I space and soft T3-II space
Definition 3.7.([18]) Let be a soft topological space over X, (G, E) be a soft closed set in X and x ∈ X such that . If there exist soft open sets (F1, E) and (F2, E) such that and , then is called a soft type-I regular space.
If a soft type-I regular space is a soft T1-I space, then it is called a soft T3-I space.
Definition 3.8.([30]) Let be a soft topological space over X, (G, E) be a nontrivial soft closed set in X and such that . If there exist soft open sets (F1, E) and (F2, E) such that and , then is called a soft type-II regular space.
If a soft type-II regular space is a soft T1-II space, then it is called a soft T3-II space.
Example 3.6.As shown in Example 3.3, every soft open set is also a soft closed set in the soft topological space .
For x and a soft closed set , we have . There is only one soft set such that . While . Then is not a soft type-I regular topological space.
For a soft point , there is only a nontrivial soft closed set such that . We have and . Then is a soft type-II regular space.
This example shows that a soft type-II regular space may not to be a soft type-I regular space.
Example 3.7.Let X = {x, y}, E = {e1, e2} and , where F (e1) = {y} , F (e2) = X.
Then is a soft topological space and is also the family of the soft closed sets in the soft topological space.
For , we have . There exists only one soft set such that . So there cannot be a soft open set (G, E) which contains and satisfies . Thus is not a soft type-I regular space.
We know that and (F, E) are two nontrivial soft closed sets in the soft topological space, and . There are four cases where the relationship between a soft point and a nontrivial soft closed set does not hold: , , and . For example, for , we have , and . The other three cases can be shown similarly. Then is a soft type-II regular space.
Thus, if the sets X and E are sets of two elements, a soft type-II regular space over X may not to be a soft type-I regular space. It is different from Proposition 3.2, Proposition 3.4 and Proposition 3.6.
Proposition 3.7.Let is a soft topological space over X. If it is a soft type-I regular space, then it is a soft type-II regular space.
Proof. Let , (K, E) be a soft closed set and . Then . Since is a soft type-I regular space, there exist two soft open sets (F, E) and (G, E) such that and . Clearly that . By Definition 3.8, is a soft type-II space.
Example 3.8. Let X = {x, y, z}, E = {e1, e2} and , where
F1 (e1) = {x, y} , F1 (e2) = {x, y},
F2 (e1) = {x, z} , F2 (e2) = {x, z},
F3 (e1) = {y, z} , F3 (e2) = {y, z},
F4 (e1) = {x} , F4 (e2) = {x},
F5 (e1) = {y} , F5 (e2) = {y},
F6 (e1) = {z} , F6 (e2) = {z} .
Then is a soft topological space.
For x and y, there exist two soft open sets (F4, E) and (F5, E) such that and . For x and z, or y and z, there are also corresponding soft open sets to separate single point soft sets generated by them. Thus is a soft T1-I space.
For x ∈ X, there are three soft closed sets (F1, E) ′, (F2, E) ′ and (F4, E) ′ that do not contain x with respect to the relationship . We explain that they meet the conditions of soft type-I regular spaces one by one. For x and (F1, E) ′, there exist two soft open sets (F4, E) and (F6, E) such that and . For x and (F2, E) ′, there exist two soft open sets (F4, E) and (F5, E) such that and . For x and (F4, E) ′, there exist two soft open sets (F4, E) and (F3, E) such that and .
Similarly, for y, z ∈ X and the corresponding soft closed sets which do not contain y or z, they also can be separated by some soft open sets. Thus is a soft type-I regular space. And then is a soft T3-I space.
For and , we have . But every soft open set (F, E) satisfying x ∈ F (e1) has x ∈ F (e2). That is to say, if , then . So is not a soft T1-II space. And then is not a soft T3-II space.
Remark 3.4.Example 3.8 shows that a soft T3-I space may not to be a soft T3-II space.
Soft T4-I space and soft T4-II space
Definition 3.9. ([18]) Let be a soft topological space over X, (F, E) and (G, E) be a soft closed set over X such that . If there exist soft open sets (F1, E) and (F2, E) such that (F, E) ⊆ (F1, E) , (G, E) ⊆ (F2, E) and , then is called a soft normal space.
Definition 3.10. ([18, 27]) Let be a soft topological space over X. Then it is said to be a soft T4-I space if it is soft normal space and soft T1-I space. If a soft normal space is a soft T1-II space, then it is called a soft T4-II space.
Example 3.9. As described in Example 3.8, the soft topological space has six soft closed sets (Fi, E) ′ (i = 1, 2, ⋯ , 6). They form six pairs of disjoint soft closed sets: (F1, E) ′ and (F2, E) ′, (F1, E) ′ and (F3, E) ′, (F1, E) ′ and (F6, E) ′, (F2, E) ′ and (F3, E) ′, (F2, E) ′ and (F5, E) ′, (F3, E) ′ and (F4, E) ′.
For the pair of soft closed set (F1, E) ′ and (F2, E) ′, there exist soft open sets (F5, E) and (F6, E) such that (F1, E) ′ ⊆ (F6, E), (F2, E) ′ ⊆ (F5, E) and . For the pair of soft closed sets (F1, E) ′ and (F6, E) ′, there exist soft open sets (F1, E) and (F6, E) such that (F1, E) ′ ⊆ (F6, E), (F6, E) ′ ⊆ (F1, E) and . Similarly, pairs of other disjoint soft closed sets also can be proved. Thus is a soft normal space.
Since is not a soft T1-II space, it is not a soft T4-II space. But it is a soft T1-I space, and then it is a soft T4-I space.
This example shows that a soft T4-I space may not to be a soft T4-II space.
Remark 3.5. If the initial universe set X and the parameter set E are sets of two elements, a soft T4-II space over X is a soft T4-I space by discussion above. Generally, if X and E are finite sets, it is worth further exploring whether the above conclusions are true, or under what conditions can these conclusions established?
Conclusion
In this paper, we gave some examples to verify differences between two types of separation axioms in soft topological spaces. We have known that a soft T0-I(T1-I, T2-I, T3-I and T4-I respectively) space over X may not to be a soft T0-II(T1-II, T2-II, T3-II and T4-II respectively) space. If the initial universe set X and the parameter set E are sets of two elements, the converse are true except for the property T3. We also showed that a soft type-I regular space is a soft type-II regular space.
Footnotes
Acknowledgments
The author gratefully thanks the reviewers of this document for their insightful comments and suggestions. Thanks are also due to the support of the National Science Foundation of China (Grant No.12071188).
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