Abstract
We introduce soft -sets on soft topological spaces and study some of their properties. We also investigate the concepts of soft -continuous and soft -open functions and discuss their relationships with soft continuous and other weaker forms of soft continuous functions. Also counterexamples are given to show the noncoincidence of these functions.
1. Introduction
Molodtsov [1] initiated a novel concept of soft set theory, which is a completely new approach for modeling vagueness and uncertainty. He successfully applied the soft set theory to several directions such as smoothness of functions, game theory, Riemann Integration, and theory of measurement. In recent years, development in the fields of soft set theory and its application has been taking place in a rapid pace. This is because of the general nature of parameterization expressed by a soft set. Shabir and Naz [2] introduced the notion of soft topological spaces which are defined over an initial universe with a fixed set of parameters. Later, Zorlutuna et al. [3], Aygunoglu and Aygun [4], and Hussain et al. continued to study the properties of soft topological space. They got many important results in soft topological spaces. Weak forms of soft open sets were first studied by Chen [5]. He investigated soft semiopen sets in soft topological spaces and studied some properties of them. Arockiarani and Arokialancy defined soft -open sets and continued to study weak forms of soft open sets in soft topological space.
In the present paper, we introduce some new concepts in soft topological spaces such as soft -open sets, soft -closed sets, and soft -continuous functions. We also study relationship between soft continuity [6], soft semicontinuity [7], and soft -continuity of functions defined on soft topological spaces. With the help of counterexamples, we show the noncoincidence of these various types of mappings.
2. Preliminaries
Definition 1 (see [1]). Let be an initial universe and let be a set of parameters. Let denote the power set of and let be a nonempty subset of . A pair is called a soft set over , where is a mapping given by . In other words, a soft set over is a parameterized family of subsets of the universe . For may be considered as the set of -approximate elements of the soft set .
Definition 2 (see [8]). A soft set over is called a null soft set, denoted by ; if , .
Definition 3 (see [8]). A soft set over is called an absolute soft set, denoted by ; if , .
Definition 4 (see [8]). The union of two soft sets of and over the common universe is the soft set , where and, for all ,
We write .
Definition 5 (see [8]). The intersection of two soft sets and over a common universe , denoted , is defined as and for all .
Definition 6 (see [8]). Let and be two soft sets over a common universe . , if , and , for all .
Definition 7 (see [2]). Let be the collection of soft sets over ; then is said to be a soft topology on if it satisfies the following axioms:(1) and 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 . Let be a soft topological space over ; then the members of are said to be soft open sets in . The relative complement of a soft set is denoted by and is defined by , where is a mapping given by for all . Let be a soft topological space over . A soft set over is said to be a soft closed set in if its relative complement belongs to . If is a soft topological space with , then is called the soft indiscrete topology on and is said to be a soft indiscrete topological space. If is a a soft topological space with is the collection of all soft sets which can be defined over , then is called the soft discrete topology on and is said to be a soft discrete topological space.
Definition 8. Let be a soft topological space over and let be a soft set over .(1)[3] The soft interior of is the soft set which is soft open and .(2)[2] The soft closure of is the soft set which is soft closed and .
Clearly is the smallest soft closed set over which contains and is the largest soft open set over which is contained in .
Throughout the paper, the spaces and (or and ) stand for soft topological spaces assumed unless otherwise stated.
3. Soft -Open Sets
Definition 9. A soft set of a soft topological space is called soft -open set if . The complement of soft -open set is called soft -closed set.
Definition 10. A soft set is called soft preopen set [9] (resp., soft semiopen [5]) in a soft topological space if (resp., ).
We will denote the family of all soft -open sets (resp., soft -closed sets and soft preopen sets) of a soft topological space by (resp., and ).
Proposition 11. (1) Arbitrary union of soft -open sets is a soft -open sets
(2) Arbitrary intersection of soft -closed sets is a soft -closed set.
Proof. (1) Let be a collection of soft -open sets. Then, for each ,. Now . Hence is a soft -open set.
(2) Follows immediately from (1) by taking complements.
Example 12. Let , , , and and let be a soft topological space. Then is a soft -open set in .
Remark 13. It is obvious that every soft open (resp., soft closed) set is a soft -open set (resp., soft -closed set). Similarly, every soft -open set is soft semiopen and soft preopen. Thus we have implications as shown in Figure 1.
The examples given below show that the converses of these implications are not true.
Example 14. Let , , and
, where
are soft sets over , defined as follows: ,
, , , ,
,
,
,
, , , ,
,
, , , ,
, , , ,
,
.
Then defines a soft topology on and thus is a soft topological space over .
Clearly the soft closed sets are .
Then, let us take , then , , , and so ; hence is soft -open but not soft open since .
Now, let us take then , , and so ; hence is soft semiopen but not soft -open.
Finally, let us take then , , and so , hence is soft preopen but not soft -open.
Definition 15. Let be a soft topological space and let be a soft set over .(1)Soft -closure of a soft set in is denoted by .(2)Soft -interior of a soft set in is denoted by .
Clearly is the smallest soft -closed set over which contains and is the largest soft -open set over which is contained in .
Proposition 16. Let be a soft topological space and let be a soft set over ; then(1);(2).
Proof. (1) Let .
This shows that .
Hence is soft -closed.
Conversely, let be soft -closed set. Since and is a soft -closed, .
Further, for all such 's.
.
(2) Similar to (1).
Proposition 17. In a soft space , the folowing hold for soft -closure.(1).(2)is soft -closed in for each soft subset of .(3), if .(4).
Proof. Easy.
Theorem 18. Let be a soft topological space and let and be two soft sets over ; then(1);(2);(3);(4) and ;(5) and ;(6);(7);(8);(9);(10);(11).
Proof. Let and be two soft sets over .(1) and .(2)Similar to (1).(3)It follows from Definition 15.(4)Since and are soft -closed sets so, and .(5)Since and are soft -open sets so, and .(6)We have and . Then by Proposition 17(3), and . Now, , . Then and imply . That is, is a soft -closed set containing . But is the smallest soft -closed set containing . Hence . So, .(7)Similar to (6)(8)We have and and . .(9)Similar to (8).(10)Since so by Proposition 16(1), .(11)Since so by Proposition 16(2), .
Theorem 19. If is any soft set in a soft topological space , then following are equivalent:(1) is soft -closed set;(2);(3);(4) is soft -open set.
Proof. (1)(2) If is soft -closed set, then
;
(2)(3);
(3)(4) It is obvious from Definition 9;
(4)(1) It is obvious from Definition 9.
4. Soft -Continuity
Definition 20 (see [10]). Let and be soft classes. Let and be mappings. Then a mapping is defined as follows: for a soft set in , , is a soft set in given by for . is called a soft image of a soft set . If , then we will write as .
Definition 21 (see [10]). Let be a mapping from a soft class to another soft class and a soft set in soft class , where . Let and be mappings. Then , , is a soft set in the soft classes , defined as for . is called a soft inverse image of . Hereafter we will write as .
Theorem 22 (see [10]). Let , , and be mappings. Then for soft sets and and a family of soft sets in the soft class , one has:(1), (2),(3) in general ,(4) in general ,(5)if , then ,(6),(7),(8) in general ,(9) in general ,(10)if , then .
Definition 23. A mapping is said to be soft mapping if and are soft topological spaces and and are mappings.
Throughout the paper, the spaces and and stand for soft topological spaces assumed unless otherwise stated.
Definition 24. A soft mapping is said to be soft -continuous if the inverse image of each soft open subset of is a soft -open set in .
Example 25. Let , , , , , , , and and let and be soft topological spaces.
Define and as , , , , , .
Let be a soft mapping. Then is a soft open in and is a soft -open in . Therefore, is a soft -continuous function.
Theorem 26. Let be a mapping from a soft space to soft space . Then the following statements are true:(1) is soft -continuous;(2)for each soft singleton in and each soft open set in and , there exists a soft -open set in such that and ;(3)the inverse image of each soft closed set in is soft -closed in ;(4), for each soft set in ;(5), for each soft set in .
Proof. (1)(2) Since is soft open in and , so and is a soft -open set in . Put . Then and .
(2)(1) Let be a soft open set in such that and thus there exists such that and . Then . Hence and therefore is soft -continuous.
(1)(3) Let be a soft closed set in . Then is soft open in . Thus ; that is, . Hence is a soft -set in .
(3)(4) Let be a soft set in . Then is a soft closed set in , so that is soft -closed in .
Therefore, we have .
(4)(5) Since be a soft set in , then is a soft set in ; thus by hypothesis we have or ; that is, .
(5)(1) Let be soft open in . Let , and . By (5) we have ; that is, . Therefore is a soft -open set in ; hence is a soft -continuous function.
Corollary 27. Let be a soft -continuous mapping. Then(1), for each ;(2), for each .
Proof. Since for each , therefore the proof follows directly from statements (4) and (5) of Theorem 26.
Definition 28. A soft mapping is called soft precontinuous (resp., soft semicontinuous [7]) if the inverse image of each soft open set in is soft preopen (resp., soft semiopen) in .
Remark 29. It is clear that every soft -continuous map is soft semicontinuous and soft precontinuous. Every soft continuous map is soft -continuous. Thus we have implications as shown in Figure 2.
The converses of these implications are not true, which is clear from the following examples.
Example 30. Let , , , and and and let be soft topological spaces.
Define and as , , , , , , .
Let us consider the soft topology given in Example 14; that is,
, , and and let mapping be a soft mapping. Then is a soft open in and is a soft -open but not soft open in . Therefore, is a soft -continuous function but not soft continuous function.
Example 31. Let , , , and and let and be soft topological spaces.
Define and as , , , , , , .
Let us consider the soft topology given in Example 14; that is,
, , and and be mapping be a soft mapping. Then is a soft open in and is a soft preopen but not soft -open in . Therefore, is a soft precontinuous function but not soft -continuous function.
Example 32. Let , , , and and let and be soft topological spaces.
Define and as , , , , , , .
Let us consider the soft topology given in Example 14; that is,
and , and be mapping be a soft mapping. Then is a soft open in and is a soft semiopen but not soft -open in .
Therefore, is a soft semicontinuous function but not soft -continuous function.
Theorem 33. Let and . Then .
Proof. Since , it follows that .
Therefore is a soft -open set of .
Theorem 34. If is a soft -continuous mapping and , then is a soft -continuous mapping.
Proof. Let in be a soft open set. Then and since is a soft preopen set in , by Theorem 26, we have . Therefore is a soft -continuous mapping.
Theorem 35. A soft function is soft -continuous if and only if for every soft set of .
Proof. Let be soft -continuous. Now is a soft closed set of ; so by soft -continuity of , is soft -closed and . But is the smallest -closed set containing ; hence . Conversely, let be any soft closed set of and by hypothesis ; hence is soft -closed. Consequently, is soft -continuous.
Theorem 36. A soft function is soft -continuous if and only if for every soft set of .
Proof. Let be soft -continuous. Now for any soft set in , is a soft open set in ; since is soft -continuity, then is soft -open and . As is the largest soft -open set contained in , .
Conversely, take a soft open set in . Then
is soft -open.
5. Soft -Open and Soft -Closed Mappings
Definition 37. A soft mapping is called soft -open (resp., soft -closed) mapping if the image of each soft open (resp., soft closed) set in is a soft -open set (resp., soft -closed set) in .
Definition 38. A soft mapping is called soft preopen (resp., soft semiopen [7]) if the image of each soft open set in is soft preopen (resp., soft semiopen) in .
Clearly a soft open map is soft -open and every soft -open map is soft preopen as well as soft -open. Similar implications hold for soft closed mappings.
Theorem 39. A soft mapping is soft -closed if and only if for each soft set in .
Proof. Let . By the definition of soft -closure, we have and so is a soft -closed set and is a soft -closed mapping.
Conversely, if is soft -closed, then is a soft -closed set containing and therefore .
Theorem 40. A soft function is soft -open if and only if for every soft set in .
Proof. If is soft -open, then .
On the other hand, take a soft open set in . Then by hypothesis, is soft -open in .
Theorem 41. Let be a soft -open (resp., soft -closed) mapping. If is a soft set in and is a soft closed (resp., soft open) set in , containing ; then there exists a soft -closed (resp., soft -open) set in , such that and .
Proof. Let . Since , we have . Since is soft -open (resp., soft -closed), then is a soft -closed set (resp., soft -open set) if and hence and .
Corollary 42. If is a soft -open mapping, then(1), for every soft set in ;(2), .
Proof. (1) is a soft closed set in , containing , for a soft set in .
By Theorem 44, there exists a soft -closed set in , and such that .
Thus
.
(2) follows easily from (1).
Theorem 43. If is a soft precontinuous and soft -open mapping, then for each .
Proof. We have .
Since is a soft -open map, we have, by Corollary 42, , .
Therefore is a soft preopen set in .
Theorem 44. If is a soft precontinuous and soft semicontinuous, then is soft -continuous.
Proof. Let be any soft open set in . Then is a soft preopen set as well as a soft semiopen set in .
We have and .
Hence is a soft -continuous mapping.
Theorem 45. If is a soft preopen mapping, then, for each soft set in , .
Proof. It follows immediately from Corollary 42.
Theorem 46. If is soft -continuous and soft preopen, then the inverse image of each soft -set is a soft -open set.
Proof. Let be any soft -open set in .
Then .
By Theorem 44 we have .
Since is a soft -continuous mapping, by Theorem 22(5), .
Hence is a soft -open set.
Corollary 47. If is soft -continuous and soft preopen mapping, then one has the following:(1)the inverse image of each soft -closed set is soft -closed,(2) for each soft set in .
Proof. It follows immediately from the previous Theorem 45.
Theorem 48. Let and be two soft mappings. If is soft preopen and soft -continuous and is soft -continuous, then is soft -continuous.
Proof. It follows immediately from Theorem 45.
Conflict of Interests
There is no conflict of interests regarding the publication of this paper.