Abstract
We introduce the concepts soft -interior and soft -closure of a soft set in soft topological spaces. We also study soft -continuous functions and discuss their relations with soft continuous and other weaker forms of soft continuous functions.
1. Introduction and Preliminary
The concept of soft sets was first introduced by Molodtsov [1] in 1999 who began to develop the basics of corresponding theory as a new approach to modeling uncertainties. In [1, 2], Molodtsov successfully applied the soft theory in several directions such as smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability, and theory of measurement.
In recent years, an increasing number of papers have been written about soft sets theory and its applications in various fields [3, 4]. Shabir and Naz [5] introduced the notion of soft topological spaces which are defined to be over an initial universe with a fixed set of parameters. In addition, Maji et al. [6] proposed several operations on soft sets, and some basic properties of these operations have been revealed so far.
In general topology, the concept of -open sets was introduced in [7] and -open sets have been referred to as semipreopen by Andrijevic [8]. In [9] this concept has been generalized to soft setting. Our motivation in this paper is to define soft -interiors and soft -closures and investigate their properties which are important for further research on soft topology. These researches not only can form the theoretical basis for further applications of topology on soft sets but also lead to the development of information system and various fields in engineering. Furthermore, we will study soft -continuous functions and obtain some characterizations of such functions.
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 [6]). A soft set over is called a null soft set, denoted by , if , .
Definition 3 (see [6]). A soft set over is called an absolute soft set, denoted by , if , .
If , then the -universal soft set is called a universal soft set, denoted by .
Definition 4 (see [5]). Let be a nonempty subset of ; then denotes the soft set over for which , for all .
Definition 5 (see [6]). The union of two soft sets of and over the common universe is the soft set , where and for all ,
We write .
Definition 6 (see [6]). The intersection of two soft sets and over a common universe , denoted by , is defined as , and for all .
Definition 7 (see [6]). Let and be two soft sets over a common universe . , if , and for all .
Definition 8 (see [5]). Let be the collection of soft sets over ; then is said to be a soft topology on if it satisfies the following axioms:(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 . 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 soft topological space with being 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 9. Let be a soft topological space over and let be a soft set over .(1)Reference [4]: the soft interior of is the soft set .(2)Reference [5]: the soft closure of is the soft set .
Clearly is the smallest soft closed set over which contains and is the largest soft open set over which is contained in .
Definition 10. A soft set of a soft topological space is said to be(a)soft open [5] if its complement is soft closed,(b)soft -open [10] if ,(c)soft preopen [9] if ,(d)soft semiopen [11] if ,(e)soft -open [9] if .
Proposition 11. (a) Every soft open set is soft -closed. (b) Every soft -open set is soft preopen. (c) Every soft -open set is soft semiopen. (d) Every soft semiopen set is soft -open. (e) Every soft preclosed set is soft -open.
Proof. The proof is obvious from Definition 10.
Remark 12. We have following implications; however, the converses of these implications are not true, in general, as shown in Figure 1.
Example 13. 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 set but not soft open set (since is not soft open set).
Now, let us take ; then , , and so ; hence, is soft semiopen set but not soft -open set.
Now, let us take ; then , , and so ; hence, is soft preopen set but not soft -open set.
Finally, let us consider as a soft set in .
Then , and so ; as a result, is soft -open set, but it is neither soft semiopen set nor soft preopen set.
2. Some Properties of Soft -Open Sets and Soft -Closed Sets
Recall that a soft set of a soft topological space is said to be soft -open [9] if . The complement of a soft -open set is called soft -closed. Soft -closure and soft -interior of a soft set are defined as follows.
Definition 14. Let be a soft topological space and let be a soft set over .(a)Soft -interior of a soft set in is denoted by .(b)Soft -closure 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 .
We will denote the family of all soft -open sets (resp., soft -closed sets) of a soft topological space by (resp., ).
Proposition 15 (see [12]). (1) Arbitrary union of soft -open sets is a soft -open set. (2) Arbitrary intersection of soft -closed sets is a soft -closed set.
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 set.
Conversely, let be soft -closed set.
Since and is a soft -closed set, .
Further, for all such ’s.
.
(2) Similar to (1).
Proposition 17. In a soft space , the following hold for soft -closure:(1).(2) is soft -closed set in for each soft subset of .(3), if .
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).
Proof. Let and be two soft sets over .(1)We have and and and .(2)Similar to (1).(3)Follows from definition.(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 .(7)Similar to (6).(8)Since , so by Proposition 16 (1), .(9)Since , so by Proposition 16 (2), .
Theorem 19. For a soft topological space the following are valid.(a).(b)If is a soft set in and is a soft preopen set in such that , then is a soft -open set.
Proof. (a) The proof is obvious. (b) Since is a soft preopen set we have .
Then , so is a soft -open set.
Definition 20 (see [13]). Let be soft topological space and let be an ordinary subset of . Then is a soft topology on and is called the induced or relative soft topology. The pair is called a soft subspace of : is called a soft open/soft closed/soft -open soft subspace if the characteristic function of , namely, , is soft open/soft closed/soft -open, respectively.
Theorem 21. Let be a soft topological space. Suppose and is a soft -open soft subspace of . Then is soft -open soft subspace in if and only if is soft -open soft subspace in .
Proof. Suppose that is soft -open soft subspace in .
Then .
But implies so that .
This implies that is soft -open in . That is, is soft -open soft subspace in .
3. Soft -Continuity
Definition 22 (see [14]). 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 23 (see [14]). Let be a mapping from a soft class to another soft class , and let be a soft set in soft class , where . Let and be mappings. Then , , is a soft set in the soft classes , defined as follows: for . is called a soft inverse image of . Hereafter we will write as .
Theorem 24 (see [14]). Let , , and be mappings. Then for soft sets , and a family of soft sets in the soft class , we have the following:(1), (2), (3) in general ,(4) in general ,(5)if , then ,(6), (7), (8) in general ,(9) in general ,(10)if , then .
Throughout the paper, the spaces and stand for soft topological spaces with ( and ) assumed unless otherwise stated. Moreover, throughout this paper, a soft mapping stands for a mapping, where , , and are assumed mappings unless otherwise stated.
Definition 25. A soft mapping is called soft -continuous [12] (resp., soft -continuous [10], soft precontinuous [10], and soft semicontinuous [15]) if the inverse image of each soft open set in is soft -open (resp., soft -open, soft preopen, and soft semiopen) set in .
Remark 26. We have the following implications; however, the converses of these implications are not true, in general, as shown in Figure 2.
Example 27. Let , , , and and let and be soft topological spaces.
Define and as , , , , , .
Let us consider the soft topology on given in Example 13; that is, , ; and mapping; is a soft mapping. Then is a soft open set in ; is a soft -open set but not soft open set in .
Therefore, is a soft -continuous function but not soft continuous function.
Example 28. Let , , , and and let and be soft topological spaces.
Let us consider the and as mapping given in Example 27 and the soft topology on given in Example 13; that is, , and mapping; is a soft mapping. Then is a soft open set in ; is a soft preopen set but not soft -open set in . Thus, is a soft precontinuous function but not soft -continuous function.
Example 29. Let , , , and and let and be soft topological spaces.
Let us consider the and as mapping given in Example 27 and the soft topology on given in Example 13; that is, , and mapping; is a soft mapping. Then is a soft open set in ; is a soft semiopen set but not soft -open set in . Hence, is a soft semicontinuous function but not soft -continuous function.
Example 30. Let , , , and and let and be soft topological spaces.
Let us consider the and as mapping given in Example 27 and the soft topology on given in Example 13; that is, , and mapping; is a soft mapping. Then is a soft open set in ; is a soft -open set, but it is neither soft semiopen set nor soft preopen set in .
Therefore, is a soft -continuous function, but it is neither soft semicontinuous function nor soft precontinuous function.
Definition 31 (see [12]). Let be a function. is called soft -irresolute if the inverse image of soft -open set in is soft -open in .
Definition 32. Let be a function. is called soft -open if the image of each soft -open set in is soft -open in .
Theorem 33. Let be a soft continuous and soft open set. Then is soft -open set.
Proof. Let be any soft -open set. Then . Therefore, , is soft -open.
This shows that is soft -open set.
Theorem 34. If is soft closed and is soft -open then is soft -open.
Proof. By hypothesis . Now , .
This shows that is soft -open set.
Theorem 35. Let be soft continuous and soft open. Then is soft -irresolute.
Proof. Let be any soft -open set in . Then . Since is soft continuous and soft open it follows that , , . This shows that is soft -open.
This shows that is soft -irresolute.
Proposition 36. A function is soft -irresolute if and only if for every soft -closed set of , is soft -closed.
Proposition 37. In a soft topological space the following are valid:(a) is soft -open .(b) is soft -closed .
Theorem 38. is soft -irresolute if and only if for every soft set of , .
Proof. Suppose that is soft -irresolute. Now is soft -closed set. By hypothesis is soft -closed set.
And . Hence, by the definition of soft -closure, .
That is .
Conversely, suppose that is soft -closed set in . Now by hypothesis . This implies so that .
That is is soft -closed set and so is soft -irresolute.
Theorem 39. is soft -irresolute if and only if for all soft sets of , .
Proof. Suppose is soft -irresolute. Now is soft -closed set so that is soft -closed set. Since , it follows from the definition of soft -closure that .
Conversely suppose that is soft -closed set in . Then .
Now by hypothesis ( .
Therefore, .
Thus, is soft -closed set and so is soft -irresolute.
The following results are easy to establish.
Proposition 40. Suppose and are both soft -irresolute. Then is soft -irresolute.
Proposition 41. Let be soft continuous and soft open. Then(a) is soft -irresolute;(b), with being a soft set in .
Definition 42. Let and be soft topological spaces. and are said to be M-soft -homeomorphic if and only if there exists such that is 1-1, onto, M soft -continuous and soft -open. Such an is called soft -homeomorphism.
Proposition 43. If is soft -homeomorphism, then , where is a soft set in .
Corollary 44. If is a soft -homeomorphism, then(a),(b),(c).
4. Conclusion
In this paper, we introduce the concept of soft -interior and soft -closure of a soft set in topological spaces and study some of their properties. We also introduce the concept of soft -open sets and soft -continuous functions in topological spaces and some of their properties have been established. We hope that the findings in this paper are just the beginning of a new structure and not only will form the theoretical basis for further applications of topology on soft sets but also will lead to the development of information system and various fields in engineering.
Conflict of Interests
There is no conflict of interests regarding the publication of this paper.