Journal of Mathematics
Volume 2016 (2016), Article ID 2480842, 7 pages
http://dx.doi.org/10.1155/2016/2480842

Research Article

Fuzzy Soft Compact Topological Spaces

Seema Mishra and Rekha Srivastava

Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi 221005, India

Received 26 July 2016; Accepted 16 November 2016

Academic Editor: Ali Jaballah

Copyright © 2016 Seema Mishra and Rekha Srivastava. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we have studied compactness in fuzzy soft topological spaces which is a generalization of the corresponding concept by R. Lowen in the case of fuzzy topological spaces. Several basic desirable results have been established. In particular, we have proved the counterparts of Alexander’s subbase lemma and Tychonoff theorem for fuzzy soft topological spaces.

1. Introduction

Molodtsov introduced the concept of a soft set in 1999 (cf. [1]) as a new approach to model uncertainties. He also applied his theory in several directions, for example, stability and regularization, game theory, and soft analysis (cf. [1]).

Maji et al. [2] introduced and studied fuzzy soft sets. Yang et al. [3] introduced interval valued fuzzy soft sets. Majumdar and Samanta [4] defined generalized fuzzy soft set.

Algebraic structures of soft sets and fuzzy soft sets have been studied by many researchers. Aktaş and Çağman [5] introduced soft groups. Feng et al. [6] gave the concept of a soft semiring and many related concepts. Aygünoğlu and Aygün [7] introduced fuzzy soft groups and Varol et al. [8] studied fuzzy soft rings.

Shabir and Naz [9] defined soft topological spaces and many related basic concepts. Aygünoğlu and Aygün [10] also studied soft topological spaces. Fuzzy soft topology was introduced by Tanay and Kandemir [11] and it was further studied by Varol and Aygün [12], Mahanta and Das [13], Mishra and Srivastava [14], and so forth.

In [15], Varol et al. have introduced a soft topology and an -fuzzy soft topology in a different way, using Šostak approach [16], and studied soft compactness and -fuzzy soft compactness.

In fuzzy topological spaces, compactness was first introduced by Chang [17], but it is well known by now that compactness in the sense of Chang does not satisfy the Tychonoff property. Lowen [18] introduced compactness in a fuzzy topological space, in another way which satisfies the Tychonoff property and has many other desirable properties.

Gain et al. [19], Osmanoğlu and Tokat [20], and Sreedevi and Ravi Shankar [21] have studied fuzzy soft compactness in a fuzzy soft topological space introduced by Tanay and Kandemir [11]. These authors have introduced fuzzy soft compactness as a generalization of Chang’s fuzzy compactness.

In this paper, we have introduced and studied fuzzy soft compactness as a generalization of Lowen’s fuzzy compactness, in a fuzzy soft topological space introduced by Varol and Aygün [12].

Several basic desirable results have been established. In particular, we have proved the counterparts of Alexander’s subbase lemma and Tychonoff theorem for fuzzy soft topological spaces.

2. Preliminaries

Throughout this paper, denotes a nonempty set, called the universe, denotes the set of parameters for the universe , and .

Definition 1 (see [22]). A fuzzy set in is a function . Now we define some basic fuzzy set operations as follows.
Let and be fuzzy sets in . Then

(1) if .(2) if .(3).(4).(5) (here denotes the complement of ).

The constant fuzzy set in , taking value , will be denoted by .

Definition 2 (see [23]). Let be an index set and be a family of fuzzy sets in . Then their union and intersection are defined, respectively, as follows:

(1).(2).

Definition 3 (see [24]). A fuzzy point in is a fuzzy set in given by

Here and are, respectively, called the support and the value of .

Definition 4 (see [18]). A fuzzy topological space is a pair , where is a nonempty set and is a family of fuzzy sets in such that the following conditions are satisfied:

(1).(2)If , then .(3)If is an arbitrary family of fuzzy sets in , then .

Then is called a fuzzy topology on and members of are called fuzzy open sets. A fuzzy set in is called fuzzy closed if .

Definition 5 (see [18]). A fuzzy set in is said to be fuzzy compact if for any family such that and for all , there exists a finite subfamily such that .

Definition 6 (see [18]). A fuzzy topological space is said to be fuzzy compact if each constant fuzzy set in is fuzzy compact.

Definition 7 (see [2]). A pair is called a fuzzy soft set over if is a mapping from to ; that is, , where is the collection of all fuzzy sets in .

Definition 8 (see [12]). A fuzzy soft set over is a mapping from to ; that is, such that , if and , otherwise, where denotes the constant fuzzy set in taking value 0.

Definition 9 (see [12]). A constant fuzzy soft set over , where , is the fuzzy soft set over such that .

From here onwards, we will denote by the collection of all fuzzy soft sets over , where is the parameters set for .

Definition 10 (see [12]). Let . Then

(1) is said to be a fuzzy soft subset of (or that is contained in ), denoted by , if (2) and are said to be equal, denoted by , if and (3) The union of and , denoted by , is the fuzzy soft set over defined by

(4) The intersection of and , denoted by , is the fuzzy soft set over defined by

 Two fuzzy soft sets and over are said to be disjoint if (5) Let be an index set and be a family of fuzzy soft sets over . Then their union and intersection are defined, respectively, as follows:

(a) (b)

(6) The complement of , denoted by , is the fuzzy soft set over , defined by

Definition 11 (see [7]). Let and be the collections of all fuzzy soft sets over the universe sets and , respectively, where and are the parameters sets for and , respectively. Let and be two maps. Then the pair is called a fuzzy soft mapping from to and is denoted by

(1)Let . Then the image of under the fuzzy soft mapping is the fuzzy soft set over , denoted by , and is defined as follows:

(2)Let . Then the inverse image of under the fuzzy soft mapping is the fuzzy soft set over , denoted by , and is defined as follows:

Proposition 12 (see [25]). Let be a fuzzy soft mapping and and be fuzzy soft sets over such that . Then .

Proposition 13 (see [25]). Let be a fuzzy soft mapping and be a family of fuzzy soft sets over . Then

(1).(2).

Proposition 14 (see [25]). Let be a fuzzy soft mapping and be a family of fuzzy soft sets over . Then

(1).(2).

Definition 15. Let and be fuzzy soft sets over such that . Then is the fuzzy soft set over given by

Definition 16 (see [12]). Let and . Then the fuzzy soft product of and , denoted by , is the fuzzy soft set over and is defined by

and, for ,

Definition 17 (see [11, 12]). A fuzzy soft topological space relative to the parameters set is a pair consisting of a nonempty set and a family of fuzzy soft sets over satisfying the following conditions:

(1).(2)If , then .(3)If , where is some index set, then .

Then is called a fuzzy soft topology over and members of are called fuzzy soft open sets. A fuzzy soft set over is called fuzzy soft closed if .

We mention here that the fuzzy soft topology defined above has been called “enriched fuzzy soft topology” in [12].

Definition 18 (see [12]). Let be a fuzzy soft topological space. Then a subfamily of is called a base for if every member of can be written as a union of members of .

Definition 19 (see [12]). Let be a fuzzy soft topological space. Then a subfamily of is called a subbase for if the family of finite intersections of its members forms a base for .

Definition 20 (see [12]). A fuzzy soft topology over is said to be generated by a subfamily of fuzzy soft sets over if every member of is a union of finite intersections of members of .

Definition 21 (see [12]). Let be a family of fuzzy soft topological spaces relative to the parameters sets , respectively, and, for each , let be a fuzzy soft mapping. Then the fuzzy soft topology over is said to be initial with respect to the family if has as subbase the set

that is, the fuzzy soft topology over is generated by .

Definition 22 (see [12]). Let be a family of fuzzy soft topological spaces relative to the parameters sets , respectively. Then their product is defined as the fuzzy soft topological space relative to the parameters set , where , , and is the fuzzy soft topology over which is initial with respect to the family , where and are the projection maps; that is, is generated by

Definition 23 (see [14]). A fuzzy soft point over is a fuzzy soft set over defined as follows:

where is the fuzzy point in with support and value ,
A fuzzy soft point is said to belong to a fuzzy soft set , denoted by , if and two fuzzy soft points and are said to be distinct if or .

Proposition 24 (see [14]). Let be a fuzzy soft topological space. Then a fuzzy soft set is fuzzy soft open iff ; there exists a basic fuzzy soft open set such that .

Definition 25 (see [26]). A family of sets is said to be of finite character iff each finite subset of a member of the family is also a member, and each set belongs to this family if each of its finite subsets belong to it.

Lemma 26 (TUKEY, [26]). Each nonempty family of sets of finite character has a maximal element.

3. Fuzzy Soft Compact Topological Spaces

Definition 27. Let be a fuzzy soft topological space relative to the parameters set . Then a fuzzy soft set over is said to be fuzzy soft compact if, for any family such that and such that , there exists a finite subfamily of such that .

Definition 28. A fuzzy soft topological space relative to the parameters set is said to be fuzzy soft compact if each constant fuzzy soft set over is fuzzy soft compact; that is, for , if there exists a family of fuzzy soft open sets over such that , then ; there exists a finite subfamily of such that .

Proposition 29. Let and be fuzzy soft topological spaces relative to parameters sets and , respectively, be a fuzzy soft continuous mapping, and be fuzzy soft compact. Then is fuzzy soft compact.

Proof. Let be such that

Since is a family of fuzzy soft open sets over and is fuzzy soft compact, so, such that , there exists a finite subfamily such that

Then applying on both sides, we get

which implies that is fuzzy soft compact.

From the fact that is surjective if and both are surjective (cf. [12]), each constant fuzzy soft set over is the image of constant fuzzy soft set over . Hence we have the following result.

Corollary 30. Let and be fuzzy soft topological spaces where is fuzzy soft compact and be a surjective fuzzy soft continuous mapping from to . Then is fuzzy soft compact.

As in the case of soft topological spaces [10], here we have the following.

Definition 31. Let be a fuzzy soft topological space relative to the parameters set . Then, for , the -parameter fuzzy topological spaces are given by , where .

The following proposition is a counterpart of Theorem  4.1 in [10].

Proposition 32. Let be a fuzzy soft topological space relative to the parameters set , which is finite. Then is fuzzy soft compact if each -parameter fuzzy topological space is fuzzy compact.

Proof. Suppose that each -parameter fuzzy topological space is fuzzy compact. Then to show that is fuzzy soft compact, consider a family of fuzzy soft open sets over such that

Then, for , by fuzzy compactness of , for , there exists a finite subfamily of such that
Now set . Then is a finite subfamily of such that . Hence , which shows that is fuzzy soft compact.

Now we consider the mappings (cf. [27]) , where is the set of all fuzzy sets in , defined as follows:

and as follows:

In view of the above, we state the following theorem proved in [27].

Theorem 33 (see [27]). is isomorphic to , where denotes the set of all fuzzy sets in .

Then it is easy to verify that if is a fuzzy soft topological space relative to the parameters set , then is a fuzzy topological space and also if is a fuzzy topological space, then is a fuzzy soft topological space relative to the parameters set , where and (cf. [27]).

Proposition 34. A fuzzy soft topological space relative to the parameters set is fuzzy soft compact iff is fuzzy compact.

Proof. First, suppose that is fuzzy soft compact. Then to show that is fuzzy compact, consider a family of fuzzy open sets in such that

Note that if , then . So by fuzzy soft compactness of , for , there exists a finite subfamily of such that

which proves the fuzzy compactness of .
Conversely, assume that is fuzzy compact. To show that is fuzzy soft compact, we have to show that is fuzzy soft compact. Let such that

Then by the fuzzy compactness of , for , there exists a finite subfamily of such that

which proves that the fuzzy soft topological space is fuzzy soft compact.

Now we prove the counterparts of the well known Alexander’s subbase lemma and the Tychonoff theorem for fuzzy soft topological spaces, the proofs of which are based on the proofs of the corresponding results given in [18] and [28], respectively.

Theorem 35. Let be a fuzzy soft topological space relative to the parameters set E. Then is fuzzy soft compact iff, for any subbase for , if there is a family such that , then, for , there exists a finite subfamily of such that .

Proof. First assume that is fuzzy soft compact. Choose such that . Now since and is fuzzy soft compact, there exists a finite subfamily of such that .
Conversely, to show that is fuzzy soft compact, we have to show that if, for , there exist and such that there does not exist any finite subfamily of , such that , then it must follow that does not contain .
Consider the family

Then is of finite character. Now it follows from Tukey’s lemma that, , there exists a maximal element containing .
Next, we show that if and such that , then there exists some , such that . For this we proceed as follows.
If we take such that , then since is maximal, for the family , there exists a finite subfamily , where , such that , which implies that if, for and ,

then

Similarly, if we take another such that , then again since is maximal, for the family , there exists a finite subfamily , where , such that , which implies that if, for and ,

then

Now we show that

as follows.
If for and , which implies that and , then, from (26) and (28), we get and . This implies that and hence . Thus, in general, if do not belong to , then does not belong to implying that there is no fuzzy soft open set containing which belong to . Thus we have shown that if do not belong to , then no fuzzy soft open set such that belongs to . Equivalently, if such that , then there exists some , such that .
Next, consider . Then, from the assumption of the theorem, we have that does not contain . Now we show . Since, and such that and , then and hence, using Proposition 24, there exist such that and . Since and is maximal, it follows that there exists some such that . Thus, , there exists such that and . Now fix and both. Then such that and , where , there exists such that

This implies that does not contain and hence does not contain . Thus, is fuzzy soft compact.

Theorem 36. If is a family of fuzzy soft topological spaces relative to the parameters sets , respectively, then the product of fuzzy soft topological space,

is fuzzy soft compact if and only if each coordinate fuzzy soft topological space is fuzzy soft compact.

Proof. Let us first assume that each coordinate space , is fuzzy soft compact. From the previous theorem, to show that is fuzzy soft compact, it is sufficient to show that for any family , if there exist such that finite subfamily of , does not contain , then it must follow that does not contain .
Let be such a family. Then, , put . Then, finite subfamily of , is a finite subfamily of . Hence, from our assumption there exist some and such that

Since the above inequality holds finite subfamily of , so, from the fuzzy soft compactness of , there exist and such that

The same inequality holds for all . Finally, if we set and , then

The converse part follows using Corollary 30 as well as the fact that are fuzzy soft continuous maps, .

4. Conclusion

Soft sets were introduced by Molodtsov in 1999 [1]. Maji et al. [2] introduced and studied fuzzy soft sets. The theory of fuzzy soft topology was initiated by Tanay and Kandemir [11] and further studied by Varol and Aygün [12], Mahanta and Das [13], and so forth. In this paper we have introduced fuzzy soft compactness in a fuzzy soft topological space, which is an extension of Lowen’s concept of fuzzy compactness in the case of fuzzy topological spaces. Several basic desirable results have been obtained. In particular the counterparts of Alexander’s subbase lemma and the Tychonoff theorem for fuzzy soft topological spaces have been proved.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

Seema Mishra gratefully acknowledges the financial support in the form of scholarship, given by Council of Scientific and Industrial Research, New Delhi (Award no. 09/013(0544)/2014-EMR-I) and Indian Institute of Technology (Banaras Hindu University).

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