Fuzzy Soft Compact Topological Spaces

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 L-fuzzy soft topology in a different way, using Šostak approach [16], and studied soft compactness and L-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 byVarol 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.


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]).
Algebraic structures of soft sets and fuzzy soft sets have been studied by many researchers.Aktas ¸and C ¸agman [5] introduced soft groups.Feng et al. [6] gave the concept of a soft semiring and many related concepts.Aygünoglu 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.Aygünoglu 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], Osmanoglu 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.

Preliminaries
Throughout this paper,  denotes a nonempty set, called the universe,  denotes the set of parameters for the universe , and  ⊆ .
Definition 4 (see [18]).A fuzzy topological space is a pair (, T), where  is a nonempty set and T is a family of fuzzy sets in  such that the following conditions are satisfied: (1)   ∈ T, ∀ ∈ [0, 1].
Then T is called a fuzzy topology on  and members of T are called fuzzy open sets.A fuzzy set in  is called fuzzy closed if   ∈ T.
Definition 5 (see [18]).A fuzzy set  in  is said to be fuzzy compact if for any family  ⊆ T such that ⋃ ∈  ⊇  and for all  > 0, there exists a finite subfamily   ⊆  such that ⋃ ∈   ⊇  −   .
Definition 6 (see [18]).A fuzzy topological space (, T) 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 .
From here onwards, we will denote by F(, ) the collection of all fuzzy soft sets over , where  is the parameters set for .
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)   ∈ , ∀ ∈ [0, 1].
( 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 B of  is called a base for  if every member of  can be written as a union of members of B.
Definition 19 (see [12]).Let (, ) be a fuzzy soft topological space.Then a subfamily S 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 S of fuzzy soft sets over  if every member of  is a union of finite intersections of members of S.
Definition 21 (see [12]).Let {(  ,   )} ∈Ω be a family of fuzzy soft topological spaces relative to the parameters sets   , respectively, and, for each  ∈ Ω, let (, )  :  → (  ,   ) be a fuzzy soft mapping.Then the fuzzy soft topology  over  is said to be initial with respect to the family {(, )  } ∈Ω if  has as subbase the set that is, the fuzzy soft topology  over  is generated by S.
Definition 22 (see [12]).Let {(  ,   )} ∈Ω 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 {(   ,    )} ∈Ω , 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 ,  ∈ (0, 1 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.

Fuzzy Soft Compact Topological Spaces
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.
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   ,  ∈ [0, 1] is fuzzy soft compact, consider a family  of fuzzy soft open sets over  such that Then, for  ∈ , by fuzzy compactness of (,   ), for  ∈ (0, ), there exists a finite subfamily    of  such that Now we consider the mappings (cf.[27]) ℎ : F(, ) →  × , where  × is the set of all fuzzy sets in  × , defined as follows: and  :  × → F(, ) as follows: In view of the above, we state the following theorem proved in [27].
Similarly, if we take another   2 ∈  such that   2 ∉   , then again since   is maximal, for the family {  2 } ∪   , there exists a finite subfamily {  2 , then Now we show that The converse part follows using Corollary 30 as well as the fact that (   ,    ) are fuzzy soft continuous maps, ∀ ∈ Ω.

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.
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 The intersection of   and   , denoted by   ⊓   , is the fuzzy soft set over  defined by (  ⊓   ) () =   () ∩   () , ∀ ∈  (3) Two fuzzy soft sets   and   over  are said to be disjoint if   ⊓   = 0  (5) Let Ω be an index set and { :  ∈ Ω} be a family of fuzzy soft sets over .Then their union ⨆ ∈Ω    and intersection ⨅ ∈Ω    are defined, respectively, as follows: ) If    ∈ , ∀ ∈ Ω, where Ω is some index set, then ⨆ ∈Ω    ∈ .
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 (  )  ∈ .
).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  ̸ =   .Then a fuzzy soft set   is fuzzy soft open iff ∀   ∈   ; there exists a basic fuzzy soft open set   such that    ∈   ⊑   .
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  ∈ [0, 1], if there exists a family  of fuzzy soft open sets over  such that ⨆   ∈   ⊒   , then ∀ ∈ (0, ); there exists a finite subfamily   of  such that ⨆   ∈    ⊒ ( − )  .