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Fuzzy Soft Compact 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. Aktas and Cagman [5] introduced soft groups. Feng et al. [6] gave the concept of a soft semiring and many related concepts. Aygiinoglu and Aygiin [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. Aygunoglu and Aygun [10] also studied soft topological spaces. Fuzzy soft topology was introduced by Tanay and Kandemir [11] and it was further studied by Varol and Aygun [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 Sostak 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], 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 Aygun [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, X denotes a nonempty set, called the universe, E denotes the set of parameters for the universe X, and A [subset or equal to] E.

Definition 1 (see [22]). A fuzzy set in X is a function f : X [right arrow] [0,1]. Now we define some basic fuzzy set operations as follows.

Let f and g be fuzzy sets in X. Then

(1) f = g if f(x) = g(x), [for all] x [member of] X.

(2) f [subset or equal to] if f(x) [less than or equal to] g(x), [for all] x [member of] X.

(3) (f [union] g)(x) = max {f(x), g(x)}, [for all]x [member of] X.

(4) (f [intersection] g)(x) = min{f(x), g(x)}, [for all]x [member of] X.

(5) [f.sup.c](x) = 1 - f(x), [for all]x [member of] X (here [f.sup.c] denotes the complement of f).

The constant fuzzy set in X, taking value [alpha] [member of] [0,1], will be denoted by [[alpha].sub.X].

Definition 2 (see [23]). Let [OMEGA] be an index set and {[f.sub.i]: i [member of] [OMEGA]} be a family of fuzzy sets in X. Then their union [Union.sub.i [member of] [OMEGA]] [f.sub.i] and intersection [[intersection].sub.i[member of][OMEGA]] [f.sub.i] are defined, respectively, as follows:

(1) ([Union.sub.i [member of] [OMEGA]] [f.sub.i])(x) = sup{[f.sub.i](x): i [member of] [OMEGA]}, [for all] x [member of] X.

(2) ([[intersection].sub.i[member of][OMEGA]] [f.sub.i])(x) = inf {[f.sub.i](x): i [member of] Q}, [for all]x [member of] X.

Definition 3 (see [24]). A fuzzy point [x.sub.[lambda]] (0 < [lambda] < 1) in X is a fuzzy set in X given by

[mathematical expression not reproducible] (1)

Here x and [lambda] are, respectively, called the support and the value of [x.sub.[lambda]].

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

(1) [a.sub.X] [member of] T, [for all][alpha] [member of] [0,1].

(2) If f, g [member of] T, then f [intersection] g [member of] T.

(3) If {[f.sub.i]: i [member of] [OMEGA]} is an arbitrary family of fuzzy sets in T, then [U.sub.i [member of] [OMEGA]] [f.sub.i] [member of] T.

Then T is called a fuzzy topology on X and members of T are called fuzzy open sets. A fuzzy set in X is called fuzzy closed if [f.sup.c] [member of] T.

Definition 5 (see [18]). A fuzzy set f in X is said to be fuzzy compact if for any family [beta] [subset or equal to] T such that [U.sub.[mu] [member of] [beta]] [mu] [contains or equal to] f and for all [member of] > 0, there exists a finite subfamily [[beta].sub.0] [subset or equal to] [beta] such that [mathematical expression not reproducible].

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

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

Definition 8 (see [12]). A fuzzy soft set [f.sub.A] over X is a mapping from E to [I.sup.X]; that is, [f.sub.A] : E [right arrow] [I.sup.X] such that [f.sub.A](e) [not equal to] [0.sub.X], if e [member of] A [subset or equal to] E and [f.sub.A](e) = [0.sub.X], otherwise, where 0X denotes the constant fuzzy set in X taking value 0.

Definition 9 (see [12]). A constant fuzzy soft set [[alpha].sub.E] over X, where [alpha] [member of] [0,1], is the fuzzysoft set over X such that [[alpha].sub.E](e) = [a.sub.X], [for all]e [member of] E.

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

Definition 10 (see [12]). Let [f.sub.A], [g.sub.B] [member of] F(X, E). Then

(1) [f.sub.A] is said to be a fuzzy soft subset of [g.sub.B] (or that [f.sub.A] is contained in [g.sub.B]), denoted by [f.sub.A] [??] [g.sub.B], if [f.sub.A](e) [subset or equal to] [g.sub.B](e), [for all]e [member of] E

(2) [f.sub.A] and [g.sub.B] are said to be equal, denoted by [f.sub.A] = [g.sub.B], if [f.sub.A] [??] [g.sub.B] and [g.sub.B] [??] [f.sub.A]

(3) The union of [f.sub.A] and [g.sub.B], denoted by [f.sub.A] [union] [g.sub.B], is the fuzzy soft set over X defined by

([f.sub.A] [intersection] [g.sub.B]) (e) = [f.sub.A] (e) [union] [g.sub.B](e), [for all]e [member of] E (2)

(4) The intersection of [f.sub.A] and [g.sub.B], denoted by [f.sub.A] [intersection] [g.sub.B], is the fuzzy soft set over X defined by

([f.sub.A] [intersection] [g.sub.B]) (e) = [f.sub.A] (e) [intersection] [g.sub.B] (e), [for all] e [member of] E (3)

Two fuzzy soft sets [f.sub.A] and [g.sub.B] over X are said to be disjoint if [f.sub.A] [intersection] [g.sub.B] = [0.sub.E]

(5) Let [OMEGA] be an index set and {[mathematical expression not reproducible]} be a family of fuzzy soft sets over X. Then their union [mathematical expression not reproducible]. and intersection [mathematical expression not reproducible] are defined, respectively, as follows:

(a) [mathematical expression not reproducible]

(b) [mathematical expression not reproducible]

(6) The complement of [f.sub.A], denoted by [f.sup.c.sub.A], is the fuzzy soft set over X, defined by

[f.sup.c.sub.A] (e) = [1.sub.X] - [f.sub.A] (e), [for all]e [member of] E (4)

Definition 11 (see [7]). Let F(X,E) and F(Y,K) be the collections of all fuzzy soft sets over the universe sets X and Y, respectively, where E and K are the parameters sets for X and Y, respectively. Let [phi] : X [right arrow] Y and [phi] : E [right arrow] K be two maps. Then the pair ([phi], [psi]) is called a fuzzy soft mapping from X to Y and is denoted by

([phi], y): F (X, E) [right arrow] F (Y, K). (5)

(1) Let [f.sub.A] [member of] F(X, E). Then the image of [f.sub.A] under the fuzzy soft mapping ([phi], [psi]) is the fuzzy soft set over Y, denoted by ([phi], [psi])[f.sub.A], and is defined as follows:

[mathematical expression not reproducible] (6)

(2) Let [g.sub.B] [member of] F(Y, K). Then the inverse image of [g.sub.B] under the fuzzy soft mapping ([phi], [psi]) is the fuzzy soft set over X, denoted by [([phi], [psi]).sup.-1] [g.sub.B], and is defined as follows:

[mathematical expression not reproducible]. (7)

Proposition 12 (see [25]). Let ([phi], [psi]) : (X, [tau]) [right arrow] (Y, [delta]) be a fuzzy soft mapping and [mathematical expression not reproducible] be fuzzy soft sets over X such that [mathematical expression not reproducible]. Then [mathematical expression not reproducible].

Proposition 13 (see [25]). Let ([phi], [psi]) : (X, [tau]) [right arrow] (Y, [delta]) be a fuzzy soft mapping and [mathematical expression not reproducible] be a family of fuzzy soft sets over X. Then

(1) [mathematical expression not reproducible].

(2) [mathematical expression not reproducible].

Proposition 14 (see [25]). Let ([phi], [psi]) : (X, [tau]) [right arrow] (Y, [delta]) be a fuzzy soft mapping and {[mathematical expression not reproducible]} be a family of fuzzy soft sets over Y. Then

(1) [mathematical expression not reproducible].

(2) [mathematical expression not reproducible].

Definition 15. Let [f.sub.A] and [g.sub.B] be fuzzy soft sets over X such that [f.sub.A] [??] [g.sub.B]. Then [f.sub.A] - [g.sub.B] is the fuzzy soft set over X given by

([f.sub.A] - [g.sub.B]) (e) = [f.sub.A] (e) - [g.sub.B] (e), [for all]e [member of] E. (8)

Definition 16 (see [12]). Let [f.sub.A] [member of] F(X, E) and [g.sub.B] [member of] F(Y, K). Then the fuzzy soft product of [f.sub.A] and [g.sub.B], denoted by [f.sub.A] x [g.sub.B], is the fuzzy soft set over X x Y and is defined by

([f.sub.A] x [g.sub.B]) (e, k) = [f.sub.A] (e) x [g.sub.B](k), [for all](e, k) [member of] E x K, (9)

and, for (x, y) [member of] X x Y,

([f.sub.A] (e) x [g.sub.B] (k)) (x, y) = min {[f.sub.A] (e) (x), [g.sub.B] (k) (y)). (10)

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

(1) [[alpha].sub.E] [member of] [tau], [for all][alpha] [member of] [0,1].

(2) If [f.sub.A], [g.sub.B] [member of] [tau], then [f.sub.A] [intersection] [g.sub.B] [member of] r.

(3) If [f.sub.A] [member of] [tau], [for all]j [member of] [OMEGA], where [OMEGA] is some index set, then [mathematical expression not reproducible].

Then [tau] is called a fuzzy soft topology over X and members of [tau] are called fuzzy soft open sets. A fuzzy soft set [g.sub.B] over X is called fuzzy soft closed if [([g.sub.B]).sup.c] [member of] [tau].

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

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

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

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

Definition 21 (see [12]). Let [{([X.sub.i], [[tau].sub.i])}.sub.i [member of] [OMEGA]] be a family of fuzzy soft topological spaces relative to the parameters sets [E.sub.i], respectively, and, for each i [member of] [OMEGA], let [([phi],'[psi]).sub.i] : X [right arrow] ([X.sub.i], [[tau].sub.i]) be a fuzzy soft mapping. Then the fuzzy soft topology [tau] over X is said to be initial with respect to the family [{[([phi], [psi]).sub.i]}.sub.i}i [member of] [OMEGA]] if [tau] has as subbase the set

[mathematical expression not reproducible]; (11)

that is, the fuzzy soft topology [tau] over X is generated by S.

Definition 22 (see [12]). Let [{([X.sub.i], [[tau].sub.i])}.sub.i [member of] [OMEGA]] be a family of fuzzy soft topological spaces relative to the parameters sets [E.sub.i], respectively. Then their product is defined as the fuzzy soft topological space (X, [tau]) relative to the parameters set E, where X = [[??].sub.i][X.sub.i], E = [[??].sub.i] [E.sub.i], and [tau] is the fuzzy soft topology over X which is initial with respect to the family [mathematical expression not reproducible] are the projection maps; that is, r is generated by

[mathematical expression not reproducible]. (12)

Definition 23 (see [14]). A fuzzy soft point [mathematical expression not reproducible] over X is a fuzzy soft set over X defined as follows:

[mathematical expression not reproducible] (13)

where [x.sub.[lambda]] is the fuzzy point in X with support x and value [lambda], [lambda] [member of] (0,1).

A fuzzy soft point [mathematical expression not reproducible] is said to belong to a fuzzy soft set [f.sub.A], denoted by [mathematical expression not reproducible], if [lambda] < [f.sub.A](e)(x) and two fuzzy soft points [mathematical expression not reproducible] are said to be distinct if x [not equal to] y or e [not equal to] e'.

Proposition 24 (see [14]). Let (X, [tau]) be a fuzzy soft topological space. Then a fuzzy soft set [f.sub.A] is fuzzy soft open [mathematical expression not reproducible]; there exists a basic fuzzy soft open set [g.sub.B] such that [mathematical expression not reproducible].

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 (X, [tau]) be a fuzzy soft topological space relative to the parameters set E. Then a fuzzy soft set [f.sub.A] over X is said to be fuzzy soft compact if, for any family [beta] [subset or equal to] [tau] such that [mathematical expression not reproducible], there exists a finite subfamily [mathematical expression not reproducible].

Definition 28. A fuzzy soft topological space (X, [tau]) relative to the parameters set E is said to be fuzzy soft compact if each constant fuzzy soft set over X is fuzzy soft compact; that is, for [alpha] [member of] [0,1], if there exists a family [beta] of fuzzy soft open sets over X such that [mathematical expression not reproducible], then [for all][epsilon] [member of] (0, [alpha]); there exists a finite subfamily [[beta].sub.0] of [beta] such that [mathematical expression not reproducible].

Proposition 29. Let (X, r) and (Y, S) be fuzzy soft topological spaces relative to parameters sets E and K, respectively, ([phi], [psi]): (X, [tau]) [right arrow] (Y, [delta]) be a fuzzy soft continuous mapping, and [f.sub.A] [member of] F(X, [delta]) be fuzzy soft compact. Then (f,y)fA is fuzzy soft compact.

Proof. Let [beta] [subset or equal to] [delta] be such that

[mathematical expression not reproducible]. (14)

Since [mathematical expression not reproducible] is a family of fuzzy soft open sets over X and [f.sub.A] is fuzzy soft compact, so, [for all][epsilon] such that [f.sub.A] [??] [[epsilon].sub.E], there exists a finite subfamily [[beta].sub.0] [subset or equal to] [beta] such that

[mathematical expression not reproducible]. (15)

Then applying ([phi], [psi]) on both sides, we get

[mathematical expression not reproducible], (16)

which implies that ([phi], [psi])[f.sub.A] is fuzzy soft compact.

From the fact that ([phi], [psi]) is surjective if [phi] and [psi] both are surjective (cf. [12]), each constant fuzzy soft set [[alpha].sub.K] over Y is the image of constant fuzzy soft set [[alpha].sub.E] over X. Hence we have the following result.

Corollary 30. Let (X, [tau]) and (Y, [delta]) be fuzzy soft topological spaces where (X, [tau]) is fuzzy soft compact and ([phi], [psi]) be a surjective fuzzy soft continuous mapping from (X, [tau]) to (Y, [delta]). Then (Y, [delta]) is fuzzy soft compact.

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

Definition 31. Let (X, [tau]) be a fuzzy soft topological space relative to the parameters set E. Then, for e [member of] E, the e-parameter fuzzy topological spaces are given by (X, [[tau].sub.e]), where [[tau].sub.e] = {[f.sub.A](e) : [f.sub.A] [member of] [tau]}.

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

Proposition 32. Let (X, [tau]) be a fuzzy soft topological space relative to the parameters set E, which is finite. Then (X, [tau]) is fuzzy soft compact if each e-parameter fuzzy topological space is fuzzy compact.

Proof. Suppose that each e-parameter fuzzy topological space is fuzzy compact. Then to show that [[alpha].sub.E], [alpha] [member of] [0,1] is fuzzy soft compact, consider a family [beta] of fuzzy soft open sets over X such that

[mathematical expression not reproducible]. (17)

Then, for e [member of] E, by fuzzy compactness of (X, [[tau].sub.e]), for [epsilon] [member of] (0, [alpha]), there exists a finite subfamily [[beta].sup.e.sub.o] of ft such that [([alpha] - [epsilon]).sub.X] [subset or equal to] [mathematical expression not reproducible].

Now set [[beta].sub.0] = [[??].sub.e [member of] E] [[beta].sup.e.sub.o]. Then [[beta].sub.o] is a finite subfamily of [beta] such that [mathematical expression not reproducible]. Hence [mathematical expression not reproducible], which shows that (X, [tau]) is fuzzy soft compact.

Now we consider the mappings (cf. [27]) h : F(X,E) [right arrow] [I.sup.XxE], where [I.sup.XxE] is the set of all fuzzy sets in X x E, defined as follows:

h([f.sub.A])(x, e) = [f.sub.A] (e)(x), [for all] [f.sub.A] [member of] F(X,E), (18)

and g : [I.sup.XxE] [right arrow] F(X, E) as follows:

[mathematical expression not reproducible]. (19)

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

Theorem 33 (see [27]). (F(X,E), [union], [intersection], [sup.c]) is isomorphic to ([I.sup.XxE] [union], [intersection], [sup.c]), where [I.sup.XxE] denotes the set of all fuzzy sets in X x E.

Then it is easy to verify that if (X, [tau]) is a fuzzy soft topological space relative to the parameters set E, then (X x E,h([tau])) is a fuzzy topological space and also if (X x E, T) is a fuzzy topological space, then (X,g(T)) is a fuzzy soft topological space relative to the parameters set E, where h([tau]) = {h([f.sub.A]): [f.sub.A] [member of] [tau]} and g(T) = {g(U): U [member of] T} (cf. [27]).

Proposition 34. A fuzzy soft topological space (X, [tau]) relative to the parameters set E is fuzzy soft compact iff (X x E, h([tau])) is fuzzy compact.

Proof. First, suppose that (X, [tau]) is fuzzy soft compact. Then to show that (X x E, h([tau])) is fuzzy compact, consider a family [beta] [subset or equal to] h([tau]) of fuzzy open sets in X x E such that

[mathematical expression not reproducible]. (20)

Note that if v [member of] [beta] [subset or equal to] h([tau]), then g(v) [member of] [tau]. So by fuzzy soft compactness of (X, [tau]), for [epsilon] [member of] (0, [alpha]), there exists a finite subfamily [[beta].sub.0] of [beta] such that

[mathematical expression not reproducible] (21)

which proves the fuzzy compactness of (X x E, h([tau])).

Conversely, assume that (X x E, h([tau])) is fuzzy compact. To show that (X, [tau]) is fuzzy soft compact, we have to show that [[alpha].sub.E], [alpha] [member of] [0,1] is fuzzy soft compact. Let [beta] [subset or equal to] [tau] such that

[mathematical expression not reproducible]. (22)

Then by the fuzzy compactness of (X x E, h([tau])), for [epsilon] [member of] (0, [alpha]), there exists a finite subfamily [[beta].sub.0] of [beta] such that

[mathematical expression not reproducible], (23)

which proves that the fuzzy soft topological space (X, [tau]) 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 (X, [tau]) be a fuzzy soft topological space relative to the parameters set E. Then (X, [tau]) is fuzzy soft compact iff, for any subbase [sigma] for [tau], if there is a family [beta] [subset] [alpha] such that [mathematical expression not reproducible], there exists a finite subfamily [mathematical expression not reproducible].

Proof. First assume that (X, [tau]) is fuzzy soft compact. Choose [mathematical expression not reproducible]. Now since [sigma] [subset or equal to] [tau] and (X, [tau]) is fuzzy soft compact, there exists a finite subfamily [mathematical expression not reproducible].

Conversely, to show that (X, [tau]) is fuzzy soft compact, we have to show that if, for [beta] [subset or equal to] [tau], there exist [alpha] and [epsilon] [member of] (0, [alpha]) such that there does not exist any finite subfamily [[beta].sub.0] of [beta], such that [mathematical expression not reproducible], then it must follow that [mathematical expression not reproducible] does not contain [[alpha].sub.E].

Consider the family

[mathematical expression not reproducible]. (24)

Then C is of finite character. Now it follows from Tukey's lemma that, [for all][beta] [member of] C, there exists a maximal element [beta]' [member of] C containing [beta].

Next, we show that if [mathematical expression not reproducible], then there exists some k, k = 1, 2, ..., n such that [mathematical expression not reproducible]. For this we proceed as follows.

If we take [mathematical expression not reproducible], then since [beta]' is maximal, for the family [mathematical expression not reproducible], there exists a finite

subfamily [mathematical expression not reproducible],

[mathematical expression not reproducible], (25)

then

[mathematical expression not reproducible]. (26)

Similarly, if we take another [mathematical expression not reproducible], then again since [beta]' is maximal, for the family [mathematical expression not reproducible], there exists a finite subfamily [mathematical expression not reproducible],

[mathematical expression not reproducible] (27)

then

[mathematical expression not reproducible]. (28)

Now we show that

[mathematical expression not reproducible], (29)

as follows.

If for [mathematical expression not reproducible], then, from (26) and (28), we get [mathematical expression not reproducible]. This implies that [mathematical expression not reproducible]. Thus, in general, if [mathematical expression not reproducible] do not belong to [beta]', then [mathematical expression not reproducible] does not belong to [beta]' implying that there is no fuzzy soft open set containing [mathematical expression not reproducible] which belong to [beta]'. Thus we have shown that if [mathematical expression not reproducible] do not belong to [beta]', then no fuzzy soft open set [f.sub.A] such that [mathematical expression not reproducible] belongs to [beta]'. Equivalently, if [f.sub.A] [member of] [beta]' such that [mathematical expression not reproducible], then there exists some k, k = 1, 2, ..., n such that [mathematical expression not reproducible].

Next, consider [beta]' [intersection] [sigma]. Then, from the assumption of the theorem, we have that [mathematical expression not reproducible] does not contain [[alpha].sub.E]. Now we show [mathematical expression not reproducible]. Since, [for all][f.sub.A] [member of] [beta]e [member of] E and [for all]x [member of] X such that [f.sub.A](e)(x) > 0 and [for all]a < [f.sub.A](e)(x), then [mathematical expression not reproducible] and hence, using Proposition 24, there exist [mathematical expression not reproducible]. Since [f.sub.A] [member of] [beta]' and [beta]' is maximal, it follows that there exists some k, k = 1, 2, ..., n such that [mathematical expression not reproducible]. Thus, [mathematical expression not reproducible]. Now fix e and x both. Then [for all][f.sub.A] [less than or equal to] [beta]' such that [f.sub.A](e)(x) > 0 and a < [f.sub.A](e)(x), where a >0, there exists [f.sup.a.sub.A] [member of] [beta]' [intersection] [sigma] such that

[mathematical expression not reproducible]. (30)

This implies that [mathematical expression not reproducible]. Thus, (X, [tau]) is fuzzy soft compact.

Theorem 36. If {([X.sub.i] [[tau].sub.i]): i [member of] [OMEGA]} is a family of fuzzy soft topological spaces relative to the parameters sets [E.sub.i], respectively, then the product of fuzzy soft topological space,

[mathematical expression not reproducible], (31)

is fuzzy soft compact if and only if each coordinate fuzzy soft topological space ([X.sub.i], [[tau].sub.i]) is fuzzy soft compact.

Proof. Let us first assume that each coordinate space ([X.sub.i], [[tau].sub.i]), i [member of] [OMEGA], is fuzzy soft compact. From the previous theorem, to show that (X, [tau]) is fuzzy soft compact, it is sufficient to show that for any family [mathematical expression not reproducible].

Let [beta] be such a family. Then, [mathematical expression not reproducible]. Then, V finite subfamily [mathematical expression not reproducible] is a finite subfamily of [beta]. Hence, from our assumption there exist some [epsilon] [member of] E and x [member of] X such that

[mathematical expression not reproducible] (32)

Since the above inequality holds [for all] finite subfamily [([[beta].sub.j]).sub.o] of [[beta].sub.j], so, from the fuzzy soft compactness of ([X.sub.j], [[tau].sub.j]), there exist [e'.sub.j] [member of] [E.sub.j] and [x'.sub.j] [member of] [X.sub.j] such that

[mathematical expression not reproducible]. (33)

The same inequality holds for all j [member of] [OMEGA]. Finally, if we set e = [([e'.sub.j]).sub.j[member of][OMEGA]] and x = [([x'.sub.j]).sub.j [member of] [OMEGA]], then

[mathematical expression not reproducible] (34)

The converse part follows using Corollary 30 as well as the fact that ([mathematical expression not reproducible]) are fuzzy soft continuous maps, [for all]j [member of] [OMEGA].

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 Aygun [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.

http://dx.doi.org/ 10.1155/2016/2480842

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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Seema Mishra and Rekha Srivastava

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

Correspondence should be addressed to Seema Mishra; seemamishra.rs.apm12@itbhu.ac.in

Received 26 July 2016; Accepted 16 November 2016

Academic Editor: Ali Jaballah
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Title Annotation:Research Article
Author:Mishra, Seema; Srivastava, Rekha
Publication:Journal of Mathematics
Article Type:Report
Date:Jan 1, 2017
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