#### Abstract

In the present study, we introduce and characterize the class of -generalized fuzzy -closed sets in a fuzzy ideal topological space in Šostak sense. Also, we show that -generalized fuzzy closed set by Kim and Park (2002) -generalized fuzzy -closed set, but the converse need not be true. Moreover, if we take , the -generalized fuzzy -closed set and -generalized fuzzy closed set are equivalent. After that, we define fuzzy upper (lower) generalized -continuous multifunctions, and some properties of these multifunctions along with their mutual relationships are studied with the help of examples. Finally, some separation axioms of -generalized fuzzy -closed sets are introduced and studied. Also, the notion of -fuzzy -connected sets is defined and studied with help of -generalized fuzzy -closed sets.

#### 1. Introduction

Zadeh [1] introduced the basic idea of a fuzzy set as an extension of classical set theory. The theory of fuzzy sets provides a framework for mathematical modeling of those real-world situations, which involve an element of imprecision, uncertainty, or vagueness in their description. This theory has found wide applications in engineering, information sciences, etc.; for details, the reader is referred to [2, 3].

A fuzzy multifunction is a fuzzy set valued function [47]. The biggest difference between fuzzy functions and fuzzy multifunctions has to do with the definition of an inverse image. For a fuzzy multifunction, there are two types of inverses, upper and lower. These two definitions of the inverse then lead to two definitions of continuity; for more details, the reader is referred to [8, 9].

In this work, we introduce and characterize the class of -generalized fuzzy -closed sets in a fuzzy ideal topological space in Šostak sense. Also, we show that -generalized fuzzy closed set -generalized fuzzy -closed set, but the converse need not be true. Moreover, if we take , the -generalized fuzzy -closed set and -generalized fuzzy closed set are equivalent. After that, we define fuzzy upper (lower) generalized -continuous multifunctions, and some properties of these multifunctions along with their mutual relationships are discussed with the help of examples. In the end, some separation axioms of -generalized fuzzy -closed sets are introduced and studied. Also, the notion of -fuzzy -connected sets is defined and studied with help of -generalized fuzzy -closed sets.

#### 2. Preliminary Assertions

In this section, we present the basic definitions and results which we need in the next sections. Throughout this work, refers to an initial universe, is the set of all fuzzy sets on , and for , for all (where and ). For , for all . The difference between [9] is defined as follows:

A fuzzy ideal on [10] is a map that satisfies the following:(i) and (ii)

The simplest fuzzy ideals on , , and are defined as follows:

In a fuzzy topological space based on the sense of Šostak [11], the closure and the interior of are denoted by and . A fuzzy set is called a -generalized fuzzy closed set [12] if whenever and . Also, is called --regular [12] iff for each -generalized fuzzy closed set implies that there exist with for such that , , and . Moreover, is called --normal [12] iff for each of the -generalized fuzzy closed sets for implies that there exist with such that and .

Definition 1 (see [9]). Let be a fuzzy ideal topological space, , and . Then, the -fuzzy local function of is defined as follows: .

Remark 1 (see [9]). (i)If we take , for each , we have(ii)If we take (resp., ), for each , we have .

Definition 2 (see [9]). Let be a fuzzy ideal topological space. Then, for each and , we define an operator as follows:Now, if , then for each .

Theorem 1 (see [9]). Let be a fuzzy ideal topological space. Then, for any fuzzy sets and , the operator satisfies the following properties:(i)(ii)(iii)If , then (iv)(v)(vi)

Definition 3 (see [13, 14]). Let be a fuzzy ideal topological space, , and . Then, is said to be(i)-fuzzy -open iff (ii)-fuzzy semi--open iff (iii)-fuzzy pre--open iff (iv)-fuzzy --open iff (v)-fuzzy --open iff (vi)-fuzzy --open iff The complement of -fuzzy -open (resp., semi--open, pre--open, --open, --open, and --open) set is a -fuzzy -closed (resp., semi--closed, pre--closed, --closed, --closed, and --closed) set.

Definition 4 (see [15, 16]). Let . Then, is called a fuzzy multifunction iff for each . The degree of membership of in is denoted by for any . The domain of is denoted by and the range of is denoted by , for any and : and .

Definition 5 (see [15]). Let be a fuzzy multifunction. Then, is called(i)Normalized iff for each , there exists such that .(ii)A crisp iff for each and .

Definition 6 (see [15]). Let be a fuzzy multifunction. Then, we have the following:(i)The image of is a fuzzy set and defined by(ii)The lower inverse of is a fuzzy set and defined by(iii)The upper inverse of is a fuzzy set and defined byAll properties of image, lower and upper, are found in [15].

#### 3. On -Generalized Fuzzy -Closed Sets

In this section, we introduce and characterize the class of -generalized fuzzy -closed sets in a fuzzy ideal topological space in Šostak sense. Also, we show that -generalized fuzzy closed set -generalized fuzzy -closed set, but the converse need not be true. Moreover, if we take , the -generalized fuzzy -closed set and -generalized fuzzy closed set are equivalent. Our fundamental definition is the following.

Definition 7. Let and . In , is called a -generalized fuzzy -closed set if whenever and . The complement of -generalized fuzzy -closed set is called -generalized fuzzy -open.

Lemma 1. Every -generalized fuzzy closed set [12] is -generalized fuzzy -closed.

Proof. It follows from Theorem 1 (2).
In general, the converse of Lemma 1 is not true as shown by Example 1.

Example 1. Define as follows:Then, is -generalized fuzzy -closed set but it is not -generalized fuzzy closed.

Lemma 2. If , the -generalized fuzzy -closed set and -generalized fuzzy closed set are equivalent.

Proof. It follows from Definition 2.
The following implications are obtained:

Remark 2. -generalized fuzzy -closed and -fuzzy --closed [13] are independent notions as shown by Examples 2 and 3.

Example 2. Let be a set and be defined as follows: and . Define as follows:Then, is a -fuzzy --closed set, but it is not -generalized fuzzy -closed.

Example 3. Define as follows:Then, is a -generalized fuzzy -closed set, but it is not -fuzzy --closed.

Remark 3. -generalized fuzzy -closed and -fuzzy --closed [14] are independent notions as shown by Examples 4 and 5.

Example 4. Let be a set and be defined as follows: , , , and . Define as follows:Then, is a -fuzzy --closed set, but it is not -generalized fuzzy -closed.

Example 5. Let be a set and be defined as follows: , , and . Define as follows:Then, is a -generalized fuzzy -closed set, but it is not -fuzzy --closed.

Definition 8. In , for any and , we define as follows: .

Theorem 2. In , for any , the operator satisfies the following:(i)(ii)(iii)If , then (iv)(v)(vi)If is -generalized fuzzy -closed, then

Proof. (i), (ii), (iii), and (vi) are easily proved from the definition of .(iv) From (ii) and (iii), . Now, we show that . Suppose that . There exist and such thatSince , by the definition of , there exists with such that . Since , we have . Again, by the definition of , we have . Hence, ; it is a contradiction for . Thus, . Then, .(v) However, and imply and . Thus, . Hence, the proof is complete.In a similar way, one can prove the following theorem.

Theorem 3. In , for any and , we define as follows: . For each , the operator satisfies the following:(i)(ii)(iii)(iv)(v)(vi)If is -generalized fuzzy -open, then

#### 4. Fuzzy Upper and Lower Generalized -Continuous Multifunctions

In this section, we define fuzzy upper (resp., lower) generalized -continuous multifunctions, and some properties of these multifunctions along with their mutual relationships are discussed with the help of examples. Moreover, we show that the notions of fuzzy upper (resp., lower) semi--continuous multifunctions [17] and fuzzy upper (resp., lower) generalized -continuous multifunctions are independent.

Definition 9. A fuzzy multifunction is called(i)Fuzzy upper generalized -continuous at a fuzzy point iff for each with , there exists a -generalized fuzzy -open set and such that (ii)Fuzzy lower generalized -continuous at a fuzzy point iff for each with , there exists a -generalized fuzzy -open set and such that (iii)Fuzzy upper (resp., lower) generalized -continuous iff it is fuzzy upper (resp., lower) generalized -continuous at every

Remark 4. If is normalized, then is fuzzy upper generalized -continuous at a fuzzy point iff for each with , there exists -generalized fuzzy -open set and such that .

Theorem 4. A fuzzy multifunction is fuzzy lower generalized -continuous if (resp., ) is -generalized fuzzy -open (resp., -generalized fuzzy -closed) for each with (resp., with ), .

Proof. Let , with , and . Then, we have as -generalized fuzzy -open. Thus, is fuzzy lower generalized -continuous. The other case follows similar lines and from .
In a similar way, one can prove the following theorem.

Theorem 5. A normalized fuzzy multifunction is fuzzy upper generalized -continuous if (resp., ) is a -generalized fuzzy -open (resp., -generalized fuzzy -closed) set for each with (resp., with ) and .

Lemma 3. Every fuzzy lower (resp., upper) generalized continuous multifunction [18] is a fuzzy lower (resp., upper) generalized -continuous multifunction.

Proof. It follows from Lemma 1.
In general, the converse of Lemma 3 is not true as shown by Example 6.

Example 6. Let , , and be a fuzzy multifunction defined by , and . Define and as follows:Then, is fuzzy lower (resp., upper) generalized -continuous but not fuzzy lower (resp., upper) generalized continuous.

Lemma 4. Let be a fuzzy multifunction (resp., normalized fuzzy multifunction). If , fuzzy lower (resp., upper) generalized -continuous multifunction and fuzzy lower (resp., upper) generalized continuous multifunction are equivalent.

Proof. It follows from Lemma 2.

Remark 5. Fuzzy lower (resp., upper) semi--continuous multifunctions [17] and fuzzy lower (resp., upper) generalized -continuous multifunctions are independent notions as shown by Examples 7 and 8.

Example 7. Let , , and be a fuzzy multifunction defined by , and . Define and as follows:Then, is fuzzy lower (resp., upper) generalized -continuous, but it is not fuzzy lower (resp., upper) semi--continuous.

Example 8. Let , , and be a fuzzy multifunction defined by , . Define and as follows: , , and . Define and as follows:Then, is fuzzy lower (resp., upper) semi--continuous, but it is not fuzzy lower (resp., upper) generalized -continuous.

Theorem 6. Let be a fuzzy ideal topological space that satisfies the condition: for , , and , If , it implies that is -generalized fuzzy -open.

Then, is fuzzy lower generalized -continuous if one of the following statements is hold:(i)(ii)(iii)

Proof. (i) (ii) For each and , let . Then, by (i), . Hence, we have .(ii) (iii) Let . Then, by (ii), . Hence, .Suppose that (iii) holds. Let with . Then, by (iii), , and by , is -generalized fuzzy -open. Thus, is fuzzy lower generalized -continuous. Hence, the proof is complete.

Corollary 1. Let and be two fuzzy multifunctions. Then, we have the following:(i)If is fuzzy lower generalized -continuous and is fuzzy lower semicontinuous [12], then is fuzzy lower generalized -continuous.(ii)If is normalized fuzzy upper generalized -continuous and is normalized fuzzy upper semicontinuous [15], then is fuzzy upper generalized -continuous.

#### 5. Some Applications of -Generalized Fuzzy -Closed Sets

In this section, some separation axioms of -generalized fuzzy -closed sets are introduced and studied. Also, the notion of -fuzzy -connected sets is defined and studied with help of -generalized fuzzy -closed sets.

Definition 10. (i) is called --regular iff for each -generalized fuzzy -closed set, implies that there exist with for such that , , and .(ii) is called --normal iff for each -generalized fuzzy -closed sets, for imply that there exist with such that and .

Lemma 5. Every --regular (resp., --normal) space [12] is --regular (resp., --normal) space.

Proof. It follows from Lemma 1.

Lemma 6. If , --regular (resp., --normal) space and --regular (resp., --normal) space are equivalent.

Proof. It follows from Lemma 2.

Theorem 7. For any , the following are equivalent:(i) is --regular(ii)If for each -generalized fuzzy -open set , there exists with such that (iii)If for each -generalized fuzzy -closed set , there exist with for such that , , and

Proof. (i) () (ii) Let for each -generalized fuzzy -open . Then, for -generalized fuzzy -closed . Since is --regular, there exist with and such that , , and . It implies that . Since , .(ii) () (iii) Let for each -generalized fuzzy -closed . Then, for -generalized fuzzy -open . By (ii), there exists with such that . Since , is -generalized fuzzy -open and . Again, by (ii), there exists with such thatIt implies . Put ; then, . So, , that is, .(iii) () (i) It is trivial.In a similar way, one can prove the following theorem.

Theorem 8. For any , the following are equivalent:(i) is --normal.(ii)If for each -generalized fuzzy -closed and -generalized fuzzy -open , there exists with such that (iii)If for each -generalized fuzzy -closed sets for , there exist with such that and

Definition 11. In , and are called -fuzzy -separated iff and . Also, any fuzzy set which cannot be expressed as the union of two -fuzzy -separated sets is called -fuzzy -connected set.

Theorem 9. For any , we have the following:(i)If , are -fuzzy -separated and such that and , then and are -fuzzy -separated(ii)If and either both are -generalized fuzzy -closed or both are -generalized fuzzy -open, then and are -fuzzy -separated(iii)If and are either both -generalized fuzzy -closed or both -generalized fuzzy -open, then