Abstract

The aim of this work is to introduce -operations on fuzzy topological spaces and to use them to study fuzzy generalized -closed sets and fuzzy generalized -open sets. Also, we introduce some characterizations and properties for these concepts. Finally we show that certain results of several publications on the concepts of weakness and strength of fuzzy generalized closed sets are considered as corollaries of the results of this research.

1. Preliminaries

The concept of fuzzy topology was first defined in 1968 by Chang [1] based on the concept of a fuzzy set introduced by Zadeh in [2]. Since then, various important notions in the classical topology such as generalized closed, generalized open set, and weaker and stronger forms of generalized closed and generalized open sets have been extended to fuzzy topological spaces. The purpose of this paper is to introduce and study the concept of -operations, and by using these operations, we will study fuzzy generalized -closed sets and fuzzy generalized -open sets in fuzzy topological spaces. Also, we show that some results in several papers [3–15] considered as corollaries from the results of this paper. Let be a fuzzy topological space (fts, for short), and let be any fuzzy set in . We define the closure of to be is fuzzy closed} and the interior of to be is fuzzy open}. A fuzzy point [16] is a fuzzy set with support and value . For a fuzzy set in , we write if and only if . Evidently, every fuzzy set can be expressed as the union of all fuzzy points which belongs to . A fuzzy point is said to be quasicoincident [17] with denoted by if and only if . A fuzzy set is said to be quasicoincident with , denoted by , if and only if there exists such that . If is not quasicoincident with , then we write . For a fuzzy set of a fts (resp., -----) denote the fuzzy closure (resp., semiclosure, precloure, -closure, -closure, -closure, semi-preclosure, -closure, -closure) and (resp., -----) denote the fuzzy interior (resp., semi-interior, preinterior, -interior, -interior, -interior, semi-preinterior, -interior, and -interior) of .

Definition 1.1 (see [17]). A fuzzy set in an fts is said to be -neighborhood of a fuzzy point if there exists a fuzzy open set with .

Definition 1.2. A fuzzy set in an fts is said to be: (1)fuzzy regular open [18] (, for short) if )) = , (2)fuzzy regular closed [18] (, for short) if )) =, (3)fuzzy regular semiopen [19](, for short) if there exists a fuzzy regular open set such that , (4)fuzzy -open [20] (o, for short) if , (5)fuzzy -closed [20] (c, for short) if , (6)fuzzy semiopen [18] (, for short) if , (7)fuzzy semiclosed [18] (, for short) if , (8)fuzzy preopen [20] (, for short) if , (9)fuzzy preclosed [20] (, for short) if , (10)fuzzy -open [21] (o, for short) if , (11)fuzzy -closed [21] (c, for short) if , (12) fuzzy semi-preopen [22] (, for short) if there exists a fuzzy preopen set such that , (13) fuzzy semi-preclosed [22] (, for short) if there exists a fuzzy preclosed set such that , (14)fuzzy -open [23] (o, for short) if , (15)fuzzy -closed [23] (c, for short) if .

Remark 1.3. From the above definition we have a diagram, Figure 1, showing all relationships between the classes of open sets. None of the implications shown in Figure 1 can be reversed in general.

Figure 1 

Definition 1.4 (see [18, 20–23]). (1) The intersection of all fuzzy -closed (resp., semiclosed, preclosed, -closed, semi-preclosed, -closed) sets containing a fuzzy set is called a fuzzy -closure (resp., semiclosure, preclosure, -closure, semi-preclosure, -closure) of .
(2) The union of all fuzzy -open (resp., semiopen, preopen, -open, semi-preopen, -open) sets contained in a fuzzy set is called a fuzzy -interior (resp., semi-interior, preinterior, -interior, semi-preinterior, -interior) of .

Definition 1.5. A fuzzy point in an fts is said to be a fuzzy cluster (resp., -cluster [24], -cluster [16]) point of a fuzzy set if and only if for every fuzzy open (resp., open, regular open) -neighborhood of , (resp., , ). The set of all fuzzy cluster (resp., fuzzy -cluster, fuzzy -cluster) points of is called the fuzzy closure (resp., -closure, -closure) of and is denoted by (resp., --. A fuzzy set is fuzzy -closed (resp., -closed) if and only if - (resp., -. The complement of a fuzzy -closed (resp., -closed) set is called fuzzy -open (resp., -open).

Definition 1.6. A fuzzy set of an fts is said to be: (1)fuzzy generalized closed [3] (briefly, -closed) if , whenever and is a fuzzy open set in , (2)fuzzy generalized -closed [10] (briefly, -closed) if -, whenever and is a fuzzy open set in , (3)fuzzy -generalized closed [13] (briefly, -closed) if -, whenever and is a fuzzy -open set in , (4)fuzzy generalized semiclosed [12] (briefly, -closed) if , whenever and is a fuzzy open set in , (5)fuzzy semigeneralized closed [5] (briefly, -closed) if , whenever and is a fuzzy semiopen set in , (6)fuzzy generalized preclosed [8] (briefly, -closed) if , whenever and is a fuzzy open set in , (7)fuzzy pregeneralized closed [6] (briefly, -closed) if , whenever and is a fuzzy preopen set in , (8)fuzzy generalized semi-preclosed [11] (briefly, -closed) if , whenever and is a fuzzy open set in , (9)fuzzy semi-pregeneralized closed [14] (briefly, -closed) if , whenever and is a fuzzy semi-preopen set in , (10)fuzzy regular generalized closed [9] (briefly, -closed) if , whenever and is a fuzzy regular open set in , (11)fuzzy generalized -closed [4] (briefly, -closed) if -, whenever and is a fuzzy open set in , (12)fuzzy -generalized closed [7] (briefly, -closed) if -, whenever and is a fuzzy -open set in ,(13)fuzzy -generalized closed [15] (briefly, -closed) if -, whenever and is a fuzzy -open set in .

The complement of a fuzzy generalized closed (resp., generalized -closed, -generalized closed, generalized semiclosed, semigeneralized closed, generalized preclosed, pre generalized closed, generalized semi-preclosed, semi-pregeneralized closed, regular generalized closed, -generalized closed, generalized -closed) set is called fuzzy generalized open (-open, for short) (resp., generalized -open (-open), -generalized open (-open), generalized semiopen (-open), semi generalized open (-open), generalized preopen (-open), pre generalized open (-open), generalized semi-preopen (-open), semi-pregeneralized open (-open), regular generalized open (-open), -generalized open (-open), generalized -open (-open)).

Definition 1.7 (see [25]). A fuzzy point in an fts is called weak (resp., strong) if (resp., .

2. -Operations

In this research, we will denote for a fuzzy open set from type by fuzzy -open and the family of all fuzzy -open sets in an fts by . Also we will denote a fuzzy open (resp., -open, semiopen, preopen, semi-preopen, -open, -open, -open, -open, and regular open) set by -open (resp., -open, s-open, p-open, sp-open, -open, -open, -open, -open, and r-open). Similarly we will denote a fuzzy closed (resp., -closed, semiclosed, preclosed, semi-preclosed, -closed, -closed, -closed, -closed, and regular closed) sets by -closed (resp., -closed, s-closed, p-closed, sp-closed, -closed, -closed, -closed, -closed, and r-closed). Let .

Definition 2.1. A fuzzy set in an fts is said to be a fuzzy --neighborhood of a fuzzy point if and only if there exists a fuzzy -open set such that . The family of all fuzzy -neighborhoods of a fuzzy point is denoted by .

Definition 2.2. A fuzzy point in an fts is said to be a fuzzy -cluster point of a fuzzy set if and only if for every fuzzy -neighborhood of a fuzzy point , . The set of all fuzzy -cluster points of a fuzzy set is called the fuzzy -closure of and is denoted by . A fuzzy set is fuzzy -closed if and only if and a fuzzy set is fuzzy -open if and only if its complement is fuzzy -closed.

Theorem 2.3. For a fuzzy set in an fts ,

Proof. The proof of this theorem is straightforward, so we omit it.

Theorem 2.4. Let and be fuzzy sets in an fts . Then the following statements are true: (1); (2) for each fuzzy set of ; (3)if , then ; (4)if is -closed, then , and if one supposes is -closed, then the converse of (4) is true; (5)if , then if and only if ; (6); (7). If the intersection of two fuzzy -open sets is fuzzy -open, then .

Proof. (1), (2), (3), and (4) are easily proved.
(5) Let . Then , and hence , which implies . Hence if and only if .
(6) Let be a fuzzy point with . Then there is a fuzzy -neighborhood of such that . From (5) there is a fuzzy -neighborhood of such that and hence . Thus . But . Therefore .
(7) It is clear.

Definition 2.5. For a fuzzy set in an fts , we define a fuzzy -interior of as follows:

Theorem 2.6. Let and be fuzzy sets in an fts . Then the following statements are true: (1); (2) for each fuzzy set of ; (3)if , then ; (4)if is -open, then , if one supposes, is -open, then the converse of (4) is true; (5)if , then if and only if ; (6); (7). If the intersection of two fuzzy -open sets is -open, then .

Proof. It is similar to that of Theorem 2.4.

Theorem 2.7. For a fuzzy set in an fts , the following statements are true: (1);(2).

Proof. It follows from the fact that the complement of a fuzzy -open set is fuzzy -closed and .

Definition 2.8. Let be a fuzzy set of an fts . A fuzzy point is said to be -boundary of a fuzzy set if and only if . By - one denotes the fuzzy set of all -boundary points of .

Theorem 2.9. Let be a fuzzy set of an fts . Then

Proof. It follows from Definition 2.8 and Theorem 2.4.

3. Generalized -Closed and Generalized -Open Sets

Definition 3.1. Let be an fts. We define the concepts of fuzzy generalized -closed and fuzzy generalized -open sets, where represents a fuzzy closure operation and represents a notion of fuzzy openness as follows: (1)A fuzzy set is said to be generalized -closed (-closed, for short) if and only if , whenever and is fuzzy -open.(2) The complement of a fuzzy generalized -closed set is said to be fuzzy generalized -open (-open, for short).

Remark 3.2. Note that each type of generalized closed set in Definition 2.8 is defined to be generalized -closed set for some and . Namely, a fuzzy set is fuzzy -closed [3] if it is -closed, -closed [10] if it is -closed, -closed [13] if it is -closed, -closed [12] if it is -closed, -closed [5] if it is -closed, -closed [8] if it is -closed, -closed [6] if it is -closed, -closed [11] if it is -closed, -closed [14] if it is -closed, -closed [4] if it is -closed, -closed [7] if it is -closed, and -closed [9] if it is r-closed.

Theorem 3.3. A fuzzy set is generalized -open if and only if , whenever and is fuzzy -closed.

Proof. It is clear.

Theorem 3.4. If is a fuzzy -closed set in an fts , then is fuzzy generalized -closed.

Proof. Let be a fuzzy -closed, and let be a fuzzy -open set in such that . Then , and hence is fuzzy generalized -closed.

Remark 3.5. In classical topology, if is a generalized -closed set in a topological space , then does not contain nonempty -closed. But in fuzzy topology this is not true in general as shown by the following example.

Example 3.6. Let , , , , and be fuzzy subsets of defined as follows: Let be a fuzzy topology on .

One may notice that the following.

(1) is a fuzzy -closed set and But contains nonempty fuzzy closed .

(2) is a fuzzy generalized closed set and

But contains nonempty closed set .

(3) is a fuzzy -closed (resp., -closed, -closed, -closed, -closed) set and and hence

But contains nonempty set which is fuzzy -closed and hence is fuzzy semiclosed, preclosed, and semi-preclosed, and so on.

Theorem 3.7. Let be an fts, and let be a fuzzy -closed set with . Then is a fuzzy -closed set.

Proof. Let be a fuzzy -open set in such that . Then . Since is fuzzy -closed, then , and hence . Thus , and hence is a fuzzy -closed set

Theorem 3.8. Let be an fts, and let be a fuzzy -open set with . Then is a fuzzy -open set.

Proof. It is similar to that of Theorem 3.7.

Theorem 3.9. Let be a fuzzy set in an fts, and let be -closed for each fuzzy set . Then is fuzzy -closed if and only if for each fuzzy point with , one has .

Proof. Let and suppose that . Since is -closed, then is fuzzy -open and . Since is fuzzy -closed, then and hence which contradict with and hence .
Conversely, let be fuzzy -open set with and let . By hypothesis , and hence there is such that . Put . Then , and hence . Since , is a fuzzy -open set and , then . Hence . Thus is fuzzy -closed.

Theorem 3.10. Let be an fts, and let be a fuzzy set in X. Then the following are equivalent: (1) is fuzzy -closed; (2)if is fuzzy -open, then is fuzzy -closed.

Proof. (1)(2). Let be fuzzy -closed and fuzzy -open with . Then . Since , then . Therefore is -closed.
(2)(1). Let be a fuzzy set with , where is fuzzy -open set in . From (2) we have is -closed, and hence . Thus is fuzzy -closed.

Theorem 3.11. Let be an fts and suppose that and are weak and strong fuzzy points, respectively. If is fuzzy -closed and is fuzzy -closed, then

Proof. Let and . Then . Since is a weak fuzzy point, then , and hence . Thus . So . Since is fuzzy -closed and is fuzzy -open, then , and hence , which is a contradiction, since if , then . But since , then , and hence , which implies . Since is fuzzy strong point, then , which is a contradiction. Thus .

Definition 3.12. Let be an fts. A fuzzy point is said to be fuzzy just--closed if the fuzzy set is a fuzzy point.

Theorem 3.13. Let be an fts. If and are two fuzzy points such that and is fuzzy -open, then is fuzzy just--closed if it is fuzzy -closed.

Proof. Let , be fuzzy -open, and let be fuzzy -closed. Then , and hence and for each . Thus is a fuzzy point. Therefore is fuzzy just--closed.

Definition 3.14. Let be an fts. A fuzzy set of is called fuzzy -nearly crisp if .

Theorem 3.15. If is fuzzy -closed and fuzzy -nearly crisp of an fts , then does not contain any nonempty fuzzy -closed set in .

Proof. Suppose that is a fuzzy -closed set in , and let be a fuzzy -closed set such that and . Then is fuzzy -open. Since is a fuzzy -closed, then , and hence , so . Therefore, , which is contradiction. Hence does not contain any nonempty fuzzy -closed set in .

Theorem 3.16. Let be an fts. Then every fuzzy -open set is fuzzy -closed if and only if every fuzzy subset of is fuzzy -closed.

Proof. Suppose that be a fuzzy -open set and be any fuzzy subset of such that . By hypothesis, is fuzzy -closed, and hence . Thus is fuzzy -closed.
Conversely, suppose that every fuzzy subset of is fuzzy -closed and is a fuzzy -open set. Since and is fuzzy -closed, then and hence . Thus is fuzzy -closed.

Theorem 3.17. If is fuzzy -open and fuzzy -nearly crisp of an fts , then , where is a fuzzy -open and .