Advances in Fuzzy Systems

Advances in Fuzzy Systems / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 8824296 | https://doi.org/10.1155/2020/8824296

Fadhil Abbas, "Fuzzy Ideal Supra Topological Spaces", Advances in Fuzzy Systems, vol. 2020, Article ID 8824296, 10 pages, 2020. https://doi.org/10.1155/2020/8824296

Fuzzy Ideal Supra Topological Spaces

Academic Editor: Antonin Dvorák
Received11 May 2020
Revised19 Oct 2020
Accepted17 Nov 2020
Published28 Nov 2020

Abstract

In this paper, we introduce the notion of fuzzy ideals in fuzzy supra topological spaces. The concept of a fuzzy s-local function is also introduced here by utilizing the s-neighbourhood structure for a fuzzy supra topological space. These concepts are discussed with a view to find new fuzzy supra topologies from the original one. The basic structure, especially a basis for such generated fuzzy supra topologies, and several relations between different fuzzy ideals and fuzzy supra topologies are also studied here. Moreover, we introduce a fuzzy set operator and study its properties. Finally, we introduce some sets of fuzzy ideal supra topological spaces (fuzzy -supra dense-in-itself sets, fuzzy -supra closedsets, fuzzy -supra perfect sets, fuzzy regular-I-supra closedsets, fuzzy-I-supra opensets, fuzzy semi-I-supra opensets, fuzzy pre-I-supra opensets, fuzzy -I-supra opensets, and fuzzy -I-supra opensets) and study some characteristics of these sets, and then, we introduce some fuzzy ideal supra continuous functions.

1. Introduction

The concept of fuzzy sets is an important concept in many fields. The concept was introduced by in 1965 [1]. The idea was welcomed because it addresses the uncertainty, something classical Cantor set theory could not address. Despite of some criticism expressed in the beginning by some specialists of mathematical logic, it has become an important subject in various fields and sciences. Zadeh writes in [1], “The notion of a fuzzy set provides a convenient point of departure for the construction of a conceptual framework which parallels in many respects the framework used in the case of ordinary sets, but is more general than the latter and potentially, may prove to have a much wider scope of applicability particularly in the fields of pattern classification and information processing.”

Fuzzy set theory provides a natural way to deal with inaccuracy and a strict mathematical framework for the study of uncertain phenomena and concepts. It can also be considered as a modeling language, well suited for situations in which fuzzy relations criteria and phenomena exist. Despite the slow growth and progress of fuzzy set theory before the mid-1970s, the theory developed greatly afterward. This was caused by the first successful application of the theory to technological processes, in particular to systems based on ambiguous control rules called fuzzy control and boosted the interest in this area considerably.

The concept of general topology is one of the most important mathematical topics and has wide applications in many applied sciences and mathematical subjects. The notion of fuzzy sets naturally plays a very significant role in the study of fuzzy topology introduced by Chang in 1968 [2]. Pu and Liu in 1980 [3] introduced the concept of quasi-coincidence and q-neighbourhoods by which the extensions of functions in fuzzy setting can very interestingly and effectively be carried out. The concept of an ideal in a topological space was first introduced by Kuratowski in 1966 [4] and Vaidyanathswamy in 1945 [5]. They also defined local functions in an ideal topological space. Furthermore, Jankovic and Hamlet in 1990 [6] studied the properties of ideal topological spaces and introduced another operator called the operator. They have also obtained a new topology from the original ideal topological space. Using the local function, they defined a Kuratowski closure operator in the new topological space.

The concept of supra topology was introduced by Mashhour et al. in 1983 [7]. It is fundamental with respect to the investigation of general topological spaces. In 2016, Al-shami [8] discussed the concepts of compactness and separation axioms on supra topological spaces. Then, Al-Shami [9, 10] and Al-shami et al. [11] presented new types of supra compact spaces using supra -open, supra semiopen, and supra preopensets. Later, the authors of [1215] employed some generalizations of supra opensets to investigate several kinds of supra limit points of a set and supra spaces . In fact, they provided many interesting examples to show the validity of the obtained results. Recently, Al-shami [16] have defined supra paracompact spaces, and Assad et al. [17] have studied operation on supra topological spaces.

Abd El-Monsef and Ramadan in 1987 [18] introduced the concept of fuzzy supra topological as a natural generalization of the notion of supra topology spaces. In addition to that, some properties of the concept of ideal supra topological spaces are obtained by Modak and Mistry in 2012 [19]. In 2015 [20], further properties of ideal supra topological spaces are investigated.

In this paper, we introduce the notion of fuzzy ideals in fuzzy supra topological spaces.

Section-wise description of the work carried out in this paper is given. Beginning with an introduction, necessary notation and preliminaries have been given.

In Section 3, the concept of a fuzzy s-local function is also introduced here by utilizing.

In Section 4, we give the s-neighbourhood structure for a fuzzy supra topological space. These concepts are discussed with a view to find new fuzzy supra topologies from the original one. The basic structure, especially a basis for such generated fuzzy supra topologies, and several relations between different fuzzy ideals and fuzzy supra topologies are also studied here.

In Section 5, we introduce a fuzzy set operator and study its properties.

In Section 6, we introduce some sets of fuzzy ideal supra topological spaces (fuzzy -supra dense-in-itself sets, fuzzy -supra closedsets, fuzzy -supra perfect sets, fuzzy regular-I-supra closedsets, fuzzy-I-supra opensets, fuzzy semi-I-supra opensets, fuzzy pre-I-supra opensets, fuzzy -I-supra opensets, and fuzzy -I-supra opensets) and study some characteristics of theses sets. Finally, in Section 7, we introduce some fuzzy ideal supra continuous functions.

2. Preliminaries

Definition 1. (see [1]). Let X be a nonempty set. A fuzzy set A in X is characterized by its membership function , and is interpreted as the degree of membership of the element x in the fuzzy set A, for each . It is clear that A is completely determined by the set of tuples .

Definition 2. (see [1]). Let A and B be fuzzy sets of the form and . Then, we define(1) for all (2) and (3)(4)(5)(6) and

Definition 3. (see [1]). For any family of fuzzy sets in X, we define(1)(2)

Definition 4. (see [21]). A fuzzy set in X is called a fuzzy point if and only if it takes the value 0 for all except one, say . If its value at x is , we denote this fuzzy point by , where the point x is called its support.

Definition 5. (see [18]). A subclass (P(X) is the collection of all fuzzy sets on X and is called a fuzzy supra topology on X if , and S is closed under arbitrary union. The pair (X, S) is called a fuzzy supra topological space, and the members of S are called fuzzy supra opensets. A fuzzy set A is fuzzy supra closed if and only if its complement is fuzzy supra open.

Definition 6. (see [18]). Let (X, S) be a fuzzy supra topological space and let A be a fuzzy set in X. Then, the fuzzy supra interior and the fuzzy supra closure of A in (X, S) are defined asrespectively.

Corollary 1. From Definition 6, is a fuzzy supra openset and is a fuzzy supra closedset.

Definition 7. (see [22]). Let (X, S) be a fuzzy supra topological space. A fuzzy set A in X is said to be quasi-coincident with a fuzzy set B if there exists , such that . In this case, we write .

Definition 8. (see [22]). A fuzzy set A in a fuzzy supra topological space (X, S) is an s-neighbourhood of a fuzzy point if there is with . The collection of all s-neighbourhoods of is called the s-neighbourhood system of .

Definition 9. (see [18]). Let and be two fuzzy supra topologies on a set X such that . Then, we say that is stronger(finer) than or is weaker(coarser) than .

Definition 10. (see [18]). Let (X, S) be a fuzzy supra topological space and . Then, is called a base for the fuzzy supra topology S if every fuzzy supra openset is a union of members of . Equivalently, is a fuzzy supra base for S if, for any fuzzy point , there exists with .

Definition 11. (see [18]). A mapping is said to be a fuzzy supra closure operator if it satisfies the following axioms:(1)(2) for every fuzzy set A in X(3) for every fuzzy sets A and B in X(4) for every fuzzy set A in X

Theorem 1. (see [18]). Let X be a nonempty set, and let the mapping be a fuzzy supra closure operator. Then, the collection is fuzzy supra topology on X induced by the fuzzy supra closure operator c.

Definition 12. (see [23]). A nonempty collection of fuzzy sets I of a set X is called a fuzzy ideal on X if and only if(1) and (heredity)(2) and (finite additivity)

3. Fuzzy S-Local Function

Definition 13. A fuzzy supra topological space (X, S) with a fuzzy ideal I on X is called a fuzzy ideal supra topological space and denoted as (X, S, I).

Definition 14. Let (X, S, I) be a fuzzy ideal supra topology and let A be any fuzzy set in X. Then, the fuzzy s-local function (I, S) of A is the union of all fuzzy points such that if and , then there is at least one for which .
In other words, we say that a fuzzy set A is fuzzy s-local in I at if there exists , such that for every , for some . We will occasionally write for , and it will cause no ambiguity.

Example 1. The simplest fuzzy ideals on X are and P(X). Then, , for any fuzzy set A in X and .

Theorem 2. Let (X, S, I) be a fuzzy ideal supra topological space, and let A and B be fuzzy sets in X. Then,(1)(2)If , then (3)If , then (4)(5)(6) is a fuzzy supra closedset(7)(8)(9)If , then (10)If , then (11)If , then (12)If , then

Proof. (1)This is obvious from the definition of fuzzy s-local function(2)Let . Then, for every . From the definition of fuzzy s-local function, if , then . Therefore, .(3)Let and let be any fuzzy points, such that . Then, there is at least one and for every , such that for some . But , then . This implies . Therefore, .(4)For any fuzzy ideal on X, . Therefore by (3) and Example 1, for any fuzzy set A in X,. Now, let . Then, for every , there is at least one , such that . Hence, , and let . Clearly, and , so that . Now, implies there is at least one , such that , for each and . This is also true for M. So, there is at least one such that , for each . Since ; therefore, . Hence, .(5)From (4), (6)Let . Then, there is at least one , such that for every for some , and implies , and we have which is a fuzzy supra openset. Therefore, is fuzzy supra closed.(7)We have and . Then, from (2), and . Hence, .(8)We have and . Then, from (2), and . Hence, .(9)We have . Then, from (2), . Now, let ; then, for every , there is at least one such that , for each . If , then for each and , , and this implies , and this is a contradiction. Then, for each and , and this implies . Therefore, . Hence, . Now, from the defined fuzzy s-local function, clear ; then, .(10)We have . Then, from (2), . So, .(11)This is obvious from the definition of fuzzy s-local function(12)This is obvious from (9)

Theorem 3. Let (X, S, I) be a fuzzy ideal supra topological space, and let A be any fuzzy set in X. Then, .

Proof.
Let . Then, for every , there is at least one , such that , for each . Now, if implies ; therefore, the proof is done. If , then , and let . Clearly, and . Now, implies there is at least one , such that for each and . This is also true for M. So, there is at least one , such that , for each . Since , then . Therefore, .

Theorem 4. Let (X, S, I) be a fuzzy ideal supra topological space. Then, the operator , defined by for any fuzzy set A in X, is a fuzzy supra closure operator, and hence, it generates a fuzzy supra topology which is finer than S.

Proof. (1)By (1) in Theorem 2, , we have (2)Clear that for every fuzzy set A(3)Let A and B be any two fuzzy sets. Then, (by (7) in Theorem 2). Hence, .(4)Let A be any fuzzy set. Since, by (2), ; then, . On the other hand, (by Theorem 3); it follows that . Hence, . Consequently, is a fuzzy supra closure operator. Also, it is easy to show that the collection is a fuzzy supra topology on X which is called the fuzzy supra topology induced by the fuzzy supra closure operator.

Proposition 1. For any fuzzy ideal on X, if for every a fuzzy set A in X. So, , and if because for every a fuzzy set A in X. So, is a fuzzy discrete supra topology on X. Since and P(X) are the tow extreme fuzzy ideal on X, for any fuzzy ideal I on X, we have . So, we can conclude by (2) in Theorem 2 that , i.e., , for any fuzzy ideal I on X. In particular, we have for any tow fuzzy ideals and on X, .

Theorem 5. For any fuzzy ideal supra topological space (X, S, I), the class is the base for the fuzzy supra topology .

Proof.
Since , then from which it follows . Also, for every we have, and , where , and . Then, . Therefore, is a base for .

Example 2. Let T be the fuzzy indiscrete supra topology on X, i.e., . So, is the only s-neighbourhoods of . Now, for a fuzzy set A if and only if for each , there is at least one , such that . This implies, for each for at least one . So, . Therefore, if and if . This implies that we have if and if for any fuzzy set A of X. Hence, . Let be the supremum fuzzy supra topology of S and , i.e., the smallest fuzzy supra topology is generated by . Then, we have the following theorem.

Theorem 6. .

Proof. Follows from the fact that forms a basis for .

Theorem 7. Let be tow fuzzy supra topologies on X. Then, for any fuzzy ideal I on X, implies(1) for every fuzzy set in X(2)

Proof. (1)Since every s-neighbourhood of any fuzzy point is also a s-neighbourhood of . Therefore, .(2)Clearly, as

Theorem 8. Let (X, S, I) be a fuzzy supra topological space. Then,(1)If (2) for every fuzzy set A in X, i.e., is a fuzzy -supra closedset.

Proof. (1)For every . Hence, , i.e., H is a fuzzy -supra closedset.(2)From (5) in Theorem 2, we have . Hence, is a fuzzy -supra closedset.

4. S-Compatible of Fuzzy Ideals with Fuzzy Supra Topology

Definition 15. Let (X, S, I) be a fuzzy supra topological space. S is said to be fuzzy S-compatible with I, denoted by , if for every fuzzy set A in X; if for all fuzzy point , there exists such that holds for every and some , and then, .

Definition 16. (see [20]). Let be any indexed family of fuzzy sets in X such that , for each , where A is a fuzzy set in X. Then, is said to be a quasicover of A if and only if for every .

Definition 17. Let be quasicover of A; if each is a fuzzy supra openset, then this quasicover with be called a fuzzy quasi-supra open cover A in X. Therefore, in either case, .

Theorem 9. Let (X, S, I) be a fuzzy ideal supra topological space. Then, the following conditions are equivalent:(1)(2)If for every fuzzy set A in X has a fuzzy quasi-supra open cover such that for each for some and for every , and then, (3)For every fuzzy set A in X, implies (4)For every fuzzy set A in X, , where , such that but

Proof. (1)(2) Let be a fuzzy quasi-supra open cover of a fuzzy set A in X such that for each for some and for every . Therefore, as is a fuzzy quasi-supra open cover of A, for each , there exists at least one such that and for every for some . Obviously, . Therefore, as .(2)(1) Clear from the fact that a collection of fuzzy supra opensets which contain at least one open s-neighbourhood of each fuzzy point of A that constitutes a fuzzy quasi-supra open cover of A.(2)(3) Let , i.e., , for every . So, a fuzzy point implies . That means, there is such that for every for some . If since , there is a fuzzy supra openset V (in S) such that , and so, the collection of such V’s for each constitutes a fuzzy quasi-supra open cover of A. Therefore, by condition (2), .(3)(1) Let for every fuzzy point , there is such that for every for some . That means . Now, there are two cases: either or , but . Let, if possible, be such that . Let . Then, the fuzzy point and also . This implies for each , and for each , there is at least one such that . Since , this contradicts the assumption for every fuzzy point of A. So, . That means, implies . Now, this is true for every fuzzy set A in X. So, for every fuzzy set A in X, . Hence, by condition (3), we have , which implies .(3)(4) Let the fuzzy point . That means, , but . So, there is such that for every for some . Since , so for every for some . Therefore, , so that either or , but . Let , be a fuzzy point such that , i.e., . So, for each and for each , there is at least one such that . Since , for each and for each , there is at least one such that . This implies . But as , and therefore . This is a contradiction. Hence, , so that implies with . Thus, we have for every fuzzy set A in X. Hence, by condition (3), .(4) (1) Let for every fuzzy set A in X, . This means for every then and then for every , there exist such that for some . This implies .

Theorem 10. Let (X, S, I) be a fuzzy ideal supra topological space. Then, the following are equivalent and implied by .(1)For every fuzzy set A in X, implies .(2)For every fuzzy set A in X, (Ã is defined as in (4) in Theorem 9).(3)For every fuzzy set A in X, .

Proof.
Clear from Theorem 9.

Theorem 11. Let (X, S, I) be a fuzzy ideal supra topological space, let . Then a fuzzy set A in X is supra closed with respect to if and only if it is the union of a fuzzy set which is supra closed with respect to S and a fuzzy set in I.

Proof. Let A be a fuzzy set in X such that it is fuzzy -supra closedset. That means , and we have . Since , . Also, is always fuzzy S-supra closedset by (3) in Theorem 2. Conversely, let A be any fuzzy set in X such that , where . This means . So, we have and implies A is a fuzzy -supra closedset.

Corollary 2. The fuzzy supra topology S is S-compatible with fuzzy ideal I on X implies , a basis for , and is itself a fuzzy topology and also .

Proof. Clear.

5. Fuzzy Set Operator

In this section, we introduce the fuzzy set operator , S-codense, and -C-fuzzy set, and we give new results.

Definition 18. Let (X, S, I) be a fuzzy ideal supra topological space. An operator is defined as follows: for every fuzzy set A in X,  = { fuzzy point: there exists such that }. We observe that . The behaviors of the operator have been discussed in the following theorem:

Theorem 12. Let (X, S, I) be a fuzzy ideal supra topological space. Let A and B be two fuzzy sets in X. Then,(1) is a fuzzy supra openset(2)(3)If , then (4)(5)(6)If , then (7)(8) if and only if (9)If , then (10)If , then (11)If , then (12)If , then

Proof. (1)Since is a fuzzy supra closedset, is a fuzzy supra openset. Hence, is a fuzzy supra openset.(2)From definition of the operator, . Then, , from (4) in Theorem 2. Hence, .(3)Let . Then, . Then, from (2) in Theorem 2, . Therefore, .(4)We have and . Then, from (3), .(5)We have and . Then, from (3), .(6)Let . Then, is a fuzzy supra closedset, and hence, . Then, . Hence, , so .(7)From (2), , and from (6), .(8)Let . Then, Therefore, . Conversely suppose that hold. Then, and . Hence, .(9)Let , and let . We observe that by heredity, and . Thus, .(10)By (9) in Theorem 2, we obtain if , then (11)This follows from (9) in Theorem 2, and (12)This follows from (9) in Theorem 2, and

Theorem 13. Let (X, S, I) be a fuzzy ideal supra topological space. If . Then, is a fuzzy supra topology for X.

Proof. Let . By (1) in Theorem 2, and . Moreover, . Therefore, we obtain that and , and thus, . Now, if , then for every , and hence, . This shows that is a fuzzy supra topology.

Definition 19. A fuzzy ideal I in a space (X, S, I) is called S-codense fuzzy ideal if . The following theorem is related to S-codense fuzzy ideal.

Theorem 14. Let (X, S, I) be a fuzzy ideal supra topological space, and let I be S-codense with S. Then, .

Proof. It is obvious that . For converse, suppose but , there is at least one and for every such that for some . That means , a contradiction to the fact that . Hence, .

Definition 20. Let (X, S, I) be a fuzzy ideal supra topological space. A fuzzy set A in X is called the -C-fuzzy set if . The collection of all -C-fuzzy sets in (X, S, I) is denoted by .

Theorem 15. Let (X, S, I) be a fuzzy ideal supra topological space. If , then .

Proof. From (6) in Theorem 12, it follows that .

Theorem 16. Let be a collection of nonempty -C-fuzzy sets in a fuzzy ideal supra topological space (X, S, I); then, .

Proof. For each . This implies that . Thus, .

6. Some Sets of a Fuzzy Ideal Supra Topological Space

Definition 21. Let (X, S, I) be a fuzzy ideal supra topological space and let A be any fuzzy set in X. Then, A is said to be(1)Fuzzy -supra dense-in-itself set if (2)Fuzzy -supra closedset if (3)Fuzzy -supra prefect set if (4)Fuzzy regular-I-supra closedset if

Theorem 17. Let (X, S, I) be a fuzzy ideal supra topological space, and let A be any fuzzy set in X. Then, the following statements hold:(1)Every fuzzy regular-I-supra closedset is a fuzzy -supra prefect set(2)Every fuzzy -supra prefect set is a fuzzy -supra closedset(3)Every fuzzy -supra prefect set is a fuzzy -supra dense-in-itself set

Proof. (1)Let A be a fuzzy regular-I-supra closedset. Then, we have . Since by (2) in Theorem 2, then . We have . Since , then . Therefore, we obtain . This shows that A is a fuzzy -supra prefect set.(2)Let A be a fuzzy -supra perfect set. Then, we have ; therefore, we obtain . This shows that A is a fuzzy -supra closedset.(3)Let A be a fuzzy -supra perfect set. Then, we have ; therefore, we obtain . This shows that A is a fuzzy -supra dense-in-itself set.

Remark 1. The converses of Theorem 17 need not be true as the following examples show.

Example 3. Let and let A and B be fuzzy sets in X defined as follows:We put . If we take , then B is a fuzzy -supra perfect set but not a fuzzy regular-I-supra closedset.