#### Abstract

In this article, we will define the new notions (e.g., -neighborhood system of point, -closure (interior) of a set, and -closed (open) set) based on fuzzy logic (i.e., fuzzifying topology). Then, we will explain the interesting properties of the above five notions in detail. Several basic results (for instance, Definition 7, Theorem 3 (iii), (v), and (vi), Theorem 5, Theorem 9, and Theorem 4.6) in classical topology are generalized in fuzzy logic. In addition to, we will show that every fuzzifying -closed set is fuzzifying -closed set (by Theorem 3 (vi)). Further, we will study the notion of fuzzifying -derived set and fuzzifying -boundary set and discuss several of their fundamental basic relations and properties. Also, we will present a new type of fuzzifying strongly -continuous mapping between two fuzzifying topological spaces. Finally, several characterizations of fuzzifying strongly -continuous mapping, fuzzifying strongly -irresolute mapping, and fuzzifying weakly -irresolute mapping along with different conditions for their existence are obtained.

#### 1. Introduction and Preliminaries

In classical topology, the notions of -open set, -closed set, and strongly -continuous mapping are presented in [1, 2]. After that, Hanafy [3] used the term -open sets instead of -open sets and studied the notions of -open sets and -continuous mapping in fuzzy topology [4]. Benchalli and Karnel [5] presented a novel form of fuzzy subset named fuzzy b-open (closed) set, and some basic properties are proved and also their relations with different fuzzy sets in fuzzy topological spaces are investigated. In 2017, Dutta and Tripathy [6] introduced a new kind of open set named fuzzy open set (i.e., which is a generalization of open set). Ying [7] extended the basic notions in classical topology to fuzzifying topology based on fuzzy logic (i.e., as considered a novel approach of fuzzy topology, which depends on the various basic relations of topological spaces and the logical analysis of topological axioms). Many researchers are interested in fuzzifying topology (such as fuzzifying semiopen sets [8], fuzzifying preopen sets [9], fuzzifying -open sets [10], fuzzifying -open sets [11], and fuzzifying -open sets [12]). Therefore, in this article, we will extend the notions of -neighborhood system of a point, -closure (interior) of a set, -open (closed) set, -derived sets, and -boundary sets in fuzzifying topology. Also, we introduce the notion of fuzzifying strongly -continuous mapping, fuzzifying strongly -irresolute mapping, and fuzzifying weakly -irresolute mapping between two fuzzifying topological spaces.

The rest of this article is arranged as follows. In this section, we briefly recall several notions: closed (open) set, closure (interior) of a set, neighborhood system of point, -closed (open) set, -closure (interior) of a set, -neighborhood system of point, continuous mapping, and -continuous mapping in fuzzifying topology which are used in the sequel. In Section 2, we define the notions of -neighborhood system of a point, -closure (interior) of a set, and -open (closed) set in fuzzifying topology. The interesting relation properties of the above notions are explained in detail. In Section 3, we present the notions -derived set and -boundary set in fuzzifying topology and introduce the characterizations of interesting properties between fuzzifying -derived set and fuzzifying -closure of a set. In Section 4, we define the fuzzifying strongly -continuous mapping, fuzzifying strongly -irresolute mapping, and fuzzifying weakly -irresolute mapping between two fuzzifying topological spaces and investigate some properties of them.

Firstly, we give the notions of a fuzzy logical [7, 13] as follows:(1)We give the symbol which is the truth value of (i.e., the set of truth values means the unit interval ).(2)We can write (i.e., is valid).(3)For (i.e., mean the whole of fuzzy subsets of a set ) have the following for :

Secondly, we present the basic notions related to fuzzifying topological space as follows.

*Definition 1. *(see [7]). (i.e., , the set of all subsets of a set ) is called a fuzzifying topological space (for short, ) if we have the following three conditions:(i)(ii)(iii)

*Definition 2. *(see [7, 13]). The several notions of are given as follows :(i) (i.e., ) is called the set of all fuzzifying closed sets if , where is the complement of (ii) (i.e., ) is called a fuzzifying neighborhood system of if (iii) is called a fuzzifying closure of if (iv) is called a fuzzifying interior of if

Noiri and Sayed [12] presented and studied the following notions in as indicated below.

*Definition 3. *(see [12]). (i) (i.e.) is called the set of all fuzzifying -open sets if i.e.,(ii) (i.e., ) is called the set of all fuzzifying -closed sets if , where is the complement of .(iii) (i.e., ) is called a fuzzifying -neighborhood system of if .(iv) is called a fuzzifying -closure of if .(v) is called a fuzzifying -interior of if .

*Definition 4. *(see [14]). (i.e., ) (a unary fuzzy predicate) is called fuzzifying continuous mappings between and ifi.e.,

*Definition 5. *(see [12]). (i.e., ) (a unary fuzzy predicate) is called fuzzifying -continuous mappings between and ifi.e.,

#### 2. On Fuzzifying -Neighborhood System

*Definition 6. * (i.e., ) is called a fuzzifying -neighborhood system of if

Theorem 1. *Let be a mapping from to s.t. and is a family of N-fuzzy (Normal fuzzy) subsets of ) having the following three properties :*(i)

*(ii)*

*(iii)*

*Proof. *(i)Case 1: if , then . Case 2: if , then . There exists such that . From Theorems 4.2 (1) and 5.2 (3) in [12], we obtain Consequently, , and so . If , then we have ; it is contradiction. Thus, , and hence, .(ii)Case 1: if , then . Case 2: if and , then and so . Thus, . Hence, .(iii). SinceThus,

Next, we will generalize the notion of -closure [2] in as follows.

*Definition 7. * is called a fuzzifying -closure of if

Theorem 2. *The following relation is holding in :*

*Proof. *

Theorem 3. *The following relations are holding in :*(i)*(ii)**(iii)(iv)**(v)**(vi)(vii)*

*Proof. *(i)(ii)From (i) above and since is normal, we have .(iii)Case 1: if , then . Case 2: if , then . Thus, .(iv).(v)Case 1: if , then . Case 2: if , then .(vi)From Theorems 5.2 (3) and 5.3 (2) in [12], we have .(vii)It is easy to get . Conversely, for every ,

Lemma 1. *.*

*Proof. *It follows from Theorem 3 (vi).

Clearly, if we consider and in .

*Definition 8. *(i) (i.e., ) is called the set of all fuzzifying -closed sets if i.e.,(ii) (i.e., is called the set of all fuzzifying -open sets if , where is the complement of .

By Definition 8, we can conclude .

*Definition 9. * is called a fuzzifying -interior of ifi.e.,

Theorem 4. *The following relations are holding in :*(i)*(ii)**(iii)(iv)**(v)**(vi)**(vii)(viii)*

*Proof. *(i)By Theorem 3 (i), we obtain . Hence, .(ii)By (i) above, we obtain .(iii)By Theorem 3 (iii), we obtain .(iv)(v)Case 1: if , then . Case 2: if , then from Theorem 1 (ii) and by (v) above.(vi)Similar to Theorem 3 (vi).(vii)By Lemma 1, we obtain .(viii)By Theorem 3 (vii), we obtain

Theorem 5. *The following two relations are holding in :*(i)*(ii)*

*Proof. *(i)(ii)Similar to (i).

Theorem 6. * (i.e., is a set) is a if we have the following three conditions:*(i)*(ii)**(iii)*

*Proof. *(i)Clear.(ii)By Theorem 4 (v) and Theorem 1 (iii), we obtain(iii)By Theorem 4 (v), we obtainThis completes the proof.

Theorem 7. *The following two relations are holding in :*(i)*(ii)*

*Proof. *(i). Firstly, we obtain . Further, assume that . Hence, , we obtain and .(ii)By (i) we obtain

Theorem 8. *The following two relations are holding in :*(i)(ii)

*Proof. *(i)By Corollary 4.1 in [12] and Theorem 3 (vi), we obtain .(ii)Follows from (i) above.

Next, we will generalize Theorem 3.8 (a) in [2] in by the following theorem.

Theorem 9. *The following relation is holding in :*

*Proof. *Firstly, we prove that . By Corollary 4.1 and Theorem 5.3 (1) in [12], we obtain Case 1: if , then Case 2: if , thenConsequently, .

Thus,This completes the proof.

Corollary 1. *The following relation is holding in :*

Theorem 10. *The following relation is holding in :*

*Proof. *

#### 3. Fuzzifying -Derived Sets and Fuzzifying -Boundary Sets

*Definition 10. * is called a fuzzifying -derived set of ifi.e.,

Theorem 11. *The following relations are holding in :*(i)*(ii)**(iii)*

*Proof. *(i).(ii)By (i) above, we obtain (i.e., is normal).(iii)Case 1: , then . Case 2: if and by Theorem 1 (ii), we obtain .

Theorem 12. *The following relation is holding in :*

*Proof. *By Theorem 7, we obtain