Advances in Fuzzy Systems

Volume 2019, Article ID 2570926, 9 pages

https://doi.org/10.1155/2019/2570926

## On Tri-*α*-Open Sets in Fuzzifying Tritopological Spaces

Mathematics Department, College of Computer Science and Mathematics, University of Mosul 41002, Mosul 41001, Iraq

Correspondence should be addressed to Tahir H. Ismail; moc.oohay@sh_rihat

Received 14 November 2018; Revised 18 January 2019; Accepted 7 February 2019; Published 15 April 2019

Academic Editor: Patricia Melin

Copyright © 2019 Barah M. Sulaiman and Tahir H. Ismail. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, we introduced and studied (1,2,3)-*α*-open set, (1,2,3)-*α*-neighborhood system, (1,2,3)-*α*-derived, (1,2,3)-*α*-closure, (1,2,3)-*α*-interior, (1,2,3)-*α*-exterior, (1,2,3)-*α*-boundary, (1,2,3)-*α*-convergence of nets, and (1,2,3)-*α*-convergence of filters in fuzzifying tritopological spaces.

#### 1. Introduction

The fuzzy set is an important concept, which was introduced for the first time in 1965 by Zadeh [1]; it was then used in many studies in various fields. Here, we are interested in fuzzy with topology. The fuzzy and fuzzifying topologies are two branches of fuzzy mathematics. The basic concepts and properties of fuzzy topologies were subedited and investigated by Chang in 1968 [2] and Wong in 1974 [3]. After that, so many works of literature have appeared for different kinds of fuzzy topological spaces for, e.g., [4–8]. In 1991-1993, Ying introduced a new approach for fuzzy topology with fuzzy logic and established some properties in fuzzifying topology [9–11]. Also, we are interested in the concept of *α*-open set which was introduced by Njåstad in 1965 [12], and the tritopological space which was first initiated by Kovar in 2000 [13]. In 2017, Tapi and Sharma introduced *α*-open sets in tritopological spaces [14]. In 1999, Khedr et al. presented semiopen sets and semicontinuity in fuzzifying topology [15]. In 2016, Allam and et al. studied semiopen sets in fuzzifying bitopological spaces [16]. We will use in this paper Ying's basic fuzzy logic formulas with appropriate set theoretical notations from [9, 10].

The following are some useful definitions and results that will be used in the rest of the present work.

If is the universe of discourse, and if satisfy the following three conditions:

(1) ;

(2) for any ;

(3) for any ;

then is a fuzzifying topology and a fuzzifying topological space [9].

The family of fuzzifying closed sets is denoted by and defined as , where is the complement of [9].

The neighborhood system of is denoted by and defined as [9].

The closure set of a set is denoted by and defined as [9].

The fuzzifying interior set of a set is denoted by and defined as [10].

The family of all fuzzifying *α*-open sets is denoted by and defined as , i.e., [17].

The family of all fuzzifying *α*-closed sets is denoted by *αℱ* and defined as [17].

The fuzzifying *α*-interior set of a set is denoted by and defined as follows: , where is *α*–neighborhood system of defined as [17].

The fuzzifying *α*-derived set of a set is denoted by and defined as , i.e., [18].

The *α*-closure set of a set is denoted by and defined as [17].

#### 2. --Open Sets in Fuzzifying Tritopological Spaces

*Definition 1. *If is a fuzzifying tritopological space (FTTS), then we have the following: (i)The family of all fuzzifying (1,2,3)-*α*-open sets is denoted by and defined as , i.e., .(ii)The family of all fuzzifying (1,2,3)-*α*-closed sets is denoted by and defined as .

Lemma 2. *Let ( be a FTTS.**If , then .*

*Proof. *If , then ⇒ then .

Lemma 3. *If is a FTTS and . Then *(i)*;*(ii)*.*

*Proof. *From Theorem 2.2-(5) in [10], we have(i).(ii).

*Theorem 4. If is a FTTS, then (i), ;(ii)for any ;(iii), ;(iv)for any .*

*Proof. *(i). (ii)From Lemma 2, we have , then , ≥ (iii)and (iv) are obvious.

*Lemma 5. If is a FTTS, then .*

*Proof. *From Theorem (2.2)-(3) in [10] and Lemma (2.1) in [15], we have ≤ ≤

*Theorem 6. If is a FTTS and , , and are the families of closed sets with respect to , , and , respectively, then (i);(ii).*

*Proof. *(i)From Theorem (2.2)-(3) in [10] and Theorem (5.2)-(3) in [9], we have = = ≤ *⊆*∧ ≤ (ii)It follows directly from (i).

*Remark 7. *The following example shows that

(i) ;

(ii) ;

(iii) ;

(iv) .

It may not be true for all FTTS .

*Example 8. *For and . Let be a fuzzifying tritopological space defined by Now, we have , , , , , , , , , . and , , , , , , , , , . and , , , , , , , , , . and , , , , , , , , , and , , , , , , , , , . . Therefore , , , and .

*Lemma 9. If is a FTTS, then .*

*Proof. *From Theorem (2.2)-(3) in [10], we have = ≤ ≤ = .

Therefore .

*Theorem 10. If is a FTTS, then .*

*Proof. *From Lemma 3-(ii), we have = =

*Lemma 11. If is a FTTS, then (i);(ii).*

*Proof. *(i)≤ ∧ (ii)From Theorem 2.2-(5) in [10], we have ≤ ≤

*Theorem 12. If is a FTTS, then (i);(ii).*

*Proof. *(i)From Theorem (2.2)-(3) in [10], we have ≤ ≤ ≤ ≤ ≤ (ii)From (i) above and Theorem (2.2)-(5) in [10]. We have = =

*3. -–Neighborhood System in Fuzzifying Tritopological Spaces*

*3. -–Neighborhood System in Fuzzifying Tritopological Spaces**Definition 13. *If is a FTTS and . Then indicates the “(1,2,3)-*α*-neighborhood system of ” and defined as , i.e., .

*Theorem 14. If is a FTTS, then .*

*Proof. *=

*Theorem 15. If ( is a FTTS and , then (i),(ii).*

*Proof. *(i)From Theorem 14 we get = .(ii)From Lemma 5 we get

*Theorem 16. If is a FTTS, the mapping , , where is the set of all normal fuzzy subset of , has the following properties:(i),(ii),(iii).*

*Proof. *(i)If , then (i) is obtained. If , then such that . Now we have . Therefore .(ii)(iii)= =

*4. --Derived Set and --Closure Operator in Fuzzifying Tritopological Space*

*4. --Derived Set and --Closure Operator in Fuzzifying Tritopological Space**Definition 17. *If is a FTTS, then indicates the “(1,2,3)-*α*-derived set of ” and defined as follows: , i.e., .

*Lemma 18. .*

*Proof. *= =

*Theorem 19. If is a FTTS and , then (i);(ii);(iii);(iv), where is the fuzzifying derived set of with respect to .*

*Proof. *(i)From Lemma 18 we have (ii)Let , then from Lemma 18 and Theorem 16-(ii) we get (iii)From Lemma 18 and Theorem 15-(ii).We have = = (iv)From Theorem 15-(ii) and Lemma (5.1) in [9] we have

*Definition 20. *If is a FTTS, then indicates the “(1,2,3)-*α*-closure set of ” and defined as , i.e., .

*Theorem 21. If is a FTTS, and , then (i);(ii);(iii) ;(iv);(v));(vi);(vii);(viii).*

*Proof. *(i)= .(ii).(iii)If and for any and if , then . If , then . Thus .(iv)From Lemma 18 and (iii) above, for any we have . If , then