Advances in Fuzzy Systems
Volume 2019 (2019), Article ID 2570926, 9 pages
https://doi.org/10.1155/2019/2570926

Research Article

On Tri-α-Open Sets in Fuzzifying Tritopological Spaces

Barah M. Sulaiman and Tahir H. Ismail

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

Correspondence should be addressed to Tahir H. Ismail; tahir_hs@yahoo.com

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

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

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 . If , then . Thus .(v)= .(vi)From Theorem 19-(iii), Lemma (8.2) in [15] and (iv) above, since , we get ⋀.(vii)If , then . From (i) above and Theorem 16-(ii) we get . Thus .(viii)If , then . Assume that , then and . Therefore . If , then . For any , we get . Thus , i.e., such that and . To prove that . If , then and . Hence we get ⟹ Contradiction. Therefore , since is arbitrary; thus .

5. -–Interior, --Exterior, and --Boundary Operators in Fuzzifying Tritopological Space

Definition 22. If is a FTTS and , then indicates the “(1,2,3)-α-interior set of ” defined as

Theorem 23. If is a FTTS, and , then

(i);(ii);(iii);(iv);(v);(vi);(vii);(viii);(ix).

Proof.

(i)⟹(ii)Let , . If , then ⟹.(iii)From Theorem 15-(ii) we have . Therefore .(iv)If , then the result holds. If , then = (v)= , = , = = (vi)From Definition 22 and Theorem 16 -(ii) the proof follows.(vii)From Theorem 21-(i) we have . Therefore .(viii)From Lemma 18 we get ). If , then . If , then . Therefore .(ix)From Theorem 21-(ix) and (vii) above we get 

Definition 24. If ( is a FTTS and . Then indicates the “(1,2,3)-α-exterior set of ” and defined as , i.e., .

Theorem 25. If is a FTTS and . Then

(i);(ii);(iii);(iv);(v);(vi);(vii);(viii);(ix).

Proof. The proofs of (i) - (vii) follow from Theorem 23.

(ix)= . By Definition 24

Definition 26. If is a FTTS and , then indicates the “(1,2,3)-α-boundary of a set ” and defined as , i.e., .

Lemma 27. If is a FTTS, and , then

Proof. = =

Theorem 28. If is a FTTS and , then

(i);(ii);(iii);(iv);(v);(vi);(vii);(viii);(ix).

Proof.

(i)From Theorem 23-(vii), we have = (ii)Since =.(iii)From (i) above and Theorem 23-(vii), we get = = .(iv)If , then . If , then = (v)From Theorem 19-(iii), Theorem 21-(v), Lemma (8.2) in [15] and (iv) above, we get ⊆(vi)From Theorem 23-(vii) and (vi) above, we get ≡(vii)From Theorem 23-(v) and (vi) above, we have ≡.(viii)From Theorem 23-(iii), we get ≤  .(ix)From (iii) above, we have 

6. --Convergence of Nets in Fuzzifying Tritopological Spaces

Definition 29. If is a FTTS, then the class of all nets in is defined as , where is a directed .

Definition 30. If is a FTTS, then the binary fuzzy predicates , , are defined as

 , ,

where stand for “ is (1,2,3)-α-convergence to ” and stand for “ is (1,2,3)-α-accumulation point of ”. Also, the binary crisp predicate is “almost in” and is “often in”.

Definition 31. Let . One has the following fuzzy sets:
is (1,2,3)-α-limit of ;
is (1,2,3)-α-adherence of .

Theorem 32. If is a FTTS, , and , then

(i);(ii);(iii);(iv), where standing for “ is a subnet of ”.

Proof.

(i). Now, since , then and this implies . Therefore (ii)If , then from Theorem 21-(i) and (i) above we have  If , then . From Theorem 21-(i) and (i) above we have ≤ (iii)From Theorem 21-(vi) and (ii) above, we get = = (iv)We have if , then , for any and any . Therefore ≤ .

Theorem 33. If is a FTTS and is a universal net, then .

Proof.

Lemma 34. If is a FTTS, then .

Proof. If and , then .
≥ =
Conversely,
= =

7. -α-Convergence of Filters in Fuzzifying Tritopological Spaces

Definition 35. If is a FTTS and is the set of all filters on , then the binary fuzzy predicates , are defined as
;
, where .

Definition 36. The fuzzy sets
are (1,2,3)-α-limit of ;
are (1,2,3)-α-adherence of .

Theorem 37. If is a FTTS, then we have the following.

(1)If and is the filter corresponding to , i.e., , then

(i);(ii).

(2)If and is the net corresponding to , i.e., , where iff  , then

(i);(ii).

Proof.

(1)

(i)(ii)= =

(2)Similar to (i) above

(i)= .(ii)= .

8. Conclusion

The main contribution of the present paper is to give characterization of tri-α-open sets in fuzzifying tritopological space. We also define the concepts of tri-α-closed sets, tri-α-neighborhood system, tri-α-interior, tri-α-closure, tri-α-derived, tri-α-boundary, tri-α-exterior, and tri-α-convergence in fuzzifying tritopological spaces and some basics of such spaces. We present some problems for future study.

(1)Study the results of the present paper by considering the quad-α-open sets in fuzzifying quad-topological spaces.(2)Investigate relations between fuzzifying quad-topology, tritopology, bitopology and fuzzifying topology.(3)Study of quad-α-separation axioms in fuzzifying quad-topological spaces.(4)Generalize the results in the present work to soft fuzzifying topology.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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