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

Authors: Barah M. Sulaiman; Tahir H. Ismail
Journal: Advances in Fuzzy Systems
 (2019)

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## 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.


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## Body

## 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.IfX is the universe of discourse, and if τ∈I(P(X)) satisfy the following three conditions:(1)τ(X)=1andτ(∅)=1;(2) for anyG,H,τ(G∩H)≥τ(G)∧τ(H);(3) for anyGλ:λ∈Λ,τ(⋃λ∈ΛGλ)≥⋀λ∈Λτ(Gλ);thenτ is a fuzzifying topology and (X,τ) a fuzzifying topological space [9].The family of fuzzifying closed sets is denoted byF and defined as G∈F≔X~G∈τ, where X~G is the complement of G [9].The neighborhood system ofx is denoted by Nx∈I(P(X)) and defined as Nx(G)=supx∈H⊆GτH [9].The closure set of a setG⊆X is denoted by cl(G)∈IX and defined as clGx=1-Nx(X~G) [9].The fuzzifying interior set of a setG⊆X is denoted by int(G)∈IX and defined as intG(x)=Nx(G) [10].The family of all fuzzifyingα-open sets is denoted by ατ and defined as G∈ατ≔∀xx∈G→x∈int(cl(int(G)), i.e., ατG=infx∈G(int(cl(intG)))(x) [17].The family of all fuzzifyingα-closed sets is denoted by αℱ and defined as G∈αF≔X~G∈ατ [17].The fuzzifyingα-interior set of a set G⊆X is denoted by αint(G)∈IX and defined as follows: αintG(x)=αNx(G), where αNx is α–neighborhood system of x defined as αNx(G)=supx∈H⊆GατH [17].The fuzzifyingα-derived set of a set G⊆X is denoted by αd(G)∈IX and defined as x∈αd(G)≔∀HH∈αNx→H⋂(G~x)≠∅), i.e., αdG(x)=infH⋂G~x=∅1-αNx(H) [18].Theα-closure set of a set G⊆X is denoted by αcl(G)∈IX and defined as αclGx=infx∉H,G⊆H1-αF(H) [17].


## 2.(1,2,3)-α-Open Sets in Fuzzifying Tritopological Spaces

Definition 1.
If(X,τ1,τ2,τ3) 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 ατ(1,2,3)∈IP(X) and defined as G∈ατ(1,2,3)≔∀x(x∈G→x∈int1(cl2(int3G))), i.e., ατ(1,2,3)G=infx∈G(int1(cl2(int3G)))(x).(ii)
The family of all fuzzifying (1,2,3)-α-closed sets is denoted by αF(1,2,3)∈IP(X) and defined as G∈αF(1,2,3)≔X~G∈ατ(1,2,3).Lemma 2.
Let (X,τ1,τ2,τ3) be a FTTS.
IfG⊆H=1, then ⊨int1(cl2(int3G))⊆int1(cl2(int3H)).Proof.
IfG⊆H=1, then int3G⊆int3H ⇒ cl2(int3G)⊆cl2(int3H) then int1(cl2(int3G))⊆int1(cl2(int3H)).Lemma 3.
If(X,τ1,τ2,τ3) is a FTTS and G⊆X. Then (i)
⊨X~int1(cl2(int3G))≡cl1(int2(cl3X~G));(ii)
⊨X~cl1(int2(cl3G))≡int1(cl2(int3X~G)).Proof.
From Theorem 2.2-(5) in [10], we have(i)
X~int1(cl2(int3G))x=cl1(X~cl2(int3G))(x)=cl1(int2(X~int3G))(x)=cl1(int2(cl3X~G))(x).(ii)
X~cl1(int2(cl3G))x=int1(X~int2(cl3G))(x)=int1(cl2(X~cl3G))(x)=int1(cl2(int3X~G))(x).Theorem 4.
If(X,τ1,τ2,τ3) is a FTTS, then (i)
ατ(1,2,3)X=1, ατ(1,2,3)∅=1;(ii)
for anyGλ:λ∈Λ,ατ(1,2,3)(⋃λ∈ΛGλ)≥⋀λ∈Λατ(1,2,3)(Gλ);(iii)
αF(1,2,3)X=1, αF(1,2,3)∅=1;(iv)
for anyGλ:λ∈Λ,αF(1,2,3)(⋂λ∈ΛGλ)≥⋀λ∈ΛαF(1,2,3)(Gλ).Proof.
(i)
α τ ( 1,2 , 3 ) X = i n f x ∈ X ( i n t 1 ( c l 2 ( i n t 3 X ) ) ) ( x ) = i n f x ∈ X ( i n t 1 ( c l 2 ( X ) ) ) ( x ) = i n f x ∈ X ( i n t 1 ( X ) ) ( x ) = i n f x ∈ X ( X ) ( x ) = 1. α τ ( 1,2 , 3 ) ∅ = i n f x ∈ ∅ ( i n t 1 ( c l 2 ( i n t 3 ∅ ) ) ) ( x ) = i n f x ∈ ∅ ( i n t 1 ( c l 2 ( ∅ ) ) ) ( x ) = i n f x ∈ ∅ ( i n t 1 ( ∅ ) ) ( x ) = i n f x ∈ ∅ ( ∅ ) ( x ) = 1 . (ii)
From Lemma2, we have [Gλ⊆⋃λ∈ΛGλ]=1, then ⊨int1(cl2(int3Gλ))⊆int1(cl2(int3(⋃λ∈ΛGλ))), α τ 1,2 , 3 ( ⋃ λ ∈ Λ G λ ) = i n f x ∈ ⋃ λ ∈ Λ G λ i n t 1 ( c l 2 ( i n t 3 ( ⋃ λ ∈ Λ G λ ) ) ) ( x ) = i n f λ ∈ Λ i n f x ∈ G λ i n t 1 ( c l 2 ( i n t 3 ( ⋃ λ ∈ Λ G λ ) ) ) ( x )≥ infλ∈Λinfx∈Gλint1(cl2(int3(Gλ)))x=⋀λ∈Λατ(1,2,3)(Gλ)(x). (iii)
and (iv) are obvious.Lemma 5.
If(X,τ1,τ2,τ3) is a FTTS, then ⊨τ1≡τ3→τ1⊆ατ1,2,3.Proof.
From Theorem (2.2)-(3) in [10] and Lemma (2.1) in [15], we have (G∈τ1)∧(G∈τ3)=(G≡int1G)∧(G≡int3G)≤ G≡int3G=[(G⊆int3G)∧(int3(G)⊆G)]≤G⊆int3G≤[cl2(G)⊆cl2(int3(G))]≤[G⊆cl2(int3(G))]≤ [int1(G)⊆int1(cl2(int3(G)))]=[G⊆int1(cl2(int3(G)))]=[G∈ατ1,2,3].Theorem 6.
If(X,τ1,τ2,τ3) is a FTTS and F1, F2, and F3 are the families of closed sets with respect to τ1, τ2, and τ3, respectively, then (i)
⊨E∈τ1∧E∈F2∧E∈τ3→E≡int1(cl2(int3(E)));(ii)
⊨E∈F1∧E∈τ2∧E∈F3→E≡cl1(int2(cl3(E))).Proof.
(i)
From Theorem (2.2)-(3) in [10] and Theorem (5.2)-(3) in [9], we have E ∈ τ 1 ∧  E ∈ F 2 ∧  E ∈ τ 3 = [ ( E ≡ i n t 1 E ) ∧ ( E ≡ c l 2 E ) ∧ ( E ≡ i n t 3 E ) ]= [((E⊆int1E)∧(nt1(E)⊆E))∧((E⊆cl2E)∧(cl2(E)⊆E))∧((E⊆int3E)∧(int3(E)⊆E))]= [(E⊆int1E)∧(E⊆cl2E)∧(E⊆int3E)∧(int1(E)⊆E)∧(cl2(E)⊆E)∧(int3(E)⊆E)]≤ [(E⊆int1E)∧(E⊆cl2E)∧(cl2E⊆cl2(int3E))∧(int1(E)⊆E)∧(cl2(E)⊆E)∧(cl2(int3E)⊆ cl2(E))]≤[(E⊆int1E)∧(E⊆cl2(int3E))∧(int1(E)⊆E)∧(cl2(int3(E))⊆E))]≤[(E⊆int1E)∧(int1(E)⊆int1(cl2(int3E)))∧ (int1(E)⊆E)∧(int1(cl2(int3E))⊆int1(E))]≤ [E⊆int1(cl2(int3E))]∧[int1(cl2(int3E))⊆E]=[E≡int1(cl2(int3(E)))]. (ii)
It follows directly from (i).Remark 7.
The following example shows that
(i)ατ1⊆ατ(1,2,3);
(ii)ατ2⊆ατ(1,2,3);
(iii)ατ3⊆ατ(1,2,3);
(iv)ατ(1,2,3)=ατ(3,2,1).
It may not be true for all FTTS(X,τ1,τ2,τ3).Example 8.
ForX={a,b,c} and B={a,b}. Let τ1,τ2,τ3 be a fuzzifying tritopological space X defined by(1)τ1A=1ifA∈∅,X,a,34ifA∈c,a,c,0Otherwise.τ2A=1ifA∈∅,X,14ifA=c,0Otherwise.τ3A=1ifA∈∅,X,b,a,c,34ifA∈c,b,c,0ifA∈a,a,b.Now, we haveint1Ba=1, int1Bb=0, int1Bc=0, cl1(int1B)a=1, cl1(int1B)b=1, cl1(int1B)c=1/4, int1(cl1(int1B))a=1, int1(cl1(int1B))b=1/4, int1(cl1(int1B))c=1/4,⇒ατ1B=infx∈B(int1(cl1(int1B)))x=1/4.andint2Ba=0, int2Bb=0, int2Bc=0, cl2(int2B)a=0, cl2(int2B)b=0, cl2(int2B)c=0,int2(cl2(int2B))a=0, int2(cl2(int2B))b=0, int2(cl2(int2B))c=0,⇒ατ2B=infx∈B(int2(cl2(int2B)))x=0.andint3Ba=0, int3Bb=1, int3Bc=0, cl3(int3B)a=0, cl3(int3B)b=1, cl3(int3B)c=0,int3(cl3(int3B))a=0, int3(cl2(int1B))b=1, int3(cl3(int3B))c=0,⇒ατ3B=infx∈B(int3(cl3(int3B)))x=0.andint3Ba=0, int3Bb=1, int3Bc=0, cl2(int3B)a=1, cl2(int3B)b=1, cl2(int3B)c=3/4,int1(cl2(int3B))a=0, int1(cl2(int3B))b=3/4, int1(cl2(int3B))c=3/4,⇒ατ1,2,3B=infx∈B(int1(cl2(int3B)))x=3/4.andint1Ba=1, int1Bb=0, int1Bc=0, cl2(int1B)a=1, cl2(int1B)b=1, cl2(int1B)c=3/4,int3(cl2(int1B))a=3/4, int3(cl2(int1B))b=1, int3(cl2(int1B))c=3/4,⇒ατ3,2,1B=infx∈B(int3(cl2(int1B)))x=3/4.∴ατ1,2,3B=ατ3,2,1B. Therefore ατ1,2,3=ατ3,2,1, ατ1⊆ατ1,2,3, ατ2⊆ατ1,2,3, and ατ3⊆ατ1,2,3.Lemma 9.
If(X,τ1,τ2,τ3) is a FTTS, then ⊨τ1≡τ3→ατ(1,2,3)≡ατ(3,2,1).Proof.
From Theorem (2.2)-(3) in [10], we have (G∈τ1)∧(G∈τ3)=(G≡int1G)∧(G≡int3G)= [(G⊆int1G)∧(int1(G)⊆G)∧(G⊆int3G)∧(int3(G)⊆G)]≤(G⊆int3G)∧(G⊆int1G)≤ [(cl2(G)⊆cl2(int3(G)))∧(cl2(G)⊆cl2(int1(G)))]≤[(G⊆cl2(int3(G)))∧(G⊆cl2(int1(G)))]≤ [(int1(G)⊆int1(cl2(int3(G))))∧(int3(G)⊆int3(cl2(int1(G))))]=[(G⊆int1(cl2(int3(G))))∧(G⊆int3(cl2(int1(G))))]= [(G∈ατ1,2,3)∧(G∈ατ3,2,1)].
Thereforeατ(1,2,3)≡ατ(3,2,1).Theorem 10.
If(X,τ1,τ2,τ3) is a FTTS, then ⊨G∈αF(1,2,3)↔∀x(x∈cl1(int2(cl3G))→x∈G).Proof.
From Lemma3-(ii), we have [∀x(x∈cl1(int2(cl3G))→x∈G))]=[∀x(x∈X~G→x∈X~cl1(int2(cl3G))]= infx∈X~G(X~cl1(int2(cl3G)))(x)=infx∈X~G(int1(cl2(int3X~G)))(x)= [X~G∈ατ(1,2,3)]=[G∈αF(1,2,3)].Lemma 11.
If(X,τ1,τ2,τ3) is a FTTS, then (i)
⊨H≡˙cl2(int3G)∧G∈ατ(1,2,3)→G⊆int1H;(ii)
⊨H≡˙int2(cl3G)∧G∈αF(1,2,3)→cl1H⊆G.Proof.
(i)
H  ≡ ˙  c l 2 ( i n t 3 G ) ∧ ( G ∈ α τ 1,2 , 3 ) = [ ( H ⊆ c l 2 i n t 3 G ⟑ c l 2 ( i n t 3 G ) ⊆ H ) ∧ ( G ⊆ i n t 1 ( c l 2 ( i n t 3 G ) ) ) ]≤ [cl2(int3G)⊆H∧(G⊆int1(cl2(int3G)))]≤[int1(cl2(int3G))⊆int1(H)∧ (G⊆int1(cl2(int3G)))]≤G⊆int1(H). (ii)
From Theorem 2.2-(5) in [10], we have H  ≡ ˙  i n t 2 ( c l 3 G ) ∧ ( G ∈ α F 1,2 , 3 ) = [ ( H ⊆ i n t 2 ( c l 3 G ) ⟑ i n t 2 ( c l 3 G ) ⊆ H ) ∧ ( X ~ G ∈ α τ 1,2 , 3 ) ]≤ [(H⊆int2(cl3G)∧(X~G⊆int1(cl2(int3X~G)))]≤[(cl1(H)⊆cl1(int2(cl3G))∧(X~G⊆X~cl1(int2(cl3G)))]≤ [(cl1(H)⊆cl1(int2(cl3G))∧(cl1(int2(cl3G))⊆G)]≤[cl1(H)⊆G].Theorem 12.
If(X,τ1,τ2,τ3) is a FTTS, then (i)
⊨∃H(H∈τ3∧H⊆G⊆int1(cl2H))→G∈ατ(1,2,3);(ii)
⊨∃K(K∈F3∧cl1(int2(K))⊆G⊆K)→G∈αF(1,2,3).Proof.
(i)
From Theorem (2.2)-(3) in [10], we have [ ∃ H   ( H ∈ τ 3  ⋀  H ⊆ G ⊆ i n t 1 ( c l 2 H ) ) ] = s u p H ∈ P ( X ) H ≡ i n t 3 ( H )  ⋀  H ⊆ G  ⋀  G ⊆ i n t 1 ( c l 2 H )  ≤ s u p H ∈ P ( X )  ( H ⊆ i n t 3 H ) ∧ ( i n t 3 ( H ) ⊆ H )  ⋀  i n t 3 ( H ) ⊆ i n t 3 ( G )  ⋀  G ⊆ i n t 1 ( c l 2 H )≤ supH∈P(X)H⊆int3(H)⋀int3(H)⊆int3(G)⋀G⊆int1(cl2H)≤ supH⊆GH⊆int3(G)⋀G⊆int1(cl2H)≤supH⊆GH⊆int3(G)⋀G⊆int1(cl2H)≤ supH⊆G([cl2H⊆cl2(int3G)]⋀G⊆int1(cl2H))≤ supH⊆G([int1(cl2H)⊆int1(cl2(int3G))]⋀G⊆int1(cl2H))≤ supH⊆G[G⊆int1(cl2(int3G))]=[G∈ατ(1,2,3)]. (ii)
From (i) above and Theorem (2.2)-(5) in [10]. We have [ G ∈ α F ( 1,2 , 3 ) ] = [ X ~ G ∈ α τ ( 1,2 , 3 ) ] ≥ [ ∃ H   ( H ∈ τ 3  ⋀  H ⊆ X ~ G ⊆ i n t 1 ( c l 2 H ) ) ]= ∃HH∈τ3⋀X~int1(cl2H)⊆G⊆X~H=∃X~HX~H∈F3⋀cl1(int2X~H)⊆G⊆X~H= ∃KK∈F3⋀cl1(int2K)⊆G⊆K.


## 3.(1,2,3)-α–Neighborhood System in Fuzzifying Tritopological Spaces

Definition 13.
If(X,τ1,τ2,τ3) is a FTTS and x∈X. Then αNx1,2,3∈I(PX) indicates the “(1,2,3)-α-neighborhood system of x” and defined as G∈αNx(1,2,3)≔∃H(H∈ατ1,2,3⋀x∈H⊆G), i.e., αNx(1,2,3)G=supx∈H⊆Gατ(1,2,3)H.Theorem 14.
If(X,τ1,τ2,τ3) is a FTTS, then ⊨G∈ατ(1,2,3)↔∀x(x∈G→∃H(H⊆int1(cl2(int3G))∧x∈H⊆G)).Proof.
[ ∀ x   ( x ∈ G → ∃ H   ( H ⊆ i n t 1 ( c l 2 ( i n t 3 G ) ) ∧  x ∈ H ⊆ G ) ) ] = i n f x ∈ G s u p x ∈ H ⊆ G α τ 1,2 , 3 ( H ) = i n f x ∈ G α N x ( 1,2 , 3 ) G= ατ(1,2,3)(G).Theorem 15.
If (X,τ1,τ2,τ3) is a FTTS and G∈P(X), then (i)
⊨G∈ατ(1,2,3)↔∀x(x∈G→∃H(H∈αNx(1,2,3)⋀H⊆G)),(ii)
⊨τ1≡τ3→Nx(1)(G)≤αNx(1,2,3)(G).Proof.
(i)
From Theorem14 we get [ ∀ x   ( x ∈ G → ∃ H   ( H ∈ α N x ( 1,2 , 3 )  ⋀  H ⊆ G ) ) ] = i n f x ∈ G s u p H ⊆ G α N x ( 1,2 , 3 ) ( H )= infx∈GsupH⊆Gsupx∈K⊆Hατ(1,2,3)(K)=infx∈Gsupx∈K⊆Gατ(1,2,3)(K)=ατ(1,2,3)G. (ii)
From Lemma5 we get α N x ( 1,2 , 3 ) G = s u p x ∈ H ⊆ G α τ ( 1,2 , 3 ) H ≥ s u p x ∈ H ⊆ G τ 1 H = N x ( 1 ) G .Theorem 16.
If(X,τ1,τ2,τ3) is a FTTS, the mapping αN(1,2,3):X→INPX, x↦αNx(1,2), where INPX is the set of all normal fuzzy subset of PX, has the following properties:(i)
⊨G∈αNx(1,2,3)→x∈G,(ii)
⊨G⊆H→(G∈αNx(1,2,3)→H∈αNx(1,2,3)),(iii)
⊨G∈αNx(1,2,3)→∃K(K∈αNx(1,2,3)⋀K⊆G⋀∀y(y∈K→K∈αNx(1,2,3)).Proof.
(i)
If[G∈αNx(1,2,3)]=0, then (i) is obtained. If [G∈αNx(1,2,3)]=supx∈H⊆Gατ1,2,3(H)>0, then ∃H0 such that x∈H0⊆G. Now we have x∈G=1. Therefore[G∈αNx(1,2,3)]≤x∈G. (ii)
[ G ∈ α N x 1,2 , 3 ] = s u p x ∈ E ⊆ G α τ 1,2 , 3 E ≤ s u p x ∈ E ⊆ H α τ 1,2 , 3 E = [ H ∈ α N x 1,2 , 3 ] . (iii)
[ ∃ K   ( K ∈ α N x ( 1,2 , 3 ) ∧ K ⊆ G ⋀ ∀ y   ( y ∈ K → K ∈ α N y ( 1,2 , 3 ) ) ]= supK⊆G(αNx(1,2,3)⋀infy∈KαNy(1,2,3)K)=supK⊆G(αNx(1,2,3)⋀ατ(1,2,3)K= supK⊆Gατ(1,2,3)K≥supx∈K⊆Gατ(1,2,3)K=αNx(1,2,3)G=[G∈αNx(1,2,3)].


## 4.(1,2,3)-α-Derived Set and (1,2,3)-α-Closure Operator in Fuzzifying Tritopological Space

Definition 17.
If(X,τ1,τ2,τ3) is a FTTS, then αd(1,2,3)G indicates the “(1,2,3)-α-derived set of G” and defined as follows: x∈αd(1,2,3)G≔∀H(H∈αNx(1,2,3)→H∩(G~x)≠∅), i.e., αd(1,2,3)Gx=infH∩G~x=∅(1-αNx(1,2,3)H).Lemma 18.
α d ( 1,2 , 3 ) G x =  1 - α N x ( 1,2 , 3 ) ( X ~ G ∪ x ).Proof.
α d ( 1,2 , 3 ) G x = 1 - s u p H ∩ G ~ x = ∅ α N x 1,2 , 3 H = 1 - s u p H ∩ G ~ x = ∅ s u p x ∈ K ⊆ H α τ 1,2 , 3 K= 1-supx∈K⊆X~G∪xsupx∈K⊆Hατ1,2,3K=1-supx∈K⊆X~G∪xατ1,2,3K= 1-αNx(1,2,3)(X~G∪x).Theorem 19.
If(X,τ1,τ2,τ3) is a FTTS and G,H∈P(X), then (i)
⊨αd(1,2,3)∅=0;(ii)
⊨G⊆H→αd(1,2,3)G⊆αd(1,2,3)H;(iii)
⊨G∈αF(1,2,3)↔αd(1,2,3)G⊆G;(iv)
⊨τ1≡τ3→αd(1,2,3)G⊆d1G, where d1G is the fuzzifying derived set of G with respect to τ1.Proof.
(i)
From Lemma18 we have α d ( 1,2 , 3 ) ∅ x =  1 - α N x ( 1,2 , 3 ) X ~ ∅ ∪ { x } =  1 - α N x ( 1,2 , 3 ) X = 1 - 1 = 0 . (ii)
LetG⊆H, then from Lemma 18 and Theorem 16-(ii) we get α d ( 1,2 , 3 ) G x = 1 - α N x ( 1,2 , 3 ) X ~ G ∪ x ≤ 1 - α N x ( 1,2 , 3 ) X ~ H ∪ x = α d ( 1,2 , 3 ) H x (iii)
From Lemma18 and Theorem 15-(ii).We have [ α d 1,2 , 3 G ⊆ G ] = i n f x ∈ X ~ G ( 1 - α d 1,2 , 3 G x ) =  i n f x ∈ X ~ G α N x 1,2 , 3 ( X ~ G ∪ x )= infx∈X~GαNx1,2,3X~G=infx∈X~Gsupx∈H⊆X~Gατ1,2,3H=ατ1,2,3X~G= αF1,2,3G=[G∈αF(1,2,3)x]. (iv)
From Theorem15-(ii) and Lemma (5.1) in [9] we have α d ( 1,2 , 3 ) G = 1 - α N x ( 1,2 , 3 ) ( X ~ G ∪ x ) ≤ 1 - α N x ( 1 ) X ~ G ∪ x = d 1 ( G ) ( x ) .Definition 20.
If(X,τ1,τ2,τ3) is a FTTS, then αcl(1,2,3)G indicates the “(1,2,3)-α-closure set of G” and defined as x∈αcl1,2,3G≔∀H(H⊇G)∧(H∈αF(1,2,3))→x∈H), i.e., αcl(1,2,3)Gx=infx∉H⊇G(1-αF(1,2,3)H).Theorem 21.
If(X,τ1,τ2,τ3) is a FTTS, G,H∈P(X) and x∈X, then (i)
αcl(1,2,3)Gx=1-αNx(1,2,3)X~G;(ii)
⊨αcl(1,2,3)∅=0;(iii)
⊨G⊆αcl(1,2,3)G;(iv)
⊨αcl(1,2,3)G=αd(1,2,3)G∪G;(v)
⊨x∈αcl(1,2,3)G↔∀H(H∈αNx(1,2,3)→G∩H≠∅);(vi)
⊨G≡αcl(1,2,3)G↔G∈αF(1,2,3)(G);(vii)
⊨G⊆H→αcl(1,2,3)G⊆αcl(1,2,3)H;(viii)
⊨H≡˙αcl(1,2,3)G→H∈αF(1,2,3).Proof.
(i)
α c l ( 1,2 , 3 ) G x = i n f x ∉ H ⊇ G ( 1 - α F ( 1,2 , 3 ) H = i n f x ∉ H ⊇ G ( 1 -  α τ ( 1,2 , 3 ) X ~ H )= 1-supx∈X~H⊆X~Gατ(1,2,3)X~H=1-Nx(1,2,3)X~G. (ii)
α c l ( 1,2 , 3 ) ∅ x =  1 - α N x 1,2 , 3 X ~ ∅ = 1 - α N x 1,2 , 3 X = 1 - s u p x ∈ G ⊆ X α τ ( 1,2 , 3 ) G = 1 - 1 = 0. (iii)
IfG∈P(X) and for any x∈X and if x∉G, then x∈G≤[x∈αcl(1,2,3)G]. If x∈G, then αcl(1,2,3)Gx=1-αNx(1,2,3)X~G=1-0=1. Thus x∈G≤x∈αcl(1,2,3)G⟹[G⊆αcl(1,2,3)G]=1. (iv)
From Lemma18 and (iii) above, for any x∈X we have [ x ∈ ( α d ( 1,2 , 3 ) G ∪ G ) ] = max ⁡ ( ( 1 - α N x ( 1,2 , 3 ) X ~ G ∪ x ) , G ( x ) ). If x∈G, then x∈αd(1,2,3)G∪G=Gx=1=[x∈αcl(1,2,3)G]. If x∉G, then x∈αd(1,2,3)G∪G=1-αNx(1,2,3)X~G=x∈αcl(1,2,3)G. Thusαcl(1,2,3)G=[αd(1,2,3)G∪G]. (v)
[ ∀ H   ( H ∈ α N x ( 1,2 , 3 ) → G ∩ H ≠ ∅ ) ] = i n f H ⊆ X ~ G ( 1 - α N x ( 1,2 , 3 ) ( H ) )= 1-αNx(1,2,3)(X~G)=[x∈αcl(1,2,3)G]. (vi)
From Theorem19-(iii), Lemma (8.2) in [15] and (iv) above, since [ G ⊆ α d ( 1,2 , 3 ) G ∪ G ] = 1, we get α F ( 1,2 , 3 ) G = [ α d ( 1,2 , 3 ) G ⊆ G ] = [ α d ( 1,2 , 3 ) G ∪ G ⊆ G ] = [ α d ( 1,2 , 3 ) G ∪ G ⊆ G ]⋀ [G⊆αd(1,2,3)G∪G]=[αd(1,2,3)G∪G≡G]=[G≡αcl(1,2,3)G]. (vii)
IfG⊆H, then X~H⊆X~G. From (i) above and Theorem 16-(ii) we get α c l 1,2 , 3 G x = 1 - α N x 1,2 , 3 X ~ H ≤ 1 - α N x 1,2 , 3 X ~ H = α c l 1,2 , 3 H x. Thusαcl(1,2,3)G⊆αcl(1,2,3)H. (viii)
IfG⊆H=0, then [H≡˙αcl(1,2,3)G]=0. Assume that G ⊆ H = 1, then [H⊆αcl(1,2,3)G]=1-supx∈H~GαNx(1,2,3)(X~G) and α c l ( 1,2 , 3 ) G ⊆ H = i n f x ∈ X ~ H α N x ( 1,2 , 3 ) ( X ~ G ). Therefore H≡˙αcl(1,2,3)G=max⁡(0,infx∈X~HαNx(1,2,3)X~G-supx∈H~GαNx1,2,3(X~G)). IfH≡˙αcl(1,2,3)G>c, then infx∈X~HαNx(1,2,3)X~G>c+supx∈H~GαNx(1,2,3)(X~G). For anyx∈X~H, we get αNx(1,2,3)X~G>c+supx∈H~GαNx(1,2,3)(X~G). Thus supx∈E⊆X~Gατ(1,2,3)E>c+supx∈H~GαNx(1,2,3)X~G, i.e., ∃Ex such that x∈Ex⊆X~G and ατ(1,2,3)Ex>c+supx∈H~GαNx(1,2,3)X~G. To prove that Ex⊆X~H. If Ex⊈X~H, then ∃x′∈Ex and x′∈X~H. Hence we get supx∈H~GαNx(1,2,3)(X~G)≥αNx′(1,2,3)X~G≥ατ(1,2,3)Ex>c+supx∈H~GαNx(1,2,3)X~H ⟹ Contradiction. Therefore αF(1,2,3)H=ατ(1,2,3)X~H=infx∈X~HαNx(1,2,3)X~H≥infx∈X~Hατ(1,2,3)Ex≥ατ(1,2,3)Ex>c+supx∈H~GαNx(1,2,3)X~G>c, since c is arbitrary; thus H≡˙αcl(1,2,3)G≤H∈αF(1,2,3).


## 5.(1,2,3)-α–Interior, (1,2,3)-α-Exterior, and (1,2,3)-α-Boundary Operators in Fuzzifying Tritopological Space

Definition 22.
If(X,τ1,τ2,τ3) is a FTTS and G∈P(X), then αint(1,2,3)G indicates the “(1,2,3)-α-interior set of G” defined as αint(1,2,3)Gx=αNx(1,2,3)Theorem 23.
If(X,τ1,τ2,τ3) is a FTTS, G,H∈P(X) and x∈X, then (i)
⊨αint(1,2,3)X≡X;(ii)
⊨αint(1,2,3)G⊆G;(iii)
⊨τ1≡τ3→int1(G)⊆αint(1,2,3)G;(iv)
⊨H∈ατ(1,2,3)⋀H⊆G→H⊆αint(1,2,3)G;(v)
⊨G≡αint(1,2,3)G↔G∈ατ(1,2,3);(vi)
⊨G⊆H→αint(1,2,3)G⊆αint(1,2,3)H;(vii)
⊨αint(1,2,3)G≡X~αcl(1,2,3)X~G;(viii)
⊨αint(1,2,3)G≡G∩(X~αd(1,2,3)X~G);(ix)
⊨H≡˙αint(1,2,3)G→H∈ατ(1,2,3).Proof.
(i)
α i n t 1,2 , 3 X x = α N x 1,2 , 3 ( X ) = 1 ⟹ αint(1,2,3)X≡X (ii)
LetG∈P(X), x∈X. If x∉G, then αint(1,2,3)Gx=αNx(1,2,3)=0 ⟹ αint(1,2,3)G⊆G. (iii)
From Theorem15-(ii) we have i n t 1 G x = N x ( 1 ) G ≤ α N x ( 1,2 , 3 ) G = α i n t ( 1,2 , 3 ) G ( x ). Therefore int1Gx⊆αint(1,2,3)G. (iv)
IfH⊈G, then the result holds. IfH⊆G, then [ H ⊆ α i n t ( 1,2 , 3 ) G ] = i n f x ∈ H α i n t ( 1,2 , 3 ) G ( x ) = i n f x ∈ H α N x ( 1,2 , 3 ) G ≥ i n f x ∈ H α N x ( 1,2 , 3 ) H = α τ ( 1,2 , 3 ) H= [H∈ατ(1,2,3)⋀H⊆G]. (v)
G ≡ α i n t ( 1,2 , 3 ) G= min⁡(infx∈Gαint(1,2,3)Gx, infx∈X~G(1-αint(1,2,3)Gx))= min⁡(infx∈GαNx(1,2,3)G, infx∈X~G(1-αNx(1,2,3)G))= infx∈GαNx(1,2,3)G=ατ(1,2,3)G= [G∈ατ(1,2,3)] (vi)
From Definition22 and Theorem 16 -(ii) the proof follows. (vii)
From Theorem21-(i) we have (X~αcl(1,2,3)X~G)x=1-(1-αNx(1,2,3)G)=αNx(1,2,3)G=αint(1,2,3)Gx. Therefore αint(1,2,3)G=X~αcl(1,2,3)X~G. (viii)
From Lemma18 we get [ G ∩ ( X ~ α d ( 1,2 , 3 ) X ~ G ) ] = m i n ⁡ ( G x , α N x ( 1,2 , 3 ) ( G ∪ x )). If x∉G, then [ G ∩ ( X ~ α d ( 1,2 , 3 ) X ~ G ) ] = 0 = α N x ( 1,2 , 3 ) G = α i n t ( 1,2 , 3 ) G x. If x∈G, then [ G ∩ ( X ~ α d ( 1,2 , 3 ) X ~ G ) ] = α N x ( 1,2 , 3 ) G = α i n t ( 1,2 , 3 ) G x. Therefore α i n t ( 1,2 , 3 ) G = G ∩ ( X ~ α d ( 1,2 , 3 ) X ~ G ). (ix)
From Theorem21-(ix) and (vii) above we get H  ≡ ˙  α i n t ( 1,2 , 3 ) G = [ X ~ H  ≡ ˙  α c l ( 1,2 , 3 ) X ~ G ] ≤ [ X ~ H ∈ α F ( 1,2 , 3 ) ] = [ H ∈ α τ ( 1,2 , 3 ) ] .Definition 24.
If (X,τ1,τ2,τ3) is a FTTS and G⊆X. Then αext(1,2,3)G indicates the “(1,2,3)-α-exterior set of G” and defined as x∈αext(1,2,3)G≔x∈αint(1,2,3)(X~G), i.e., αext(1,2,3)Gx=αint(1,2,3)(X~G)(x).Theorem 25.
If(X,τ1,τ2,τ3) is a FTTS and G⊆X. Then (i)
⊨αext(1,2,3)∅≡X;(ii)
⊨αext(1,2,3)G⊆X~G;(iii)
⊨τ1≡τ3→αext1G⊆αext(1,2,3)G;(iv)
⊨G∈αF(1,2,3)↔αext(1,2,3)G≡X~G;(v)
⊨H∈αF(1,2,3)∧G⊆H→X~H⊆αext(1,2,3)G;(vi)
⊨H⊆G→αext(1,2,3)H⊆αext(1,2,3)G;(vii)
⊨αext(1,2,3)G≡(X~G)∩(X~αd(1,2,3)G);(viii)
⊨αext(1,2,3)G≡X~αcl(1,2,3)G);(ix)
⊨x∈αext(1,2,3)G↔∃H(x∈H∈ατ(1,2,3)⋀H∩G=∅).Proof.
The proofs of (i) - (vii) follow from Theorem23.(ix)
∃Hx∈H∈ατ(1,2,3)⋀H∩G=∅=supx∈H⊆X~Gατ1,2,3H=αNx1,2,3X~G= αint(1,2,3)X~Gx=αext(1,2,3)Gx. By Definition 24Definition 26.
If(X,τ1,τ2,τ3) is a FTTS and G⊆X, then αb(1,2,3)G indicates the “(1,2,3)-α-boundary of a set G” and defined as x∈αb(1,2,3)G≔(x∉αint(1,2,3)(G))⋀(x∉αint(1,2,3)X~G), i.e., x∈αb(1,2,3)G(x)≔min(1-αint(1,2,3)(G)(x))⋀(1-αint(1,2,3)X~G(x)).Lemma 27.
If(X,τ1,τ2,τ3) is a FTTS, G∈PX and x∈X, then ⊨x∈αb(1,2,3)(A)↔∀H(H∈αNx(1,2,3)→(H∩G≠∅)⋀(H∩X~G)≠∅).Proof.
[ ∀ H   ( H ∈ α N x ( 1,2 , 3 ) → H ∩ G ≠ ∅  ⋀  H ∩ X ~ G ) ≠ ∅ ]= min(infH⊆G(1-αNx(1,2,3)H),infH⊆X~G(1-αNx(1,2,3)H))=min(1-αNx(1,2,3)G,1-αNx(1,2,3)X~G)= min(1-αint(1,2,3)G(x),1-αint(1,2,3)X~G(x))=[x∈αb(1,2,3)G].Theorem 28.
If(X,τ1,τ2,τ3) is a FTTS and G∈PX, then (i)
⊨αb(1,2,3)G≡αcl(1,2,3)G∩αcl(1,2,3)X~G;(ii)
⊨αb(1,2,3)G≡αb(1,2,3)(X~G);(iii)
⊨x~αb(1,2,3)G≡αint(1,2,3)G∪αint(1,2,3)X~G;(iv)
⊨αcl(1,2,3)(G)≡G∪αb(1,2,3)G;(v)
⊨αb(1,2,3)G⊆G↔G∈αF(1,2,3);(vi)
⊨αint(1,2,3)(G)≡G∩(X~αb(1,2,3)G);(vii)
⊨(αb(1,2,3)G∩G≡∅)↔A∈ατ(1,2,3);(viii)
⊨τ1≡τ3→αb(1,2,3)(G)⊆b1(G);(ix)
⊨X~αb(1,2,3)G≡αint(1,2,3)G∪αext(1,2,3)X~G.Proof.
(i)
From Theorem23-(vii), we have ( α c l ( 1,2 , 3 ) G ∩ α c l ( 1,2 , 3 ) X ~ G x ) = min ⁡ ( α c l ( 1,2 , 3 ) G x , α c l ( 1,2 , 3 ) X  ~ G x )= min⁡(1-αint(1,2,3)Gx,1-αint(1,2,3)X~Gx)=αb(1,2,3)(G)(x). (ii)
Sinceαb(1,2,3)G(x)=min(1-αNx(1,2,3)G(x),1-αNx(1,2,3)X~G(x)) =min(1-αNx(1,2,3)X~G(x),1-αNx(1,2,3)G(x))=αb(1,2,3)X~G(x). (iii)
From (i) above and Theorem23-(vii), we get X ~ α b ( 1,2 , 3 ) G ≡ X ~ ( α c l ( 1,2 , 3 ) G ∩ α c l ( 1,2 , 3 ) X ~ G )= (X~αcl(1,2,3)G)∪(X~αcl(1,2,3)(X~G)= αint(1,2,3)(X~G)∪αint(1,2,3)(G). (iv)
Ifx∈G, then αcl(1,2,3)(G)(x)=1=(G∪αb(1,2,3)(G))(x). If x∉G, then (G∪αb(1,2,3)G)x=αb(1,2,3)Gx=min(1-αint(1,2,3)Gx,1-αint(1,2,3)X~Gx)= 1-αint(1,2,3)X~Gx=αcl(1,2,3)(G)(x). (v)
From Theorem19-(iii), Theorem 21-(v), Lemma (8.2) in [15] and (iv) above, we get G ∈ α F ( 1,2 , 3 ) ↔ α d ( 1,2 , 3 ) G ⊆ G  ↔ G ∪ α d ( 1,2 , 3 ) ( G ) ⊆ G  ↔ α c l ( 1,2 , 3 ) ( G )⊆ G↔G∪αb(1,2,3)(G)⊆G↔αb(1,2,3)(G)⊆G (vi)
From Theorem23-(vii) and (vi) above, we get α i n t ( 1,2 , 3 ) ( G ) ≡ X ~ α c l ( 1,2 , 3 ) ( X ~ G ) ≡ X ~ ( X ~ G ∪ α b ( 1,2 , 3 ) X ~ G )≡ G∩(X~αb(1,2,3)X~G)≡G∩(X~αb(1,2,3)G). (vii)
From Theorem23-(v) and (vi) above, we have (αb(1,2,3)(G)∩G≡∅)↔(X~αb(1,2,3)(G))∪(X~G))≡ X↔G⊆X~αb(1,2,3)G↔G∩(X~αb(1,2,3)G)≡G↔αint(1,2,3)G≡G↔G∈ατ(1,2,3). (viii)
From Theorem23-(iii), we get αb(1,2,3)Gx=min⁡(1-αint(1,2,3)G(x),1-αint(1,2,3)(X~G)(x))≤ min⁡(1-int1Gx,1-int1GX~Gx=αb1(G) α b ( 1,2 , 3 ) G ⊆ b 1 G x. (ix)
From (iii) above, we have X ~ α b ( 1,2 , 3 ) G ≡ α i n t ( 1,2 , 3 ) G ∪ α i n t ( 1,2 , 3 ) X ~ G ≡ α i n t ( 1,2 , 3 ) G ∪ α e x t ( 1,2 , 3 ) G .


## 6.(1,2,3)-α-Convergence of Nets in Fuzzifying Tritopological Spaces

Definition 29.
If(X,τ1,τ2,τ3) is a FTTS, then the class of all nets in X is defined as NX={SsuchthatS:D→X, where (D,≥) is a directed set}.Definition 30.
If(X,τ1,τ2,τ3) is a FTTS, then the binary fuzzy predicates ⊳1,2,31,2,3α, ∝1,2,3α∈I(N(X)×X), are defined asS⊳1,2,3αx≔∀G(G∈αNx1,2,3→S⊂~G),S∝1,2,3αx≔∀G(G∈αNx1,2,3→S⊏~G),S∈N(X),
whereS⊳1,2,3αx stand for “S is (1,2,3)-α-convergence to x” and S∝1,2,3αx stand for “x is (1,2,3)-α-accumulation point of S”. Also, the binary crisp predicate ⊂~ is “almost in” and ⊏~ is “often in”.Definition 31.
LetT∈N(X). One has the following fuzzy sets:
l i m 1,2 , 3 α T ( x ) = [ T  ⊳ 1,2 , 3 α  x ] is (1,2,3)-α-limit of T;
a d h 1,2 , 3 α T ( x ) = [ T  ∝ 1,2 , 3 α  x ] is (1,2,3)-α-adherence of T.Theorem 32.
If(X,τ1,τ2,τ3) is a FTTSx∈X, G∈PX, and S∈N(X), then (i)
⊨∃S((S⊆G~{x})∧(S⊳1,2,3αx))→x∈αd(1,2,3)(G);(ii)
⊨∃S((S⊆G)⋀(S⊳1,2,3αx))→x∈αcl(1,2,3)(G);(iii)
⊨G∈αF(1,2,3)→∀S(S⊆G→lim1,2,3αS⊆G);(iv)
⊨∃T((T<S)⋀(T⊳1,2,3αx))→S∝1,2,3αx, where T<S standing for “T is a subnet of S”.Proof.
(i)
[ ∃ S   ( ( S ⊆ G ~ { x } )  ⋀  ( S  ⊳ 1,2 , 3 α  x ) ) ] = s u p S ⊆ G ~ { x } i n f S ⊄ ~ G ( 1 - α N x 1,2 , 3 H ). Now, since S⊆G~{x}, then S⊈(X~G)∪{x} and this implies S⊄~(X~G)∪{x}. Therefore i n f S ⊄ ~ G ( 1 - α N x 1,2 , 3 H ) ≤ 1 - α N x 1,2 , 3 X ~ G ∪ x = [ x ∈ α d ( 1,2 , 3 ) G ] . (ii)
Ifx∈G, then from Theorem 21-(i) and (i) above we have [ ∃ S   ( ( S ⊆ G )  ⋀  ( S  ⊳ 1,2 , 3 α  x ) ) ] = s u p S ⊆ G i n f S ⊄ ~ G ( 1 - α N x 1,2 , 3 H ) ≤ 1 - α N x 1,2 , 3 X ~ G = [ x ∈ α c l ( 1,2 , 3 ) G ] . Ifx∉G, then G~x=G. From Theorem 21-(i) and (i) above we have [ ∃ S   ( S ⊆ G  ⋀  ( S  ⊳ 1,2 , 3 α  x ) ) ] = [ ∃ S   ( S ⊆ G ~ x  ⋀  ( S  ⊳ 1,2 , 3 α  x ) ) ]≤ 1-αNx1,2,3X~G=αcl(1,2,3)G=[x∈αcl(1,2,3)(G)]. (iii)
From Theorem21-(vi) and (ii) above, we get [ G ∈ α F ( 1,2 , 3 ) ]  =  [ G ≡ α c l ( 1,2 , 3 ) ( G ) ] = [ G ⊆ α c l ( 1,2 , 3 ) ( G ) ]  ⋀  [ α c l ( 1,2 , 3 ) ( G ) ⊆ G ]  ≤  [ α c l ( 1,2 , 3 ) ( G ) ⊆ G ]= [X~G⊆X~αcl(1,2,3)(G)]=infx∈X~G(1-αcl(1,2,3)Gx)≤infx∈X~G(1-supS⊆GinfS⊄~H(1-αNx1,2,3(H)))= infx∉GinfS⊆G(1-infS⊄~H(1-αNx1,2,3H))=[∀S(S⊆G→lim1,2,3αS⊆G)]. (iv)
We have ifS⊄~G, then S⊏~G, for any S∈NX and any G⊆X. Therefore [ ∃ T   ( ( T < S ) ∧ ( T  ⊳ 1,2 , 3 α  x ) ) ] = s u p T < S i n f T ⊄ ~ G ( 1 - α N x 1,2 , 3 ( G ) ) = i n f T ⊄ ~ G ( 1 - i n f T < S α N x 1,2 , 3 ( G ) )≤ infT⊄~G(1-αNx1,2,3(G))≤infS⊄~G(1-αNx1,2,3(G))=infS⊏~G(1-αNx1,2,3(G))=[S∝1,2,3αx].Theorem 33.
If(X,τ1,τ2,τ3) is a FTTS and T is a universal net, then ⊨lim1,2,3αT=adh1,2,3αT.Proof.
l i m 1,2 , 3 α T x = i n f T ⊄ ~ G ( 1 - α N x 1,2 , 3 G ) =  i n f T ⊏ ~ G ( 1 - α N x 1,2 , 3 ( G ) ) =  a d h 1,2 , 3 α T ( x ) .Lemma 34.
If(X,τ1,τ2,τ3) is a FTTS, then ⊨(T⊳1,2,3αx))↔∀G(x∈G∈ατ(1,2,3)→T⊂~G).Proof.
IfH⊆G and T⊄~G, then T⊄~H.
[ T  ⊳ 1,2 , 3 α  x ) ] = i n f T ⊄ ~ G ( 1 - α N x 1,2 , 3 ( G ) ) =  1 - s u p T ⊄ ~ G s u p x ∈ H ⊆ G α τ ( 1,2 , 3 ) H≥ 1-supT⊄~H,x∈Hατ(1,2,3)H=infT⊄~H,x∈H(1-ατ(1,2,3)H)= [∀G(x∈G∈ατ(1,2,3)→T⊂~G].
Conversely,
[ ∀ G   ( x ∈ G ∈ α τ ( 1,2 , 3 ) → T  ⊂ ~  G ) ] = i n f T ⊄ ~ G , x ∈ G ( 1 - α τ ( 1,2 , 3 ) G )= infT⊄~G,x∈G(1-infx∈GsupH⊆GαNx1,2,3(H))≥1-supT⊄~G,x∈GαNx1,2,3H= infT⊄~G,x∈G(1-αNx1,2,3H)=[T⊳1,2,3αx].


## 7.(1,2,3)-α-Convergence of Filters in Fuzzifying Tritopological Spaces

Definition 35.
If(X,τ1,τ2,τ3) is a FTTS and F(X) is the set of all filters on X, then the binary fuzzy predicates ⊳1,2,3α, ∝1,2,3α∈I(FX×X) are defined as
K  ⊳ 1,2 , 3 α  x ≔ ∀ G   ( G ∈ α N x 1,2 , 3 → G ∈ K );
K  ∝ 1,2 , 3 α  x  ≔ ∀ G   ( G ∈ K → x ∈ α c l ( 1,2 , 3 ) G ), where K∈F(X).Definition 36.
The fuzzy sets
l i m 1,2 , 3 α K ( x ) = [ K  ⊳ 1,2 , 3 α  x ] are (1,2,3)-α-limit of K;
a d h ( 1,2 , 3 α K ( x ) = [ K  ∝ 1,2 , 3 α  x ] are (1,2,3)-α-adherence of K.Theorem 37.
If(X,τ1,τ2,τ3) is a FTTS, then we have the following.(1)
IfT∈N(X) and KT is the filter corresponding to T, i.e., KT={G:T⊂~G}, then
(i)
⊨lim1,2,3αKT=lim1,2,3αT;(ii)
⊨adh1,2,3αKT=adh1,2,3αT.(2)
IfK∈F(X) and TK is the net corresponding to K, i.e., TK:D→X,(x,G)↦x,(x,G)∈D, where D=x,G:x∈G∈K,x,G≥y,H iff  G⊆H, then
(i)
⊨lim1,2,3αTK=lim1,2,3αK;(ii)
⊨adh1,2,3αTK=adh1,2,3αK.Proof.
(1)
(i)
l i m 1,2 , 3 α K T x = i n f G ∉ K T ( 1 - α N x 1,2 , 3 ( G ) ) = i n f T ⊄ ~ G ( 1 - α N x 1,2 , 3 ( G ) ) = l i m 1,2 , 3 α T . (ii)
a d h 1,2 , 3 α K T = i n f G ∈ K T α c l ( 1,2 , 3 ) ( G ) ( x ) = i n f T ⊂ ~ G ( 1 - α N x 1,2 , 3 X ~ G )= infT⊄~X~G(1-αNx1,2,3X~G)=infT⊏~X~G(1-αNx1,2,3X~G)= adh1,2,3αT. (2)
Similar to (i) above
(i)
l i m 1,2 , 3 α T K = [ T K  ⊳ 1,2 , 3 α  x ] = i n f T K ⊄ ~ G ( 1 - α N x 1,2 , 3 ( G ) ) = i n f G ∉ K ( 1 - α N x 1,2 , 3 ( G ) )= lim1,2,3αK. (ii)
a d h 1,2 , 3 α T K ( x ) = [ T K ∝ 1,2 , 3 α x ] = i n f T K  ⊏ ~  G ( 1 - α N x 1,2 , 3 ( G ) ) =  i n f X ~ G ∈ K α c l ( 1,2 , 3 ) ( X ~ G )= adh1,2,3αK.


## 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.



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