International Journal of Mathematical Analysis
Vol. 13, 2019, no. 4, 191 – 203
HIKARI Ltd, www.m-hikari.com
https://doi.org/10.12988/ijma.2019.9319
On Tri 𝛂-Separation Axioms in Fuzzifying
Tri-Topological Spaces
Barah M. Sulaiman and Tahir H. Ismail
Mathematics Department
College of Computer Science and Mathematics
University of Mosul, Iraq
This article is distributed under the Creative Commons by-nc-nd Attribution License.
Copyright © 2019 Hikari Ltd.
Abstract
The present article introduce 𝛼𝑇0(1,2,3)
(Kolmogorov), 𝛼𝑇1(1,2,3)
(Fréchet),
𝛼𝑇2(1,2,3)
(Hausdorff), 𝛼ℛ(1,2,3)(𝛼-regular), 𝛼𝒩(1,2,3)(𝛼-normal),𝛼𝑅0(1,2,3)
, 𝛼𝑅1(1,2,3)
and 𝛼𝑅2(1,2,3)
separation axioms in fuzzifying tri-topological spaces and studying
the relation among them and also some of their properties.
Keywords: Fuzzifying Tri topology; Fuzzifying tri 𝛼-separation axioms
1 Introduction
Ying (1991-1993) introduced the concept of the term “fuzzifying topologyˮ [7-
9]. Wuyts and Lowen (1983) studied "separation axioms in fuzzy topological
spaces" [6]. Shen (1993) introduced and studied 𝑇0, 𝑇1, 𝑇2 (Hausdorff), 𝑇3
(regularity), 𝑇4 (normality)-separation axioms in fuzzifying topology [3]. Khedr et
al. (2001) studied “separation axioms in fuzzifying topologyˮ [2]. Sayed (2014)
presented "α-separation axioms based on Łukasiewicz logic" [4]. Allam et al.
(2015) studied “semi separation axioms in fuzzifying bitopological spacesˮ [1].
We use the fundamentals of fuzzy logic with consonant set theoretical notations
which are introduced by Ying (1991-1993) [7-9] throughout this paper.
Definition 1.1 [5]
If (𝑋, 𝜏1, 𝜏2, 𝜏3) is a fuzzifying tri-topological space (FTTS),
192 Barah M. Sulaiman and Tahir H. Ismail
(i) The family of fuzzifying (1,2,3) α-open sets in 𝑋, symbolized as 𝛼𝜏(1,2,3) ∈
ℑ(𝑃(𝑋)), and defined as
𝐸 ∈ 𝛼𝜏(1,2,3) ≔ ∀ 𝑥 (𝑥 ∈ 𝐸 → 𝑥 ∈ 𝑖𝑛𝑡1(𝑐𝑙2(𝑖𝑛𝑡3(𝐸)))),
i.e., 𝛼𝜏(1,2,3)(𝐸) = 𝑖𝑛𝑓𝑥∈𝐸
(𝑖𝑛𝑡1(𝑐𝑙2(𝑖𝑛𝑡3(𝐸))))(𝑥).
(ii) The family of fuzzifying (1,2,3) α-closed sets in 𝑋, symbolized as 𝛼ℱ(1,2,3),
and defined by 𝐹 ∈ 𝛼ℱ(1,2,3) ≔ 𝑋~𝐹 ∈ 𝛼𝜏(1,2,3).
(iii) The (1,2,3) α-neighborhood system of 𝑥, denoted by 𝛼𝑁𝑥(1,2,3)
and defined as
𝐸 ∈ 𝛼𝑁𝑥(1,2,3)
≔ ∃ 𝐹 (𝐹 ∈ 𝛼𝜏(1,2,3) ⋀ 𝑥 ∈ 𝐹 ⊆ 𝐸);
i.e. 𝛼𝑁𝑥(1,2,3)(𝐸) = 𝑠𝑢𝑝
𝑥∈𝐹⊆𝐸𝛼𝜏(1,2,3)(𝐹).
(iv) The (1,2,3) α-derived set of E ⊆ X, denoted by 𝛼𝑑(1,2,3)(𝐸) and defined as
𝑥 ∈ 𝛼𝑑(1,2,3)(𝐸) ≔ ∀ 𝐹 (𝐹 ∈ 𝛼𝑁𝑥(1,2,3)
→ 𝐹 ∩ (𝐸 − {𝑥}) ≠ ∅),
i.e., 𝛼𝑑(1,2,3)(𝐸)(𝑥) = 𝑖𝑛𝑓𝐹∩(𝐸−{𝑥})≠∅
(1 − 𝛼𝑁𝑥(1,2,3)(𝐹)).
(v) The (1,2,3) α-closure set of 𝐸 ⊆ 𝑋, denoted by 𝛼𝑐𝑙(1,2,3)(𝐸) and defined as
𝑥 ∈ 𝛼𝑐𝑙(1,2,3)(𝐸) ≔ ∀ 𝐹 (𝐹 ⊇ 𝐸) ∩ (𝐹 ∈ 𝛼ℱ(1,2,3)) → 𝑥 ∈ 𝐹),
i.e., 𝛼𝑐𝑙(1,2,3)(𝐸)(𝑥) = 𝑖𝑛𝑓𝑥∉𝐹⊇𝐸
(1 − 𝛼ℱ(1,2,3)(𝐹)).
(vi) The (1,2,3) α-interior set of 𝐸 ⊆ 𝑋, denoted by 𝛼𝑖𝑛𝑡(1,2,3)(𝐸) and defined as
𝛼𝑖𝑛𝑡(1,2,3)(𝐸)(𝑥) = 𝛼𝑁𝑥(1,2,3)
(𝐸).
(vii) The (1,2,3) α-exterior set of 𝐸 ⊆ 𝑋, denoted by 𝛼𝑒𝑥𝑡(1,2,3)(𝐸) and defined as
𝑥 ∈ 𝛼𝑒𝑥𝑡(1,2,3)(𝐸) ≔ 𝑥 ∈ 𝛼𝑖𝑛𝑡(1,2,3)(𝑋~𝐸)(𝑥),
i.e. 𝛼𝑒𝑥𝑡(1,2,3)(𝐸)(𝑥) = 𝛼𝑖𝑛𝑡(1,2,3)(𝑋~𝐸)(𝑥).
(viii) The (1,2,3) α-boundary set of 𝐸 ⊆ 𝑋, denoted by 𝛼𝑏(1,2,3)(𝐸) and defined as
𝑥 ∈ 𝛼𝑏(1,2,3)(𝐸) ≔ (𝑥 ∉ 𝛼𝑖𝑛𝑡(1,2,3)(𝐸)) ⋀ (𝑥 ∉ 𝛼𝑖𝑛𝑡(1,2,3)(𝑋~𝐸)),
i.e. 𝛼𝑏(1,2,3)(𝐸)(𝑥) ≔ 𝑚𝑖𝑛(1 − 𝛼𝑖𝑛𝑡(1,2,3)(𝐸)(𝑥)) ⋀ (1 −
𝛼𝑖𝑛𝑡(1,2,3)(𝑋~𝐸)(𝑥)).
2 Tri 𝛂-Separation axioms in fuzzifying tri-topological spaces
Remark 2.2 We consider the following notations:
𝛼𝒦𝑥,𝑦(1,2,3)
≔ ∃ 𝐺 ((𝐺 ∈ 𝛼𝑁𝑥(1,2,3)
⋀ 𝑦 ∉ 𝐺) ⋁ (𝐺 ∈ 𝛼𝑁𝑦(1,2,3)
⋀ 𝑥 ∉ 𝐺));
𝛼ℋ𝑥,𝑦(1,2,3)
≔ ∃ 𝐻 ∃ 𝐸 (𝐻 ∈ 𝛼𝑁𝑥(1,2,3)
⋀ 𝐸 ∈ 𝛼𝑁𝑦(1,2,3)
⋀ 𝑦 ∉ 𝐻 ⋀ 𝑥 ∉ 𝐸);
𝛼ℳ𝑥,𝑦(1,2,3)
≔ ∃ 𝐻 ∃ 𝐸 (𝐻 ∈ 𝛼𝑁𝑥(1,2,3)
⋀ 𝐸 ∈ 𝛼𝑁𝑦(1,2,3)
⋀ 𝐻⋂𝐸 = ∅).
Definition 2.3 If 𝛺 is the class of all FTTSs. The predicates 𝛼𝑇𝑖(1,2,3)
, 𝛼𝑅𝑖(1,2,3)
∈
ℑ(𝛺), 𝑖 = 0,1,2, are defined as follow
(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇0(1,2,3)
≔ ∀ 𝑥 ∀ 𝑦 (𝑥 ∈ 𝑋 ⋀ 𝑦 ∈ 𝑋 ⋀ 𝑥 ≠ 𝑦 → 𝛼𝒦𝑥,𝑦(1,2,3)
);
(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇1(1,2,3)
≔ ∀ 𝑥 ∀ 𝑦 (𝑥 ∈ 𝑋 ⋀ 𝑦 ∈ 𝑋 ⋀ 𝑥 ≠ 𝑦 → 𝛼ℋ𝑥,𝑦(1,2,3)
);
(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇2(1,2,3)
≔ ∀ 𝑥 ∀ 𝑦 (𝑥 ∈ 𝑋 ⋀ 𝑦 ∈ 𝑋 ⋀ 𝑥 ≠ 𝑦 → 𝛼ℳ𝑥,𝑦(1,2,3)
);
On tri 𝛂-separation axioms in fuzzifying tri-topological spaces 193
(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅0(1,2,3)
≔ ∀ 𝑥 ∀ 𝑦 (𝑥 ∈ 𝑋 ⋀ 𝑦 ∈ 𝑋 ⋀ 𝑥 ≠ 𝑦 → (𝛼𝒦𝑥,𝑦(1,2,3)
→
𝛼ℋ𝑥,𝑦(1,2,3)
);
(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅1(1,2,3)
≔ ∀ 𝑥 ∀ 𝑦 (𝑥 ∈ 𝑋 ⋀ 𝑦 ∈ 𝑋 ⋀ 𝑥 ≠ 𝑦 → (𝛼𝒦𝑥,𝑦(1,2,3)
→
𝛼ℳ𝑥,𝑦(1,2,3)
);
(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅2(1,2,3)
≔ ∀ 𝑥 ∀ 𝑦 (𝑥 ∈ 𝑋 ⋀ 𝑦 ∈ 𝑋 ⋀ 𝑥 ≠ 𝑦 → (𝛼ℋ𝑥,𝑦(1,2,3)
→
𝛼ℳ𝑥,𝑦(1,2,3)
).
Definition 2.4 If 𝛺 is the class of all FTTSs. The predicates 𝛼ℛ(1,2,3), 𝛼𝒩(1,2,3) ∈ℑ(Ω), are defined as follow
(1) (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼ℛ(1,2,3) ≔ ∀ 𝑥 ∀ 𝑈 (𝑥 ∈ 𝑋 ⋀ 𝑈 ∈ 𝛼ℱ(1,2,3) ⋀ 𝑥 ∉ 𝑈 →
∃ 𝐺 ∃ 𝐻 (𝐺 ∈ 𝛼𝑁𝑥(1,2,3)
⋀ 𝐻 ∈ 𝛼𝜏(1,2,3) ⋀ 𝑈 ⊆ 𝐻 ⋀ 𝐺⋂𝐻 = ∅));
(2) (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝒩(1,2,3) ≔ ∀ 𝐺 ∀ 𝐻 (𝐺 ∈ 𝛼ℱ(1,2,3) ⋀ 𝐻 ∈ 𝛼ℱ(1,2,3) ⋀ 𝐺⋂𝐻 =
∅) → ∃ 𝑈 ∃ 𝑉 (𝑈 ∈ 𝛼𝜏(1,2,3) ⋀ 𝑉 ∈ 𝛼𝜏(1,2,3)⋀ 𝐺 ⊆ 𝑉 ⋀𝐻 ⊆ 𝑈 ⋀ 𝑈⋂𝑉 = ∅).
Definition 2.5 If 𝛺 is the class of all FTTSs. The predicates 𝛼𝑇3(1,2,3)
, 𝛼𝑇4(1,2,3)
∈ℑ(𝛺) are defined as follow
(1) 𝛼𝑇3(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) ≔ 𝛼ℛ(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) ⟑ 𝛼𝑇1
(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3);
(2) 𝛼𝑇4(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) ≔ 𝛼𝒩(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) ⟑ 𝛼𝑇1
(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3).
Remark 2.6 If (𝑋, 𝜏1, 𝜏2, 𝜏3) is a FTTS. Note that
(1) 𝛼𝑇𝑖(1,2,3)
= 𝛼𝑇𝑖(3,2,1)
, 𝑖 = 0,1,2,3,4;
(2) 𝛼𝑅𝑖(1,2,3)
= 𝛼𝑅𝑖(3,2,1)
, 𝑖 = 0,1,2.
Lemma 2.7 If (𝑋, 𝜏1, 𝜏2, 𝜏3) is a FTTS. Then
(1) ⊨ 𝛼ℳ𝑥,𝑦(1,2,3)
→ 𝛼ℋ𝑥,𝑦(1,2,3)
;
(2) ⊨ 𝛼ℋ𝑥,𝑦(1,2,3)
→ 𝛼𝒦𝑥,𝑦(1,2,3)
;
(3) ⊨ 𝛼ℳ𝑥,𝑦(1,2,3)
→ 𝛼𝒦𝑥,𝑦(1,2,3)
.
Proof.
(1) [ 𝛼𝑀𝑥,𝑦(1,2,3)
] = 𝑠𝑢𝑝𝐵⋂𝐶=∅
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐵), 𝛼𝑁𝑦
(1,2,3)(𝐶)) ≤
𝑠𝑢𝑝𝑦∉𝐵,𝑥∉𝐶
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐵), 𝛼𝑁𝑦
(1,2,3)(𝐶)) = [𝛼ℋ𝑥,𝑦(1,2,3)
].
(2) [ 𝛼𝒦𝑥,𝑦(1,2,3)
] = 𝑚𝑎𝑥(𝑠𝑢𝑝 𝑦∉𝐴
𝛼𝑁𝑥(1,2,3)(𝐴), 𝑠𝑢𝑝
𝑥∉𝐴 𝛼𝑁𝑦
(1,2,3)(𝐴))
≥ 𝑠𝑢𝑝𝑦∉𝐴
𝛼𝑁𝑥(1,2,3)(𝐴) ≥
𝑠𝑢𝑝𝑦∉𝐴,𝑥∉𝐵
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐴), 𝛼𝑁𝑦
(1,2,3)(𝐵)) = [𝛼ℋ𝑥,𝑦(1,2,3)
].
(3) is concluded from (1) and (2) above.
Theorem 2.8 If (𝑋, 𝜏1, 𝜏2, 𝜏3) is a FTTS. Then
194 Barah M. Sulaiman and Tahir H. Ismail
⊨ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇0(1,2,3)
↔ ∀ 𝑥 ∀ 𝑦 (𝑥 ∈ 𝑋 ⋀ 𝑦 ∈ 𝑋 ⋀ 𝑥 ≠ 𝑦 → 𝑥 ∉𝛼𝑐𝑙(1,2,3)({𝑦})⋁𝑦 ∉ 𝛼𝑐𝑙(1,2,3)({𝑥})).
Proof.
𝛼𝑇0(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)
= 𝑖𝑛𝑓𝑥≠𝑦
𝑚𝑎𝑥(𝑠𝑢𝑝𝑦∉𝐴
𝛼𝑁𝑥(1,2,3)(𝐴), 𝑠𝑢𝑝
𝑥∉𝐴 𝛼𝑁𝑦
(1,2,3)(𝐴))
= 𝑖𝑛𝑓𝑥≠𝑦
𝑚𝑎𝑥(𝛼𝑁𝑥(1,2,3)(𝑋~{𝑦}), 𝛼𝑁𝑦
(1,2,3)(𝑋~{𝑥}))
= 𝑖𝑛𝑓𝑥≠𝑦
𝑚𝑎𝑥(1 − 𝛼𝑐𝑙(1,2,3)({𝑦})(𝑥),1 − 𝛼𝑐𝑙(1,2,3)({𝑥})(𝑦))
= [∀ 𝑥 ∀ 𝑦 (𝑥 ∈ 𝑋 ⋀ 𝑦 ∈ 𝑋 ⋀ 𝑥 ≠ 𝑦 → 𝑥 ∉ 𝛼𝑐𝑙(1,2,3)({𝑦})⋁𝑦 ∉ 𝛼𝑐𝑙(1,2,3)({𝑥}))].
Theorem 2.9 If (𝑋, 𝜏1, 𝜏2, 𝜏3) is a FTTS. Then
⊨ ∀ 𝑥 ({𝑥} ∈ 𝛼ℱ(1,2,3)) ↔ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇1(1,2,3)
.
Proof.
𝛼𝑇1(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)
= 𝑖𝑛𝑓𝑥1≠𝑥2
𝑚𝑖𝑛( 𝑠𝑢𝑝𝑥2∉𝐴
𝛼𝑁𝑥1
(1,2,3)(𝐴), 𝑠𝑢𝑝𝑥1∉𝐵
𝛼𝑁𝑥2
(1,2,3)(𝐵)) =
𝑖𝑛𝑓𝑥1≠𝑥2
𝑚𝑖𝑛(𝛼𝑁𝑥1
(1,2,3)(𝑋~{𝑥2}), 𝛼𝑁𝑥2
(1,2,3)(𝑋~{𝑥1})) ≤
𝑖𝑛𝑓𝑥1≠𝑥2
𝛼𝑁𝑥1
(1,2,3)(𝑋~{𝑥2}) = 𝑖𝑛𝑓𝑥2∈𝑋
𝑖𝑛𝑓𝑥1∈𝑋~{𝑥2}
𝛼𝑁𝑥1
(1,2,3)(𝑋~{𝑥2})
= 𝑖𝑛𝑓𝑥2∈𝑋
𝛼𝜏(1,2,3)(𝑋~{𝑥2}) = 𝑖𝑛𝑓𝑥∈𝑋
𝛼𝜏(1,2,3)(𝑋~{𝑥}) = 𝑖𝑛𝑓𝑥∈𝑋
𝛼ℱ(1,2,3)({𝑥}).
Now, for any 𝑥1, 𝑥2 ∈ 𝑋 with 𝑥1 ≠ 𝑥2.
[∀ 𝑥 ({𝑥} ∈ 𝛼ℱ(1,2,3))]
= 𝑖𝑛𝑓𝑥∈𝑋
[{𝑥} ∈ 𝛼ℱ(1,2,3)] = 𝑖𝑛𝑓𝑥∈𝑋
𝛼𝜏(1,2,3)(𝑋~{𝑥}) = 𝑖𝑛𝑓𝑥∈𝑋
𝑖𝑛𝑓𝑦∈𝑋~{𝑥}
𝛼𝑁𝑦(1,2,3)(𝑋~{𝑥})
≤ 𝑖𝑛𝑓𝑦∈𝑋~{𝑥2}
𝛼𝑁𝑦(1,2,3)(𝑋~{𝑥2}) ≤ 𝛼𝑁𝑥2
(1,2,3)(𝑋~{𝑥2}) = 𝑠𝑢𝑝𝑥2∉𝐴
𝛼𝑁𝑥1
(1,2,3)(𝐴).
By the same way, we have
[∀ 𝑥 ({𝑥} ∈ 𝛼ℱ(1,2,3))] ≤ 𝑠𝑢𝑝𝑥1∉𝐴
𝛼𝑁𝑥2
(1,2,3)(𝐵). So
[∀ 𝑥 ({𝑥} ∈ 𝛼ℱ(1,2,3))] ≤ 𝑖𝑛𝑓𝑥1≠𝑥2
𝑚𝑖𝑛( 𝑠𝑢𝑝𝑥2∉𝐴
𝛼𝑁𝑥1
(1,2,3)(𝐴), 𝑠𝑢𝑝𝑥1∉𝐵
𝛼𝑁𝑥2
(1,2,3)(𝐵))
= 𝛼𝑇1(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3).
Therefore 𝛼𝑇1(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) = [∀ 𝑥 ({𝑥} ∈ 𝛼ℱ(1,2,3))].
Definition 2.10 If (𝑋, 𝜏1, 𝜏2, 𝜏3) is a FTTS, we define
(1) 𝛼ℛ(1) (1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) ≔ ∀ 𝑥 ∀ 𝑈 (𝑥 ∈ 𝑋 ⋀ 𝑈 ∈ 𝛼ℱ(1,2,3) ⋀ 𝑥 ∉ 𝑈 →
∃ 𝐺 (𝐺 ∈ 𝛼𝑁𝑥(1,2,3)
⋀ 𝛼𝑐𝑙(1,2,3)(𝐺)⋂𝑈 = ∅));
(2) 𝛼ℛ(2) (1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) ≔ ∀ 𝑥 ∀ 𝑈 (𝑥 ∈ 𝑋 ⋀ 𝑈 ∈ 𝛼𝜏(1,2,3) ⋀ 𝑥 ∈ 𝑈 →
∃ 𝐺 ∃ 𝐻 (𝐺 ∈ 𝛼𝑁𝑥(1,2,3)
⋀ 𝐻 ∈ 𝛼𝜏(1,2,3) ⋀ 𝐺 ⊆ 𝑈 ⋀ 𝐺⋂𝐻 = ∅)).
On tri 𝛂-separation axioms in fuzzifying tri-topological spaces 195
Theorem 2.11 If (𝑋, 𝜏1, 𝜏2, 𝜏3) is a FTTS. Then
⊨ 𝛼ℛ(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) ↔ 𝛼ℛ(𝑖) (1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3), 𝑖 = 1,2.
Proof.
(a) [ 𝛼ℛ(1) (1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)]
= 𝑖𝑛𝑓𝑥∉𝑈
𝑚𝑖𝑛(1,1 − 𝛼ℱ(1,2,3)(𝑈)
+ 𝑠𝑢𝑝𝐺∈𝑃(𝑋)
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐺), 𝑖𝑛𝑓
𝑦∈𝑈 (1 − 𝛼𝑐𝑙(1,2,3)(𝐺)(𝑦))))
= 𝑖𝑛𝑓𝑥∉𝑈
𝑚𝑖𝑛(1,1 − 𝛼ℱ(1,2,3)(𝑈)
+ 𝑠𝑢𝑝𝐺∈𝑃(𝑋)
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐺), 𝑖𝑛𝑓
𝑦∈𝑈 𝛼𝑁𝑦
(1,2,3)(𝑋~𝐺)))
= 𝑖𝑛𝑓𝑥∉𝑈
𝑚𝑖𝑛(1,1 − 𝛼ℱ(1,2,3)(𝑈) +
𝑠𝑢𝑝𝐺⋂𝑈=∅,𝐺∈𝑃(𝑋)
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐺), 𝑖𝑛𝑓
𝑦∈𝑈 𝛼𝑁𝑦
(1,2,3)(𝑋~𝐺)))
= 𝑖𝑛𝑓𝑥∉𝑈
𝑚𝑖𝑛(1,1 − 𝛼ℱ(1,2,3)(𝑈)
+ 𝑠𝑢𝑝𝐺⋂𝐻=∅,𝐺∈𝑃(𝑋)
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐺), 𝑖𝑛𝑓
𝑦∈𝑈 𝑠𝑢𝑝𝑦∈𝐻⊆𝑋~𝐺
𝛼𝜏(1,2,3)(𝐻)))
= 𝑖𝑛𝑓𝑥∉𝑈
𝑚𝑖𝑛(1,1 − 𝛼ℱ(1,2,3)(𝑈)
+ 𝑠𝑢𝑝𝐺⋂𝐻=∅,𝐺∈𝑃(𝑋)
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐺), 𝑠𝑢𝑝
𝐺⋂𝐻=∅,𝑈 ⊆𝐻 𝛼𝜏(1,2,3)(𝐻)))
= 𝑖𝑛𝑓𝑥∉𝑈
𝑚𝑖𝑛(1,1 − 𝛼ℱ(1,2,3)(𝑈)
+ 𝑠𝑢𝑝𝐺⋂𝐻=∅,𝐺∈𝑃(𝑋)
𝑠𝑢𝑝𝐺⋂𝐻=∅,𝑈 ⊆𝐻
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐺), 𝛼𝜏(1,2,3)(𝐻)))
= 𝑖𝑛𝑓𝑥∉𝑈
𝑚𝑖𝑛(1,1 − 𝛼ℱ(1,2,3)(𝑈) + 𝑠𝑢𝑝𝐺⋂𝐻=∅,𝑈 ⊆𝐻
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐺), 𝛼𝜏(1,2,3)(𝐻)))
= [𝛼ℛ(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)].
(b) [ 𝛼ℛ(2) (1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)]
= 𝑖𝑛𝑓𝑥∈𝑈
𝑚𝑖𝑛(1,1 − 𝛼𝜏(1,2,3)(𝑈) + 𝑠𝑢𝑝𝐺⋂𝐻=∅,𝐺 ⊆𝑈
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐺), 𝛼𝜏(1,2,3)(𝐻)))
= 𝑖𝑛𝑓𝑥∉𝑋~𝑈
𝑚𝑖𝑛(1,1 − 𝛼ℱ(1,2,3)(𝑋~𝑈)
+ 𝑠𝑢𝑝𝐺⋂𝑋~𝑈=∅,𝐻 ⊆𝑋~𝑈
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐺), 𝛼𝜏(1,2,3)(𝐻)))
= [𝛼ℛ(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)].
Definition 2.12 If (𝑋, 𝜏1, 𝜏2, 𝜏3) is a FTTS, we define
(1) 𝛼𝒩(1) (1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) ≔ ∀ 𝐺 ∀ 𝐻 (𝐺 ∈ 𝛼ℱ(1,2,3) ⋀ 𝐻 ∈ 𝛼𝜏(1,2,3) ⋀ 𝐺 ⊆
𝐻 → ∃ 𝑈 ∃ 𝑉 (𝑈 ∈ 𝛼ℱ(1,2,3) ⋀ 𝑉 ∈ 𝛼𝜏(1,2,3) ⋀ 𝑈 ⊆ 𝑉 ⋀ 𝑉⋂𝐻 = ∅));
(2) 𝛼𝒩(2) (1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) ≔ ∀ 𝐺 ∀ 𝐻 (𝐺 ∈ 𝛼ℱ(1,2,3) ⋀ 𝐻 ∈ 𝛼ℱ(1,2,3) ⋀ 𝐺⋂𝐻 =
∅ → ∃ 𝑈 (𝑈 ∈ 𝛼𝜏(1,2,3) ⋀ 𝐺 ⊆ 𝑈 ⋀ 𝛼𝑐𝑙(1,2,3)(𝑈)⋂𝐻 = ∅)).
Theorem 2.13 If (𝑋, 𝜏1, 𝜏2, 𝜏3) is a FTTS. Then
196 Barah M. Sulaiman and Tahir H. Ismail
⊨ 𝛼𝒩(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) ↔ 𝛼𝒩(𝑖) (1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3), 𝑖 = 1,2.
Proof.
(a) [ 𝛼𝒩(1) (1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)]
= 𝑖𝑛𝑓𝐺⊆𝐻
𝑚𝑖𝑛 (1,1 − 𝑚𝑖𝑛 (𝛼ℱ(1,2,3)(𝐺), 𝛼𝜏(1,2,3)(𝐻))
+ 𝑠𝑢𝑝𝐸⊆𝐹,𝐹⋂𝐻=∅
𝑚𝑖𝑛 (𝛼ℱ(1,2,3)(𝐸), 𝛼𝜏(1,2,3)(𝐹)))
= 𝑖𝑛𝑓𝐺⋂𝑋~𝐻=∅
𝑚𝑖𝑛 (1,1 − 𝑚𝑖𝑛 (𝛼ℱ(1,2,3)(𝐺), 𝛼ℱ(1,2,3)(𝑋~𝐻))
+ 𝑠𝑢𝑝𝑋~𝐸⋂𝐹=∅,𝐺⊆𝑋~𝐸,𝐹⊆𝑋~𝐻
𝑚𝑖𝑛 (𝛼𝜏(1,2,3)(𝑋~𝐸), 𝛼𝜏(1,2,3)(𝐹)))
= [𝛼𝒩(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)]. (b) is analogous to the proof of (a) of Theorem (2.11).
3 Relations among 𝛂-separation axioms in fuzzifying tri-
topological spaces
Theorem 3.1 If (𝑋, 𝜏1, 𝜏2, 𝜏3) is a FTTS. Then
(1) ⊨ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇1(1,2,3)
→ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇0(1,2,3)
;
(2) ⊨ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇2(1,2,3)
→ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇𝑖(1,2,3)
, 𝑖 = 0,1.
Proof. From Lemma (2.7), it is clear.
Theorem 3.2 If (𝑋, 𝜏1, 𝜏2, 𝜏3) is a FTTS. Then
(1) ⊨ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅1(1,2,3)
→ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅𝑖(1,2,3)
, 𝑖 = 0,2;
(2) ⊨ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇1(1,2,3)
→ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅0(1,2,3)
;
(3) ⊨ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇2(1,2,3)
→ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅𝑖(1,2,3)
, 𝑖 = 0,1,2.
Proof. (1) (a) From (1) of Lemma (2.7), we have
𝛼𝑅1(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) = 𝑖𝑛𝑓
𝑥≠𝑦 𝑚𝑖𝑛 (1,1 − [𝛼𝒦𝑥,𝑦
(1,2,3)] + [𝛼ℳ𝑥,𝑦
(1,2,3)])
≤ 𝑖𝑛𝑓𝑥≠𝑦
𝑚𝑖𝑛 (1,1 − [𝛼𝒦𝑥,𝑦(1,2,3)
] + [𝛼ℋ𝑥,𝑦(1,2,3)
])
= 𝛼𝑅0(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)
(b) From (2) of Lemma (2.7), we have
𝛼𝑅1(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) = 𝑖𝑛𝑓
𝑥≠𝑦 𝑚𝑖𝑛 (1,1 − [𝛼𝒦𝑥,𝑦
(1,2,3)] + [𝛼ℳ𝑥,𝑦
(1,2,3)])
≤ 𝑖𝑛𝑓𝑥≠𝑦
𝑚𝑖𝑛 (1,1 − [𝛼ℋ𝑥,𝑦(1,2,3)
] + [𝛼ℳ𝑥,𝑦(1,2,3)
])
= 𝛼𝑅2(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)
(2) Using Lemma 2.2 in [2], we have
𝛼𝑇1(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) = 𝑖𝑛𝑓
𝑥≠𝑦 [𝛼ℋ𝑥,𝑦
(1,2,3)]
≤ 𝑖𝑛𝑓𝑥≠𝑦
[𝛼𝒦𝑥,𝑦(1,2,3)
→ 𝛼ℋ𝑥,𝑦(1,2,3)
]
On tri 𝛂-separation axioms in fuzzifying tri-topological spaces 197
= 𝛼𝑅0(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3).
(3) (a) From (2) above and (2) of Theorem (3.1), we have
𝛼𝑇2(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) = 𝑖𝑛𝑓
𝑥≠𝑦 [𝛼ℳ𝑥,𝑦
(1,2,3)] ≤ 𝑖𝑛𝑓
𝑥≠𝑦 [𝛼𝒦𝑥,𝑦
(1,2,3)→ 𝛼ℳ𝑥,𝑦
(1,2,3)]
= 𝛼𝑅1(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)
= 𝑖𝑛𝑓𝑥≠𝑦
𝑚𝑖𝑛 (1,1 − [𝛼𝒦𝑥,𝑦(1,2,3)
] + [𝛼ℳ𝑥,𝑦(1,2,3)
])
≤ 𝑖𝑛𝑓𝑥≠𝑦
𝑚𝑖𝑛 (1,1 − [𝛼𝒦𝑥,𝑦(1,2,3)
] + [𝛼ℋ𝑥,𝑦(1,2,3)
])
= 𝛼𝑅0(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3).
(b) Using Lemma 2.2 in [2], we have
𝛼𝑇2(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) = 𝑖𝑛𝑓
𝑥≠𝑦 [𝛼ℳ𝑥,𝑦
(1,2,3)]
≤ 𝑖𝑛𝑓𝑥≠𝑦
[𝛼𝒦𝑥,𝑦(1,2,3)
→ 𝛼ℳ𝑥,𝑦(1,2,3)
]
= 𝛼𝑅1(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3).
(c) Using Lemma 2.2 in [2], we have
𝛼𝑇2(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) = 𝑖𝑛𝑓
𝑥≠𝑦 [𝛼ℳ𝑥,𝑦
(1,2,3)]
≤ 𝑖𝑛𝑓𝑥≠𝑦
[𝛼ℋ𝑥,𝑦(1,2,3)
→ 𝛼ℳ𝑥,𝑦(1,2,3)
]
= 𝛼𝑅2(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3).
Theorem 3.3 If (𝑋, 𝜏1, 𝜏2, 𝜏3) is a FTTS. Then
⊨ 𝛼ℛ(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) ⟑ 𝛼𝑇1(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) → 𝛼𝑇2
(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3).
Proof. It suffices to show that
[𝛼𝑇2(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)] ≥ 𝑚𝑎𝑥(0, 𝛼ℛ(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3))] +
[𝛼𝑇1(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)] − 1).
Since [𝛼𝑇2(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)] ≥ 0.
Then from Theorem (3.2), we have
[𝛼𝑇1(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)] = 𝑖𝑛𝑓
𝑥∈𝑋 𝛼ℱ(1,2,3)({𝑥}) = 𝑖𝑛𝑓
𝑥∈𝑋 𝛼𝜏(1,2,3)(𝑋~{𝑥 })
So [𝛼ℛ(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)] + [𝛼𝑇1(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)]
= 𝑖𝑛𝑓𝑥∉𝑈
𝑚𝑖𝑛(1,1 − 𝛼𝜏(1,2,3)(𝑋~𝑈)
+ 𝑠𝑢𝑝𝐺⋂𝐻=∅,𝑈⊆𝐻
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐺), 𝛼𝜏(1,2,3)(𝐻))) + 𝑖𝑛𝑓
𝑧∈𝑋 𝛼𝜏(1,2,3)(𝑋~{𝑧})
≤ 𝑖𝑛𝑓𝑥∈𝑋,𝑥≠𝑦
𝑖𝑛𝑓𝑦∈𝑋
𝑚𝑖𝑛(1,1 − 𝛼𝜏(1,2,3)(𝑋~{𝑦}) +
𝑠𝑢𝑝𝐺⋂𝐻=∅,𝑦∈𝐻
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐺), 𝛼𝜏(1,2,3)(𝐻))) + 𝑖𝑛𝑓
𝑧∈𝑋 𝛼𝜏(1,2,3)(𝑋~{𝑧}))
≤ 𝑖𝑛𝑓𝑥∈𝑋,𝑥≠𝑦
𝑖𝑛𝑓𝑦∈𝑋
𝑚𝑖𝑛(1,1 − 𝛼𝜏(1,2,3)(𝑋~{𝑦}) +
198 Barah M. Sulaiman and Tahir H. Ismail
𝑠𝑢𝑝𝐺⋂𝐻=∅,𝑦∈𝐻
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐺), 𝛼𝑁𝑦
(1,2,3)(𝐻))) + 𝛼𝜏(1,2,3)(𝑋~{𝑦}))
= 𝑖𝑛𝑓𝑥∈𝑋,𝑥≠𝑦
𝑖𝑛𝑓𝑦∈𝑋
(𝑚𝑖𝑛(1,1 + 𝑠𝑢𝑝𝐺⋂𝐻=∅
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐺), 𝛼𝑁𝑦
(1,2,3)(𝐻))))
= 𝑖𝑛𝑓𝑥∈𝑋,𝑥≠𝑦
𝑖𝑛𝑓𝑦∈𝑋
(1 + 𝑠𝑢𝑝𝐺⋂𝐻=∅
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐺), 𝛼𝑁𝑦
(1,2,3)(𝐻)))
= 1 + 𝑖𝑛𝑓𝑥≠𝑦
𝑠𝑢𝑝𝐺⋂𝐻=∅
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐺), 𝛼𝑁𝑦
(1,2,3)(𝐻))
= 1 + [𝛼𝑇2(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)].
Thus
[𝛼𝑇2(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)] ≥ 𝑚𝑎𝑥(0, 𝛼ℛ(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3))] +
[𝛼𝑇1(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)] − 1).
Corollary 3.4 If (𝑋, 𝜏1, 𝜏2, 𝜏3) is a FTTS. Then
(1) ⊨ 𝛼𝑇3(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) → 𝛼𝑇2
(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3).
(2) ⊨ 𝛼𝑇3(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) → 𝛼𝑅𝑖
(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3), 𝑖 = 0,1,2.
Theorem 3.5 If (𝑋, 𝜏1, 𝜏2, 𝜏3) is a FTTS. Then
⊨ 𝛼𝑇4(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) → 𝛼ℛ(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3).
Proof.
𝛼𝑇4(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) = 𝑚𝑎𝑥(0, [𝛼𝒩(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3))] +
[𝛼𝑇1(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)] − 1),
now we prove that
[𝛼ℛ(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)] ≥ [𝛼𝒩(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)] + [𝛼𝑇1(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)] −
1.
In fact
[𝛼𝒩(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)] + [𝛼𝑇1(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)]
= 𝑖𝑛𝑓𝑈⋂𝑉=∅
𝑚𝑖𝑛 (1,1 − 𝑚𝑖𝑛 (𝛼ℱ(1,2,3)(𝑈), 𝛼ℱ(1,2,3)(𝑉))
+ 𝑠𝑢𝑝𝐺⋂𝐻=∅,𝑈⊆𝐻,𝑉⊆𝐺
𝑚𝑖𝑛 (𝛼𝜏(1,2,3)(𝐺), 𝛼𝜏(1,2,3)(𝐻))) + 𝑖𝑛𝑓𝑧∈𝑋
𝛼𝜏(1,2,3)(𝑋~{𝑧})
= 𝑖𝑛𝑓𝑈⋂𝑉=∅
𝑚𝑖𝑛 (1,1 − 𝑚𝑖𝑛 (𝛼𝜏(1,2,3)(𝑋~𝑈), 𝛼𝜏(1,2,3)(𝑋~𝑉))
+ 𝑠𝑢𝑝𝐺⋂𝐻=∅,𝑈⊆𝐻,𝑉⊆𝐺
𝑚𝑖𝑛 (𝛼𝜏(1,2,3)(𝐺), 𝛼𝜏(1,2,3)(𝐻))) + 𝑖𝑛𝑓𝑧∈𝑋
𝛼𝜏(1,2,3)(𝑋~{𝑧})
≤ 𝑖𝑛𝑓 𝑥∉𝑈
𝑚𝑖𝑛 (1,1 − 𝑚𝑖𝑛 (𝛼𝜏(1,2,3)(𝑋~𝑈), 𝛼𝜏(1,2,3)(𝑋~{𝑥}))
+ 𝑠𝑢𝑝𝐺⋂𝐻=∅,𝑈⊆𝐻
𝑚𝑖𝑛 (𝛼𝑁𝑥(1,2,3)(𝐺), 𝛼𝜏(1,2,3)(𝐻)) + 𝑖𝑛𝑓
𝑧∈𝑋 𝛼𝜏(1,2,3)(𝑋~{𝑧})
= 𝑖𝑛𝑓 𝑥∉𝑈
𝑚𝑖𝑛 (1, 𝑚𝑎𝑥 (1 − 𝛼𝜏(1,2,3)(𝑋~𝑈)
+ 𝑠𝑢𝑝𝐺⋂𝐻=∅,𝑈⊆𝐻
𝑚𝑖𝑛 (𝛼𝑁𝑥(1,2,3)(𝐺), 𝛼𝜏(1,2,3)(𝐻)),1 − 𝛼𝜏(1,2,3)(𝑋~{𝑥})
+ 𝑠𝑢𝑝𝐺⋂𝐻=∅,𝑈⊆𝐻
𝑚𝑖𝑛 (𝛼𝑁𝑥(1,2,3)(𝐺), 𝛼𝜏(1,2,3)(𝐻))) + 𝑖𝑛𝑓
𝑧∈𝑋 𝛼𝜏(1,2,3)(𝑋~{𝑧})
On tri 𝛂-separation axioms in fuzzifying tri-topological spaces 199
= 𝑖𝑛𝑓 𝑥∉𝑈
𝑚𝑎𝑥 (𝑚𝑖𝑛 (1,1 − 𝛼𝜏(1,2,3)(𝑋~𝑈)
+ 𝑠𝑢𝑝𝐺⋂𝐻=∅,𝑈⊆𝐻
𝑚𝑖𝑛 (𝛼𝑁𝑥(1,2,3)(𝐺), 𝛼𝜏(1,2,3)(𝐻))), 𝑚𝑖𝑛 (1,1 − 𝛼𝜏(1,2,3)(𝑋~{𝑥})
+ 𝑠𝑢𝑝𝐺⋂𝐻=∅,𝑈⊆𝐻
𝑚𝑖𝑛 (𝛼𝑁𝑥(1,2,3)(𝐺), 𝛼𝜏(1,2,3)(𝐻))) + 𝑖𝑛𝑓
𝑧∈𝑋 𝛼𝜏(1,2,3)(𝑋~{𝑧})
≤ 𝑖𝑛𝑓 𝑥∉𝑈
𝑚𝑎𝑥(𝑚𝑖𝑛 (1,1 − 𝛼𝜏(1,2,3)(𝑋~𝑈)
+ 𝑠𝑢𝑝𝐺⋂𝐻=∅,𝑈⊆𝐻
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐺), 𝛼𝜏(1,2,3)(𝐻)))
+𝛼𝜏(1,2,3)(𝑋~{𝑥}), 𝑚𝑖𝑛 (1,1 − 𝛼𝜏(1,2,3)(𝑋~{𝑥}))
+ 𝑠𝑢𝑝𝐺⋂𝐻=∅,𝑈⊆𝐻
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐺), 𝛼𝜏(1,2,3)(𝐻))) + 𝛼𝜏(1,2,3)(𝑋~{𝑥}))
≤ 𝑖𝑛𝑓 𝑥∉𝑈
𝑚𝑎𝑥(𝑚𝑖𝑛 (1,1 − 𝛼𝜏(1,2,3)(𝑋~𝑈)
+ 𝑠𝑢𝑝𝐺⋂𝐻=∅,𝑈⊆𝐻
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐺), 𝛼𝜏(1,2,3)(𝐻))) + 𝛼𝜏(1,2,3)(𝑋~{𝑥}),
1 + 𝑠𝑢𝑝𝐺⋂𝐻=∅,𝑈⊆𝐻
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐺), 𝛼𝜏(1,2,3)(𝐻)))
≤ 𝑖𝑛𝑓 𝑥∉𝑈
𝑚𝑖𝑛 (1,1 − 𝛼𝜏(1,2,3)(𝑋~𝑈)
+ 𝑠𝑢𝑝𝐺⋂𝐻=∅,𝑈⊆𝐻
𝑚𝑖𝑛(𝛼𝑁𝑥(1,2,3)(𝐺), 𝛼𝜏(1,2,3)(𝐻))) + 1
= [𝛼ℛ(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)] + 1.
Theorem 3.6 If (𝑋, 𝜏1, 𝜏2, 𝜏3) is a FTTS. Then
(1) ⊨ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇1(1,2,3)
→ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅0(1,2,3)
⋀ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈
𝛼𝑇0(1,2,3)
;
(2) If 𝛼𝑇0(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) = 1, then
⊨ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇1(1,2,3)
↔ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅0(1,2,3)
⋀ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈
𝛼𝑇0(1,2,3)
.
Proof. (1) Follows from (1) of Theorem (3.1) and (2) of Theorem (3.2).
(2) Since 𝛼𝑇0(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) = 1, then for every 𝑥, 𝑦 ∈ 𝑋 such that 𝑥 ≠ 𝑦, we
have [𝛼𝒦𝑥,𝑦(1,2,3)
] = 1. So
𝛼𝑅0(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) ⋀ 𝛼𝑇0
(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)
= 𝛼𝑅0(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3)
= 𝑖𝑛𝑓𝑥≠𝑦
𝑚𝑖𝑛(1,1 − [𝛼𝒦𝑥,𝑦(1,2,3)
] + [𝛼ℋ𝑥,𝑦(1,2,3)
])
= 𝑖𝑛𝑓𝑥≠𝑦
[𝛼ℋ𝑥,𝑦(1,2,3)
] = 𝛼𝑇1(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3).
Theorem 3.7 If (𝑋, 𝜏1, 𝜏2, 𝜏3) is a FTTS. Then
200 Barah M. Sulaiman and Tahir H. Ismail
(1) ⊨ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇2(1,2,3)
→ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅1(1,2,3)
⋀ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈
𝛼𝑇0(1,2,3)
;
(2) If 𝛼𝑇0(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) = 1, then
⊨ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇2(1,2,3)
↔ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅1(1,2,3)
⋀ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈
𝛼𝑇0(1,2,3)
.
Proof. (1) Follows from (3) and (4) of Theorems (3.1) and (3.2) respectively.
(2) Likewise from (2) theorem 3.6.
Theorem 3.8 If (𝑋, 𝜏1, 𝜏2, 𝜏3) is a FTTS. Then
(1) ⊨ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇2(1,2,3)
→ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅0(1,2,3)
⋀ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈
𝛼𝑇1(1,2,3)
;
(2) If 𝛼𝑇1(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) = 1, then
⊨ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇2(1,2,3)
↔ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅0(1,2,3)
⋀ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈
𝛼𝑇1(1,2,3)
.
Proof. (1) Follows from (2) and (3) of Theorems (3.1) and (3.2) respectively.
(2) Likewise from (3) Theorem 3.6.
Remark 3.9 If (𝑋, 𝜏1, 𝜏2, 𝜏3) is a FTTS. Then we have
(1) ⊨ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇1(1,2,3)
→ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅0(1,2,3)
⋀ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈
𝛼𝑇0(1,2,3)
;
(2) ⊨ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇2(1,2,3)
→ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅1(1,2,3)
⋀ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈
𝛼𝑇0(1,2,3)
.
(3) ⊨ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇2(1,2,3)
→ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅0(1,2,3)
⋀ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈
𝛼𝑇1(1,2,3)
.
Theorem 3.10 If (𝑋, 𝜏1, 𝜏2, 𝜏3) is a FTTS. Then
(1) (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅0(1,2,3)
⟑ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇0(1,2,3)
→ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈
𝛼𝑇1(1,2,3)
;
(2) If 𝛼𝑇0(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) = 1, then
⊨ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅0(1,2,3)
⟑ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇0(1,2,3)
↔ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈
𝛼𝑇1(1,2,3)
.
Proof.
(1) [(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅0(1,2,3)
⟑ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇0(1,2,3)
]
= 𝑚𝑎𝑥(0, 𝛼𝑅0(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) + 𝛼𝑇0
(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) − 1)
= 𝑚𝑎𝑥(0, 𝑖𝑛𝑓𝑥≠𝑦
𝑚𝑖𝑛(1,1 − [𝛼𝒦𝑥,𝑦(1,2,3)
] + [𝛼ℋ𝑥,𝑦(1,2,3)
]) + 𝑖𝑛𝑓𝑥≠𝑦
[𝛼𝒦𝑥,𝑦(1,2,3)
] − 1)
On tri 𝛂-separation axioms in fuzzifying tri-topological spaces 201
≤ 𝑚𝑎𝑥(0, 𝑖𝑛𝑓𝑥≠𝑦
(𝑚𝑖𝑛(1,1 − [𝛼𝒦𝑥,𝑦(1,2,3)
] + [𝛼ℋ𝑥,𝑦(1,2,3)
]) + [𝛼𝒦𝑥,𝑦(1,2,3)
] − 1)
≤ 𝑚𝑎𝑥(0, 𝑖𝑛𝑓𝑥≠𝑦
(1 − [𝛼𝒦𝑥,𝑦(1,2,3)
] + [𝛼ℋ𝑥,𝑦(1,2,3)
] + [𝛼𝒦𝑥,𝑦(1,2,3)
] − 1)
= 𝑖𝑛𝑓𝑥≠𝑦
[𝛼ℋ𝑥,𝑦(1,2,3)
] = 𝛼𝑇1(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3).
(2) Follows from (2) Theorem (3.6).
Theorem 3.11 If (𝑋, 𝜏1, 𝜏2, 𝜏3) is a FTTS. Then
(1) (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅1(1,2,3)
⟑ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇0(1,2,3)
→ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈
𝛼𝑇2(1,2,3)
;
(2) If 𝛼𝑇0(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) = 1, then
⊨ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅1(1,2,3)
⟑ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇0(1,2,3)
↔ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈
𝛼𝑇2(1,2,3)
.
Proof.
(1) [(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅1(1,2,3)
⟑ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇0(1,2,3)
]
= 𝑚𝑎𝑥(0, 𝛼𝑅1(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) + 𝛼𝑇0
(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3) − 1)
= 𝑚𝑎𝑥(0, 𝑖𝑛𝑓𝑥≠𝑦
𝑚𝑖𝑛(1,1 − [𝛼𝒦𝑥,𝑦(1,2,3)
] + [𝛼ℳ𝑥,𝑦(1,2,3)
]) + 𝑖𝑛𝑓𝑥≠𝑦
[𝛼𝒦𝑥,𝑦(1,2,3)
] − 1)
≤ 𝑚𝑎𝑥(0, 𝑖𝑛𝑓𝑥≠𝑦
(𝑚𝑖𝑛(1,1 − [𝛼𝒦𝑥,𝑦(1,2,3)
] + [𝛼ℳ𝑥,𝑦(1,2,3)
]) + [𝛼𝒦𝑥,𝑦(1,2,3)
] − 1)
≤ 𝑚𝑎𝑥(0, 𝑖𝑛𝑓𝑥≠𝑦
(1 − [𝛼𝒦𝑥,𝑦(1,2,3)
] + [𝛼ℳ𝑥,𝑦(1,2,3)
] + [𝛼𝒦𝑥,𝑦(1,2,3)
] − 1)
= 𝑖𝑛𝑓𝑥≠𝑦
[𝛼ℳ𝑥,𝑦(1,2,3)
] = 𝛼𝑇2(1,2,3)(𝑋, 𝜏1, 𝜏2, 𝜏3).
(2) Follows from (2) Theorem (3.6).
Theorem 3.12 If (𝑋, 𝜏1, 𝜏2, 𝜏3) is a FTTS. Then
(1) ⊨ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇0(1,2,3)
→ ((𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅0(1,2,3)
→ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈
𝛼𝑇1(1,2,3)
;
(2) ⊨ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅0(1,2,3)
→ ((𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇0(1,2,3)
→ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈
𝛼𝑇1(1,2,3)
;
(3) ⊨ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇0(1,2,3)
→ ((𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅1(1,2,3)
→ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈
𝛼𝑇2(1,2,3)
;
(4) ⊨ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅1(1,2,3)
→ ((𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇0(1,2,3)
→ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈
𝛼𝑇2(1,2,3)
.
Proof. (1) From (2) Theorem (3.1) and (3) Theorem (3.2), we have
[(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇0(1,2,3)
→ ((𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅0(1,2,3)
→ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈
𝛼𝑇1(1,2,3)
]
202 Barah M. Sulaiman and Tahir H. Ismail
= 𝑚𝑖𝑛(1,1 − [(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇0(1,2,3)
] + 𝑚𝑖𝑛(1,1 − [(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈
𝛼𝑅0(1,2,3)
] + [(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇1(1,2,3)
]))
= 𝑚𝑖𝑛(1,1 − [(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇0(1,2,3)
] + 1 − [(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅0(1,2,3)
] +
[(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇1(1,2,3)
]))
= 𝑚𝑖𝑛(1,1 − ([(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅0(1,2,3)
] + [(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇0(1,2,3)
] − 1) +
[(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇1(1,2,3)
]) = 1.
(2) From (1) Theorem (3.1) and (3) Theorem (3.6), we have
[(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅0(1,2,3)
→ ((𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇0(1,2,3)
→ (𝑋, 𝜏1, 𝜏2, 𝜏3) ∈
𝛼𝑇1(1,2,3)
]
= 𝑚𝑖𝑛(1,1 − [(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅0(1,2,3)
] + 𝑚𝑖𝑛(1,1 − [(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈
𝛼𝑇0(1,2,3)
] + [(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇1(1,2,3)
]))
= 𝑚𝑖𝑛(1,1 − [(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅0(1,2,3)
] + 1 − [(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇0(1,2,3)
] +
[(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇1(1,2,3)
]))
= 𝑚𝑖𝑛(1,1 − ([(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑅0(1,2,3)
] + [(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇0(1,2,3)
] − 1) +
[(𝑋, 𝜏1, 𝜏2, 𝜏3) ∈ 𝛼𝑇1(1,2,3)
]) = 1.
(3) and (4) are likewise (2) and (3) above
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Received: April 3, 2019; Published: May 1, 2019