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

References

[1] A.A. Allam, A.M. Zahran, A.K. Mousa, and H.M. Binshahnah, On semi

separation axioms in fuzzifying bitopological spaces, Journal of Advanced Studies

in Topology, 6(4) (2015), 152–163. https://doi.org/10.20454/jast.2015.1016

[2] F.H. Khedr, F.M. Zeyada and O.R. Sayed, On separation axioms in fuzzifying

topology, Fuzzy Sets and Systems, 119 (2001), 439–458.

https://doi.org/10.1016/s0165-0114(99)00077-9

[3] J. Shen, Separation axiom in fuzzifying topology, Fuzzy Sets and Systems, 57

(1993), 111–123. https://doi.org/10.1016/0165-0114(93)90124-z

[4] O.R. Sayed, α-separation axioms based on Łukasiewicz logic, Hacettepe

Journal of Mathematics and Statistics, 43(2), (2014), 269-287.

[5] B.M. Sulaiman and T.H. Ismail, On tri α-open sets in fuzzifying tri-topological

spaces, Advances in Fuzzy Systems Journal, (2019) (2019), 1-9.

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

[6] P. Wuyts and R. Lowen, On Separation Axioms in Fuzzy Topological Spaces,

Fuzzy Neighborhood Spaces, and Fuzzy Uniform Spaces, Journal of

Mathematical Analysis and Applications, 93 (1983), 27-41.

https://doi.org/10.1016/0022-247x(83)90217-2

On tri 𝛂-separation axioms in fuzzifying tri-topological spaces 203

[7] M.S. Ying, A new approach for fuzzy topology (I), Fuzzy Sets and Systems, 39

(1991), 302-321. https://doi.org/10.1016/0165-0114(91)90100-5

[8] M.S. Ying, A new approach for fuzzy topology (II), Fuzzy Sets and Systems,

47 (1992), 221-232. https://doi.org/10.1016/0165-0114(92)90181-3

[9] M.S. Ying, A new approach for fuzzy topology (III), Fuzzy Sets and Systems,

55 (1993), 193-207. https://doi.org/10.1016/0165-0114(93)90132-2

Received: April 3, 2019; Published: May 1, 2019