on tri ð-separation axioms in fuzzifying tri-topological ...jan 04, 2019  · al. (2001) studied...
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International Journal of Mathematical Analysis
Vol. 13, 2019, no. 4, 191 â 203
HIKARI Ltd, www.m-hikari.com
https://doi.org/10.12988/ijma.2019.9319
On Tri ð-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 ðŒð0(1,2,3)
(Kolmogorov), ðŒð1(1,2,3)
(Fréchet),
ðŒð2(1,2,3)
(Hausdorff), ðŒâ(1,2,3)(ðŒ-regular), ðŒð©(1,2,3)(ðŒ-normal),ðŒð 0(1,2,3)
, ðŒð 1(1,2,3)
and ðŒð 2(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 ðŒ-separation axioms
1 Introduction
Ying (1991-1993) introduced the concept of the term âfuzzifying topologyË® [7-
9]. Wuyts and Lowen (1983) studied "separation axioms in fuzzy topological
spaces" [6]. Shen (1993) introduced and studied ð0, ð1, ð2 (Hausdorff), ð3
(regularity), ð4 (normality)-separation axioms in fuzzifying topology [3]. Khedr et
al. (2001) studied âseparation axioms in fuzzifying topologyË® [2]. Sayed (2014)
presented "α-separation axioms based on Åukasiewicz logic" [4]. Allam et al.
(2015) studied âsemi separation axioms in fuzzifying bitopological spacesË® [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 (ð, ð1, ð2, ð3) 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 ð, symbolized as ðŒð(1,2,3) â
â(ð(ð)), and defined as
ðž â ðŒð(1,2,3) â â ð¥ (ð¥ â ðž â ð¥ â ððð¡1(ðð2(ððð¡3(ðž)))),
i.e., ðŒð(1,2,3)(ðž) = ðððð¥âðž
(ððð¡1(ðð2(ððð¡3(ðž))))(ð¥).
(ii) The family of fuzzifying (1,2,3) α-closed sets in ð, symbolized as ðŒâ±(1,2,3),
and defined by ð¹ â ðŒâ±(1,2,3) â ð~ð¹ â ðŒð(1,2,3).
(iii) The (1,2,3) α-neighborhood system of ð¥, denoted by ðŒðð¥(1,2,3)
and defined as
ðž â ðŒðð¥(1,2,3)
â â ð¹ (ð¹ â ðŒð(1,2,3) â ð¥ â ð¹ â ðž);
i.e. ðŒðð¥(1,2,3)(ðž) = ð ð¢ð
ð¥âð¹âðžðŒð(1,2,3)(ð¹).
(iv) The (1,2,3) α-derived set of E â X, denoted by ðŒð(1,2,3)(ðž) and defined as
ð¥ â ðŒð(1,2,3)(ðž) â â ð¹ (ð¹ â ðŒðð¥(1,2,3)
â ð¹ â© (ðž â {ð¥}) â â ),
i.e., ðŒð(1,2,3)(ðž)(ð¥) = ðððð¹â©(ðžâ{ð¥})â â
(1 â ðŒðð¥(1,2,3)(ð¹)).
(v) The (1,2,3) α-closure set of ðž â ð, denoted by ðŒðð(1,2,3)(ðž) and defined as
ð¥ â ðŒðð(1,2,3)(ðž) â â ð¹ (ð¹ â ðž) â© (ð¹ â ðŒâ±(1,2,3)) â ð¥ â ð¹),
i.e., ðŒðð(1,2,3)(ðž)(ð¥) = ðððð¥âð¹âðž
(1 â ðŒâ±(1,2,3)(ð¹)).
(vi) The (1,2,3) α-interior set of ðž â ð, denoted by ðŒððð¡(1,2,3)(ðž) and defined as
ðŒððð¡(1,2,3)(ðž)(ð¥) = ðŒðð¥(1,2,3)
(ðž).
(vii) The (1,2,3) α-exterior set of ðž â ð, denoted by ðŒðð¥ð¡(1,2,3)(ðž) and defined as
ð¥ â ðŒðð¥ð¡(1,2,3)(ðž) â ð¥ â ðŒððð¡(1,2,3)(ð~ðž)(ð¥),
i.e. ðŒðð¥ð¡(1,2,3)(ðž)(ð¥) = ðŒððð¡(1,2,3)(ð~ðž)(ð¥).
(viii) The (1,2,3) α-boundary set of ðž â ð, denoted by ðŒð(1,2,3)(ðž) and defined as
ð¥ â ðŒð(1,2,3)(ðž) â (ð¥ â ðŒððð¡(1,2,3)(ðž)) â (ð¥ â ðŒððð¡(1,2,3)(ð~ðž)),
i.e. ðŒð(1,2,3)(ðž)(ð¥) â ððð(1 â ðŒððð¡(1,2,3)(ðž)(ð¥)) â (1 â
ðŒððð¡(1,2,3)(ð~ðž)(ð¥)).
2 Tri ð-Separation axioms in fuzzifying tri-topological spaces
Remark 2.2 We consider the following notations:
ðŒðŠð¥,ðŠ(1,2,3)
â â ðº ((ðº â ðŒðð¥(1,2,3)
â ðŠ â ðº) â (ðº â ðŒððŠ(1,2,3)
â ð¥ â ðº));
ðŒâð¥,ðŠ(1,2,3)
â â ð» â ðž (ð» â ðŒðð¥(1,2,3)
â ðž â ðŒððŠ(1,2,3)
â ðŠ â ð» â ð¥ â ðž);
ðŒâ³ð¥,ðŠ(1,2,3)
â â ð» â ðž (ð» â ðŒðð¥(1,2,3)
â ðž â ðŒððŠ(1,2,3)
â ð»âðž = â ).
Definition 2.3 If ðº is the class of all FTTSs. The predicates ðŒðð(1,2,3)
, ðŒð ð(1,2,3)
â
â(ðº), ð = 0,1,2, are defined as follow
(ð, ð1, ð2, ð3) â ðŒð0(1,2,3)
â â ð¥ â ðŠ (ð¥ â ð â ðŠ â ð â ð¥ â ðŠ â ðŒðŠð¥,ðŠ(1,2,3)
);
(ð, ð1, ð2, ð3) â ðŒð1(1,2,3)
â â ð¥ â ðŠ (ð¥ â ð â ðŠ â ð â ð¥ â ðŠ â ðŒâð¥,ðŠ(1,2,3)
);
(ð, ð1, ð2, ð3) â ðŒð2(1,2,3)
â â ð¥ â ðŠ (ð¥ â ð â ðŠ â ð â ð¥ â ðŠ â ðŒâ³ð¥,ðŠ(1,2,3)
);
On tri ð-separation axioms in fuzzifying tri-topological spaces 193
(ð, ð1, ð2, ð3) â ðŒð 0(1,2,3)
â â ð¥ â ðŠ (ð¥ â ð â ðŠ â ð â ð¥ â ðŠ â (ðŒðŠð¥,ðŠ(1,2,3)
â
ðŒâð¥,ðŠ(1,2,3)
);
(ð, ð1, ð2, ð3) â ðŒð 1(1,2,3)
â â ð¥ â ðŠ (ð¥ â ð â ðŠ â ð â ð¥ â ðŠ â (ðŒðŠð¥,ðŠ(1,2,3)
â
ðŒâ³ð¥,ðŠ(1,2,3)
);
(ð, ð1, ð2, ð3) â ðŒð 2(1,2,3)
â â ð¥ â ðŠ (ð¥ â ð â ðŠ â ð â ð¥ â ðŠ â (ðŒâð¥,ðŠ(1,2,3)
â
ðŒâ³ð¥,ðŠ(1,2,3)
).
Definition 2.4 If ðº is the class of all FTTSs. The predicates ðŒâ(1,2,3), ðŒð©(1,2,3) ââ(Ω), are defined as follow
(1) (ð, ð1, ð2, ð3) â ðŒâ(1,2,3) â â ð¥ â ð (ð¥ â ð â ð â ðŒâ±(1,2,3) â ð¥ â ð â
â ðº â ð» (ðº â ðŒðð¥(1,2,3)
â ð» â ðŒð(1,2,3) â ð â ð» â ðºâð» = â ));
(2) (ð, ð1, ð2, ð3) â ðŒð©(1,2,3) â â ðº â ð» (ðº â ðŒâ±(1,2,3) â ð» â ðŒâ±(1,2,3) â ðºâð» =
â ) â â ð â ð (ð â ðŒð(1,2,3) â ð â ðŒð(1,2,3)â ðº â ð âð» â ð â ðâð = â ).
Definition 2.5 If ðº is the class of all FTTSs. The predicates ðŒð3(1,2,3)
, ðŒð4(1,2,3)
ââ(ðº) are defined as follow
(1) ðŒð3(1,2,3)(ð, ð1, ð2, ð3) â ðŒâ(1,2,3)(ð, ð1, ð2, ð3) â ðŒð1
(1,2,3)(ð, ð1, ð2, ð3);
(2) ðŒð4(1,2,3)(ð, ð1, ð2, ð3) â ðŒð©(1,2,3)(ð, ð1, ð2, ð3) â ðŒð1
(1,2,3)(ð, ð1, ð2, ð3).
Remark 2.6 If (ð, ð1, ð2, ð3) is a FTTS. Note that
(1) ðŒðð(1,2,3)
= ðŒðð(3,2,1)
, ð = 0,1,2,3,4;
(2) ðŒð ð(1,2,3)
= ðŒð ð(3,2,1)
, ð = 0,1,2.
Lemma 2.7 If (ð, ð1, ð2, ð3) is a FTTS. Then
(1) âš ðŒâ³ð¥,ðŠ(1,2,3)
â ðŒâð¥,ðŠ(1,2,3)
;
(2) âš ðŒâð¥,ðŠ(1,2,3)
â ðŒðŠð¥,ðŠ(1,2,3)
;
(3) âš ðŒâ³ð¥,ðŠ(1,2,3)
â ðŒðŠð¥,ðŠ(1,2,3)
.
Proof.
(1) [ ðŒðð¥,ðŠ(1,2,3)
] = ð ð¢ððµâð¶=â
ððð(ðŒðð¥(1,2,3)(ðµ), ðŒððŠ
(1,2,3)(ð¶)) â€
ð ð¢ððŠâðµ,ð¥âð¶
ððð(ðŒðð¥(1,2,3)(ðµ), ðŒððŠ
(1,2,3)(ð¶)) = [ðŒâð¥,ðŠ(1,2,3)
].
(2) [ ðŒðŠð¥,ðŠ(1,2,3)
] = ððð¥(ð ð¢ð ðŠâðŽ
ðŒðð¥(1,2,3)(ðŽ), ð ð¢ð
ð¥âðŽ ðŒððŠ
(1,2,3)(ðŽ))
⥠ð ð¢ððŠâðŽ
ðŒðð¥(1,2,3)(ðŽ) â¥
ð ð¢ððŠâðŽ,ð¥âðµ
ððð(ðŒðð¥(1,2,3)(ðŽ), ðŒððŠ
(1,2,3)(ðµ)) = [ðŒâð¥,ðŠ(1,2,3)
].
(3) is concluded from (1) and (2) above.
Theorem 2.8 If (ð, ð1, ð2, ð3) is a FTTS. Then
194 Barah M. Sulaiman and Tahir H. Ismail
âš (ð, ð1, ð2, ð3) â ðŒð0(1,2,3)
â â ð¥ â ðŠ (ð¥ â ð â ðŠ â ð â ð¥ â ðŠ â ð¥ âðŒðð(1,2,3)({ðŠ})âðŠ â ðŒðð(1,2,3)({ð¥})).
Proof.
ðŒð0(1,2,3)(ð, ð1, ð2, ð3)
= ðððð¥â ðŠ
ððð¥(ð ð¢ððŠâðŽ
ðŒðð¥(1,2,3)(ðŽ), ð ð¢ð
ð¥âðŽ ðŒððŠ
(1,2,3)(ðŽ))
= ðððð¥â ðŠ
ððð¥(ðŒðð¥(1,2,3)(ð~{ðŠ}), ðŒððŠ
(1,2,3)(ð~{ð¥}))
= ðððð¥â ðŠ
ððð¥(1 â ðŒðð(1,2,3)({ðŠ})(ð¥),1 â ðŒðð(1,2,3)({ð¥})(ðŠ))
= [â ð¥ â ðŠ (ð¥ â ð â ðŠ â ð â ð¥ â ðŠ â ð¥ â ðŒðð(1,2,3)({ðŠ})âðŠ â ðŒðð(1,2,3)({ð¥}))].
Theorem 2.9 If (ð, ð1, ð2, ð3) is a FTTS. Then
âš â ð¥ ({ð¥} â ðŒâ±(1,2,3)) â (ð, ð1, ð2, ð3) â ðŒð1(1,2,3)
.
Proof.
ðŒð1(1,2,3)(ð, ð1, ð2, ð3)
= ðððð¥1â ð¥2
ððð( ð ð¢ðð¥2âðŽ
ðŒðð¥1
(1,2,3)(ðŽ), ð ð¢ðð¥1âðµ
ðŒðð¥2
(1,2,3)(ðµ)) =
ðððð¥1â ð¥2
ððð(ðŒðð¥1
(1,2,3)(ð~{ð¥2}), ðŒðð¥2
(1,2,3)(ð~{ð¥1})) â€
ðððð¥1â ð¥2
ðŒðð¥1
(1,2,3)(ð~{ð¥2}) = ðððð¥2âð
ðððð¥1âð~{ð¥2}
ðŒðð¥1
(1,2,3)(ð~{ð¥2})
= ðððð¥2âð
ðŒð(1,2,3)(ð~{ð¥2}) = ðððð¥âð
ðŒð(1,2,3)(ð~{ð¥}) = ðððð¥âð
ðŒâ±(1,2,3)({ð¥}).
Now, for any ð¥1, ð¥2 â ð with ð¥1 â ð¥2.
[â ð¥ ({ð¥} â ðŒâ±(1,2,3))]
= ðððð¥âð
[{ð¥} â ðŒâ±(1,2,3)] = ðððð¥âð
ðŒð(1,2,3)(ð~{ð¥}) = ðððð¥âð
ððððŠâð~{ð¥}
ðŒððŠ(1,2,3)(ð~{ð¥})
†ððððŠâð~{ð¥2}
ðŒððŠ(1,2,3)(ð~{ð¥2}) †ðŒðð¥2
(1,2,3)(ð~{ð¥2}) = ð ð¢ðð¥2âðŽ
ðŒðð¥1
(1,2,3)(ðŽ).
By the same way, we have
[â ð¥ ({ð¥} â ðŒâ±(1,2,3))] †ð ð¢ðð¥1âðŽ
ðŒðð¥2
(1,2,3)(ðµ). So
[â ð¥ ({ð¥} â ðŒâ±(1,2,3))] †ðððð¥1â ð¥2
ððð( ð ð¢ðð¥2âðŽ
ðŒðð¥1
(1,2,3)(ðŽ), ð ð¢ðð¥1âðµ
ðŒðð¥2
(1,2,3)(ðµ))
= ðŒð1(1,2,3)(ð, ð1, ð2, ð3).
Therefore ðŒð1(1,2,3)(ð, ð1, ð2, ð3) = [â ð¥ ({ð¥} â ðŒâ±(1,2,3))].
Definition 2.10 If (ð, ð1, ð2, ð3) is a FTTS, we define
(1) ðŒâ(1) (1,2,3)(ð, ð1, ð2, ð3) â â ð¥ â ð (ð¥ â ð â ð â ðŒâ±(1,2,3) â ð¥ â ð â
â ðº (ðº â ðŒðð¥(1,2,3)
â ðŒðð(1,2,3)(ðº)âð = â ));
(2) ðŒâ(2) (1,2,3)(ð, ð1, ð2, ð3) â â ð¥ â ð (ð¥ â ð â ð â ðŒð(1,2,3) â ð¥ â ð â
â ðº â ð» (ðº â ðŒðð¥(1,2,3)
â ð» â ðŒð(1,2,3) â ðº â ð â ðºâð» = â )).
On tri ð-separation axioms in fuzzifying tri-topological spaces 195
Theorem 2.11 If (ð, ð1, ð2, ð3) is a FTTS. Then
âš ðŒâ(1,2,3)(ð, ð1, ð2, ð3) â ðŒâ(ð) (1,2,3)(ð, ð1, ð2, ð3), ð = 1,2.
Proof.
(a) [ ðŒâ(1) (1,2,3)(ð, ð1, ð2, ð3)]
= ðððð¥âð
ððð(1,1 â ðŒâ±(1,2,3)(ð)
+ ð ð¢ððºâð(ð)
ððð(ðŒðð¥(1,2,3)(ðº), ððð
ðŠâð (1 â ðŒðð(1,2,3)(ðº)(ðŠ))))
= ðððð¥âð
ððð(1,1 â ðŒâ±(1,2,3)(ð)
+ ð ð¢ððºâð(ð)
ððð(ðŒðð¥(1,2,3)(ðº), ððð
ðŠâð ðŒððŠ
(1,2,3)(ð~ðº)))
= ðððð¥âð
ððð(1,1 â ðŒâ±(1,2,3)(ð) +
ð ð¢ððºâð=â ,ðºâð(ð)
ððð(ðŒðð¥(1,2,3)(ðº), ððð
ðŠâð ðŒððŠ
(1,2,3)(ð~ðº)))
= ðððð¥âð
ððð(1,1 â ðŒâ±(1,2,3)(ð)
+ ð ð¢ððºâð»=â ,ðºâð(ð)
ððð(ðŒðð¥(1,2,3)(ðº), ððð
ðŠâð ð ð¢ððŠâð»âð~ðº
ðŒð(1,2,3)(ð»)))
= ðððð¥âð
ððð(1,1 â ðŒâ±(1,2,3)(ð)
+ ð ð¢ððºâð»=â ,ðºâð(ð)
ððð(ðŒðð¥(1,2,3)(ðº), ð ð¢ð
ðºâð»=â ,ð âð» ðŒð(1,2,3)(ð»)))
= ðððð¥âð
ððð(1,1 â ðŒâ±(1,2,3)(ð)
+ ð ð¢ððºâð»=â ,ðºâð(ð)
ð ð¢ððºâð»=â ,ð âð»
ððð(ðŒðð¥(1,2,3)(ðº), ðŒð(1,2,3)(ð»)))
= ðððð¥âð
ððð(1,1 â ðŒâ±(1,2,3)(ð) + ð ð¢ððºâð»=â ,ð âð»
ððð(ðŒðð¥(1,2,3)(ðº), ðŒð(1,2,3)(ð»)))
= [ðŒâ(1,2,3)(ð, ð1, ð2, ð3)].
(b) [ ðŒâ(2) (1,2,3)(ð, ð1, ð2, ð3)]
= ðððð¥âð
ððð(1,1 â ðŒð(1,2,3)(ð) + ð ð¢ððºâð»=â ,ðº âð
ððð(ðŒðð¥(1,2,3)(ðº), ðŒð(1,2,3)(ð»)))
= ðððð¥âð~ð
ððð(1,1 â ðŒâ±(1,2,3)(ð~ð)
+ ð ð¢ððºâð~ð=â ,ð» âð~ð
ððð(ðŒðð¥(1,2,3)(ðº), ðŒð(1,2,3)(ð»)))
= [ðŒâ(1,2,3)(ð, ð1, ð2, ð3)].
Definition 2.12 If (ð, ð1, ð2, ð3) is a FTTS, we define
(1) ðŒð©(1) (1,2,3)(ð, ð1, ð2, ð3) â â ðº â ð» (ðº â ðŒâ±(1,2,3) â ð» â ðŒð(1,2,3) â ðº â
ð» â â ð â ð (ð â ðŒâ±(1,2,3) â ð â ðŒð(1,2,3) â ð â ð â ðâð» = â ));
(2) ðŒð©(2) (1,2,3)(ð, ð1, ð2, ð3) â â ðº â ð» (ðº â ðŒâ±(1,2,3) â ð» â ðŒâ±(1,2,3) â ðºâð» =
â â â ð (ð â ðŒð(1,2,3) â ðº â ð â ðŒðð(1,2,3)(ð)âð» = â )).
Theorem 2.13 If (ð, ð1, ð2, ð3) is a FTTS. Then
196 Barah M. Sulaiman and Tahir H. Ismail
âš ðŒð©(1,2,3)(ð, ð1, ð2, ð3) â ðŒð©(ð) (1,2,3)(ð, ð1, ð2, ð3), ð = 1,2.
Proof.
(a) [ ðŒð©(1) (1,2,3)(ð, ð1, ð2, ð3)]
= ððððºâð»
ððð (1,1 â ððð (ðŒâ±(1,2,3)(ðº), ðŒð(1,2,3)(ð»))
+ ð ð¢ððžâð¹,ð¹âð»=â
ððð (ðŒâ±(1,2,3)(ðž), ðŒð(1,2,3)(ð¹)))
= ððððºâð~ð»=â
ððð (1,1 â ððð (ðŒâ±(1,2,3)(ðº), ðŒâ±(1,2,3)(ð~ð»))
+ ð ð¢ðð~ðžâð¹=â ,ðºâð~ðž,ð¹âð~ð»
ððð (ðŒð(1,2,3)(ð~ðž), ðŒð(1,2,3)(ð¹)))
= [ðŒð©(1,2,3)(ð, ð1, ð2, ð3)]. (b) is analogous to the proof of (a) of Theorem (2.11).
3 Relations among ð-separation axioms in fuzzifying tri-
topological spaces
Theorem 3.1 If (ð, ð1, ð2, ð3) is a FTTS. Then
(1) âš (ð, ð1, ð2, ð3) â ðŒð1(1,2,3)
â (ð, ð1, ð2, ð3) â ðŒð0(1,2,3)
;
(2) âš (ð, ð1, ð2, ð3) â ðŒð2(1,2,3)
â (ð, ð1, ð2, ð3) â ðŒðð(1,2,3)
, ð = 0,1.
Proof. From Lemma (2.7), it is clear.
Theorem 3.2 If (ð, ð1, ð2, ð3) is a FTTS. Then
(1) âš (ð, ð1, ð2, ð3) â ðŒð 1(1,2,3)
â (ð, ð1, ð2, ð3) â ðŒð ð(1,2,3)
, ð = 0,2;
(2) âš (ð, ð1, ð2, ð3) â ðŒð1(1,2,3)
â (ð, ð1, ð2, ð3) â ðŒð 0(1,2,3)
;
(3) âš (ð, ð1, ð2, ð3) â ðŒð2(1,2,3)
â (ð, ð1, ð2, ð3) â ðŒð ð(1,2,3)
, ð = 0,1,2.
Proof. (1) (a) From (1) of Lemma (2.7), we have
ðŒð 1(1,2,3)(ð, ð1, ð2, ð3) = ððð
ð¥â ðŠ ððð (1,1 â [ðŒðŠð¥,ðŠ
(1,2,3)] + [ðŒâ³ð¥,ðŠ
(1,2,3)])
†ðððð¥â ðŠ
ððð (1,1 â [ðŒðŠð¥,ðŠ(1,2,3)
] + [ðŒâð¥,ðŠ(1,2,3)
])
= ðŒð 0(1,2,3)(ð, ð1, ð2, ð3)
(b) From (2) of Lemma (2.7), we have
ðŒð 1(1,2,3)(ð, ð1, ð2, ð3) = ððð
ð¥â ðŠ ððð (1,1 â [ðŒðŠð¥,ðŠ
(1,2,3)] + [ðŒâ³ð¥,ðŠ
(1,2,3)])
†ðððð¥â ðŠ
ððð (1,1 â [ðŒâð¥,ðŠ(1,2,3)
] + [ðŒâ³ð¥,ðŠ(1,2,3)
])
= ðŒð 2(1,2,3)(ð, ð1, ð2, ð3)
(2) Using Lemma 2.2 in [2], we have
ðŒð1(1,2,3)(ð, ð1, ð2, ð3) = ððð
ð¥â ðŠ [ðŒâð¥,ðŠ
(1,2,3)]
†ðððð¥â ðŠ
[ðŒðŠð¥,ðŠ(1,2,3)
â ðŒâð¥,ðŠ(1,2,3)
]
On tri ð-separation axioms in fuzzifying tri-topological spaces 197
= ðŒð 0(1,2,3)(ð, ð1, ð2, ð3).
(3) (a) From (2) above and (2) of Theorem (3.1), we have
ðŒð2(1,2,3)(ð, ð1, ð2, ð3) = ððð
ð¥â ðŠ [ðŒâ³ð¥,ðŠ
(1,2,3)] †ððð
ð¥â ðŠ [ðŒðŠð¥,ðŠ
(1,2,3)â ðŒâ³ð¥,ðŠ
(1,2,3)]
= ðŒð 1(1,2,3)(ð, ð1, ð2, ð3)
= ðððð¥â ðŠ
ððð (1,1 â [ðŒðŠð¥,ðŠ(1,2,3)
] + [ðŒâ³ð¥,ðŠ(1,2,3)
])
†ðððð¥â ðŠ
ððð (1,1 â [ðŒðŠð¥,ðŠ(1,2,3)
] + [ðŒâð¥,ðŠ(1,2,3)
])
= ðŒð 0(1,2,3)(ð, ð1, ð2, ð3).
(b) Using Lemma 2.2 in [2], we have
ðŒð2(1,2,3)(ð, ð1, ð2, ð3) = ððð
ð¥â ðŠ [ðŒâ³ð¥,ðŠ
(1,2,3)]
†ðððð¥â ðŠ
[ðŒðŠð¥,ðŠ(1,2,3)
â ðŒâ³ð¥,ðŠ(1,2,3)
]
= ðŒð 1(1,2,3)(ð, ð1, ð2, ð3).
(c) Using Lemma 2.2 in [2], we have
ðŒð2(1,2,3)(ð, ð1, ð2, ð3) = ððð
ð¥â ðŠ [ðŒâ³ð¥,ðŠ
(1,2,3)]
†ðððð¥â ðŠ
[ðŒâð¥,ðŠ(1,2,3)
â ðŒâ³ð¥,ðŠ(1,2,3)
]
= ðŒð 2(1,2,3)(ð, ð1, ð2, ð3).
Theorem 3.3 If (ð, ð1, ð2, ð3) is a FTTS. Then
âš ðŒâ(1,2,3)(ð, ð1, ð2, ð3) â ðŒð1(1,2,3)(ð, ð1, ð2, ð3) â ðŒð2
(1,2,3)(ð, ð1, ð2, ð3).
Proof. It suffices to show that
[ðŒð2(1,2,3)(ð, ð1, ð2, ð3)] ⥠ððð¥(0, ðŒâ(1,2,3)(ð, ð1, ð2, ð3))] +
[ðŒð1(1,2,3)(ð, ð1, ð2, ð3)] â 1).
Since [ðŒð2(1,2,3)(ð, ð1, ð2, ð3)] ⥠0.
Then from Theorem (3.2), we have
[ðŒð1(1,2,3)(ð, ð1, ð2, ð3)] = ððð
ð¥âð ðŒâ±(1,2,3)({ð¥}) = ððð
ð¥âð ðŒð(1,2,3)(ð~{ð¥ })
So [ðŒâ(1,2,3)(ð, ð1, ð2, ð3)] + [ðŒð1(1,2,3)(ð, ð1, ð2, ð3)]
= ðððð¥âð
ððð(1,1 â ðŒð(1,2,3)(ð~ð)
+ ð ð¢ððºâð»=â ,ðâð»
ððð(ðŒðð¥(1,2,3)(ðº), ðŒð(1,2,3)(ð»))) + ððð
ð§âð ðŒð(1,2,3)(ð~{ð§})
†ðððð¥âð,ð¥â ðŠ
ððððŠâð
ððð(1,1 â ðŒð(1,2,3)(ð~{ðŠ}) +
ð ð¢ððºâð»=â ,ðŠâð»
ððð(ðŒðð¥(1,2,3)(ðº), ðŒð(1,2,3)(ð»))) + ððð
ð§âð ðŒð(1,2,3)(ð~{ð§}))
†ðððð¥âð,ð¥â ðŠ
ððððŠâð
ððð(1,1 â ðŒð(1,2,3)(ð~{ðŠ}) +
198 Barah M. Sulaiman and Tahir H. Ismail
ð ð¢ððºâð»=â ,ðŠâð»
ððð(ðŒðð¥(1,2,3)(ðº), ðŒððŠ
(1,2,3)(ð»))) + ðŒð(1,2,3)(ð~{ðŠ}))
= ðððð¥âð,ð¥â ðŠ
ððððŠâð
(ððð(1,1 + ð ð¢ððºâð»=â
ððð(ðŒðð¥(1,2,3)(ðº), ðŒððŠ
(1,2,3)(ð»))))
= ðððð¥âð,ð¥â ðŠ
ððððŠâð
(1 + ð ð¢ððºâð»=â
ððð(ðŒðð¥(1,2,3)(ðº), ðŒððŠ
(1,2,3)(ð»)))
= 1 + ðððð¥â ðŠ
ð ð¢ððºâð»=â
ððð(ðŒðð¥(1,2,3)(ðº), ðŒððŠ
(1,2,3)(ð»))
= 1 + [ðŒð2(1,2,3)(ð, ð1, ð2, ð3)].
Thus
[ðŒð2(1,2,3)(ð, ð1, ð2, ð3)] ⥠ððð¥(0, ðŒâ(1,2,3)(ð, ð1, ð2, ð3))] +
[ðŒð1(1,2,3)(ð, ð1, ð2, ð3)] â 1).
Corollary 3.4 If (ð, ð1, ð2, ð3) is a FTTS. Then
(1) âš ðŒð3(1,2,3)(ð, ð1, ð2, ð3) â ðŒð2
(1,2,3)(ð, ð1, ð2, ð3).
(2) âš ðŒð3(1,2,3)(ð, ð1, ð2, ð3) â ðŒð ð
(1,2,3)(ð, ð1, ð2, ð3), ð = 0,1,2.
Theorem 3.5 If (ð, ð1, ð2, ð3) is a FTTS. Then
âš ðŒð4(1,2,3)(ð, ð1, ð2, ð3) â ðŒâ(1,2,3)(ð, ð1, ð2, ð3).
Proof.
ðŒð4(1,2,3)(ð, ð1, ð2, ð3) = ððð¥(0, [ðŒð©(1,2,3)(ð, ð1, ð2, ð3))] +
[ðŒð1(1,2,3)(ð, ð1, ð2, ð3)] â 1),
now we prove that
[ðŒâ(1,2,3)(ð, ð1, ð2, ð3)] ⥠[ðŒð©(1,2,3)(ð, ð1, ð2, ð3)] + [ðŒð1(1,2,3)(ð, ð1, ð2, ð3)] â
1.
In fact
[ðŒð©(1,2,3)(ð, ð1, ð2, ð3)] + [ðŒð1(1,2,3)(ð, ð1, ð2, ð3)]
= ððððâð=â
ððð (1,1 â ððð (ðŒâ±(1,2,3)(ð), ðŒâ±(1,2,3)(ð))
+ ð ð¢ððºâð»=â ,ðâð»,ðâðº
ððð (ðŒð(1,2,3)(ðº), ðŒð(1,2,3)(ð»))) + ðððð§âð
ðŒð(1,2,3)(ð~{ð§})
= ððððâð=â
ððð (1,1 â ððð (ðŒð(1,2,3)(ð~ð), ðŒð(1,2,3)(ð~ð))
+ ð ð¢ððºâð»=â ,ðâð»,ðâðº
ððð (ðŒð(1,2,3)(ðº), ðŒð(1,2,3)(ð»))) + ðððð§âð
ðŒð(1,2,3)(ð~{ð§})
†ððð ð¥âð
ððð (1,1 â ððð (ðŒð(1,2,3)(ð~ð), ðŒð(1,2,3)(ð~{ð¥}))
+ ð ð¢ððºâð»=â ,ðâð»
ððð (ðŒðð¥(1,2,3)(ðº), ðŒð(1,2,3)(ð»)) + ððð
ð§âð ðŒð(1,2,3)(ð~{ð§})
= ððð ð¥âð
ððð (1, ððð¥ (1 â ðŒð(1,2,3)(ð~ð)
+ ð ð¢ððºâð»=â ,ðâð»
ððð (ðŒðð¥(1,2,3)(ðº), ðŒð(1,2,3)(ð»)),1 â ðŒð(1,2,3)(ð~{ð¥})
+ ð ð¢ððºâð»=â ,ðâð»
ððð (ðŒðð¥(1,2,3)(ðº), ðŒð(1,2,3)(ð»))) + ððð
ð§âð ðŒð(1,2,3)(ð~{ð§})
On tri ð-separation axioms in fuzzifying tri-topological spaces 199
= ððð ð¥âð
ððð¥ (ððð (1,1 â ðŒð(1,2,3)(ð~ð)
+ ð ð¢ððºâð»=â ,ðâð»
ððð (ðŒðð¥(1,2,3)(ðº), ðŒð(1,2,3)(ð»))), ððð (1,1 â ðŒð(1,2,3)(ð~{ð¥})
+ ð ð¢ððºâð»=â ,ðâð»
ððð (ðŒðð¥(1,2,3)(ðº), ðŒð(1,2,3)(ð»))) + ððð
ð§âð ðŒð(1,2,3)(ð~{ð§})
†ððð ð¥âð
ððð¥(ððð (1,1 â ðŒð(1,2,3)(ð~ð)
+ ð ð¢ððºâð»=â ,ðâð»
ððð(ðŒðð¥(1,2,3)(ðº), ðŒð(1,2,3)(ð»)))
+ðŒð(1,2,3)(ð~{ð¥}), ððð (1,1 â ðŒð(1,2,3)(ð~{ð¥}))
+ ð ð¢ððºâð»=â ,ðâð»
ððð(ðŒðð¥(1,2,3)(ðº), ðŒð(1,2,3)(ð»))) + ðŒð(1,2,3)(ð~{ð¥}))
†ððð ð¥âð
ððð¥(ððð (1,1 â ðŒð(1,2,3)(ð~ð)
+ ð ð¢ððºâð»=â ,ðâð»
ððð(ðŒðð¥(1,2,3)(ðº), ðŒð(1,2,3)(ð»))) + ðŒð(1,2,3)(ð~{ð¥}),
1 + ð ð¢ððºâð»=â ,ðâð»
ððð(ðŒðð¥(1,2,3)(ðº), ðŒð(1,2,3)(ð»)))
†ððð ð¥âð
ððð (1,1 â ðŒð(1,2,3)(ð~ð)
+ ð ð¢ððºâð»=â ,ðâð»
ððð(ðŒðð¥(1,2,3)(ðº), ðŒð(1,2,3)(ð»))) + 1
= [ðŒâ(1,2,3)(ð, ð1, ð2, ð3)] + 1.
Theorem 3.6 If (ð, ð1, ð2, ð3) is a FTTS. Then
(1) âš (ð, ð1, ð2, ð3) â ðŒð1(1,2,3)
â (ð, ð1, ð2, ð3) â ðŒð 0(1,2,3)
â (ð, ð1, ð2, ð3) â
ðŒð0(1,2,3)
;
(2) If ðŒð0(1,2,3)(ð, ð1, ð2, ð3) = 1, then
âš (ð, ð1, ð2, ð3) â ðŒð1(1,2,3)
â (ð, ð1, ð2, ð3) â ðŒð 0(1,2,3)
â (ð, ð1, ð2, ð3) â
ðŒð0(1,2,3)
.
Proof. (1) Follows from (1) of Theorem (3.1) and (2) of Theorem (3.2).
(2) Since ðŒð0(1,2,3)(ð, ð1, ð2, ð3) = 1, then for every ð¥, ðŠ â ð such that ð¥ â ðŠ, we
have [ðŒðŠð¥,ðŠ(1,2,3)
] = 1. So
ðŒð 0(1,2,3)(ð, ð1, ð2, ð3) â ðŒð0
(1,2,3)(ð, ð1, ð2, ð3)
= ðŒð 0(1,2,3)(ð, ð1, ð2, ð3)
= ðððð¥â ðŠ
ððð(1,1 â [ðŒðŠð¥,ðŠ(1,2,3)
] + [ðŒâð¥,ðŠ(1,2,3)
])
= ðððð¥â ðŠ
[ðŒâð¥,ðŠ(1,2,3)
] = ðŒð1(1,2,3)(ð, ð1, ð2, ð3).
Theorem 3.7 If (ð, ð1, ð2, ð3) is a FTTS. Then
200 Barah M. Sulaiman and Tahir H. Ismail
(1) âš (ð, ð1, ð2, ð3) â ðŒð2(1,2,3)
â (ð, ð1, ð2, ð3) â ðŒð 1(1,2,3)
â (ð, ð1, ð2, ð3) â
ðŒð0(1,2,3)
;
(2) If ðŒð0(1,2,3)(ð, ð1, ð2, ð3) = 1, then
âš (ð, ð1, ð2, ð3) â ðŒð2(1,2,3)
â (ð, ð1, ð2, ð3) â ðŒð 1(1,2,3)
â (ð, ð1, ð2, ð3) â
ðŒð0(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 (ð, ð1, ð2, ð3) is a FTTS. Then
(1) âš (ð, ð1, ð2, ð3) â ðŒð2(1,2,3)
â (ð, ð1, ð2, ð3) â ðŒð 0(1,2,3)
â (ð, ð1, ð2, ð3) â
ðŒð1(1,2,3)
;
(2) If ðŒð1(1,2,3)(ð, ð1, ð2, ð3) = 1, then
âš (ð, ð1, ð2, ð3) â ðŒð2(1,2,3)
â (ð, ð1, ð2, ð3) â ðŒð 0(1,2,3)
â (ð, ð1, ð2, ð3) â
ðŒð1(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 (ð, ð1, ð2, ð3) is a FTTS. Then we have
(1) âš (ð, ð1, ð2, ð3) â ðŒð1(1,2,3)
â (ð, ð1, ð2, ð3) â ðŒð 0(1,2,3)
â (ð, ð1, ð2, ð3) â
ðŒð0(1,2,3)
;
(2) âš (ð, ð1, ð2, ð3) â ðŒð2(1,2,3)
â (ð, ð1, ð2, ð3) â ðŒð 1(1,2,3)
â (ð, ð1, ð2, ð3) â
ðŒð0(1,2,3)
.
(3) âš (ð, ð1, ð2, ð3) â ðŒð2(1,2,3)
â (ð, ð1, ð2, ð3) â ðŒð 0(1,2,3)
â (ð, ð1, ð2, ð3) â
ðŒð1(1,2,3)
.
Theorem 3.10 If (ð, ð1, ð2, ð3) is a FTTS. Then
(1) (ð, ð1, ð2, ð3) â ðŒð 0(1,2,3)
â (ð, ð1, ð2, ð3) â ðŒð0(1,2,3)
â (ð, ð1, ð2, ð3) â
ðŒð1(1,2,3)
;
(2) If ðŒð0(1,2,3)(ð, ð1, ð2, ð3) = 1, then
âš (ð, ð1, ð2, ð3) â ðŒð 0(1,2,3)
â (ð, ð1, ð2, ð3) â ðŒð0(1,2,3)
â (ð, ð1, ð2, ð3) â
ðŒð1(1,2,3)
.
Proof.
(1) [(ð, ð1, ð2, ð3) â ðŒð 0(1,2,3)
â (ð, ð1, ð2, ð3) â ðŒð0(1,2,3)
]
= ððð¥(0, ðŒð 0(1,2,3)(ð, ð1, ð2, ð3) + ðŒð0
(1,2,3)(ð, ð1, ð2, ð3) â 1)
= ððð¥(0, ðððð¥â ðŠ
ððð(1,1 â [ðŒðŠð¥,ðŠ(1,2,3)
] + [ðŒâð¥,ðŠ(1,2,3)
]) + ðððð¥â ðŠ
[ðŒðŠð¥,ðŠ(1,2,3)
] â 1)
On tri ð-separation axioms in fuzzifying tri-topological spaces 201
†ððð¥(0, ðððð¥â ðŠ
(ððð(1,1 â [ðŒðŠð¥,ðŠ(1,2,3)
] + [ðŒâð¥,ðŠ(1,2,3)
]) + [ðŒðŠð¥,ðŠ(1,2,3)
] â 1)
†ððð¥(0, ðððð¥â ðŠ
(1 â [ðŒðŠð¥,ðŠ(1,2,3)
] + [ðŒâð¥,ðŠ(1,2,3)
] + [ðŒðŠð¥,ðŠ(1,2,3)
] â 1)
= ðððð¥â ðŠ
[ðŒâð¥,ðŠ(1,2,3)
] = ðŒð1(1,2,3)(ð, ð1, ð2, ð3).
(2) Follows from (2) Theorem (3.6).
Theorem 3.11 If (ð, ð1, ð2, ð3) is a FTTS. Then
(1) (ð, ð1, ð2, ð3) â ðŒð 1(1,2,3)
â (ð, ð1, ð2, ð3) â ðŒð0(1,2,3)
â (ð, ð1, ð2, ð3) â
ðŒð2(1,2,3)
;
(2) If ðŒð0(1,2,3)(ð, ð1, ð2, ð3) = 1, then
âš (ð, ð1, ð2, ð3) â ðŒð 1(1,2,3)
â (ð, ð1, ð2, ð3) â ðŒð0(1,2,3)
â (ð, ð1, ð2, ð3) â
ðŒð2(1,2,3)
.
Proof.
(1) [(ð, ð1, ð2, ð3) â ðŒð 1(1,2,3)
â (ð, ð1, ð2, ð3) â ðŒð0(1,2,3)
]
= ððð¥(0, ðŒð 1(1,2,3)(ð, ð1, ð2, ð3) + ðŒð0
(1,2,3)(ð, ð1, ð2, ð3) â 1)
= ððð¥(0, ðððð¥â ðŠ
ððð(1,1 â [ðŒðŠð¥,ðŠ(1,2,3)
] + [ðŒâ³ð¥,ðŠ(1,2,3)
]) + ðððð¥â ðŠ
[ðŒðŠð¥,ðŠ(1,2,3)
] â 1)
†ððð¥(0, ðððð¥â ðŠ
(ððð(1,1 â [ðŒðŠð¥,ðŠ(1,2,3)
] + [ðŒâ³ð¥,ðŠ(1,2,3)
]) + [ðŒðŠð¥,ðŠ(1,2,3)
] â 1)
†ððð¥(0, ðððð¥â ðŠ
(1 â [ðŒðŠð¥,ðŠ(1,2,3)
] + [ðŒâ³ð¥,ðŠ(1,2,3)
] + [ðŒðŠð¥,ðŠ(1,2,3)
] â 1)
= ðððð¥â ðŠ
[ðŒâ³ð¥,ðŠ(1,2,3)
] = ðŒð2(1,2,3)(ð, ð1, ð2, ð3).
(2) Follows from (2) Theorem (3.6).
Theorem 3.12 If (ð, ð1, ð2, ð3) is a FTTS. Then
(1) âš (ð, ð1, ð2, ð3) â ðŒð0(1,2,3)
â ((ð, ð1, ð2, ð3) â ðŒð 0(1,2,3)
â (ð, ð1, ð2, ð3) â
ðŒð1(1,2,3)
;
(2) âš (ð, ð1, ð2, ð3) â ðŒð 0(1,2,3)
â ((ð, ð1, ð2, ð3) â ðŒð0(1,2,3)
â (ð, ð1, ð2, ð3) â
ðŒð1(1,2,3)
;
(3) âš (ð, ð1, ð2, ð3) â ðŒð0(1,2,3)
â ((ð, ð1, ð2, ð3) â ðŒð 1(1,2,3)
â (ð, ð1, ð2, ð3) â
ðŒð2(1,2,3)
;
(4) âš (ð, ð1, ð2, ð3) â ðŒð 1(1,2,3)
â ((ð, ð1, ð2, ð3) â ðŒð0(1,2,3)
â (ð, ð1, ð2, ð3) â
ðŒð2(1,2,3)
.
Proof. (1) From (2) Theorem (3.1) and (3) Theorem (3.2), we have
[(ð, ð1, ð2, ð3) â ðŒð0(1,2,3)
â ((ð, ð1, ð2, ð3) â ðŒð 0(1,2,3)
â (ð, ð1, ð2, ð3) â
ðŒð1(1,2,3)
]
202 Barah M. Sulaiman and Tahir H. Ismail
= ððð(1,1 â [(ð, ð1, ð2, ð3) â ðŒð0(1,2,3)
] + ððð(1,1 â [(ð, ð1, ð2, ð3) â
ðŒð 0(1,2,3)
] + [(ð, ð1, ð2, ð3) â ðŒð1(1,2,3)
]))
= ððð(1,1 â [(ð, ð1, ð2, ð3) â ðŒð0(1,2,3)
] + 1 â [(ð, ð1, ð2, ð3) â ðŒð 0(1,2,3)
] +
[(ð, ð1, ð2, ð3) â ðŒð1(1,2,3)
]))
= ððð(1,1 â ([(ð, ð1, ð2, ð3) â ðŒð 0(1,2,3)
] + [(ð, ð1, ð2, ð3) â ðŒð0(1,2,3)
] â 1) +
[(ð, ð1, ð2, ð3) â ðŒð1(1,2,3)
]) = 1.
(2) From (1) Theorem (3.1) and (3) Theorem (3.6), we have
[(ð, ð1, ð2, ð3) â ðŒð 0(1,2,3)
â ((ð, ð1, ð2, ð3) â ðŒð0(1,2,3)
â (ð, ð1, ð2, ð3) â
ðŒð1(1,2,3)
]
= ððð(1,1 â [(ð, ð1, ð2, ð3) â ðŒð 0(1,2,3)
] + ððð(1,1 â [(ð, ð1, ð2, ð3) â
ðŒð0(1,2,3)
] + [(ð, ð1, ð2, ð3) â ðŒð1(1,2,3)
]))
= ððð(1,1 â [(ð, ð1, ð2, ð3) â ðŒð 0(1,2,3)
] + 1 â [(ð, ð1, ð2, ð3) â ðŒð0(1,2,3)
] +
[(ð, ð1, ð2, ð3) â ðŒð1(1,2,3)
]))
= ððð(1,1 â ([(ð, ð1, ð2, ð3) â ðŒð 0(1,2,3)
] + [(ð, ð1, ð2, ð3) â ðŒð0(1,2,3)
] â 1) +
[(ð, ð1, ð2, ð3) â ðŒð1(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