On Tri ๐-Separation Axioms in Fuzzifying Tri-Topological ...Jan 04, 2019 ย ยท al. (2001) studied...
Transcript of On Tri ๐-Separation Axioms in Fuzzifying Tri-Topological ...Jan 04, 2019 ย ยท al. (2001) studied...
International Journal of Mathematical Analysis
Vol. 13, 2019, no. 4, 191 โ 203
HIKARI Ltd, www.m-hikari.com
https://doi.org/10.12988/ijma.2019.9319
On Tri ๐-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 ยฉ 2019 Hikari Ltd.
Abstract
The present article introduce ๐ผ๐0(1,2,3)
(Kolmogorov), ๐ผ๐1(1,2,3)
(Frรฉchet),
๐ผ๐2(1,2,3)
(Hausdorff), ๐ผโ(1,2,3)(๐ผ-regular), ๐ผ๐ฉ(1,2,3)(๐ผ-normal),๐ผ๐ 0(1,2,3)
, ๐ผ๐ 1(1,2,3)
and ๐ผ๐ 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 ๐ผ-separation axioms
1 Introduction
Ying (1991-1993) introduced the concept of the term โfuzzifying topologyหฎ [7-
9]. Wuyts and Lowen (1983) studied "separation axioms in fuzzy topological
spaces" [6]. Shen (1993) introduced and studied ๐0, ๐1, ๐2 (Hausdorff), ๐3
(regularity), ๐4 (normality)-separation axioms in fuzzifying topology [3]. Khedr et
al. (2001) studied โseparation axioms in fuzzifying topologyหฎ [2]. Sayed (2014)
presented "ฮฑ-separation axioms based on ลukasiewicz logic" [4]. Allam et al.
(2015) studied โsemi separation axioms in fuzzifying bitopological spacesหฎ [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 (๐, ๐1, ๐2, ๐3) is a fuzzifying tri-topological space (FTTS),
192 Barah M. Sulaiman and Tahir H. Ismail
(i) The family of fuzzifying (1,2,3) ฮฑ-open sets in ๐, symbolized as ๐ผ๐(1,2,3) โ
โ(๐(๐)), and defined as
๐ธ โ ๐ผ๐(1,2,3) โ โ ๐ฅ (๐ฅ โ ๐ธ โ ๐ฅ โ ๐๐๐ก1(๐๐2(๐๐๐ก3(๐ธ)))),
i.e., ๐ผ๐(1,2,3)(๐ธ) = ๐๐๐๐ฅโ๐ธ
(๐๐๐ก1(๐๐2(๐๐๐ก3(๐ธ))))(๐ฅ).
(ii) The family of fuzzifying (1,2,3) ฮฑ-closed sets in ๐, symbolized as ๐ผโฑ(1,2,3),
and defined by ๐น โ ๐ผโฑ(1,2,3) โ ๐~๐น โ ๐ผ๐(1,2,3).
(iii) The (1,2,3) ฮฑ-neighborhood system of ๐ฅ, denoted by ๐ผ๐๐ฅ(1,2,3)
and defined as
๐ธ โ ๐ผ๐๐ฅ(1,2,3)
โ โ ๐น (๐น โ ๐ผ๐(1,2,3) โ ๐ฅ โ ๐น โ ๐ธ);
i.e. ๐ผ๐๐ฅ(1,2,3)(๐ธ) = ๐ ๐ข๐
๐ฅโ๐นโ๐ธ๐ผ๐(1,2,3)(๐น).
(iv) The (1,2,3) ฮฑ-derived set of E โ X, denoted by ๐ผ๐(1,2,3)(๐ธ) and defined as
๐ฅ โ ๐ผ๐(1,2,3)(๐ธ) โ โ ๐น (๐น โ ๐ผ๐๐ฅ(1,2,3)
โ ๐น โฉ (๐ธ โ {๐ฅ}) โ โ ),
i.e., ๐ผ๐(1,2,3)(๐ธ)(๐ฅ) = ๐๐๐๐นโฉ(๐ธโ{๐ฅ})โ โ
(1 โ ๐ผ๐๐ฅ(1,2,3)(๐น)).
(v) The (1,2,3) ฮฑ-closure set of ๐ธ โ ๐, denoted by ๐ผ๐๐(1,2,3)(๐ธ) and defined as
๐ฅ โ ๐ผ๐๐(1,2,3)(๐ธ) โ โ ๐น (๐น โ ๐ธ) โฉ (๐น โ ๐ผโฑ(1,2,3)) โ ๐ฅ โ ๐น),
i.e., ๐ผ๐๐(1,2,3)(๐ธ)(๐ฅ) = ๐๐๐๐ฅโ๐นโ๐ธ
(1 โ ๐ผโฑ(1,2,3)(๐น)).
(vi) The (1,2,3) ฮฑ-interior set of ๐ธ โ ๐, denoted by ๐ผ๐๐๐ก(1,2,3)(๐ธ) and defined as
๐ผ๐๐๐ก(1,2,3)(๐ธ)(๐ฅ) = ๐ผ๐๐ฅ(1,2,3)
(๐ธ).
(vii) The (1,2,3) ฮฑ-exterior set of ๐ธ โ ๐, denoted by ๐ผ๐๐ฅ๐ก(1,2,3)(๐ธ) and defined as
๐ฅ โ ๐ผ๐๐ฅ๐ก(1,2,3)(๐ธ) โ ๐ฅ โ ๐ผ๐๐๐ก(1,2,3)(๐~๐ธ)(๐ฅ),
i.e. ๐ผ๐๐ฅ๐ก(1,2,3)(๐ธ)(๐ฅ) = ๐ผ๐๐๐ก(1,2,3)(๐~๐ธ)(๐ฅ).
(viii) The (1,2,3) ฮฑ-boundary set of ๐ธ โ ๐, denoted by ๐ผ๐(1,2,3)(๐ธ) and defined as
๐ฅ โ ๐ผ๐(1,2,3)(๐ธ) โ (๐ฅ โ ๐ผ๐๐๐ก(1,2,3)(๐ธ)) โ (๐ฅ โ ๐ผ๐๐๐ก(1,2,3)(๐~๐ธ)),
i.e. ๐ผ๐(1,2,3)(๐ธ)(๐ฅ) โ ๐๐๐(1 โ ๐ผ๐๐๐ก(1,2,3)(๐ธ)(๐ฅ)) โ (1 โ
๐ผ๐๐๐ก(1,2,3)(๐~๐ธ)(๐ฅ)).
2 Tri ๐-Separation axioms in fuzzifying tri-topological spaces
Remark 2.2 We consider the following notations:
๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
โ โ ๐บ ((๐บ โ ๐ผ๐๐ฅ(1,2,3)
โ ๐ฆ โ ๐บ) โ (๐บ โ ๐ผ๐๐ฆ(1,2,3)
โ ๐ฅ โ ๐บ));
๐ผโ๐ฅ,๐ฆ(1,2,3)
โ โ ๐ป โ ๐ธ (๐ป โ ๐ผ๐๐ฅ(1,2,3)
โ ๐ธ โ ๐ผ๐๐ฆ(1,2,3)
โ ๐ฆ โ ๐ป โ ๐ฅ โ ๐ธ);
๐ผโณ๐ฅ,๐ฆ(1,2,3)
โ โ ๐ป โ ๐ธ (๐ป โ ๐ผ๐๐ฅ(1,2,3)
โ ๐ธ โ ๐ผ๐๐ฆ(1,2,3)
โ ๐ปโ๐ธ = โ ).
Definition 2.3 If ๐บ is the class of all FTTSs. The predicates ๐ผ๐๐(1,2,3)
, ๐ผ๐ ๐(1,2,3)
โ
โ(๐บ), ๐ = 0,1,2, are defined as follow
(๐, ๐1, ๐2, ๐3) โ ๐ผ๐0(1,2,3)
โ โ ๐ฅ โ ๐ฆ (๐ฅ โ ๐ โ ๐ฆ โ ๐ โ ๐ฅ โ ๐ฆ โ ๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
);
(๐, ๐1, ๐2, ๐3) โ ๐ผ๐1(1,2,3)
โ โ ๐ฅ โ ๐ฆ (๐ฅ โ ๐ โ ๐ฆ โ ๐ โ ๐ฅ โ ๐ฆ โ ๐ผโ๐ฅ,๐ฆ(1,2,3)
);
(๐, ๐1, ๐2, ๐3) โ ๐ผ๐2(1,2,3)
โ โ ๐ฅ โ ๐ฆ (๐ฅ โ ๐ โ ๐ฆ โ ๐ โ ๐ฅ โ ๐ฆ โ ๐ผโณ๐ฅ,๐ฆ(1,2,3)
);
On tri ๐-separation axioms in fuzzifying tri-topological spaces 193
(๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 0(1,2,3)
โ โ ๐ฅ โ ๐ฆ (๐ฅ โ ๐ โ ๐ฆ โ ๐ โ ๐ฅ โ ๐ฆ โ (๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
โ
๐ผโ๐ฅ,๐ฆ(1,2,3)
);
(๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 1(1,2,3)
โ โ ๐ฅ โ ๐ฆ (๐ฅ โ ๐ โ ๐ฆ โ ๐ โ ๐ฅ โ ๐ฆ โ (๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
โ
๐ผโณ๐ฅ,๐ฆ(1,2,3)
);
(๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 2(1,2,3)
โ โ ๐ฅ โ ๐ฆ (๐ฅ โ ๐ โ ๐ฆ โ ๐ โ ๐ฅ โ ๐ฆ โ (๐ผโ๐ฅ,๐ฆ(1,2,3)
โ
๐ผโณ๐ฅ,๐ฆ(1,2,3)
).
Definition 2.4 If ๐บ is the class of all FTTSs. The predicates ๐ผโ(1,2,3), ๐ผ๐ฉ(1,2,3) โโ(ฮฉ), are defined as follow
(1) (๐, ๐1, ๐2, ๐3) โ ๐ผโ(1,2,3) โ โ ๐ฅ โ ๐ (๐ฅ โ ๐ โ ๐ โ ๐ผโฑ(1,2,3) โ ๐ฅ โ ๐ โ
โ ๐บ โ ๐ป (๐บ โ ๐ผ๐๐ฅ(1,2,3)
โ ๐ป โ ๐ผ๐(1,2,3) โ ๐ โ ๐ป โ ๐บโ๐ป = โ ));
(2) (๐, ๐1, ๐2, ๐3) โ ๐ผ๐ฉ(1,2,3) โ โ ๐บ โ ๐ป (๐บ โ ๐ผโฑ(1,2,3) โ ๐ป โ ๐ผโฑ(1,2,3) โ ๐บโ๐ป =
โ ) โ โ ๐ โ ๐ (๐ โ ๐ผ๐(1,2,3) โ ๐ โ ๐ผ๐(1,2,3)โ ๐บ โ ๐ โ๐ป โ ๐ โ ๐โ๐ = โ ).
Definition 2.5 If ๐บ is the class of all FTTSs. The predicates ๐ผ๐3(1,2,3)
, ๐ผ๐4(1,2,3)
โโ(๐บ) are defined as follow
(1) ๐ผ๐3(1,2,3)(๐, ๐1, ๐2, ๐3) โ ๐ผโ(1,2,3)(๐, ๐1, ๐2, ๐3) โ ๐ผ๐1
(1,2,3)(๐, ๐1, ๐2, ๐3);
(2) ๐ผ๐4(1,2,3)(๐, ๐1, ๐2, ๐3) โ ๐ผ๐ฉ(1,2,3)(๐, ๐1, ๐2, ๐3) โ ๐ผ๐1
(1,2,3)(๐, ๐1, ๐2, ๐3).
Remark 2.6 If (๐, ๐1, ๐2, ๐3) is a FTTS. Note that
(1) ๐ผ๐๐(1,2,3)
= ๐ผ๐๐(3,2,1)
, ๐ = 0,1,2,3,4;
(2) ๐ผ๐ ๐(1,2,3)
= ๐ผ๐ ๐(3,2,1)
, ๐ = 0,1,2.
Lemma 2.7 If (๐, ๐1, ๐2, ๐3) is a FTTS. Then
(1) โจ ๐ผโณ๐ฅ,๐ฆ(1,2,3)
โ ๐ผโ๐ฅ,๐ฆ(1,2,3)
;
(2) โจ ๐ผโ๐ฅ,๐ฆ(1,2,3)
โ ๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
;
(3) โจ ๐ผโณ๐ฅ,๐ฆ(1,2,3)
โ ๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
.
Proof.
(1) [ ๐ผ๐๐ฅ,๐ฆ(1,2,3)
] = ๐ ๐ข๐๐ตโ๐ถ=โ
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐ต), ๐ผ๐๐ฆ
(1,2,3)(๐ถ)) โค
๐ ๐ข๐๐ฆโ๐ต,๐ฅโ๐ถ
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐ต), ๐ผ๐๐ฆ
(1,2,3)(๐ถ)) = [๐ผโ๐ฅ,๐ฆ(1,2,3)
].
(2) [ ๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
] = ๐๐๐ฅ(๐ ๐ข๐ ๐ฆโ๐ด
๐ผ๐๐ฅ(1,2,3)(๐ด), ๐ ๐ข๐
๐ฅโ๐ด ๐ผ๐๐ฆ
(1,2,3)(๐ด))
โฅ ๐ ๐ข๐๐ฆโ๐ด
๐ผ๐๐ฅ(1,2,3)(๐ด) โฅ
๐ ๐ข๐๐ฆโ๐ด,๐ฅโ๐ต
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐ด), ๐ผ๐๐ฆ
(1,2,3)(๐ต)) = [๐ผโ๐ฅ,๐ฆ(1,2,3)
].
(3) is concluded from (1) and (2) above.
Theorem 2.8 If (๐, ๐1, ๐2, ๐3) is a FTTS. Then
194 Barah M. Sulaiman and Tahir H. Ismail
โจ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐0(1,2,3)
โ โ ๐ฅ โ ๐ฆ (๐ฅ โ ๐ โ ๐ฆ โ ๐ โ ๐ฅ โ ๐ฆ โ ๐ฅ โ๐ผ๐๐(1,2,3)({๐ฆ})โ๐ฆ โ ๐ผ๐๐(1,2,3)({๐ฅ})).
Proof.
๐ผ๐0(1,2,3)(๐, ๐1, ๐2, ๐3)
= ๐๐๐๐ฅโ ๐ฆ
๐๐๐ฅ(๐ ๐ข๐๐ฆโ๐ด
๐ผ๐๐ฅ(1,2,3)(๐ด), ๐ ๐ข๐
๐ฅโ๐ด ๐ผ๐๐ฆ
(1,2,3)(๐ด))
= ๐๐๐๐ฅโ ๐ฆ
๐๐๐ฅ(๐ผ๐๐ฅ(1,2,3)(๐~{๐ฆ}), ๐ผ๐๐ฆ
(1,2,3)(๐~{๐ฅ}))
= ๐๐๐๐ฅโ ๐ฆ
๐๐๐ฅ(1 โ ๐ผ๐๐(1,2,3)({๐ฆ})(๐ฅ),1 โ ๐ผ๐๐(1,2,3)({๐ฅ})(๐ฆ))
= [โ ๐ฅ โ ๐ฆ (๐ฅ โ ๐ โ ๐ฆ โ ๐ โ ๐ฅ โ ๐ฆ โ ๐ฅ โ ๐ผ๐๐(1,2,3)({๐ฆ})โ๐ฆ โ ๐ผ๐๐(1,2,3)({๐ฅ}))].
Theorem 2.9 If (๐, ๐1, ๐2, ๐3) is a FTTS. Then
โจ โ ๐ฅ ({๐ฅ} โ ๐ผโฑ(1,2,3)) โ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐1(1,2,3)
.
Proof.
๐ผ๐1(1,2,3)(๐, ๐1, ๐2, ๐3)
= ๐๐๐๐ฅ1โ ๐ฅ2
๐๐๐( ๐ ๐ข๐๐ฅ2โ๐ด
๐ผ๐๐ฅ1
(1,2,3)(๐ด), ๐ ๐ข๐๐ฅ1โ๐ต
๐ผ๐๐ฅ2
(1,2,3)(๐ต)) =
๐๐๐๐ฅ1โ ๐ฅ2
๐๐๐(๐ผ๐๐ฅ1
(1,2,3)(๐~{๐ฅ2}), ๐ผ๐๐ฅ2
(1,2,3)(๐~{๐ฅ1})) โค
๐๐๐๐ฅ1โ ๐ฅ2
๐ผ๐๐ฅ1
(1,2,3)(๐~{๐ฅ2}) = ๐๐๐๐ฅ2โ๐
๐๐๐๐ฅ1โ๐~{๐ฅ2}
๐ผ๐๐ฅ1
(1,2,3)(๐~{๐ฅ2})
= ๐๐๐๐ฅ2โ๐
๐ผ๐(1,2,3)(๐~{๐ฅ2}) = ๐๐๐๐ฅโ๐
๐ผ๐(1,2,3)(๐~{๐ฅ}) = ๐๐๐๐ฅโ๐
๐ผโฑ(1,2,3)({๐ฅ}).
Now, for any ๐ฅ1, ๐ฅ2 โ ๐ with ๐ฅ1 โ ๐ฅ2.
[โ ๐ฅ ({๐ฅ} โ ๐ผโฑ(1,2,3))]
= ๐๐๐๐ฅโ๐
[{๐ฅ} โ ๐ผโฑ(1,2,3)] = ๐๐๐๐ฅโ๐
๐ผ๐(1,2,3)(๐~{๐ฅ}) = ๐๐๐๐ฅโ๐
๐๐๐๐ฆโ๐~{๐ฅ}
๐ผ๐๐ฆ(1,2,3)(๐~{๐ฅ})
โค ๐๐๐๐ฆโ๐~{๐ฅ2}
๐ผ๐๐ฆ(1,2,3)(๐~{๐ฅ2}) โค ๐ผ๐๐ฅ2
(1,2,3)(๐~{๐ฅ2}) = ๐ ๐ข๐๐ฅ2โ๐ด
๐ผ๐๐ฅ1
(1,2,3)(๐ด).
By the same way, we have
[โ ๐ฅ ({๐ฅ} โ ๐ผโฑ(1,2,3))] โค ๐ ๐ข๐๐ฅ1โ๐ด
๐ผ๐๐ฅ2
(1,2,3)(๐ต). So
[โ ๐ฅ ({๐ฅ} โ ๐ผโฑ(1,2,3))] โค ๐๐๐๐ฅ1โ ๐ฅ2
๐๐๐( ๐ ๐ข๐๐ฅ2โ๐ด
๐ผ๐๐ฅ1
(1,2,3)(๐ด), ๐ ๐ข๐๐ฅ1โ๐ต
๐ผ๐๐ฅ2
(1,2,3)(๐ต))
= ๐ผ๐1(1,2,3)(๐, ๐1, ๐2, ๐3).
Therefore ๐ผ๐1(1,2,3)(๐, ๐1, ๐2, ๐3) = [โ ๐ฅ ({๐ฅ} โ ๐ผโฑ(1,2,3))].
Definition 2.10 If (๐, ๐1, ๐2, ๐3) is a FTTS, we define
(1) ๐ผโ(1) (1,2,3)(๐, ๐1, ๐2, ๐3) โ โ ๐ฅ โ ๐ (๐ฅ โ ๐ โ ๐ โ ๐ผโฑ(1,2,3) โ ๐ฅ โ ๐ โ
โ ๐บ (๐บ โ ๐ผ๐๐ฅ(1,2,3)
โ ๐ผ๐๐(1,2,3)(๐บ)โ๐ = โ ));
(2) ๐ผโ(2) (1,2,3)(๐, ๐1, ๐2, ๐3) โ โ ๐ฅ โ ๐ (๐ฅ โ ๐ โ ๐ โ ๐ผ๐(1,2,3) โ ๐ฅ โ ๐ โ
โ ๐บ โ ๐ป (๐บ โ ๐ผ๐๐ฅ(1,2,3)
โ ๐ป โ ๐ผ๐(1,2,3) โ ๐บ โ ๐ โ ๐บโ๐ป = โ )).
On tri ๐-separation axioms in fuzzifying tri-topological spaces 195
Theorem 2.11 If (๐, ๐1, ๐2, ๐3) is a FTTS. Then
โจ ๐ผโ(1,2,3)(๐, ๐1, ๐2, ๐3) โ ๐ผโ(๐) (1,2,3)(๐, ๐1, ๐2, ๐3), ๐ = 1,2.
Proof.
(a) [ ๐ผโ(1) (1,2,3)(๐, ๐1, ๐2, ๐3)]
= ๐๐๐๐ฅโ๐
๐๐๐(1,1 โ ๐ผโฑ(1,2,3)(๐)
+ ๐ ๐ข๐๐บโ๐(๐)
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐บ), ๐๐๐
๐ฆโ๐ (1 โ ๐ผ๐๐(1,2,3)(๐บ)(๐ฆ))))
= ๐๐๐๐ฅโ๐
๐๐๐(1,1 โ ๐ผโฑ(1,2,3)(๐)
+ ๐ ๐ข๐๐บโ๐(๐)
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐บ), ๐๐๐
๐ฆโ๐ ๐ผ๐๐ฆ
(1,2,3)(๐~๐บ)))
= ๐๐๐๐ฅโ๐
๐๐๐(1,1 โ ๐ผโฑ(1,2,3)(๐) +
๐ ๐ข๐๐บโ๐=โ ,๐บโ๐(๐)
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐บ), ๐๐๐
๐ฆโ๐ ๐ผ๐๐ฆ
(1,2,3)(๐~๐บ)))
= ๐๐๐๐ฅโ๐
๐๐๐(1,1 โ ๐ผโฑ(1,2,3)(๐)
+ ๐ ๐ข๐๐บโ๐ป=โ ,๐บโ๐(๐)
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐บ), ๐๐๐
๐ฆโ๐ ๐ ๐ข๐๐ฆโ๐ปโ๐~๐บ
๐ผ๐(1,2,3)(๐ป)))
= ๐๐๐๐ฅโ๐
๐๐๐(1,1 โ ๐ผโฑ(1,2,3)(๐)
+ ๐ ๐ข๐๐บโ๐ป=โ ,๐บโ๐(๐)
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐บ), ๐ ๐ข๐
๐บโ๐ป=โ ,๐ โ๐ป ๐ผ๐(1,2,3)(๐ป)))
= ๐๐๐๐ฅโ๐
๐๐๐(1,1 โ ๐ผโฑ(1,2,3)(๐)
+ ๐ ๐ข๐๐บโ๐ป=โ ,๐บโ๐(๐)
๐ ๐ข๐๐บโ๐ป=โ ,๐ โ๐ป
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐บ), ๐ผ๐(1,2,3)(๐ป)))
= ๐๐๐๐ฅโ๐
๐๐๐(1,1 โ ๐ผโฑ(1,2,3)(๐) + ๐ ๐ข๐๐บโ๐ป=โ ,๐ โ๐ป
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐บ), ๐ผ๐(1,2,3)(๐ป)))
= [๐ผโ(1,2,3)(๐, ๐1, ๐2, ๐3)].
(b) [ ๐ผโ(2) (1,2,3)(๐, ๐1, ๐2, ๐3)]
= ๐๐๐๐ฅโ๐
๐๐๐(1,1 โ ๐ผ๐(1,2,3)(๐) + ๐ ๐ข๐๐บโ๐ป=โ ,๐บ โ๐
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐บ), ๐ผ๐(1,2,3)(๐ป)))
= ๐๐๐๐ฅโ๐~๐
๐๐๐(1,1 โ ๐ผโฑ(1,2,3)(๐~๐)
+ ๐ ๐ข๐๐บโ๐~๐=โ ,๐ป โ๐~๐
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐บ), ๐ผ๐(1,2,3)(๐ป)))
= [๐ผโ(1,2,3)(๐, ๐1, ๐2, ๐3)].
Definition 2.12 If (๐, ๐1, ๐2, ๐3) is a FTTS, we define
(1) ๐ผ๐ฉ(1) (1,2,3)(๐, ๐1, ๐2, ๐3) โ โ ๐บ โ ๐ป (๐บ โ ๐ผโฑ(1,2,3) โ ๐ป โ ๐ผ๐(1,2,3) โ ๐บ โ
๐ป โ โ ๐ โ ๐ (๐ โ ๐ผโฑ(1,2,3) โ ๐ โ ๐ผ๐(1,2,3) โ ๐ โ ๐ โ ๐โ๐ป = โ ));
(2) ๐ผ๐ฉ(2) (1,2,3)(๐, ๐1, ๐2, ๐3) โ โ ๐บ โ ๐ป (๐บ โ ๐ผโฑ(1,2,3) โ ๐ป โ ๐ผโฑ(1,2,3) โ ๐บโ๐ป =
โ โ โ ๐ (๐ โ ๐ผ๐(1,2,3) โ ๐บ โ ๐ โ ๐ผ๐๐(1,2,3)(๐)โ๐ป = โ )).
Theorem 2.13 If (๐, ๐1, ๐2, ๐3) is a FTTS. Then
196 Barah M. Sulaiman and Tahir H. Ismail
โจ ๐ผ๐ฉ(1,2,3)(๐, ๐1, ๐2, ๐3) โ ๐ผ๐ฉ(๐) (1,2,3)(๐, ๐1, ๐2, ๐3), ๐ = 1,2.
Proof.
(a) [ ๐ผ๐ฉ(1) (1,2,3)(๐, ๐1, ๐2, ๐3)]
= ๐๐๐๐บโ๐ป
๐๐๐ (1,1 โ ๐๐๐ (๐ผโฑ(1,2,3)(๐บ), ๐ผ๐(1,2,3)(๐ป))
+ ๐ ๐ข๐๐ธโ๐น,๐นโ๐ป=โ
๐๐๐ (๐ผโฑ(1,2,3)(๐ธ), ๐ผ๐(1,2,3)(๐น)))
= ๐๐๐๐บโ๐~๐ป=โ
๐๐๐ (1,1 โ ๐๐๐ (๐ผโฑ(1,2,3)(๐บ), ๐ผโฑ(1,2,3)(๐~๐ป))
+ ๐ ๐ข๐๐~๐ธโ๐น=โ ,๐บโ๐~๐ธ,๐นโ๐~๐ป
๐๐๐ (๐ผ๐(1,2,3)(๐~๐ธ), ๐ผ๐(1,2,3)(๐น)))
= [๐ผ๐ฉ(1,2,3)(๐, ๐1, ๐2, ๐3)]. (b) is analogous to the proof of (a) of Theorem (2.11).
3 Relations among ๐-separation axioms in fuzzifying tri-
topological spaces
Theorem 3.1 If (๐, ๐1, ๐2, ๐3) is a FTTS. Then
(1) โจ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐1(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐0(1,2,3)
;
(2) โจ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐2(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐๐(1,2,3)
, ๐ = 0,1.
Proof. From Lemma (2.7), it is clear.
Theorem 3.2 If (๐, ๐1, ๐2, ๐3) is a FTTS. Then
(1) โจ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 1(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐ ๐(1,2,3)
, ๐ = 0,2;
(2) โจ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐1(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 0(1,2,3)
;
(3) โจ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐2(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐ ๐(1,2,3)
, ๐ = 0,1,2.
Proof. (1) (a) From (1) of Lemma (2.7), we have
๐ผ๐ 1(1,2,3)(๐, ๐1, ๐2, ๐3) = ๐๐๐
๐ฅโ ๐ฆ ๐๐๐ (1,1 โ [๐ผ๐ฆ๐ฅ,๐ฆ
(1,2,3)] + [๐ผโณ๐ฅ,๐ฆ
(1,2,3)])
โค ๐๐๐๐ฅโ ๐ฆ
๐๐๐ (1,1 โ [๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
] + [๐ผโ๐ฅ,๐ฆ(1,2,3)
])
= ๐ผ๐ 0(1,2,3)(๐, ๐1, ๐2, ๐3)
(b) From (2) of Lemma (2.7), we have
๐ผ๐ 1(1,2,3)(๐, ๐1, ๐2, ๐3) = ๐๐๐
๐ฅโ ๐ฆ ๐๐๐ (1,1 โ [๐ผ๐ฆ๐ฅ,๐ฆ
(1,2,3)] + [๐ผโณ๐ฅ,๐ฆ
(1,2,3)])
โค ๐๐๐๐ฅโ ๐ฆ
๐๐๐ (1,1 โ [๐ผโ๐ฅ,๐ฆ(1,2,3)
] + [๐ผโณ๐ฅ,๐ฆ(1,2,3)
])
= ๐ผ๐ 2(1,2,3)(๐, ๐1, ๐2, ๐3)
(2) Using Lemma 2.2 in [2], we have
๐ผ๐1(1,2,3)(๐, ๐1, ๐2, ๐3) = ๐๐๐
๐ฅโ ๐ฆ [๐ผโ๐ฅ,๐ฆ
(1,2,3)]
โค ๐๐๐๐ฅโ ๐ฆ
[๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
โ ๐ผโ๐ฅ,๐ฆ(1,2,3)
]
On tri ๐-separation axioms in fuzzifying tri-topological spaces 197
= ๐ผ๐ 0(1,2,3)(๐, ๐1, ๐2, ๐3).
(3) (a) From (2) above and (2) of Theorem (3.1), we have
๐ผ๐2(1,2,3)(๐, ๐1, ๐2, ๐3) = ๐๐๐
๐ฅโ ๐ฆ [๐ผโณ๐ฅ,๐ฆ
(1,2,3)] โค ๐๐๐
๐ฅโ ๐ฆ [๐ผ๐ฆ๐ฅ,๐ฆ
(1,2,3)โ ๐ผโณ๐ฅ,๐ฆ
(1,2,3)]
= ๐ผ๐ 1(1,2,3)(๐, ๐1, ๐2, ๐3)
= ๐๐๐๐ฅโ ๐ฆ
๐๐๐ (1,1 โ [๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
] + [๐ผโณ๐ฅ,๐ฆ(1,2,3)
])
โค ๐๐๐๐ฅโ ๐ฆ
๐๐๐ (1,1 โ [๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
] + [๐ผโ๐ฅ,๐ฆ(1,2,3)
])
= ๐ผ๐ 0(1,2,3)(๐, ๐1, ๐2, ๐3).
(b) Using Lemma 2.2 in [2], we have
๐ผ๐2(1,2,3)(๐, ๐1, ๐2, ๐3) = ๐๐๐
๐ฅโ ๐ฆ [๐ผโณ๐ฅ,๐ฆ
(1,2,3)]
โค ๐๐๐๐ฅโ ๐ฆ
[๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
โ ๐ผโณ๐ฅ,๐ฆ(1,2,3)
]
= ๐ผ๐ 1(1,2,3)(๐, ๐1, ๐2, ๐3).
(c) Using Lemma 2.2 in [2], we have
๐ผ๐2(1,2,3)(๐, ๐1, ๐2, ๐3) = ๐๐๐
๐ฅโ ๐ฆ [๐ผโณ๐ฅ,๐ฆ
(1,2,3)]
โค ๐๐๐๐ฅโ ๐ฆ
[๐ผโ๐ฅ,๐ฆ(1,2,3)
โ ๐ผโณ๐ฅ,๐ฆ(1,2,3)
]
= ๐ผ๐ 2(1,2,3)(๐, ๐1, ๐2, ๐3).
Theorem 3.3 If (๐, ๐1, ๐2, ๐3) is a FTTS. Then
โจ ๐ผโ(1,2,3)(๐, ๐1, ๐2, ๐3) โ ๐ผ๐1(1,2,3)(๐, ๐1, ๐2, ๐3) โ ๐ผ๐2
(1,2,3)(๐, ๐1, ๐2, ๐3).
Proof. It suffices to show that
[๐ผ๐2(1,2,3)(๐, ๐1, ๐2, ๐3)] โฅ ๐๐๐ฅ(0, ๐ผโ(1,2,3)(๐, ๐1, ๐2, ๐3))] +
[๐ผ๐1(1,2,3)(๐, ๐1, ๐2, ๐3)] โ 1).
Since [๐ผ๐2(1,2,3)(๐, ๐1, ๐2, ๐3)] โฅ 0.
Then from Theorem (3.2), we have
[๐ผ๐1(1,2,3)(๐, ๐1, ๐2, ๐3)] = ๐๐๐
๐ฅโ๐ ๐ผโฑ(1,2,3)({๐ฅ}) = ๐๐๐
๐ฅโ๐ ๐ผ๐(1,2,3)(๐~{๐ฅ })
So [๐ผโ(1,2,3)(๐, ๐1, ๐2, ๐3)] + [๐ผ๐1(1,2,3)(๐, ๐1, ๐2, ๐3)]
= ๐๐๐๐ฅโ๐
๐๐๐(1,1 โ ๐ผ๐(1,2,3)(๐~๐)
+ ๐ ๐ข๐๐บโ๐ป=โ ,๐โ๐ป
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐บ), ๐ผ๐(1,2,3)(๐ป))) + ๐๐๐
๐งโ๐ ๐ผ๐(1,2,3)(๐~{๐ง})
โค ๐๐๐๐ฅโ๐,๐ฅโ ๐ฆ
๐๐๐๐ฆโ๐
๐๐๐(1,1 โ ๐ผ๐(1,2,3)(๐~{๐ฆ}) +
๐ ๐ข๐๐บโ๐ป=โ ,๐ฆโ๐ป
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐บ), ๐ผ๐(1,2,3)(๐ป))) + ๐๐๐
๐งโ๐ ๐ผ๐(1,2,3)(๐~{๐ง}))
โค ๐๐๐๐ฅโ๐,๐ฅโ ๐ฆ
๐๐๐๐ฆโ๐
๐๐๐(1,1 โ ๐ผ๐(1,2,3)(๐~{๐ฆ}) +
198 Barah M. Sulaiman and Tahir H. Ismail
๐ ๐ข๐๐บโ๐ป=โ ,๐ฆโ๐ป
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐บ), ๐ผ๐๐ฆ
(1,2,3)(๐ป))) + ๐ผ๐(1,2,3)(๐~{๐ฆ}))
= ๐๐๐๐ฅโ๐,๐ฅโ ๐ฆ
๐๐๐๐ฆโ๐
(๐๐๐(1,1 + ๐ ๐ข๐๐บโ๐ป=โ
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐บ), ๐ผ๐๐ฆ
(1,2,3)(๐ป))))
= ๐๐๐๐ฅโ๐,๐ฅโ ๐ฆ
๐๐๐๐ฆโ๐
(1 + ๐ ๐ข๐๐บโ๐ป=โ
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐บ), ๐ผ๐๐ฆ
(1,2,3)(๐ป)))
= 1 + ๐๐๐๐ฅโ ๐ฆ
๐ ๐ข๐๐บโ๐ป=โ
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐บ), ๐ผ๐๐ฆ
(1,2,3)(๐ป))
= 1 + [๐ผ๐2(1,2,3)(๐, ๐1, ๐2, ๐3)].
Thus
[๐ผ๐2(1,2,3)(๐, ๐1, ๐2, ๐3)] โฅ ๐๐๐ฅ(0, ๐ผโ(1,2,3)(๐, ๐1, ๐2, ๐3))] +
[๐ผ๐1(1,2,3)(๐, ๐1, ๐2, ๐3)] โ 1).
Corollary 3.4 If (๐, ๐1, ๐2, ๐3) is a FTTS. Then
(1) โจ ๐ผ๐3(1,2,3)(๐, ๐1, ๐2, ๐3) โ ๐ผ๐2
(1,2,3)(๐, ๐1, ๐2, ๐3).
(2) โจ ๐ผ๐3(1,2,3)(๐, ๐1, ๐2, ๐3) โ ๐ผ๐ ๐
(1,2,3)(๐, ๐1, ๐2, ๐3), ๐ = 0,1,2.
Theorem 3.5 If (๐, ๐1, ๐2, ๐3) is a FTTS. Then
โจ ๐ผ๐4(1,2,3)(๐, ๐1, ๐2, ๐3) โ ๐ผโ(1,2,3)(๐, ๐1, ๐2, ๐3).
Proof.
๐ผ๐4(1,2,3)(๐, ๐1, ๐2, ๐3) = ๐๐๐ฅ(0, [๐ผ๐ฉ(1,2,3)(๐, ๐1, ๐2, ๐3))] +
[๐ผ๐1(1,2,3)(๐, ๐1, ๐2, ๐3)] โ 1),
now we prove that
[๐ผโ(1,2,3)(๐, ๐1, ๐2, ๐3)] โฅ [๐ผ๐ฉ(1,2,3)(๐, ๐1, ๐2, ๐3)] + [๐ผ๐1(1,2,3)(๐, ๐1, ๐2, ๐3)] โ
1.
In fact
[๐ผ๐ฉ(1,2,3)(๐, ๐1, ๐2, ๐3)] + [๐ผ๐1(1,2,3)(๐, ๐1, ๐2, ๐3)]
= ๐๐๐๐โ๐=โ
๐๐๐ (1,1 โ ๐๐๐ (๐ผโฑ(1,2,3)(๐), ๐ผโฑ(1,2,3)(๐))
+ ๐ ๐ข๐๐บโ๐ป=โ ,๐โ๐ป,๐โ๐บ
๐๐๐ (๐ผ๐(1,2,3)(๐บ), ๐ผ๐(1,2,3)(๐ป))) + ๐๐๐๐งโ๐
๐ผ๐(1,2,3)(๐~{๐ง})
= ๐๐๐๐โ๐=โ
๐๐๐ (1,1 โ ๐๐๐ (๐ผ๐(1,2,3)(๐~๐), ๐ผ๐(1,2,3)(๐~๐))
+ ๐ ๐ข๐๐บโ๐ป=โ ,๐โ๐ป,๐โ๐บ
๐๐๐ (๐ผ๐(1,2,3)(๐บ), ๐ผ๐(1,2,3)(๐ป))) + ๐๐๐๐งโ๐
๐ผ๐(1,2,3)(๐~{๐ง})
โค ๐๐๐ ๐ฅโ๐
๐๐๐ (1,1 โ ๐๐๐ (๐ผ๐(1,2,3)(๐~๐), ๐ผ๐(1,2,3)(๐~{๐ฅ}))
+ ๐ ๐ข๐๐บโ๐ป=โ ,๐โ๐ป
๐๐๐ (๐ผ๐๐ฅ(1,2,3)(๐บ), ๐ผ๐(1,2,3)(๐ป)) + ๐๐๐
๐งโ๐ ๐ผ๐(1,2,3)(๐~{๐ง})
= ๐๐๐ ๐ฅโ๐
๐๐๐ (1, ๐๐๐ฅ (1 โ ๐ผ๐(1,2,3)(๐~๐)
+ ๐ ๐ข๐๐บโ๐ป=โ ,๐โ๐ป
๐๐๐ (๐ผ๐๐ฅ(1,2,3)(๐บ), ๐ผ๐(1,2,3)(๐ป)),1 โ ๐ผ๐(1,2,3)(๐~{๐ฅ})
+ ๐ ๐ข๐๐บโ๐ป=โ ,๐โ๐ป
๐๐๐ (๐ผ๐๐ฅ(1,2,3)(๐บ), ๐ผ๐(1,2,3)(๐ป))) + ๐๐๐
๐งโ๐ ๐ผ๐(1,2,3)(๐~{๐ง})
On tri ๐-separation axioms in fuzzifying tri-topological spaces 199
= ๐๐๐ ๐ฅโ๐
๐๐๐ฅ (๐๐๐ (1,1 โ ๐ผ๐(1,2,3)(๐~๐)
+ ๐ ๐ข๐๐บโ๐ป=โ ,๐โ๐ป
๐๐๐ (๐ผ๐๐ฅ(1,2,3)(๐บ), ๐ผ๐(1,2,3)(๐ป))), ๐๐๐ (1,1 โ ๐ผ๐(1,2,3)(๐~{๐ฅ})
+ ๐ ๐ข๐๐บโ๐ป=โ ,๐โ๐ป
๐๐๐ (๐ผ๐๐ฅ(1,2,3)(๐บ), ๐ผ๐(1,2,3)(๐ป))) + ๐๐๐
๐งโ๐ ๐ผ๐(1,2,3)(๐~{๐ง})
โค ๐๐๐ ๐ฅโ๐
๐๐๐ฅ(๐๐๐ (1,1 โ ๐ผ๐(1,2,3)(๐~๐)
+ ๐ ๐ข๐๐บโ๐ป=โ ,๐โ๐ป
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐บ), ๐ผ๐(1,2,3)(๐ป)))
+๐ผ๐(1,2,3)(๐~{๐ฅ}), ๐๐๐ (1,1 โ ๐ผ๐(1,2,3)(๐~{๐ฅ}))
+ ๐ ๐ข๐๐บโ๐ป=โ ,๐โ๐ป
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐บ), ๐ผ๐(1,2,3)(๐ป))) + ๐ผ๐(1,2,3)(๐~{๐ฅ}))
โค ๐๐๐ ๐ฅโ๐
๐๐๐ฅ(๐๐๐ (1,1 โ ๐ผ๐(1,2,3)(๐~๐)
+ ๐ ๐ข๐๐บโ๐ป=โ ,๐โ๐ป
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐บ), ๐ผ๐(1,2,3)(๐ป))) + ๐ผ๐(1,2,3)(๐~{๐ฅ}),
1 + ๐ ๐ข๐๐บโ๐ป=โ ,๐โ๐ป
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐บ), ๐ผ๐(1,2,3)(๐ป)))
โค ๐๐๐ ๐ฅโ๐
๐๐๐ (1,1 โ ๐ผ๐(1,2,3)(๐~๐)
+ ๐ ๐ข๐๐บโ๐ป=โ ,๐โ๐ป
๐๐๐(๐ผ๐๐ฅ(1,2,3)(๐บ), ๐ผ๐(1,2,3)(๐ป))) + 1
= [๐ผโ(1,2,3)(๐, ๐1, ๐2, ๐3)] + 1.
Theorem 3.6 If (๐, ๐1, ๐2, ๐3) is a FTTS. Then
(1) โจ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐1(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 0(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ
๐ผ๐0(1,2,3)
;
(2) If ๐ผ๐0(1,2,3)(๐, ๐1, ๐2, ๐3) = 1, then
โจ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐1(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 0(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ
๐ผ๐0(1,2,3)
.
Proof. (1) Follows from (1) of Theorem (3.1) and (2) of Theorem (3.2).
(2) Since ๐ผ๐0(1,2,3)(๐, ๐1, ๐2, ๐3) = 1, then for every ๐ฅ, ๐ฆ โ ๐ such that ๐ฅ โ ๐ฆ, we
have [๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
] = 1. So
๐ผ๐ 0(1,2,3)(๐, ๐1, ๐2, ๐3) โ ๐ผ๐0
(1,2,3)(๐, ๐1, ๐2, ๐3)
= ๐ผ๐ 0(1,2,3)(๐, ๐1, ๐2, ๐3)
= ๐๐๐๐ฅโ ๐ฆ
๐๐๐(1,1 โ [๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
] + [๐ผโ๐ฅ,๐ฆ(1,2,3)
])
= ๐๐๐๐ฅโ ๐ฆ
[๐ผโ๐ฅ,๐ฆ(1,2,3)
] = ๐ผ๐1(1,2,3)(๐, ๐1, ๐2, ๐3).
Theorem 3.7 If (๐, ๐1, ๐2, ๐3) is a FTTS. Then
200 Barah M. Sulaiman and Tahir H. Ismail
(1) โจ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐2(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 1(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ
๐ผ๐0(1,2,3)
;
(2) If ๐ผ๐0(1,2,3)(๐, ๐1, ๐2, ๐3) = 1, then
โจ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐2(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 1(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ
๐ผ๐0(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 (๐, ๐1, ๐2, ๐3) is a FTTS. Then
(1) โจ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐2(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 0(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ
๐ผ๐1(1,2,3)
;
(2) If ๐ผ๐1(1,2,3)(๐, ๐1, ๐2, ๐3) = 1, then
โจ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐2(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 0(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ
๐ผ๐1(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 (๐, ๐1, ๐2, ๐3) is a FTTS. Then we have
(1) โจ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐1(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 0(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ
๐ผ๐0(1,2,3)
;
(2) โจ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐2(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 1(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ
๐ผ๐0(1,2,3)
.
(3) โจ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐2(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 0(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ
๐ผ๐1(1,2,3)
.
Theorem 3.10 If (๐, ๐1, ๐2, ๐3) is a FTTS. Then
(1) (๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 0(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐0(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ
๐ผ๐1(1,2,3)
;
(2) If ๐ผ๐0(1,2,3)(๐, ๐1, ๐2, ๐3) = 1, then
โจ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 0(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐0(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ
๐ผ๐1(1,2,3)
.
Proof.
(1) [(๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 0(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐0(1,2,3)
]
= ๐๐๐ฅ(0, ๐ผ๐ 0(1,2,3)(๐, ๐1, ๐2, ๐3) + ๐ผ๐0
(1,2,3)(๐, ๐1, ๐2, ๐3) โ 1)
= ๐๐๐ฅ(0, ๐๐๐๐ฅโ ๐ฆ
๐๐๐(1,1 โ [๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
] + [๐ผโ๐ฅ,๐ฆ(1,2,3)
]) + ๐๐๐๐ฅโ ๐ฆ
[๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
] โ 1)
On tri ๐-separation axioms in fuzzifying tri-topological spaces 201
โค ๐๐๐ฅ(0, ๐๐๐๐ฅโ ๐ฆ
(๐๐๐(1,1 โ [๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
] + [๐ผโ๐ฅ,๐ฆ(1,2,3)
]) + [๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
] โ 1)
โค ๐๐๐ฅ(0, ๐๐๐๐ฅโ ๐ฆ
(1 โ [๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
] + [๐ผโ๐ฅ,๐ฆ(1,2,3)
] + [๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
] โ 1)
= ๐๐๐๐ฅโ ๐ฆ
[๐ผโ๐ฅ,๐ฆ(1,2,3)
] = ๐ผ๐1(1,2,3)(๐, ๐1, ๐2, ๐3).
(2) Follows from (2) Theorem (3.6).
Theorem 3.11 If (๐, ๐1, ๐2, ๐3) is a FTTS. Then
(1) (๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 1(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐0(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ
๐ผ๐2(1,2,3)
;
(2) If ๐ผ๐0(1,2,3)(๐, ๐1, ๐2, ๐3) = 1, then
โจ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 1(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐0(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ
๐ผ๐2(1,2,3)
.
Proof.
(1) [(๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 1(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐0(1,2,3)
]
= ๐๐๐ฅ(0, ๐ผ๐ 1(1,2,3)(๐, ๐1, ๐2, ๐3) + ๐ผ๐0
(1,2,3)(๐, ๐1, ๐2, ๐3) โ 1)
= ๐๐๐ฅ(0, ๐๐๐๐ฅโ ๐ฆ
๐๐๐(1,1 โ [๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
] + [๐ผโณ๐ฅ,๐ฆ(1,2,3)
]) + ๐๐๐๐ฅโ ๐ฆ
[๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
] โ 1)
โค ๐๐๐ฅ(0, ๐๐๐๐ฅโ ๐ฆ
(๐๐๐(1,1 โ [๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
] + [๐ผโณ๐ฅ,๐ฆ(1,2,3)
]) + [๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
] โ 1)
โค ๐๐๐ฅ(0, ๐๐๐๐ฅโ ๐ฆ
(1 โ [๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
] + [๐ผโณ๐ฅ,๐ฆ(1,2,3)
] + [๐ผ๐ฆ๐ฅ,๐ฆ(1,2,3)
] โ 1)
= ๐๐๐๐ฅโ ๐ฆ
[๐ผโณ๐ฅ,๐ฆ(1,2,3)
] = ๐ผ๐2(1,2,3)(๐, ๐1, ๐2, ๐3).
(2) Follows from (2) Theorem (3.6).
Theorem 3.12 If (๐, ๐1, ๐2, ๐3) is a FTTS. Then
(1) โจ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐0(1,2,3)
โ ((๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 0(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ
๐ผ๐1(1,2,3)
;
(2) โจ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 0(1,2,3)
โ ((๐, ๐1, ๐2, ๐3) โ ๐ผ๐0(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ
๐ผ๐1(1,2,3)
;
(3) โจ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐0(1,2,3)
โ ((๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 1(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ
๐ผ๐2(1,2,3)
;
(4) โจ (๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 1(1,2,3)
โ ((๐, ๐1, ๐2, ๐3) โ ๐ผ๐0(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ
๐ผ๐2(1,2,3)
.
Proof. (1) From (2) Theorem (3.1) and (3) Theorem (3.2), we have
[(๐, ๐1, ๐2, ๐3) โ ๐ผ๐0(1,2,3)
โ ((๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 0(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ
๐ผ๐1(1,2,3)
]
202 Barah M. Sulaiman and Tahir H. Ismail
= ๐๐๐(1,1 โ [(๐, ๐1, ๐2, ๐3) โ ๐ผ๐0(1,2,3)
] + ๐๐๐(1,1 โ [(๐, ๐1, ๐2, ๐3) โ
๐ผ๐ 0(1,2,3)
] + [(๐, ๐1, ๐2, ๐3) โ ๐ผ๐1(1,2,3)
]))
= ๐๐๐(1,1 โ [(๐, ๐1, ๐2, ๐3) โ ๐ผ๐0(1,2,3)
] + 1 โ [(๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 0(1,2,3)
] +
[(๐, ๐1, ๐2, ๐3) โ ๐ผ๐1(1,2,3)
]))
= ๐๐๐(1,1 โ ([(๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 0(1,2,3)
] + [(๐, ๐1, ๐2, ๐3) โ ๐ผ๐0(1,2,3)
] โ 1) +
[(๐, ๐1, ๐2, ๐3) โ ๐ผ๐1(1,2,3)
]) = 1.
(2) From (1) Theorem (3.1) and (3) Theorem (3.6), we have
[(๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 0(1,2,3)
โ ((๐, ๐1, ๐2, ๐3) โ ๐ผ๐0(1,2,3)
โ (๐, ๐1, ๐2, ๐3) โ
๐ผ๐1(1,2,3)
]
= ๐๐๐(1,1 โ [(๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 0(1,2,3)
] + ๐๐๐(1,1 โ [(๐, ๐1, ๐2, ๐3) โ
๐ผ๐0(1,2,3)
] + [(๐, ๐1, ๐2, ๐3) โ ๐ผ๐1(1,2,3)
]))
= ๐๐๐(1,1 โ [(๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 0(1,2,3)
] + 1 โ [(๐, ๐1, ๐2, ๐3) โ ๐ผ๐0(1,2,3)
] +
[(๐, ๐1, ๐2, ๐3) โ ๐ผ๐1(1,2,3)
]))
= ๐๐๐(1,1 โ ([(๐, ๐1, ๐2, ๐3) โ ๐ผ๐ 0(1,2,3)
] + [(๐, ๐1, ๐2, ๐3) โ ๐ผ๐0(1,2,3)
] โ 1) +
[(๐, ๐1, ๐2, ๐3) โ ๐ผ๐1(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