한국지능시스템학회 논문지 2011, Vol. 21, No. 2, pp. 250-253 DOI : 10.5391/JKIIS.2011.21.2.250
250
접수일자 : 2010년 3월 22일 완료일자 : 2010년 9월 30일
Fuzzy r-Minimal 구조에서 Fuzzy Weak r-M Continuity의 특성 연구
Remarks on Fuzzy Weak r-M Continuity on Fuzzy r-Minimal Structures
민원근․김명환
Won Keun Min and Myeong Hwan Kim 강원대학교 수학과
요 약
본 논문에서는 일반화 된 fuzzy r-minimal 열린 집합들을 이용하여 fuzzy 약 r-M 연속함수의 특성을 조사한다.
Abstract
In this paper, we study characterizations of fuzzy weakly r-M continuous function on r-minimal structures in terms of generalized fuzzy r-minimal open sets.
Key Words : fuzzy r-minimal structure, fuzzy weakly r-M continuous, fuzzy r-minimal semiopen, fuzzy r-minimal preopen, r-minimal regular open, r-minimal -open.
1. 서 론
The concept of fuzzy set was introduced by Zadeh [12]. Chang [2] defined fuzzy topological spaces using fuzzy sets. In [3], Chattopadhyay, Hazra and Samanta introduced a smooth fuzzy topological space which is a generalization of fuzzy topological space. In [11], Yoo et al. introduced the concept of fuzzy -minimal space which is an extension of the smooth fuzzy topological space. The concepts of fuzzy -open sets, fuzzy -semiopen sets, fuzzy -preopen sets, fuzzy --open sets and fuzzy -regular open sets were introduced in [1, 4, 5, 6], which are kinds of fuzzy -minimal structures. The concept of fuzzy -M continuity was also introduced and investigated in [11]. In [8], the au- thor introduced and studied the concept of fuzzy weak -M continuity which is a generalization of fuzzy -M continuity. In this paper, we study characterizations of fuzzy weakly -M continuous function on -minimal structures in terms of generalized fuzzy -minimal open sets [7, 9]: fuzzy -minimal semiopen sets, fuzzy -preopen sets, fuzzy -minimal -open sets and fuzzy
-minimal regular open sets.
2. Preliminaries
Let be the unit interval of the real line. A member of is called a fuzzy set of . By and
, we denote constant maps on X with value and , respectively. For any ∈, denotes the comple- ment -. All other notations are standard notations of fuzzy set theory.
A fuzzy point in is a fuzzy set is defined as follows
=
i f i f ≠
A fuzzy point is said to belong to a fuzzy set in , denoted by ∈, if ≤ for ∈.
A fuzzy set in is the union of all fuzzy points which belong to .
Let → be a mapping and ∈ and ∈. Then is a fuzzy set in , defined by
∈ i f ≠ ∅
otherwise
for ∈ and is a fuzzy set in , defined by
∈.
A fuzzy topology [3, 4] on is a map → which satisfies the following properties:
Fuzzy r-Minimal 구조에서 Fuzzy Weak r-M Continuity의 특성 연구
251 (1) ( )=( )=1.
(2) (∩)≥()∧() for , ∈. (3) (∪)≥ ∧() for ∈.
The pair is alled a fuzzy topological space.
And ∈ is said to be fuzzy -open (resp., fuz- zy -closed) if ()≥ (resp., ()≥r).
Definition 2.1 ([11]). Let X be a nonempty set and ∈
(0,1]=. A fuzzy family → on X is said to have a fuzzy -minimal structure if the family
={∈ : ()≥ }
contains and .
Then the (,) is called a fuzzy -minimal space (simply -FMS). Every member of is called a fuzzy
-minimal open set. A fuzzy set is called a fuzzy -minimal closed set if the complement of (simply,
) is a fuzzy -minimal open set.
Let (,) be an -FMS's and ∈(0,1]=. The fuz- zy -minimal closure and the fuzzy -minimal interior of [11], denoted by mC(,) and mI(, ), re- spectively, are defined as
mC(, ) =∩{∈: ∈ and ⊆};
mI(,)=∪{∈: ∈ and ⊆}.
Theorem 2.2 ([11]). Let (,) be an -FMS and ,
∈. Then the following properties hold:
(1) mI(,)⊆ and if A is a fuzzy -minimal open set, then mI(,)=.
(2) ⊆mC(,) and if is a fuzzy -minimal closed set, then mC(,)=.
(3) If ⊆, then mI(,)⊆mI(,) and mC(,)⊆
mC(,).
(4) mI(∩,)⊆mI(,)∩mI(,) and mC(,)∪
mC(,)⊆mC(∪,).
(5) mI(mI(,),)=mI(,) and mC(mC(,),)= mC (,).
(6) -mC(,)=mI( -,) and -mI(,)=mC( -,
).
Let (,) and (,) be two -FMS's. Then a function → is said to be
(1) fuzzy -M continuous [11] if for every ∈,
is in ;
(2) fuzzy weakly -M continuous [8] if for each fuzzy point in and each fuzzy -minimal open set containing (), there exists a fuzzy
-minimal open set containing such that ()⊆mC(, ).
3. Characterizations of Fuzzy Weakly
-M Continuous Functions
Theorem 3.1 ([8]). Let → be a function be- tween -FMS's (,) and (,). Then the fol- lowing statements are equivalent:
(1) is fuzzy weakly -M continuous.
(2) ()⊆mI( (mC(,)),) for each fuzzy -minimal open set in .
(3) mC( (mI(,)),)⊆ () for each fuzzy -minimal closed set in .
(4) mC( (),)⊆ (mC(,)) for each fuzzy -minimal open set in .
Let X be a nonempty set and → a fuzzy family on X. The fuzzy family is said to have the property () [11] if for ∈ (∈),
(∪)≥ ∧().
Theorem 3.2 ([11]). Let (,) be an -FMS with the property (). Then
(1) mI(,)= if and only if is fuzzy -minimal open for ∈.
(2) mC(,)= if and only if is fuzzy -minimal closed for ∈.
Lemma 3.3 ([8]). Let → be a function between
-FMS's (,) and (,). If have property (), then the following statements are equivalent:
(1) is fuzzy weakly -M continuous.
(2) mC( (mI(,)),)⊆ () for each fuzzy -minimal closed set in .
(3) mC( (mI(mC(,),)),)⊆ (mC(,)) for each ∈.
(4) (mI(,))⊆mI( (mC(mI(,r),)),) for each
∈.
(5) mC( (),)⊆ (mC(,)) for a fuzzy -minimal open set in .
Definition 3.4. Let (,) be an -FMS's and ∈
. Then a fuzzy set is said to be
(1) fuzzy -minimal semiopen [7] if ⊆mC(mI(,),
);
(2) fuzzy -minimal preopen [9] if ⊆mI(mC(,),
);
(3) fuzzy -minimal -open [9] if ⊆mC(mI(mC(,
),),).
(4) fuzzy -minimal regular open [9] (resp., fuzzy -minimal regular closed if =mI(mC(,),) (resp., =mC(mI(,),)).
A fuzzy set is called a fuzzy -minimal semi- closed (resp., fuzzy -minimal preclosed, fuzzy -minimal -closed) set if the complement of is a fuzzy -minimal semiopen (resp., fuzzy -minimal pre-
한국지능시스템학회 논문지 2011, Vol. 21, No. 2
252
open, fuzzy -minimal -open) set.
Theorem 3.5. Let → be a function between -FMS's (,) and (,). If has the property (), then is fuzzy weakly -M continuous if and on- ly if mC( (mI(mC(,),)),)⊆ (mC(,)) for each fuzzy -minimal semiopen set in .
Proof. Suppose is fuzzy weakly -M continuous and let be a fuzzy -minimal semiopen set in . Then ⊆mC(mI(,),) and by the property (), mC(mI(,),) is fuzzy -minimal closed. So by Lemma 3.3 (3) and mC(,)=mC(mI(,),), we have mC( (mI(mC(,),)),)=mC( (mI(mC(mI(,),),
)),)⊆ (mC(mI(,),))⊆ (mC(,)). Hence mC ( (mI(mC(,),)),)⊆ (mC(,)).
For the converse, let be a fuzzy -minimal open set in . Then is also -minimal semiopen and since
⊆mI(mC(,),), by hypothesis, mC( (),)⊆mC ( (mI(mC(,),)),)⊆ (mC(,)). Hence by Lemma 3.3 (4), is fuzzy weakly -M continuous.
Theorem 3.6. Let → be a function between -FMS's (,) and (,). If has the property (), then is fuzzy weakly -M continuous if and on- ly if mC( (mI(mC(,),)),)⊆ (mC(,)) for each fuzzy -minimal preopen set in .
Proof. Let be a fuzzy weakly -M continuous function and be a fuzzy -minimal preopen set in . From the property (), mC(mI(,),) is fuzzy -minimal closed and so mC(,)=mC(mI(,),). From
⊆mC(mI(,),), we have mC( (mI(mC(,),)),)
=mC( (mI(mC(mI(,),),)),)
⊆ (mC(mI(,),))
⊆ (mC(,)).
Hence we have the result.
For the converse, let be a fuzzy -minimal open set in . Then by the property (), is also -minimal preopen, and since ⊆mI(mC(,),), it fol- lows mC( (),)⊆mC( (mI(mC(,),)),)⊆ (mC(,)). So mC( (),)⊆ (mC(,)) Hence by Lemma 3.3, is fuzzy weakly -M continuous.
Theorem 3.7. Let → be a function between -FMS's (,) and (,). If has the property (), then the following statements are equivalent:
(1) is fuzzy weakly -M continuous.
(2) mC( (),)⊆ (mC(,)) for each fuzzy
-minimal preopen set in .
(3) ()⊆mI( (mC(,)),) for each fuzzy
-minimal preopen set in .
Proof. (1) ⇒ (2) Let be a fuzzy weakly -M con- tinuous function and be a fuzzy -minimal preopen set in . Suppose ∉ (mC(,)); then there ex- ists a fuzzy -minimal open set containing () such that ∩=∅. Since is a fuzzy -minimal preopen set, ∩mC(,)=∅. From fuzzy weak -M continuity and ()∈, there exists a fuzzy -minimal open set containing such that ()⊆
mC(,). Then since ∩mC(,)=∅, ∩ ()=∅
and so ()∩=∅. This implies ∉mC( (),).
Consequently, mC( (),)⊆ (mC(,)).
(2) ⇒ (3) For a fuzzy -minimal preopen set in
, from (2), it follows
()⊆ (mI(mC(,),))
= - (mC( -mC(,),))
⊆ -mC( ( -mC(,)),)
mI( (mC(,)),).
(3) ⇒ (1) Let be a fuzzy -minimal open set in
. Then by the property (), is also -minimal preopen. Hence by Theorem 3.2 and Lemma 3.3, is fuzzy weakly -M continuous.
Theorem 3.8. Let → be a function between -FMS's (,) and (,). If has the property (), then is fuzzy weakly -M continuous if and on- ly if mC( (mI(,)),)⊆ () for each fuzzy -minimal regular closed set in .
Proof. Let be a fuzzy weakly -M continuous function and let be a fuzzy -minimal regular closed set in . Since =mC(mI(,),) is fuzzy -minimal closed, by Theorem 3.2 and Lemma 3.3, we have mC ( (mI(,)),)⊆ ().
For the converse, let be a fuzzy -minimal open set in . Then since mC(,)=mC(mI(mC(,),),), mC(,) is -minimal regular closed. By hypothesis, mC( (,))⊆mC( (mI(mC(,),)),)⊆ (mC(,
)), that is, mC( (,))⊆ (mC(,)). Hence by Lemma 3.3, is fuzzy weakly -M continuous.
Theorem 3.9. Let → be a function between -FMS's (,) and (,). If has the property (), then is fuzzy weakly -M continuous if and on- ly if mC( (mI(mC(,),)),)⊆ (mC(,)) for each fuzzy -minimal -open set in .
Proof. If is a fuzzy weakly -M continuous func- tion, then by Lemma 3.3, easily the result is obtained.
For the converse, let be a fuzzy -minimal open set in . Then is also a fuzzy -minimal -open set. From ⊆mC(mI(mC(,),),) and mC(,)=
Fuzzy r-Minimal 구조에서 Fuzzy Weak r-M Continuity의 특성 연구
253 mC(mI(mC(,),),), it follows mC( (),)⊆mC
( (mC(mI(mC(,),),)),)⊆ (mC(,)). Hence
is fuzzy weakly -M continuous.
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저 자 소 개
민원근(Won Keun Min)
1988년～현재 : 강원대학교 수학과 교수
관심분야 : 퍼지 위상, 퍼지 이론, 일반 위상 Phone : 033-250-8419
Fax : 033-252-7289
E-mail : [email protected]
김명환(Myeong Hwan Kim)
1979년～현재 : 강원대학교 수학과 교수
관심분야 : 퍼지 이론.
Phone : 033-250-8415 Fax : 033-252-7289
E-mail : [email protected]
