한국지능시스템학회 논문지 2009, Vol. 19, No. 3, pp. 421-424
421 접수일자 : 2009년 1월 13일
완료일자 : 2009년 4월 7일
Fuzzy Semi-Weakly r-Semicontinuous 함수에 관한 연구
Fuzzy Semi-Weakly r-Semicontinuous Mappings민원근 Won Keun Min
강원대학교 수학과
요 약
Fuzzy semi-weakly r-semicontinuous 함수의 개념을 소개하며 특성을 조사한다. 본 논문에서 소개된 함수와 fuzzy r-semicontinuity와 fuzzy weakly r-semicontinuity의 관계를 밝힌다.
Abstract
In this paper, we introduce the concept of fuzzy semi-weakly r-semicontinuous mappings on a fuzzy topological space and study characterizations for such mappings. And we investigate the relationships among fuzzy r-semicontinuity, fuzzy semi-weakly r-semicontinuity and fuzzy weakly r-semicontinuity.
Key Words : fuzzy semi-weakly r-semicontinuous, fuzzy weakly r-semicontinuous, fuzzy S-weakly r-continuous, fuzzy r-irresolute.
1. 서 론
Chang [1] defined fuzzy topological spaces using fuz- zy sets introduced by Zadeh [10]. In [3, 4], Chattopadhyay, Hazra and Samanta introduced the con- cept of smooth fuzzy topological spaces which are a generalization of fuzzy topological spaces.
Lee and Kim [8] introduced and studied the concept of fuzzy weakly -semicontinuous mappings in fuzzy topological spaces defined by Chattopadhyay, which is a generalized concept of fuzzy weakly semicontinuous mappings defined in the Chang's fuzzy topological spaces.
In this paper, we introduce and study the concept of fuzzy semi-weakly -semicontinuous mappings on the fuzzy topological space which is a generalization of fuz- zy -irresolute mappings. In particular, we investigate the relationships among fuzzy -semicontinuity, fuzzy weakly -semicontinuity and fuzzy semi-weakly -semicontinuity.
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
sup∈ 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:
(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).
The -closure and the -interior of , denoted by
한국지능시스템학회 논문지 2009, Vol. 19, No. 3
422
cl( ) and int( ), respectively, are defined as
cl( )=∩{∈: ⊆ and is fuzzy -closed},
int( )=∪{∈: ⊆ and is fuzzy -open}.
Definition 2.1 ([6]). Let be a fuzzy set in an FTS
and ∈(0,1]=. Then is said to be fuzzy -semiopen if there is a fuzzy -open set in such that ⊆⊆cl(,).
Let ∈ in an FTS and ∈(0,1] =. The fuzzy -semi-closure and the fuzzy -semi -interior of , denoted by scl( ) and sint( ), re- spectively, are defined as
scl( )=∩{∈: ⊆ and is fuzzy -semiclosed},
sint( )=∪{∈: ⊆ and is fuzzy -semiopen}.
Definition 2.2 ([6, 7, 8, 9]). Let → be a map- ping from FTS's and . Then f is said to be (1) fuzzy -irresolute [7] if for each fuzzy -semiopen
set of , is a fuzzy -semiopen set in
,
(2) fuzzy -semicontinous [6] if for each fuzzy -semiopen set of , is a fuzzy -semiopen set in ,
(3) fuzzy weakly -semicontinuous [8] if for each fuzzy
-open set of , ⊆sint( (scl(,)),), (4) fuzzy S-weakly -continuous [9] if ⊆sint
( (cl(,)),) for each fuzzy -open set of .
3. Main Results
Definition 3.1. Let → be a mapping from FTS's and and ∈(0,1]=. Then is said to be fuzzy semi-weakly -semicontinuous if ⊆sint ( (scl(, )), ) for each fuzzy -semiopen set of
.
Remark 3.2. Every fuzzy -semicontinuous mapping is fuzzy semi-weakly -semicontinuous but the converse is not always true.
Example 3.3. Let X=I and let and be fuzzy sets of X defined as
(x)=-
x +1, for x∈,
(x)=-
x + 1, for x∈.
Define a fuzzy topology T : → by
()=
i f
i f
and a fuzzy topology → by
()=
i f
i f
Consider the identity mapping → .
We know that every fuzzy set containing is fuz- zy
-semiopen in the FTS and scl(,
)= .
Hence the mapping f is a fuzzy semi-weakly
-semicontinuous mapping but it is not fuzzy
-semicontinuous.
Remark 3.4. Every fuzzy semi-weakly -semi con- tinuous mapping is fuzzy weakly -semicontinuous but the converse is not always true.
Example 3.5. Let and let , and be fuz- zy sets of X defined as
(x)=
, for x∈,
(x)=
, for x∈,
(x)=
, for x∈.
Define a fuzzy topology T : → by
()=
i f
i f
and a fuzzy topology → by
()=
i f
i f
Consider the identity mapping → .
Then obviously f is fuzzy weakly
-semicontinuous.
Consider a fuzzy semiopen set in defined as
()=
for ∈. Then sint( (scl(,
)),
)=sint(,
)= - in the FTS and
()=> -. Hence the mapping is not a fuzzy
Fuzzy Semi-Weakly r-Semicontinuous 함수에 관한 연구
423 semi-weakly
-semicontinuous mapping.
Now the following implications are obtained.
fuzzy -irresolute ⇒ fuzzy weakly -semicont.
⇒ fuzzy semi-weakly -semicont. ⇒ fuzzy weakly -semicont. ⇒ fuzzy S-weakly -cont.
Theorem 3.6. Let → be a mapping on FTS's and (∈). Then f is a fuzzy semi-weakly -semicontinuous mapping if and only if for every fuzzy point and each fuzzy -semiopen set V containing (), there exists a fuzzy -semiopen set containing such that ()⊆scl(, ).
Proof. Suppose is a fuzzy semi-weakly -semi -continuous mapping. Let be a fuzzy point in and
a fuzzy -semiopen set containing (). Then there exists a fuzzy -semiopen set such that ()∈⊆
. From the hypothesis, it follows
()⊆sint( (scl(,)),)⊆sint( (scl(V,)),).
Set =sint( (scl(,)),). Since is a fuzzy -semiopen set such that ()⊆⊆sint( (scl(,
)),)⊆ (scl(,)), we have ()⊆scl(,).
For the converse, let be a fuzzy -semiopen set in
. For each ∈ (), there exists a fuzzy
-semiopen set
containing such that (
)⊆
scl(,). This implies
(V)⊆ ∪{
: ∈ ()}⊆ (scl(,)).
Since ∪{
: ∈ ()} is a fuzzy -semiopen set containing (), we have ()⊆sint( (scl (,)),). Hence is a fuzzy semi-weakly -semi con- tinuous function.
Theorem 3.7. ([7]) Let be a fuzzy set in an FTS
and ∈. Then we have (1) -scl(, )=sint( -, ), (2) -sint(, )=scl( -, ).
Theorem 3.8. Let → be a mapping on FTS's and (∈). Then the follow- ing statements are equivalent:
(1) is fuzzy semi-weakly -semicontinuous.
(2) scl( (sint(, )), )⊆ () for each fuzzy -semiclosed set in .
(3) scl( (sint(, )), )⊆ (scl(, )) for each fuz- zy set in .
(4) (sint(, ))⊆sint( (scl(, )), ) for each fuzzy set in .
(5) scl( (), ) ⊆ (scl(, )) for a fuzzy -semiopen set in .
Proof. (1) ⇒ (2) Let be any fuzzy -semiclosed set of . Then from the hypothesis and Theorem 3.7, it follows
( -)⊆sint( (scl( -, )), )
=sint( ( -sint(, )), )
=sint( - (sint(, )), )
= -scl( (sint(, )), ).
Hence we have scl( (sint(,)),)⊆ ().
(2) ⇒ (3) For ∈, since scl(,) is a fuzzy -semiclosed set in , from (2), it follows
scl( (sint(, )), ) ⊆ scl( (sint(scl(, ), )) ⊆ (scl(, )).
(3) ⇒ (4) For ∈, from Theorem 3.7 and (3),
(sint(,))= ( -scl( -,))
= -( (scl( -, )))
⊆ -scl( (sint( -, )), )
=sint( (scl(, )), ).
This implies
(sint(,))⊆sint( (scl(,)), ).
(4) ⇒ (5) Let be any fuzzy -semiopen set of . From (4) and scl( -, )= -, it follows
- (scl(V, ))= (sint( -, ))
⊆sint( (scl( -, )), )
=sint( ( -), )
=sint( - (), )
= -scl( (), ).
Hence we have scl( (), )⊆ (scl(, )).
(5) ⇒ (1) Let be a fuzzy -semiopen set in . Then from ⊆sint(scl(, ), ), it follows
(V)⊆ (sint(scl(, ), ))
= - (scl( -scl(, ), ))
⊆ -scl( ( -scl(, )), )
=sint( (scl(, )), ).
Hence is a fuzzy semi-weakly -semicontinuous mapping.
References
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[4] R. N. Hazra, S. K. Samanta, and K. C.
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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
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E-mail : [email protected]
