Fuzzy r-Compactness on Fuzzy r-Minimal Spaces - Semantic Scholar
International Journal of Fuzzy Logic and Intelligent Systems, vol. 9, no. 4, December 2009 pp. 281-284
Fuzzy
r -Compactness on Fuzzy r -Minimal Spaces Jung Il Kim
1
, Won Keun Min
2
3
and Young Ho Yoo
Department of Statics, Kangwon National University, Chuncheon, 200-701, Korea Department of Mathematics, Kangwon National University, Chuncheon, 200-701, Korea Department of Statics, Kangwon National University, Chuncheon, 200-701, Korea
Abstract
r-minimal structure which is an extension of smooth fuzzy topological spaces r-M continuity. In this paper, we introduce the concepts of fuzzy r -minimal compactness, almost fuzzy r -minimal compactness and nearly fuzzy r -minimal compactness on fuzzy r -minimal spaces and investigate the relationships between fuzzy r -M continuous mappings and such types of fuzzy r -minimal compactness. In [8], we introduced the concept of fuzzy
and fuzzy topological spaces in Chang's sense. And we also introduced and studied the fuzzy
r-minimal spaces, fuzzy r-M open mapping, fuzzy r-M r-minimal compact and nearly fuzzy r-minimal compact
Key words : fuzzy fuzzy
1. Introduction
continuous, fuzzy
r-minimal compact, almost
A smooth fuzzy topology [7] on X is a map T
: IX → I
which satises the following properties: The concept of fuzzy set was introduced by Zadeh [9]. Chang [2] dened fuzzy topological spaces using fuzzy sets. In [3, 7], Chattopadhyay, Hazra and Samanta introduced smooth fuzzy topological spaces which are a generalization of fuzzy topological spaces. In [8], we introduced the concept of fuzzy
r-minimal
T (˜ 0) = T (˜ 1 ) = 1. (2) T (µ1 ∧ µ2 ) ≥ T (µ1 ) ∧ T (µ2 ). (3) T (∨µi ) ≥ ∧T (µi ). The pair (X, T ) is called a smooth (1)
fuzzy topological
space.
space which is an extension of the smooth fuzzy topological space.
The concepts of fuzzy
semiopen sets, fuzzy
r-preopen
r-open sets, fuzzy rr-fuzzy β -open sets
sets,
r-regular open sets are introduced in [1, 4, 5, 6], r-minimal structures. We also introduced and studied the concepts of fuzzy r -M continuity, fuzzy r -M open maps and fuzzy r -M closed maps. In this paper, we introduce the concepts of fuzzy r -minimal compactness, almost fuzzy r -minimal compactness and nearly fuzzy r -minimal compactness on fuzzy r -minimal spaces and investigate the relationships between fuzzy r -M continuous mappings and such types of fuzzy r -minimal comand fuzzy
which are a kind of fuzzy
A be a fuzzy set in a smooth fuzzy topological (X, T ) and r ∈ I . Then A is said to be fuzzy rsemiopen [5] (resp., fuzzy r -preopen [4], r -fuzzy β -open [1]) if A ⊆ cl(int(A, r), r) (resp., A ⊆ int(cl(A, r), r), A ⊆ cl(int(cl(A, r), r), r)). Let
spaces
Denition 2.1. ([8]) Let
(0, 1] = I0 .
X
A fuzzy family
be a nonempty set and
M : IX → I
on
X
r ∈
is said to
have a fuzzy r-minimal structure if the family
Mr = {A ∈ I X | M(A) ≥ r}
pactness. contains
˜ 0 and ˜ 1.
ber
(X, M) is called a fuzzy r-minimal space r-FMS). Every member of Mr is called a fuzzy rminimal open set. A fuzzy set A is called a fuzzy r-minimal c closed set if the complement of A (simply, A ) is a fuzzy r-minimal open set.
other notations are standard notations of fuzzy set theory.
minimal closure and the fuzzy
2. Preliminaries
I be the unit interval [0, 1] of the real line. A memµ of I X is called a fuzzy set of X . By ˜ 0 and ˜ 1 we denote constant maps on X with value 0 and 1, respectively. X c For any µ ∈ I , µ denotes the complement ˜ 1 − µ. All Let
Then the
(simply
Let
(X, M)
be an
r-FMS
and r ∈ I0 . The fuzzy r r-minimal interior of A [8],
Manuscript received Feb. 9, 2009; revised Aug. 20, 2009. Corresponding Author : [email protected] (Won Keun Min)
281
International Journal of Fuzzy Logic and Intelligent Systems, vol. 9, no. 4, December 2009
denoted by
mC(A, r)
and
mI(A, r),
respectively, are de-
ned as
Denition 3.4. Let in
mC(A, r) = ∩{B ∈ I
X
c
: B ∈ Mr
mI(A, r) = ∪{B ∈ I X : B ∈ Mr Theorem 2.2. ([8]) Let
and
and
X
(X, M)
be an
r-FMS.
A fuzzy set
A
is said to be almost fuzzy r-minimal compact if for ev-
r-minimal open cover A = {Ai ∈ I X : i ∈ J} of A, there exists J0 = {j1 , j2 , · · · , jn } ⊆ J such that A ⊆ ∪i∈J0 mC(Ai , r).
ery fuzzy
A ⊆ B},
B ⊆ A}.
(X, M) be an r-FMS and A, B in
IX . mI(A, r) ⊆ A and if A is a fuzzy r-minimal open mI(A, r) = A. (2) A ⊆ mC(A, r) and if A is a fuzzy r -minimal closed set, then mC(A, r) = A. (3) If A ⊆ B , then mI(A, r) ⊆ mI(B, r) and mC(A, r) ⊆ mC(B, r). (4) mI(A, r) ∩ mI(B, r) ⊇ mI(A ∩ B, r) and mC(A, r) ∪ mC(B, r) ⊆ mC(A ∪ B, r). (5) mI(mI(A, r), r) = mI(A, r) and mC(mC(A, r), r) = mC(A, r). (6) ˜ 1−mC(A, r) = mI(˜ 1−A, r) and ˜ 1−mI(A, r) = ˜ mC(1 − A, r).
Theorem 3.5. Let (X, M) be an r -FMS. If a fuzzy set A in
X is fuzzy r-minimal compact, then it is also almost fuzzy r-minimal compact.
(1)
set, then
Denition 2.3. ([8]) Let FMS's. Then
f :X→Y
(X, M)
and
(Y, N )
be two
(1) fuzzy r-M continuous mapping if for every
f −1 (A) is in Mr , (2) fuzzy r -M open Nr .
r-
is said to be
if for every
Proof. Obvious. In Theorem 3.5, the converse is not always true as shown the next example.
X = I and n ∈ N − {1}. An be fuzzy sets dened as follows 0.8, if x = 0, nx, if 0 < x ≤ n1 , An (x) = 1 1, if n < x ≤ 1; ½ 1, if x = 0, A1 (x) = 1 2 , otherwise.
Example 3.6. Let
A ∈ Nr ,
A ∈ Mr , f (A)
Consider a fuzzy
X
A1
and
r-minimal structure M : I X → I
on
as follows
is in
M(A) =
3. Fuzzy
Let
r-Minimal Compactness
4 5, n n+1 , 2 3,
0,
A=˜ 0, ˜ 1, A = An , if A = A1 , otherwise.
if if
A = {An : n ∈ N } be a fuzzy 21 -minimal open cover of X . Then there does not exist a nite subcover of A. Thus X is not fuzzy 12 -minimal compact. But X is Let
(X, M) be an r-FMS and A = {Ai ∈ I X : i ∈ J}. A is called a fuzzy r-minimal cover if ∪{Ai : i ∈ J} = ˜ 1. It is a fuzzy r-minimal open cover if each Ai is a fuzzy r -minimal open set. A subcover of a fuzzy r -minimal cover A is a subfamily of it which also is a fuzzy r -minimal cover. Denition 3.1. Let
Denition 3.2. Let in
X
(X, M)
be an
r-FMS.
A fuzzy set
A
is said to be fuzzy r-minimal compact if every fuzzy
r-minimal open cover A = {Ai ∈ Mr : i ∈ J} of A has a nite subcover.
f : (X, M) → (Y, N ) be a fuzzy r-M r-FMS's. If A is a fuzzy rminimal compact set, then f (A) is also a fuzzy r -minimal
Theorem 3.3. Let
continuous mapping on two compact set.
{Bi ∈ I Y : i ∈ J} be a fuzzy r-minimal open cover of f (A) in Y . Then since f is a fuzzy r -M −1 continuous mapping, {f (Bi ) : i ∈ J} is a fuzzy rminimal open cover of A in X . By fuzzy r -minimal compactness, there exists J0 = {j1 , j2 , · · · , jn } ⊆ J such that A ⊆ ∪i∈J0 f −1 (Bi ). Hence f (A) ⊆ ∪i∈J0 Bi .
Proof. Let
282
almost fuzzy
1 2 -minimal compact.
f : X → Y be a mapping on two r-FMS's (X, M) and (Y, N ). (1) f is fuzzy r -M continuous. −1 (2) f (B) is a fuzzy r-minimal closed set, for each fuzzy r -minimal closed set B in Y . X (3) f (mC(A, r)) ⊆ mC(f (A), r) for A ∈ I . −1 −1 (4) mC(f (B), r) ⊆ f (mC(B, r)) for B ∈ I Y . −1 (5) f (mI(B, r)) ⊆ mI(f −1 (B), r) for B ∈ I Y . Then (1) ⇔ (2) ⇒ (3) ⇔ (4) ⇔ (5). Theorem 3.7. ([8]) Let
f : (X, M) → (Y, N ) be a fuzzy rr-FMS's. If A is an almost fuzzy r -minimal compact set, then f (A) is also an almost fuzzy r -minimal compact set. Theorem 3.8. Let
M
continuous mapping on two
{Bi ∈ I Y : i ∈ J} be a fuzzy r-minimal open −1 cover of f (A) in Y . Then {f (Bi ) : i ∈ J} is a fuzzy rminimal open cover of A in X . By almost fuzzy r -minimal compact, there exists J0 = {j1 , j2 , · · · , jn } ⊆ J such that
Proof. Let
Fuzzzy Functions and Fuzzy Partially Ordered set
A ⊆ ∪i∈J0 mC(f −1 (Bi ), r).
From Theorem 3.7, it fol-
lows
∪i∈J0 mC(f −1 (Bi , r))
⊆ ∪i∈J0 f −1 (mC(Bi , r)) = f −1 (∪i∈J0 mC(Bi , r)).
Hence
f (A) ⊆ ∪i∈J0 mC(Bi , r).
f : X → Y be a mapping on two r-FMS's (X, M) and (Y, N ). Then (1) f is fuzzy r -M open. X (2) f (mI(A), r) ⊆ mI(f (A), r) for A ∈ I . −1 −1 (3) mI(f (B), r) ⊆ f (mI(B), r) for B ∈ I Y . Then (1) ⇒ (2) ⇔ (3).
Theorem 3.12. ([8]) Let
f : (X, M) → (Y, N ) r-M continuous and fuzzy r-M open on two rFMS's. If A is a nearly fuzzy r -minimal compact set, then f (A) is a nearly fuzzy r-minimal compact set. Theorem 3.13. Let a mapping
be fuzzy
(X, M)
Denition 3.9. Let in
X
be an
r-FMS.
A fuzzy set
A
is said to be nearly fuzzy r-minimal compact if for
r-minimal open cover A = {Ai : i ∈ J} A, there exists J0 = {j1 , j2 , · · · , jn } ⊆ J such that A ⊆ ∪i∈J0 mI(mC(Ai , r), r). every fuzzy
of
X = I . Consider the fuzzy miniM dened in Example 3.6. The fuzzy set ˜ 1 is
Example 3.10. (1) Let mal structure
1 2 -minimal compact set but it is not nearly 1 fuzzy -minimal compact in (X, M). 2
an almost fuzzy
(2) Let
X = I.
Consider fuzzy sets for
½ σn (x) =
1 n x, x−1 − 1−n ,
½
1, 1 2,
α(x) = ½
1 2,
β(x) =
1,
if if
if if
if if
{Bi ∈ I Y : i ∈ J} be a fuzzy r-minimal open f (A) in Y . Then {f −1 (Bi ) : i ∈ J} is a fuzzy rminimal open cover of A in X . By nearly fuzzy r -minimal compactness, there exists J0 = {j1 , j2 , · · · , jn } ⊆ J such −1 that A ⊆ ∪i∈J0 mI(mC(f (Bi ), r), r). From Theorem
Proof. Let cover of
3.7 and Theorem 3.12, it follows
f (A) ⊆ ∪i∈J0 f (mI(mC(f −1 (Bi ), r), r)) ⊆ ∪i∈J0 mI(f (mC(f −1 (Bi ), r)), r) ⊆ ∪i∈J0 mI(f (f −1 (mC(Bi , r))), r) ⊆ ∪i∈J0 mI(mC(Bi , r), r).
0 < n < 1,
0 ≤ x ≤ n, n < x ≤ 1;
x = 0, 0 < x ≤ 1;
f (A)
Hence
is a nearly fuzzy
r-minimal
compact
set. Remark 3.14. In Theorem 3.13, the fuzzy
0 ≤ x < 1, x = 1.
and fuzzy
r -M
openness of the mapping
r-M continuity f are necessary
conditions as shown in the next example. And consider a fuzzy minimal structure
N (µ) =
Then fuzzy
X
X = I.
Example 3.15. Let
max({1
− n, n}),
1, 0, is nearly fuzzy
1 2 -minimal compact.
½
µ = αn , if µ = α, β, ˜ 0, ˜ 1, otherwise. if
σn (x) =
1 n x, x−1 − 1−n ,
½ α(x) =
1 2 -minimal compact but not
½ β(x) =
(X, M) be an r-FMS. If a fuzzy set A in X is fuzzy r-minimal compact, then it is nearly fuzzy r-minimal compact.
Theorem 3.11. Let
½ γ(x) = ½
Proof. For any a fuzzy from
Theorem
2.2,
mI(mC(U, r), r).
it
r-minimal follows U
U in X , mI(U, r) ⊆
open set
=
Thus we get the result.
In Theorem 3.11, the converse implication is not true always true as shown in the Example 3.10. Hence the following implications are obtained:
r-minimal compact ⇒ nearly fuzzy r-minimal compact ⇒ almost fuzzy r -minimal compact fuzzy
Consider fuzzy sets for
0 < n < 1,
η(x) =
1, 0,
if
0, 1,
if
0, 1,
if
1, 0,
if
if
if
if
if
if if
0 ≤ x ≤ n, n < x ≤ 1;
x = 0, 0 < x ≤ 1; 0 ≤ x < 1, x = 1; x = 0, 0 < x ≤ 1; 0 ≤ x < 1, x = 1.
And consider fuzzy minimal structures
L(µ) =
max({1
− n, n}),
1, 0,
M(µ) =
µ = αn , µ = α, β, γ, η, ˜ 0, ˜ 1, otherwise;
if if
max({1
1, 0,
− n, n}),
µ = αn , µ = α, β, ˜ 0, ˜ 1, otherwise;
if if
283
International Journal of Fuzzy Logic and Intelligent Systems, vol. 9, no. 4, December 2009
N (µ) =
[5] -, max({1
− n, n}),
1, 0,
µ = αn , µ=˜ 0, ˜ 1, otherwise. if
if
f : (X, L) → (X, M) be the identity mapping. 1 It is obvious that f is fuzzy -M continuous. X is nearly 2 1 fuzzy -minimal compact on (X, L) but f (X) is not nearly 2 1 fuzzy -minimal compact on (X, M). 2 Let
Now let ping. Then
f : (X, N ) → (X, M) be the identity mapf is fuzzy 21 -M open. Consider a fuzzy set A
dened as follows
½ A(x) =
r-continuous
Fuzzy
and
fuzzy
27, pp. 53-63, 2001. [6] , Fuzzy
r-continuous
r-regular open sets and fuzzy almost
maps , Bull. Korean Math. Soc., vol.
39, pp. 91-108, 2002. [7] A. A. Ramadan, Smooth topological spaces, Fuzzy Sets and Systems, vol. 48, pp. 371-375, 1992. [8] Young Ho Yoo, Won Keun Min and Jung IL Kim.
r-Minimal
Fuzzy
Structures and Fuzzy
r-Minimal
Spaces, Far East Journal of MAthematical Sciences,
1, 0,
if if
0 < x < 1, x = 0, 1.
1 Then A is nearly fuzzy -minimal compact on (X, N ) 2 1 but f (A) is not nearly fuzzy -minimal compact (X, M). 2
References
[1] S. E. Abbas, Fuzzy
β -irresolute functions, Applied
Mathematics and Computation, vol. 157, pp. 369-
vol. 33, no. 2, pp. 193-205, 2009. [9] L. A. Zadeh, Fuzzy sets, Information and Control, vol. 8, pp. 338-353, 1965.
1
Jung Il Kim
Professor of Kangwon National University Research Area: Fuzzy logic, Statics E-mail : [email protected]
380, 2004. [2] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., vol. 24, pp. 182-190, 1968. [3] K. C. Chattopadhyay, R. N. Hazra, and S. K. Samanta, Gradation of openness : Fuzzy topology, Fuzzy Sets and Systems, vol. 49, pp. 237-242, 1992.
Won Keun Min
2
Professor of Kangwon National University Research Area: Fuzzy topology, General topology E-mail : [email protected]
3
Young Ho Yoo
r-preopen and fuzzy r-precontinuous maps, Bull. Korean Math. Soc., vol.
Professor of Kangwon National University
36, pp. 91-108, 1999.
E-mail : [email protected]
[4] S. J. Lee and E. P. Lee, Fuzzy
284
r-
semicontinuous maps , Int. J. Math. Math. Sci., vol.
Research Area: Fuzzy logic, Statics
Fuzzy
r -Compactness on Fuzzy r -Minimal Spaces Jung Il Kim
1
, Won Keun Min
2
3
and Young Ho Yoo
Department of Statics, Kangwon National University, Chuncheon, 200-701, Korea Department of Mathematics, Kangwon National University, Chuncheon, 200-701, Korea Department of Statics, Kangwon National University, Chuncheon, 200-701, Korea
Abstract
r-minimal structure which is an extension of smooth fuzzy topological spaces r-M continuity. In this paper, we introduce the concepts of fuzzy r -minimal compactness, almost fuzzy r -minimal compactness and nearly fuzzy r -minimal compactness on fuzzy r -minimal spaces and investigate the relationships between fuzzy r -M continuous mappings and such types of fuzzy r -minimal compactness. In [8], we introduced the concept of fuzzy
and fuzzy topological spaces in Chang's sense. And we also introduced and studied the fuzzy
r-minimal spaces, fuzzy r-M open mapping, fuzzy r-M r-minimal compact and nearly fuzzy r-minimal compact
Key words : fuzzy fuzzy
1. Introduction
continuous, fuzzy
r-minimal compact, almost
A smooth fuzzy topology [7] on X is a map T
: IX → I
which satises the following properties: The concept of fuzzy set was introduced by Zadeh [9]. Chang [2] dened fuzzy topological spaces using fuzzy sets. In [3, 7], Chattopadhyay, Hazra and Samanta introduced smooth fuzzy topological spaces which are a generalization of fuzzy topological spaces. In [8], we introduced the concept of fuzzy
r-minimal
T (˜ 0) = T (˜ 1 ) = 1. (2) T (µ1 ∧ µ2 ) ≥ T (µ1 ) ∧ T (µ2 ). (3) T (∨µi ) ≥ ∧T (µi ). The pair (X, T ) is called a smooth (1)
fuzzy topological
space.
space which is an extension of the smooth fuzzy topological space.
The concepts of fuzzy
semiopen sets, fuzzy
r-preopen
r-open sets, fuzzy rr-fuzzy β -open sets
sets,
r-regular open sets are introduced in [1, 4, 5, 6], r-minimal structures. We also introduced and studied the concepts of fuzzy r -M continuity, fuzzy r -M open maps and fuzzy r -M closed maps. In this paper, we introduce the concepts of fuzzy r -minimal compactness, almost fuzzy r -minimal compactness and nearly fuzzy r -minimal compactness on fuzzy r -minimal spaces and investigate the relationships between fuzzy r -M continuous mappings and such types of fuzzy r -minimal comand fuzzy
which are a kind of fuzzy
A be a fuzzy set in a smooth fuzzy topological (X, T ) and r ∈ I . Then A is said to be fuzzy rsemiopen [5] (resp., fuzzy r -preopen [4], r -fuzzy β -open [1]) if A ⊆ cl(int(A, r), r) (resp., A ⊆ int(cl(A, r), r), A ⊆ cl(int(cl(A, r), r), r)). Let
spaces
Denition 2.1. ([8]) Let
(0, 1] = I0 .
X
A fuzzy family
be a nonempty set and
M : IX → I
on
X
r ∈
is said to
have a fuzzy r-minimal structure if the family
Mr = {A ∈ I X | M(A) ≥ r}
pactness. contains
˜ 0 and ˜ 1.
ber
(X, M) is called a fuzzy r-minimal space r-FMS). Every member of Mr is called a fuzzy rminimal open set. A fuzzy set A is called a fuzzy r-minimal c closed set if the complement of A (simply, A ) is a fuzzy r-minimal open set.
other notations are standard notations of fuzzy set theory.
minimal closure and the fuzzy
2. Preliminaries
I be the unit interval [0, 1] of the real line. A memµ of I X is called a fuzzy set of X . By ˜ 0 and ˜ 1 we denote constant maps on X with value 0 and 1, respectively. X c For any µ ∈ I , µ denotes the complement ˜ 1 − µ. All Let
Then the
(simply
Let
(X, M)
be an
r-FMS
and r ∈ I0 . The fuzzy r r-minimal interior of A [8],
Manuscript received Feb. 9, 2009; revised Aug. 20, 2009. Corresponding Author : [email protected] (Won Keun Min)
281
International Journal of Fuzzy Logic and Intelligent Systems, vol. 9, no. 4, December 2009
denoted by
mC(A, r)
and
mI(A, r),
respectively, are de-
ned as
Denition 3.4. Let in
mC(A, r) = ∩{B ∈ I
X
c
: B ∈ Mr
mI(A, r) = ∪{B ∈ I X : B ∈ Mr Theorem 2.2. ([8]) Let
and
and
X
(X, M)
be an
r-FMS.
A fuzzy set
A
is said to be almost fuzzy r-minimal compact if for ev-
r-minimal open cover A = {Ai ∈ I X : i ∈ J} of A, there exists J0 = {j1 , j2 , · · · , jn } ⊆ J such that A ⊆ ∪i∈J0 mC(Ai , r).
ery fuzzy
A ⊆ B},
B ⊆ A}.
(X, M) be an r-FMS and A, B in
IX . mI(A, r) ⊆ A and if A is a fuzzy r-minimal open mI(A, r) = A. (2) A ⊆ mC(A, r) and if A is a fuzzy r -minimal closed set, then mC(A, r) = A. (3) If A ⊆ B , then mI(A, r) ⊆ mI(B, r) and mC(A, r) ⊆ mC(B, r). (4) mI(A, r) ∩ mI(B, r) ⊇ mI(A ∩ B, r) and mC(A, r) ∪ mC(B, r) ⊆ mC(A ∪ B, r). (5) mI(mI(A, r), r) = mI(A, r) and mC(mC(A, r), r) = mC(A, r). (6) ˜ 1−mC(A, r) = mI(˜ 1−A, r) and ˜ 1−mI(A, r) = ˜ mC(1 − A, r).
Theorem 3.5. Let (X, M) be an r -FMS. If a fuzzy set A in
X is fuzzy r-minimal compact, then it is also almost fuzzy r-minimal compact.
(1)
set, then
Denition 2.3. ([8]) Let FMS's. Then
f :X→Y
(X, M)
and
(Y, N )
be two
(1) fuzzy r-M continuous mapping if for every
f −1 (A) is in Mr , (2) fuzzy r -M open Nr .
r-
is said to be
if for every
Proof. Obvious. In Theorem 3.5, the converse is not always true as shown the next example.
X = I and n ∈ N − {1}. An be fuzzy sets dened as follows 0.8, if x = 0, nx, if 0 < x ≤ n1 , An (x) = 1 1, if n < x ≤ 1; ½ 1, if x = 0, A1 (x) = 1 2 , otherwise.
Example 3.6. Let
A ∈ Nr ,
A ∈ Mr , f (A)
Consider a fuzzy
X
A1
and
r-minimal structure M : I X → I
on
as follows
is in
M(A) =
3. Fuzzy
Let
r-Minimal Compactness
4 5, n n+1 , 2 3,
0,
A=˜ 0, ˜ 1, A = An , if A = A1 , otherwise.
if if
A = {An : n ∈ N } be a fuzzy 21 -minimal open cover of X . Then there does not exist a nite subcover of A. Thus X is not fuzzy 12 -minimal compact. But X is Let
(X, M) be an r-FMS and A = {Ai ∈ I X : i ∈ J}. A is called a fuzzy r-minimal cover if ∪{Ai : i ∈ J} = ˜ 1. It is a fuzzy r-minimal open cover if each Ai is a fuzzy r -minimal open set. A subcover of a fuzzy r -minimal cover A is a subfamily of it which also is a fuzzy r -minimal cover. Denition 3.1. Let
Denition 3.2. Let in
X
(X, M)
be an
r-FMS.
A fuzzy set
A
is said to be fuzzy r-minimal compact if every fuzzy
r-minimal open cover A = {Ai ∈ Mr : i ∈ J} of A has a nite subcover.
f : (X, M) → (Y, N ) be a fuzzy r-M r-FMS's. If A is a fuzzy rminimal compact set, then f (A) is also a fuzzy r -minimal
Theorem 3.3. Let
continuous mapping on two compact set.
{Bi ∈ I Y : i ∈ J} be a fuzzy r-minimal open cover of f (A) in Y . Then since f is a fuzzy r -M −1 continuous mapping, {f (Bi ) : i ∈ J} is a fuzzy rminimal open cover of A in X . By fuzzy r -minimal compactness, there exists J0 = {j1 , j2 , · · · , jn } ⊆ J such that A ⊆ ∪i∈J0 f −1 (Bi ). Hence f (A) ⊆ ∪i∈J0 Bi .
Proof. Let
282
almost fuzzy
1 2 -minimal compact.
f : X → Y be a mapping on two r-FMS's (X, M) and (Y, N ). (1) f is fuzzy r -M continuous. −1 (2) f (B) is a fuzzy r-minimal closed set, for each fuzzy r -minimal closed set B in Y . X (3) f (mC(A, r)) ⊆ mC(f (A), r) for A ∈ I . −1 −1 (4) mC(f (B), r) ⊆ f (mC(B, r)) for B ∈ I Y . −1 (5) f (mI(B, r)) ⊆ mI(f −1 (B), r) for B ∈ I Y . Then (1) ⇔ (2) ⇒ (3) ⇔ (4) ⇔ (5). Theorem 3.7. ([8]) Let
f : (X, M) → (Y, N ) be a fuzzy rr-FMS's. If A is an almost fuzzy r -minimal compact set, then f (A) is also an almost fuzzy r -minimal compact set. Theorem 3.8. Let
M
continuous mapping on two
{Bi ∈ I Y : i ∈ J} be a fuzzy r-minimal open −1 cover of f (A) in Y . Then {f (Bi ) : i ∈ J} is a fuzzy rminimal open cover of A in X . By almost fuzzy r -minimal compact, there exists J0 = {j1 , j2 , · · · , jn } ⊆ J such that
Proof. Let
Fuzzzy Functions and Fuzzy Partially Ordered set
A ⊆ ∪i∈J0 mC(f −1 (Bi ), r).
From Theorem 3.7, it fol-
lows
∪i∈J0 mC(f −1 (Bi , r))
⊆ ∪i∈J0 f −1 (mC(Bi , r)) = f −1 (∪i∈J0 mC(Bi , r)).
Hence
f (A) ⊆ ∪i∈J0 mC(Bi , r).
f : X → Y be a mapping on two r-FMS's (X, M) and (Y, N ). Then (1) f is fuzzy r -M open. X (2) f (mI(A), r) ⊆ mI(f (A), r) for A ∈ I . −1 −1 (3) mI(f (B), r) ⊆ f (mI(B), r) for B ∈ I Y . Then (1) ⇒ (2) ⇔ (3).
Theorem 3.12. ([8]) Let
f : (X, M) → (Y, N ) r-M continuous and fuzzy r-M open on two rFMS's. If A is a nearly fuzzy r -minimal compact set, then f (A) is a nearly fuzzy r-minimal compact set. Theorem 3.13. Let a mapping
be fuzzy
(X, M)
Denition 3.9. Let in
X
be an
r-FMS.
A fuzzy set
A
is said to be nearly fuzzy r-minimal compact if for
r-minimal open cover A = {Ai : i ∈ J} A, there exists J0 = {j1 , j2 , · · · , jn } ⊆ J such that A ⊆ ∪i∈J0 mI(mC(Ai , r), r). every fuzzy
of
X = I . Consider the fuzzy miniM dened in Example 3.6. The fuzzy set ˜ 1 is
Example 3.10. (1) Let mal structure
1 2 -minimal compact set but it is not nearly 1 fuzzy -minimal compact in (X, M). 2
an almost fuzzy
(2) Let
X = I.
Consider fuzzy sets for
½ σn (x) =
1 n x, x−1 − 1−n ,
½
1, 1 2,
α(x) = ½
1 2,
β(x) =
1,
if if
if if
if if
{Bi ∈ I Y : i ∈ J} be a fuzzy r-minimal open f (A) in Y . Then {f −1 (Bi ) : i ∈ J} is a fuzzy rminimal open cover of A in X . By nearly fuzzy r -minimal compactness, there exists J0 = {j1 , j2 , · · · , jn } ⊆ J such −1 that A ⊆ ∪i∈J0 mI(mC(f (Bi ), r), r). From Theorem
Proof. Let cover of
3.7 and Theorem 3.12, it follows
f (A) ⊆ ∪i∈J0 f (mI(mC(f −1 (Bi ), r), r)) ⊆ ∪i∈J0 mI(f (mC(f −1 (Bi ), r)), r) ⊆ ∪i∈J0 mI(f (f −1 (mC(Bi , r))), r) ⊆ ∪i∈J0 mI(mC(Bi , r), r).
0 < n < 1,
0 ≤ x ≤ n, n < x ≤ 1;
x = 0, 0 < x ≤ 1;
f (A)
Hence
is a nearly fuzzy
r-minimal
compact
set. Remark 3.14. In Theorem 3.13, the fuzzy
0 ≤ x < 1, x = 1.
and fuzzy
r -M
openness of the mapping
r-M continuity f are necessary
conditions as shown in the next example. And consider a fuzzy minimal structure
N (µ) =
Then fuzzy
X
X = I.
Example 3.15. Let
max({1
− n, n}),
1, 0, is nearly fuzzy
1 2 -minimal compact.
½
µ = αn , if µ = α, β, ˜ 0, ˜ 1, otherwise. if
σn (x) =
1 n x, x−1 − 1−n ,
½ α(x) =
1 2 -minimal compact but not
½ β(x) =
(X, M) be an r-FMS. If a fuzzy set A in X is fuzzy r-minimal compact, then it is nearly fuzzy r-minimal compact.
Theorem 3.11. Let
½ γ(x) = ½
Proof. For any a fuzzy from
Theorem
2.2,
mI(mC(U, r), r).
it
r-minimal follows U
U in X , mI(U, r) ⊆
open set
=
Thus we get the result.
In Theorem 3.11, the converse implication is not true always true as shown in the Example 3.10. Hence the following implications are obtained:
r-minimal compact ⇒ nearly fuzzy r-minimal compact ⇒ almost fuzzy r -minimal compact fuzzy
Consider fuzzy sets for
0 < n < 1,
η(x) =
1, 0,
if
0, 1,
if
0, 1,
if
1, 0,
if
if
if
if
if
if if
0 ≤ x ≤ n, n < x ≤ 1;
x = 0, 0 < x ≤ 1; 0 ≤ x < 1, x = 1; x = 0, 0 < x ≤ 1; 0 ≤ x < 1, x = 1.
And consider fuzzy minimal structures
L(µ) =
max({1
− n, n}),
1, 0,
M(µ) =
µ = αn , µ = α, β, γ, η, ˜ 0, ˜ 1, otherwise;
if if
max({1
1, 0,
− n, n}),
µ = αn , µ = α, β, ˜ 0, ˜ 1, otherwise;
if if
283
International Journal of Fuzzy Logic and Intelligent Systems, vol. 9, no. 4, December 2009
N (µ) =
[5] -, max({1
− n, n}),
1, 0,
µ = αn , µ=˜ 0, ˜ 1, otherwise. if
if
f : (X, L) → (X, M) be the identity mapping. 1 It is obvious that f is fuzzy -M continuous. X is nearly 2 1 fuzzy -minimal compact on (X, L) but f (X) is not nearly 2 1 fuzzy -minimal compact on (X, M). 2 Let
Now let ping. Then
f : (X, N ) → (X, M) be the identity mapf is fuzzy 21 -M open. Consider a fuzzy set A
dened as follows
½ A(x) =
r-continuous
Fuzzy
and
fuzzy
27, pp. 53-63, 2001. [6] , Fuzzy
r-continuous
r-regular open sets and fuzzy almost
maps , Bull. Korean Math. Soc., vol.
39, pp. 91-108, 2002. [7] A. A. Ramadan, Smooth topological spaces, Fuzzy Sets and Systems, vol. 48, pp. 371-375, 1992. [8] Young Ho Yoo, Won Keun Min and Jung IL Kim.
r-Minimal
Fuzzy
Structures and Fuzzy
r-Minimal
Spaces, Far East Journal of MAthematical Sciences,
1, 0,
if if
0 < x < 1, x = 0, 1.
1 Then A is nearly fuzzy -minimal compact on (X, N ) 2 1 but f (A) is not nearly fuzzy -minimal compact (X, M). 2
References
[1] S. E. Abbas, Fuzzy
β -irresolute functions, Applied
Mathematics and Computation, vol. 157, pp. 369-
vol. 33, no. 2, pp. 193-205, 2009. [9] L. A. Zadeh, Fuzzy sets, Information and Control, vol. 8, pp. 338-353, 1965.
1
Jung Il Kim
Professor of Kangwon National University Research Area: Fuzzy logic, Statics E-mail : [email protected]
380, 2004. [2] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., vol. 24, pp. 182-190, 1968. [3] K. C. Chattopadhyay, R. N. Hazra, and S. K. Samanta, Gradation of openness : Fuzzy topology, Fuzzy Sets and Systems, vol. 49, pp. 237-242, 1992.
Won Keun Min
2
Professor of Kangwon National University Research Area: Fuzzy topology, General topology E-mail : [email protected]
3
Young Ho Yoo
r-preopen and fuzzy r-precontinuous maps, Bull. Korean Math. Soc., vol.
Professor of Kangwon National University
36, pp. 91-108, 1999.
E-mail : [email protected]
[4] S. J. Lee and E. P. Lee, Fuzzy
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