Download $\ alpha $-compact fuzzy topological spaces

Mathematica Bohemica
Samajh, Singh Thakur; Ratnesh Kumar Saraf
α-compact fuzzy topological spaces
Mathematica Bohemica, Vol. 120 (1995), No. 3, 299–303
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120(1995)
MATHEMATICA BOHEMICA
No. 3, 299-303
a-COMPACT FUZZY TOPOLOGICAL SPACES
S.S.
THAKUR, R.K.
SARAF, Jabalpur
(Received March 23, 1994)
Summary. The purpose of this paper is to introduce and discuss the concept of acompactness for fuzzy topological spaces.
Keywords: fuzzy topological spaces, compactness, a-compactness, a-open, fuzzy acontinuity
AMS classification: 54A40
1.
INTRODUCTION
The concept of fuzzy sets and fuzzy set operations was first introduced by Zadeh
in his classical paper [7]. Subsequently several authors have applied various basic
concepts from general topology to fuzzy sets and developed a theory of fuzzy topological spaces. The concept of a-compactness for topological spaces has been discussed
in [5]. The purpose of this paper is to introduce and study a-compactness for fuzzy
topological spaces, thus filling a gap in the existing literature on fuzzy topological
spaces.
2.
DEFINITIONS
Throughout this paper (X, T) will denote a fuzzy topological space. If A is a fuzzy
set in a fuzzy topological space then the closure and interior of A will be as usual
denoted by C\A and Int A, respectively. We now introduce the following definitions.
D e f i n i t i o n 2 . 1 . Let A be a fuzzy subset of a fuzzy topological space X. A is
said to be fuzzy a-open if A C Int CI Int A. The set of all fuzzy a-open subsets of X
will be denoted by Fa{X).
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Definition 2.2. In a fuzzy topological space (X, T), a family v of fuzzy subsets
of X is called an a-covering of X iff v covers X and v C Fa(X).
Definition 2 . 3 . A fuzzy topological space (X, T) is said to be a-compact if every
a-open cover of X has a finite subcover.
D e f i n i t i o n 2.4. Let (X,T) and (Y, S) be fuzzy topological spaces. A mapping
/ : X -4 Y is called fuzzy a-continuous if the inverse image of each fuzzy open set in
Y is fuzzy a-open in X.
Definition 2 . 5 . A mapping / : X -> Y is said to be fuzzy a-irresolute if the
inverse image of every fuzzy a-open set in Y is fuzzy a-open in X.
D e f i n i t i o n 2.6. Let (X, T) and (Y, S) be fuzzy topological spaces and let T { be
a fuzzy topology on X which has Fa(X) as a subbase. A mapping / : X -4 Y is
called fuzzy ^-continuous if / : (X,T() -4 (Y,S) is fuzzy continuous; / is said to be
fuzzy ^'-continuous if / : (X,T{) -4 (Y,S{) is fuzzy continuous.
3.
RESULTS
T h e o r e m 3 . 1 . Let (X, T) and (Y, S) be fuzzy topological spaces and let T ? be a
fuzzy topology on X which has Fa(X) as a subbase. If f: (X,T) -¥ (Y,S) is fuzzy
a-continuous, then f is fuzzy
^-continuous.
P r o o f . Let / be fuzzy a-continuous and let V 6 S. Then / - 1 ( V ) e
Fa(X)
and so / - 1 ( V ) £ 7$. Thus / is fuzzy ^-continuous and this completes the proof.
D
T h e o r e m 3.2. Let (X,T) and (Y,S) be fuzzy topological spaces. Let T$ and 5 {
be respectively the fuzzy topologies on X and Y which have Fa(X) and Fa(Y) as
subbases. If f: (X, T) -4 (Y, S) is fuzzy a-irresolute then f is fuzzy £'-continuous.
Proof.
Let / be fuzzy a-irresolute and let V e S { . Then
V = ( J C p | S;„. \
i 4=i
'
and
where S,„. 6 Fa(Y,
S),
l
rЧv)^r ({j(n^)-ö(Qr^))Since / is fuzzy a-irresolute, / - 1 ( S i „ ; ) 6 Fa(X,T).
T { and thus / is fuzzy ^'-continuous.
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This implies that /
X
(K) e
E
T h e o r e m 3 . 3 . A fuzzy topological space X is a-compact if and only if every
family of fuzzy a-closed subsets of X with finite intersection property has non-empty
intersection.
Proof.
Evident.
D
T h e o r e m 3.4. Let (X,T) be a fuzzy topological space and T( a fuzzy topology
on X which has Fa(X) as a subbase. Then (X,T) is a-compact if and only if (X ,T()
is compact.
P r o o f . Let (X,T()
is a-compact.
be compact. Then, since Fa(X)
C T ? , it follows that (X, T)
D
T h e o r e m 3.5. Let (X, T) be a fuzzy topological space which is a-compact.
each T^-closed fuzzy set in X is a-compact.
Then
P r o o f . Let U be any T r closed fuzzy set in X. Let {Vpi: ft e / } be a T r o p e n
cover of U. Since X - U is T r o p e n , {V0i: ft e / } U (X - U) is a T r o p e n cover of
X. Since X is T r c o m p a c t , by Theorem 3.4 there exists a finite subset Io C I such
that
X = \J{VPi : d e I0} U(XU).
This implies that
U C \J{V0I:
Pi e Io}.
Hence U is a-compact relative to X and this completes the proof.
D
T h e o r e m 3.6. Let a fuzzy topological space (X,T) be a-compact. Then every
family ofT(-closed fuzzy subsets ofX with finite intersection property has non-empty
intersection.
P r o o f . Let X be a-compact. Let U = {B0i : ft e / } be any family of T r closed
fuzzy subsets of X with finite intersection property. Suppose fli-Bft : ft e / } = 0.
Then {X - B0i : ft e / } is a T r o p e n cover of X. Hence it must contain a
finite subcover {X - BSij: j = l , 2 , . . . , n } for X. This implies that
f]{B0ij:
j = 1,2,... ,n} = 0 and contradicts the assumption that U has finite intersection
property.
D
T h e o r e m 3.7. Let X and Y be fuzzy topological spaces and let f: X -¥ Y be
fuzzy £' -continuous. If a fuzzy subset G of X is a-compact relative to X, then f(G)
is a-compact relative to Y.
P r o o f . Let {Vp.: ft e / } be a cover of f(G) by 5 r o p e n fuzzy sets in Y. Then
{/" (Vp^: Pi e / } is a cover of G by T r o p e n fuzzy sets in X. G is a-compact
301
relative to X. Hence by Theorem 3.4, G is T ? -compact.
subset I0 C I such that
Gc(J{r1(VA):/9,e/o}
So there exists a finite
and so
/ ( G ) C {V0i: A 6 / „ } .
Hence / ( G ) is T^-compact relative to Y. Thus / ( G ) is a-compact relative to Y and
this completes the proof.
D
Corollary 3.8. Iff: (X,T) -» (Y, S) is a fuzzy £'-continuous
and X is a-compact, then Y is a-compact.
surjective
function
Corollary 3.9. If f: (X,T) -> (Y, S) is a fuzzy a-irresolute
and X is a-compact then Y is a-compact.
surjective
function
T h e o r e m 3.10. Let A and B be fuzzy subsets of a fuzzy topological space X
such that A is a-compact relative to X and B is T^-closed in X. Then An B is
a-compact relative to X.
P r o o f . Let {Vp{: ft e 1} be a cover of A n B by Tj-open fuzzy subsets of X.
Since X - B is a T^-open fuzzy set,
{VPi
:/3ieI}U(X-B)
is a cover of A. A is a-compact and thus T^-compact relative to X.
exists a finite subset I0 C I such that
A C {}{VPi: fc e I0} U(X-
Hence there
B).
Therefore
AnB
Cljffis.: A €Io}.
Hence A n B is T^-compact. Therefore A n B is a-compact and this completes the
proof.
D
References
[1] C.L. Chang: Fuzzy topological spaces. J. Math. Anal. 24 (1968), 182-190.
[2] O. Njdsted: On some classes of nearly open sets. Pacific J. Math. 15 (1976), 961-970.
[3] S. N. Maheshwari and S. S. Thakur: On a-irresolute mapping. Tamkang J. Math. 11
(1980), 209-214.
[4] S. N. Maheshwari and S. S. Thakur: On a-continuous mapping. J. Ind. Acad. Math. 7
(1985), 46-50.
[5] S. N. Maheshwari and S. S. Thakur: On a-compact spaces. Bull. Inst. Math. Acad.
Sinica 13 (1985), 341-347.
[6] A.S. Mashhour, I. A. Hassanian, S.N. El-deeb: a-continuous and a-open mappings.
Acta Math. Sci. Hungar. 41 (1983), 213-218.
[7] L. A. Zadeh: Fuzzy sets. Inform and control 8 (1965), 338-353.
Authors' addresses: S. S. Thakur, Department of Applied Mathematics, Government
Engineering College, Jabalpur (M.P.), India 482011, R.K. Saraf Department of Mathematics, Government Autonomous Science College, Jabalpur (M.P.), India 482001.
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