Download a hit-and-miss hyperspace topology on the space of fuzzy sets

A HIT-AND-MISS HYPERSPACE
TOPOLOGY ON THE SPACE OF FUZZY
SETS
Anil Kumar
Department of Applied Sciences
Lingayas Institute of Management and Technology
Faridabad
INDIA
E-mail: [email protected]
J.V.Ramani
Department of mathematics
Agra college, Agra
INDIA
E-mail: [email protected]
Abstract
A hit-and-miss type Vietoris topology is defined on the space of
non-empty closed fuzzy sets of a fuzzy topological space and it is
proved that if this space is compact then the fuzzy topological space
is also compact.
2000 Mathematics Subject Classification: Primary 54A40.
Keywords and phrases: fuzzy topological space, hyperspace
1
Introduction
The concept of a fuzzy set was introduced by L.A.Zadeh in [12] and later
the concept of a fuzzy topology by C.L.Chang in [5]. After this introduction
numerous authors have investigated how the properties of topological spaces
can be extended to fuzzy topological spaces like compactness, connectedness
and separation, etc., which can be found in [4, 7, 8, 9, 11]. On the other hand,
topologies on the spaces of non-empty closed subsets of a metric space(or
a Banach space or a topological space) have been investigated by G.Beer
and his colloborators with regard to the problems in convex analysis and
optimization [1, 2, 3, 6]. Of these topologies(called the hyperspace topologies)
Vietoris topology on the space of non-empty closed subsets of a metric space
occupies a central place among all the other hyperspace topologies. Vietoris
topology is also called as a hit-and-miss type of topology as it is generated
by the sets which hit a finite number of open sets and misses a closed set.
More precisely, if (X, d) is a metric space and CL(X) denotes the space of
non-empty closed subsets of X then the Vietoris topology τV on CL(X) has
basic open sets of the form
[U ; V1 , V2 , . . . Vn ] = {A ∈ CL(X) | A ⊂ U, A ∩ Vi 6= φ, i = 1, 2, . . . , n}
where n ∈ N and U, V1 , V2 , . . . , Vn are open sets in (X, d).
In [10] L. Vietoris proved that the hyperspace 2X of all non-empty closed
subsets of a topological space X equipped with the Vietoris topology is compact iff X is compact. He also proved in the same paper that the connectedness of 2X in Vietoris topology is equivalent to that of X. In this paper
we define a topology of hit-and-miss type on the space of non-empty closed
fuzzy sets of a T1 -fuzzy toplogical space and prove that if this hyperspace
topology is compact then the fuzzy topological space must be compact.
2
Notations
In this note X will always denote a non-empty (ordinary) set. Any function
f from X to I = [0, 1] is called a fuzzy set in X or a fuzzy subset of X.
For any x ∈ X, A(x) is called the grade of membership of x in X. The set
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{x ∈ X | A(x) > 0} is called the support of A and denoted by SuppA. If a
fuzzy set A takes only the values 0 and 1 then the set is called a crisp set.
The fuzzy set A(x) ≡ 1 will be denoted by X. Sometimes we will also denote
this by 1 or I. The fuzzy set A(x) ≡ 0 will be denoted by 0 and sometimes
by φ.
Let A = {Ai | i ∈ Γ} be a family of fuzzy sets. Then we have (∪Ai )(x) =
W
V
( i Ai )(x) = sup {Ai (x)} for each x ∈ X, (∩Ai )(x) = ( i Ai )(x) =
Inf {Ai (x)} for each x ∈ X.
We recall the definition of a fuzzy topological space from [5]. A fuzzy
topology is a family T of fuzzy sets in X satisfying the following conditions:
φ, X ∈ T , for A, B ∈ T we have A ∩ B ∈ T , and ∪Ai ∈ T for Ai ∈ T for
all i ∈ Λ. In this case we say (X, T ) is a fuzzy topological space and T is
called a fuzzy topology on X. Every member of T is called a fuzzy open set.
A fuzzy set in X is called a fuzzy point iff it takes the value 0 for all x ∈ X
except one, say y ∈ X. If its value at y is λ (0 < λ ≤ 1), then we denote this
fuzzy point by Pyλ . Two fuzzy sets A and B in X are said to be intersecting
iff there exist a x ∈ X such that (A ∩ B)(x) 6= 0. In this case we also write
A ∩ B 6= φ. (X, T ) is called a fuzzy T1 -space iff every fuzzy point is a closed
set([7]).
A family A of fuzzy sets is a cover of a fuzzy set C iff C ⊂ ∪ {A | A ∈ A}.
It is an open cover of C iff each member of A is an open fuzzy set. A fuzzy
topological space (X, T ) is said to be compact iff each open cover of X has a
finite subcover([5]). This compactness is also called quasi fuzzy compactness.
We now make the following definition required for this paper.
Definition 2.1 Let (X, T ) be a fuzzy topological space and let C(X) denote
the collection of all the non-empty closed fuzzy sets in X. That is φ ∈
/ C(X).
Consider the topology on C(X) generated by the sets of the form
[U ; V1 , V2 , . . . Vn ] = {A ∈ C(X) | A ⊂ U, A ∩ Vi 6= φ, i = 1, 2, . . . , n}
where n ∈ N and U, V1 , V2 , . . . , Vn are open sets in (X, T ). We denote this
topology on C(X) by V.
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3
Main Theorem
We start with the following
Proposition 3.1 If (X, T ) has a finite open covering then C(X) also has a
finite open covering in V.
Proof. Let X ≤ m
k=1 Tk , Tk ∈ T , k = 1, . . . , m be a open covering of
X. That is 1 = I(x) ≤ max {T1 (x), . . . , Tm (x)} for each x ∈ X. That
is max {T1 (x), . . . , Tm (x)} = 1 for each x ∈ X. Let A ∈ C(X). Then
A 6= 0. That is, there exists a y ∈ X such that A(y) = λ 6= 0. Now
max {T1 (x), . . . , Tm (x)} = 1 for each x ∈ X. Therefore for this y ∈ X
there exists at least one j, 1 ≤ j ≤ m with Tj (y) = 1. This gives that
(A ∩ Tj )(y) = min(A(y), Tj (y)) = min(λ, 1) 6= 0. That is A ∩ Tj 6= φ. Also
A(x) ≤ I(x) for all x ∈ X. This means that A ∈ [I; Tj ] for some j. Hence
S
C(X) ⊂ m
i=1 [I; Ti ].
Now we come to the main theorem of the paper.
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Theorem 3.2 Let (X, T ) be a T1 -fuzzy topological space. Then if (C(X), V)
is compact then (X, T ) is also compact.
Proof. Let X ≤ α∈Λ Tα , Tα ∈ T for all α ∈ Λ. That is Supα∈Λ {Tα (x)} = 1,
for each x ∈ X. This gives Tα (x) ≤ I(x) = 1 for all x ∈ X, for all α ∈ Λ.
Let A ∈ C(X). Then there exists a x ∈ X such that A(x) = λ 6= 0. If
λ 6= 1 then we can find a α0 such that 1 − λ < Tα0 (x) ≤ 1. That is
A ∧ Tα0 6= φ. That is A ∈ [I; Tα0 ]. If λ = 1 then SupTα (x) = 1 = A(x).
That is Tα (x) ≤ A(x). Or A ∧ Tα1 6= φ for some α1 . That is A ∈ [I; Tα1 ].
S
Thus C(X) ⊂ α∈Λ [I; Tα ] which is an open cover of C(X). Since C(X) is
S
compact, we have C(X) ⊂ ni=1 [I; Ti ]. As (X, T ) is a T1 -space, every fuzzy
point is in C(X). Let x ∈ X and 0 < λ < 1. Consider the fuzzy point Pxλ .
S
Then Pxλ ∈ C(X) ⊂ ni=1 [I; Ti ]. That is Pxλ ∈ [I; Tj ] for some j. We get
Pxλ ∧ Tj 6= φ. That is, min(λ, Tj (x)) 6= 0. If Tj (x) = 1 we are done. Suppose
Tj (x) < 1. If λ = P (x) < Tj (x) then 1 ≤ Tj (x) as this is true for all λ → 1.
If Tj (x) ≤ P (x) = λ then Tj (x) = 0 because 0 < λ, which gives Tj (x) = 0,
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a contradiction. Hence we must have Tj (x) = 1. This gives ni=1 Ti (x) = 1.
W
Hence X ≤ ni=1 Ti . That is (X, T ) is fuzzy compact.
W
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References
[1] G. Beer, Conjugate convex functions and the epi-distance topology,
Proc. Amer. Math. Soc., 108(1990) 117-126.
[2] G. Beer and R. Lucchetti, Convex optimization and the epi-distance
topology, Trans. Amer. Math. Soc., 327(1991) 795-813.
[3] G. Beer and R. Lucchetti, Weak topologies for the closed subsets of a
metrizable space, Trans. Amer. Math. Soc., 335(1993) 805-822.
[4] G. Balasubramanian and V. Chandrasekhar, Fuzzy α-connectedness
and fuzzy α-disconnectedness in fuzzy topological spaces, Vesnik
Matematicki., 56(2004) 47-56.
[5] C. L. Chang, Fuzzy topological spaces, J. math. Anal. Appl., 24(1968)
182-169.
[6] J.J. Charatonik, Recent Research in Hyperspace Theory, , Extracta
Mathematicae Vol.18,no.2(2003) 235-262.
[7] Pu. Pao-Ming and Liu. Yiing-Ming, Fuzzy Topology I. neighbourhood
structure of a fuzzy point and moore-smith convergence, J. Math. Anal.
Appl., 76(1980) 571-599.
[8] Pu. Pao-Ming and Liu. Yiing-Ming, Fuzzy Topology II. Product and
Quotient spaces, J. Math. Anal. Appl., 77(1980) 20-37.
[9] C. K. Wong, Fuzzy points and local properties of fuzzy topology, J.
Math. Anal. Appl., 46(1974) 316-328.
[10] L. Vietoris, Bereiche zweiter Ordnung, Monatshefte fur Mathematik
und Physik., 32(1922) 258-280.
[11] V. Gregori and Anna Vidal, Fuzziness in Chang’s fuzzy topological
spaces, Rend.Inst. Mat.Univ.Trieste., 30(1999) 111-121.
[12] L. A. Zadeh, Fuzzy sets, Inform. and contr., 8(1965) 338-353.
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