Download properties of fuzzy metric space and its applications

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CHAPTER
(V)
PROPERTIES OF FUZZY METRIC
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SPACE AND ITS APPLICATIONS
CHAPTER -V
PROPERTIES OF FUZZY METRIC SPACE AND ITS
APPLICATIONS
A metric space is a set of points and a prescribed
quantitative measure of the degree of closeness of pairs of
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points in this space. The real number system and the
coordinate plane of analytical geometry are familiar example
of metric space.
We prove that the topology induced by any fuzzy metric
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space is metrizable we also show that every separable fuzzy
metric space admits a precompact fuzzy metric and that a
fuzzy metric space Ps compact if and only if it is precompact
and complete.
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We also defined a Hausdroff topology on a fuzzy metric
space introduced by Kramosil and Michalak [154] and prove
some known result of metric space including Baire’s theorem
for fuzzy metric spaces.
5.1
Introduction
5.2
Fuzzy Metric Space
5.3
Theorems Related to Fuzzy Metric Space
5.4
Topology Induced by a fuzzy metric
5.5
Theorems of topology induced by a fuzzy metric
5.6
Fuzzy Compactness.
5.7
Conclusion
5.1
Introduction
The theory of fuzzy sets was introduced by L. Zadeh in
1965 [118]. Many authors have introduced the concept of
fuzzy metric spaces in different ways [1-3]. In this paper; we
modify the concept of fuzzy metric space introduced by
Kramosil and Michalek [154] and define a Hausdroff topology
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on this fuzzy metric space.
We show that every metric induces a fuzzy metric.
Further we modify the definition of Cauchy sequence in [131],
because of the fact that even R is not complete with the
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definition given in [131].
We define F-bounded ness on a fuzzy metric space and
prove that compactness implies F-bounded ness. We prove
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that every closed ball is a closed set in a fuzzy metric space.
Finally: we prove Baire’s theorem for fuzzy metric spaces.
5.2
Fuzzy Metric Space- Now firstly we define fuzzy metric space-
5.2.1 Definition- A binary operation∗ : [0:1] × [0:1] → [0.1]
Is continuous t-norm if ([0:1]. ∗ )
Is a topological monoid with unit 1 such that
a ∗ b≤c∗d
whenever a ≤ c and b ≤ d.
(a: b : c: d ∈ [0:1]).
5.2.2 Definition-The 3-tuple (X:M:∗) is said to be a fuzzy metric
space if X is an arbitrary set ∗ is a continues t-norm and m is
a fuzzy set on a
3.
4.
5.
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1.
2.
X2 × [0: ∞) satisfying the following conditions:m (x:y:0) = 0
m (x: y : t) = 1 For all t > o if and only if
X=Y
m (x: y : t) = m (y: x : t)
m (x: y : t) ∗ m (y: z : s) ≤ m (x : z : t +s)
m ( x : y:.) : [ o. ∞ ) → [0:1] is left continuous.
x : y: z ∈ X and t : s >0
5.2.3 Definition- The 3-tuple (X:M ∗ ) is said to be a fuzzy metric
space if X is an arbitrary set ∗ is a continuous t-norm and M
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is a fuzzy set on
X2 x (0:∞)
Satisfying the following conditions :M (x : y : t) > 0
2:-
M (x : y : t) = 1 if and only if
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1:-
x=y
3:-
M (x : y : t) = M (y : x : t)
4:-
M (x : y : t) ∗ M (y : z : s ) ≤ M (x : z : t + s)
5:-
M (x : y:.) : (0 : ∞) → [0:1] is continuous.
x : y : z ∈ X and t : s > 0.
5.2.4 Definition-Let (F x , µ x ) be a fuzzy metric space then an element A
∈ F x is said to be an F- adherent point of a subset G of F x if every
F-open sphere with centre A contains at least one element of G.
F-adherent points are of two types:
(i)
F- Limit Points or F- accumulations points:
(ii)
F- isolated points:
5.2.5 Definition-Let (F X, µ x ) be a fuzzy metric space. Then an element
A∈F x is said to be an F-limit point of F-accumulation point of a
subset G of F x if every F-open sphere with centre A contains atleast one element of G other then A.
An F-limit point of G is not necessarily a point of G.
5.2.6 Definition- An F-adherent point A of a subset G of F x is called an
F- isolated point if there exists at-least F-open sphere with centre
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A which contains no point element of G is an F- isolated point then
G is called to be an F-isolated set.
Let (F x, µ x ) be a fuzzy metric space and let G ⊆ F x Then the distance
between A and G where A∈ F x defined as
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d(A,G)=inf. {µ × (A,y):y∈G}.
5.2.7 Definition- As Hohle observed in [235] the fact that the
topology associated to a probabilistic metric space is metrizable
means that. from this topological point of view. Probabilistic metric
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spaces are always equivalent to ordinary metric spaces, and the
problem of topologization of probabilistic metric space is not
satlsfactorily solved. He proposed many-value3d topologies as
suitable tools for this purpose. Hence in [251], we endowed George
and veeramani’s fuzzy metric (which has close relation to
probabilistic metric) with many-valued structuesfuzzifying topology
and fuzzifying uniformity. The aim of this paper is to go on studying
the properties of George and Veeramani’s fuzzy metric. We will give
the
concept
of
convewrgence
degree
and
generalize
the
convergencwe and compactness theories in metric spaces to
Veeramani’s fuzzy metric space.
Since the value M(x,y,t) can be thought as the degree of the
nearness between x and y with respect to t, in this section, we will
give
the
definitions
of
degree
convergence
and
study
the
relationship between them.
Let (X,M) be a fuzzy metric space, x ∈X and {x n } be sequence.
The degree to which {x n } converges to x is defined by
∧∨∧
M(x n ,x,ε).
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Con({x n },x) =
ε>0N∈N n>N
The degree to which {x n } accumulates to x is defined by
∧∨∧
M(x n ,x,ε).
ε>0N∈N n>N
The degree to which {x n } is a Cauchy sequence is defined by
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Ad({x n },x) =
Cauchy({x n },x) =
∧∨∧
M(x n ,x,ε).
ε>0N∈N n,m>N
5.2.8 Definition- In this section. We want to generalized the
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compactness in metric spaces to fuzzy setting according to the
above convergence theory.
Let (X,M) be a fuzzy metric space. The degree to which (X,M)
is compact is defined by
Comp(M) =
∧∨
Ad({x n ),x)
{x n }x∈X
The degree to which (X,M) is sequently compact is
defined by
Comp(M) =
∧∨∨
Con({x nk ),x).
{x n }x nk }x∈X
5.3 Theorems Related to Fuzzy Metric Space- Now we define some
theorems related to fizzy metric space.
5.3.1 Theorem- The element A of a fuzzy metric space (F x .µ x ) is an Fadherent point of the subset G of F x iff
d (A,G)=0
Proof: we have d(A,G)= inf. {µ x (A.y): y∈G} Therefore d(A,G)=0 ⇒
every F-open sphere S (A,r) contains element of G. which implies A
is an F-adherent point of G. conversely if A is an F-adherent point
of G. then wither A is an F-isolated point of G case d(A.G) =O hence
5.3.2 Theorem
Let X=R Define
a ∗ b = ab
m (x : y : t) = exp. (|x-y|)
t
-1
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and
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the theorem is proved.
for all x : y ∈ X and t ∈ (o: ∞ ). Then (x : m : ∗ ) is a fuzzy
metric space.
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Proof1.
clearly m (x : y : t) = 1 if and only if x = y.
If x = y.
1. m (x : y : t) = m (y : x : t)
2. To prove
m (x : y : t ). M ( y: z : S) ≤ m (x : z : t + S)
we know that
i.e.
[x-z] ≤ t+S |x-y| +
t
|x-z|
t+S
≤ |x-z |
t
t+S | y – z |
S
+ | y-z |
S
therefore
exp
≤ exp.
|x-z|
t+S
|x-y|
S
exp
|y-z|
S
thus
m (x: y :t). m (y :z :s) ≤ m (x :z :t+S).
4.
m ( x :y: .) : (o:∞) → [0:1] is continuous. Hence (x: m: ∗) is a
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fuzzy metric space.
5.4 Topology Induced by a Fuzzy Metric- Topology induced by a
Fuzzy metric is defined by the following way-
5.4.1 Definition-Let (x : m : ∗) be a fuzzy metric space we define
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open ball B (x : r : t ) with centre
x∈X
and radius r;
o < r < 1 : t > o as
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B (x : r : t ) = { y ∈ x : m ( x : y : t ) > 1- r }
5.4.2 Definition-Let (X : M : ∗) be a fuzzy metric space. A subset A
of X is said to be F-bounded if and only if there exist
t>o
and o < r <l
such that
m ( x : y : t) > 1-r
for all x. y ∈ A.
Remark-Let (x : m : ∗ ) be a fuzzy metric space induced by a metric
d on x. then A ⊆ x is F-bounded if and only if it is bounded.
5.4.3 Definition-A sequence {x n } in a fuzzy metric space (x : m : ∗ )
is a Cauchy sequence if and only if lim m (x n+p : x n : t) = 1 :
P>o:t:>on→∞
A fuzzy metric space in which every Cauchy sequence is
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convergent is called a complete fuzzy metric space.
Note-We note that with the above definition: even R fails to be
complete. For example.
Consider
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S n = 1 + 1 + 1 + …………… + 1
2 3
n
in ( R:M): where
m (x:y: t) =
t
t + d (x:y)
.
a is a metric on R.
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Now
M ( S n+p : S n : t)
=
t
.
t + |S n+p -S n |
Now.
=
t
.
t + ( 1 + 1) + ( 1 + 2 ) + …….(1 + P)
n
n
n
Therefore
Lim m (S n+p : S n :t) = 1
n→∞
thus {S n } is a Cauchy sequence in the fuzzy metric space R. if
R is fuzzy complete then there exits
x∈R
such that
m (S n : x : t) → |
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as n → ∞
From this if follows that
→1
t
t + | Sn – x |
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Further
as n → ∞
| Sn – x 1 → 0
as n → ∞
And so → x in R which is not true.
Hence to make R complete fuzzy metric space we redefine
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Cauchy sequence as follows.
5.4.4 Definition- A sequence {x n } in a fuzzy metric space (x : M ∗ )
is a Cauchy sequence if and only if for each ε >o; t > o there
exits
no ∈ N
such that
m (x n : x m : t) > 1-ε for all n : m ≥ n o .
5.4.5 Definition- Let (x : m : ∗ ) be a fuzzy metric space. Then we
define a closed ball with centre
x ∈X
and radius r: o < r < 1 : t > o as
B [ x : r : t ] = { y ∈ X ; M ( x : y : t ) ≥ 1- r }.
We need the following lemma to prove Baire’s theorem for
fuzzy metric space.
5.5 Theorems- now we defined some theorems :5.5.1 Theorem Every open ball is an open set.
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Proof consider an open ball B (x : r : t ) now
Y ∈ B. (x : r : t ) ⇒ m ( x : y : t ) > 1 – r
Since
m (x : y : t) > 1-r
We can find a to ; 0 < t 0 < t such that
m (x : y : t 0 ) > 1-r.
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Let
= m (x : y : t 0 ) > 1-r
since
r o > 1-r. we can find a s;
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o<s<l
such that
r o > 1 – s > 1 – r.
now for a given r o and S such that
r o > 1-S we can find r 1 ;
o < r 1 <1
such that
r o ∗ r 1 ≥ 1-S.
now consider the ball
B (y. 1-r 1 ; t-t 0 )
We claim
B (y : 1-r 1 : t – t 0 ) ⊂ B (x : r: t)
Now
z ∈ B (y : 1-r 1 : t-t 0 ) ⇒ m (y : z : t-t 0 ) > r 1
Therefore
m (x : z : t) ≥ m (x : y : t 1 ) ∗ m (y : z : t – t 0 )
≥ r o ∗ r 1 ≥ 1-s
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> 1-r
z ∈ B (x : r : t) and hence
Therefore
B ( y : 1-r 1 : t-t 0 ) ⊂ B (x: r: t).
5.5.2 Theorem- Every fuzzy metric space is Hausdroff.
Let (x : m : ∗ ) be the given fuzzy metric space.
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PROOF-
Let x : y be two distinct points of x. then
O < m (x : y : t) <l
m ( x : y : t) = r; for some r;
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Let
O < r <l.
For each r o ;
r < r o < l;
We can find a r 1 such that
r1 ∗ r1 ≥ ro.
Now consider
The open balls B (x : 1-r 1 : 1 t )
2
and
B (y : 1-r 1 : 1 t).
2
Clearly
B (x : 1-r 1 : 1 t) ∩ B (y : 1-r 1 : 1 t ) = Ø
2
2
For if there exists
z ∈ B (x : 1- r 1 ; 1 t) ∩ B (y : 1-r 1 ; 1 t)
2
2
Then
r = m (x : y : t)
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≥ m (x : z : 1 t) ∗ m (z : y : 1 t)
2
2
≥ r1 ∗ r1 ≥ ro
> r
Which is a contradiction therefore
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(X : M: ∗ ) is Hausdroff.
5.5.3 Theorem- Every compact subset A of a fuzzy metric space X
is F-bounded.
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Proof- Given A is a compact subset of X. Fix
t > o and o < r <l.
consider an open cover
{ B (x : r : t ) : x ∈ A } of A.
Since A is compact; there exist
x 1 : x 2 : ……….. x n ∈ A .
Such that
A ⊆ ∪ B (x : r : t ).
Let x : y : ∈ A.
Then
x ∈ B ( x i : r : t)
and y ∈ B ( x i : r : t)
for some i : j .
therefore
m (x, x i , t) > 1-r
m ( y : x j : t ) > 1-r
and
now let
α = min { m (x i : x j : t) : ≤ i : j ≤ n }
Then α > o
Now
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m ( x : y : 3 t ) ≥ m ( x : x i : t ) ∗ m ( x i : x j : t)
∗ m ( xj : y : t )
≥ ( 1-r ) ∗ ( 1-r) ∗ α .
Taking t’ = 3t and
( 1-r ) ∗ ( 1-r ) ∗ α > 1-S : o < s < 1
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We have
m ( x : y : t’) > 1-S
for all x : y ∈ A
hence A is F- bounded.
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5.5.4 Theorem- Let (x : m : ∗ ) be a fuzzy metric space and T be
the topology induced by the fuzzy metric. Then for a sequence
{x n ) in X. x n → x if and only if
m (x n: x :t) → 1 as n → ∞
Proof- Fix t > o.
Suppose x n → x. then for o<r <l. then there exits n o ∈ N
such that
X n ∈ B (x : r : t)
for all n ≥ n o .
It follows that
m (x n : x : t) > 1-r
and hence 1- m (x n : x : t ) < r
therefore
m ( x n : x : t) → 1 as n → ∞.
Conversely : if for each t > o:
m (x n : x t ) → 1 as n → ∞
then for o < r <l.
there exits
n0 ∈ N
such that
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1-m (x n : x : t) <r
For all n ≥ n o .
It follows that
m (x n : x : t) > 1-r
thus
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for all n ≥ n 0 .
x n ∈ B (x : r : t)
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for all n ≥ n o . and hence x n → x.
5.5.5 Lemma- Every closed ball is closed set.
Proof Let y ∈ B (x : r : t). since x is first countable: there
exists a sequence {y n } in B [x : r : t] such that
y n → y.
therefore
m ( y n : y: t) → 1 for all t.
for a given ε > o
(x : y : t + ε ) ≥ m (x : y n : t) ∗
m (y n : y: ε )
Hence
m ( x : y : t + ε ) ≥ lim m (x : y n : t)
n→∞
∗ lim (y n : y : ε )
n→∞
≥ ( 1 – r ) ∗ 1 = 1-r
if m (x : y n : t) is bounded: {y n } has a subsequence: which we
again denote by { y n } for which
n→∞
In particular
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lim m (x: y n :t) exists.
For n ∈ N
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Take ∈ = 1
n
then m ( x : y : t + 1) ≥ 1-r
n
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hence
m (x : y : t) = lim m( x : y : t + 1/n) ≥ 1-r
n→∞
Thus
y ∈ B. [ x : r : t]
Therefore B [ x : r : t ] is a closed set.
5.5.6
Baire’s theorem- let X be a complete fuzzy metric space. Then
the intersection of a countable number of dense open sets in
dense.
Proof- Let X be the given complete fuzzy metric space. Let B o be a
non-empty open set.
Let
D 1 : D 2 : D 3 ………… be dense open sets in X since D 1 is dense
in X:
Bo ∩ D1 = Ø .
Let
X 1 ∈ B o ∩ D 1.
B o ∩ D 1 . is open. There exist
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Since
O < r 1 <1:
t 1 >o such that
B(x 1 : r 1 : t 1 ) ⊂ B o ∩ D 1 .
r 1' < r 1
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Choose
and t 1' = min {t 1 : 1}
such that
B [ x1 : r1’ : t1’ ] ⊂ Bo ∩ D1.
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Let
B 1 = B (x 1 : r 1 ’ : t i ’) .
Since D 2 in dense in X:
B1 ∩ D2 ≠ Ø
Let
X
2
∈ B1 ∩ D2
Since B 1 ∩ D 2 is open there exits o< r 2 < 1 and t 2 >o
2
Such that
B ( x 2 :r 2 : t 2 ) ⊂ B ∩ D 2.
Choose
r2' < r2
and t 2 ' = min { t 2 : 1 }
2
Such that B [ x 2 : r 2' , t 2' ] ⊂ B 1 ∩ D 2.
Let
B 2 = B (x 2 : (x 2 : r 2 ’: t 2 ' )
Similarly proceeding by induction we can find a x n ∈ B n -1∩
Dn.
Since
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B n-1 ∩ D n is open
There exits
and
t n >o
O < rn < 1
n
such that
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B (x n : r n: t n ) ⊂ B n-1 ∩ D n .
Choose
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r n ’ < r n : t n ’ = min { t n : 1 }
n
such that
B [ x n : r n ’ : t n ’ ] ⊂ B n-1 ∩ D n .
Let B n = B (x n : r n ’ : t n ’ ).
Now we claim that {x n } is a Cauchy sequence.
For a given t > o : ε > o :
Choose n o such that
1 < t and 1 < ∈
no
no
then for n ≥ n o ; m ≥ n.
m ( x n : x m : t) ≥ M (x n : x m : 1/n)
≥ 1- 1
n
≥ 1- 1
n
≥ 1–∈
Therefore { x n } is a Cauchy sequence since X is complete.
X n → x.
x k ∈ B [ x n : r n ’ : t n ’]
But
For all K ≥ n. and by the previous result B [ x n : r n ’ :t n ’ ] is a
closed set.
Hence
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x ∈ B [ x n: r n ’ :t n ’] ⊂ B n-1 ∩ Dn
for all n.
therefore
∞
B o ∩ (∩ n=1 Dn) ≠ Ø.
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∞
Hence ∩ n=1 Dn dense in X.
5.6 Fuzzy Compactness - in this Section we define fuzzy
compactness.
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5.6.1 Definition - A fuzzy metric space (x : m : ∗ ) is called
precompact if for each r; with o < r < 1 and each t > o
There is a finite subset of A of X: such that
X=
∪ a∈A B (a : r : t).
in this case we say that M is a precompact fuzzy metric on X.
A fuzzy metric space (x : m : ∗ ) is called compact if (x: T m )
is a compact topological space.
5.6.2 LEMMA - A fuzzy metric space is precompact if and only if
every sequence has a Cauchy sequence.
Proof-suppose that (x : m : ∗ ) is a precompact fuzzy metric space.
Let (x n ) n∈N be a sequence in X. for each m∈N there is a finite
subset A m of X such that
∪ a∈Am B (a : 1 : 1 ) .
m n
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X=
hence : for m=1; there exists an a 1 ∈ A 1 and a subsequence
(x 1(n) ) n∈N of (x n ) n∈N such that
x 1 (n) ∈ B (a 1 :1:1) for every n∈N.
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similarly there exits a n a 2 ∈ A 2 and a subsequence
(x 2 (n)) n∈N of (x 1(n) ) n ∈ N such that
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X 2 (n) ∈ B (a 2 :1 : 1 )
2 2
For every n∈N.
Following this process:
For m ∈ N ; m > 1 there is a n a m ∈ Am and a
Subsequence (x m(n) ) n∈N of
(x( m-1)(n) ) n∈N such that
X m(n) ∈ B. (a m : 1 : 1 ) for every n∈N.
m n
now consider the subsequence (x n(n) ) n∈N of (x n ) n∈N. given r
with o<r<1 and t > o there is an n o ∈N such that
(1-(1/n o )) ∗ (1-(1/n o ))>1-r
And
2 < t.
no
Then for every K:
m ≥ n o we have
m ( x k(k) : x m(m) ; t)
≥ m (x k(k) ; x m(m) ; 2/n o
≥ m (x k(k) : a no : 1/n o )
∗ m ( a no : x m(m) : 1/n o )
≥ (1-1/n o ∗ (1- 1/n o ) > 1-r
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Hence (x n(n) ) n∈N is a Cauchy sequence in (x : m : ∗ )
Conversely : suppose that (x : m : ∗ ) is a non- precompact
fuzzy metric space. Then there exist r ;
With o<r<1 and t >o.
Such that for each finite subset A of X;
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X = ∪ a∈A B (a : r: t )
Fix X 1 ∈ X there is
X 2 ∈ X\ B (x 1 : r : t) .
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Moreover : there is
X3 ∈ X \
∪2 k=1 (x k :r:t) .
Following this process we construct a sequence (x n ) n∈N
of distinct points in x: such that
Xn+1 ∉
∪n k=1 B (x k : r : t)
For every n∈N.
Therefore (x n ) n∈N has no Cauchy subsequence.
5.6.3 Theorem-
A Fuzzy metric space is compact if and only if it is
precompact and complete.
Proof suppose that (x:m:∗) is a complete fuzzy metric space for
each r ; with o< r <1 and each t >o the open cover
{ B(x:r:t): x∈X} of X: has a finite sub cover. Hence (x:m:∗) is
precompact.
On the other hand: every Cauchy sequence (x n ) n∈N in
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(x:m:∗) has a cluster point y∈x.
And all (x : m : ∗ ) be a fuzzy metric space. If a Cauchy
sequence clusters to a point x ∈ x : there the sequence
converges to x.
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Then (x n ) n∈N converges to y. thus (x:m:∗) is complete.
Conversely let (x n ) n∈N be a sequence in X and from the
lemma as discuses
before and the completeness of (x:m:∗) if
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follows that (x n ) n∈N has a cluster point.
Since by (x:T m ) is metrizable and every sequentially
compact metrizable space is compact we conclude that (x:m:∗)
is compact.
5.7 Conclusion-
in this work we firstly discussed fuzzy metric
space. We shown that the topology induced by any fuzzy metric
space is metrizable. Firstly we define a fuzzy metric space with the
help of a lot definitions and theorems. we have also discussed
Hausdroff
topology
on
a
fuzzy
compactness of a fuzzy metric space.
*******
metric
space
and
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last