Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
la r CHAPTER (V) PROPERTIES OF FUZZY METRIC Es te 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 la r 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 te 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. Es 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 la r 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 te definition given in [131]. We define F-bounded ness on a fuzzy metric space and prove that compactness implies F-bounded ness. We prove Es 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. la r 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 te 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 Es 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 la r 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 te 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 Es 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). la r 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 te 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 Es 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 te and la r the theorem is proved. for all x : y ∈ X and t ∈ (o: ∞ ). Then (x : m : ∗ ) is a fuzzy metric space. Es 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 la r 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 te open ball B (x : r : t ) with centre x∈X and radius r; o < r < 1 : t > o as Es 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 la r 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 te 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. Es 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) → | la r as n → ∞ From this if follows that →1 t t + | Sn – x | te 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 Es 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. la r 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. ro te Let = m (x : y : t 0 ) > 1-r since r o > 1-r. we can find a s; Es 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 la r > 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. te 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; Es 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) la r ≥ m (x : z : 1 t) ∗ m (z : y : 1 t) 2 2 ≥ r1 ∗ r1 ≥ ro > r Which is a contradiction therefore te (X : M: ∗ ) is Hausdroff. 5.5.3 Theorem- Every compact subset A of a fuzzy metric space X is F-bounded. Es 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 la r 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 te We have m ( x : y : t’) > 1-S for all x : y ∈ A hence A is F- bounded. Es 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 la r 1-m (x n : x : t) <r For all n ≥ n o . It follows that m (x n : x : t) > 1-r thus te for all n ≥ n 0 . x n ∈ B (x : r : t) Es 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 la r lim m (x: y n :t) exists. For n ∈ N te Take ∈ = 1 n then m ( x : y : t + 1) ≥ 1-r n Es 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 la r 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 te Choose and t 1' = min {t 1 : 1} such that B [ x1 : r1’ : t1’ ] ⊂ Bo ∩ D1. Es 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 la r B n-1 ∩ D n is open There exits and t n >o O < rn < 1 n such that te B (x n : r n: t n ) ⊂ B n-1 ∩ D n . Choose Es 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 la r x ∈ B [ x n: r n ’ :t n ’] ⊂ B n-1 ∩ Dn for all n. therefore ∞ B o ∩ (∩ n=1 Dn) ≠ Ø. te ∞ Hence ∩ n=1 Dn dense in X. 5.6 Fuzzy Compactness - in this Section we define fuzzy compactness. Es 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 la r 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. te 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 Es 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 la r 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; te X = ∪ a∈A B (a : r: t ) Fix X 1 ∈ X there is X 2 ∈ X\ B (x 1 : r : t) . Es 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 la r (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. te 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 Es 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 at last