On Weak G -Completeness for Fuzzy Metric Spaces
aa r X i v : . [ m a t h . GN ] J u l ON WEAK G -COMPLETENESS FOR FUZZY METRIC SPACES SUGATA ADHYA AND A. DEB RAY
Abstract.
In this paper, we provide equivalent characterizations of weak G -completefuzzy metric spaces. Since such spaces are complete, we also characterize fuzzy metricspaces that have weak G -complete fuzzy metric completions. Moreover we establishanalogous results for classical metric spaces. AMS Subject Classification:
Keywords: (Fuzzy) metric space, weak G -complete.1. Introduction
Grabiec [9] introduced G -Cauchy sequence as a weaker form of Cauchy sequence inthe fuzzy context. He employed it to establish the celebrated Banach Contraction Prin-ciple for fuzzy metric spaces proposed by Kramosil and Michalek [16]. G -Cauchynesswas later adopted for fuzzy metrics in the context of Georege and Veeramani [7]. Theassociated notion of completeness, known as G -completeness, has been extensively usedto study fixed point theorems in fuzzy metric spaces. For details, one may consult[2, 6, 9, 17, 18]. G -Cauchyness, being weaker than the usual Cauchyness, leads to a stronger com-pleteness. Unfortunately, G -completeness is even more stronger than it is desired tobe, so that even a compact fuzzy metric space fails to be G -complete. To overcome thisdrawback Gregori, Mi˜nana and Sapena introduced the notion of weak G -completeness[11]. They adopted and studied this new notion both for metric and fuzzy metric set-tings. In particular, they generalized Grabiecs Banach Contraction Principle. Recently,in [10] the authors characterized weak G -completeness by means of nested sequencesof non-empty closed sets in the classical metric context.It is worth noting, at this stage, that the class of weak G -complete (fuzzy) metricspaces lie between the classes of compact and complete (fuzzy) metric spaces. Metricspaces lying in this intermediate class have been an active research area in classicalanalysis over the years. Atsuji spaces [3, 5] and cofinally complete space [5] are examplesof such metric spaces. Further, the spaces lying in this intermediate class demandconvergence of a class of sequences broader than the class of Cauchy sequences. Thus forAtsuji spaces we obtain the class of pseudo-Cauchy sequences [5] whereas for cofinallycomplete spaces we obtain the class of cofinally Cauchy sequences [4].The aim of this paper is to provide new characterizations for weak G -complete fuzzymetric spaces. Here we characterize weak G -complete fuzzy metric spaces by meansof the fuzzy metric analogue of pseudo-Cauchy and cofinally Cauchy sequences. Sincea weak G -complete fuzzy metric space is complete, in what follows, we characterizethose fuzzy metrics that have weak G -complete fuzzy metric completions. We also provide the classical metric analogue of our fuzzy metric characterizations for weak G -completeness. 2. Preliminaries
Throughout the paper the only notion of fuzzy metric we will be working on is theone due to George and Veeramani [7, 8] that goes as follows:
Definition 1.
A fuzzy metric space is an ordered triple (
X, M, ∗ ) where X is anonempty set, ∗ is a continuous t -norm and M : X × X × (0 , ∞ ) → [0 ,
1] is a mappingsuch that, for all x, y, z ∈ X and s, t > , the following conditions hold:a) M ( x, y, t ) > , b) M ( x, y, t ) = 1 ⇐⇒ x = y, c) M ( x, y, t ) = M ( y, x, t ) , d) M ( x, y, t ) ∗ M ( y, z, s ) ≤ M ( x, z, t + s ) , e) M ( x, y, . ) : (0 , ∞ ) → [0 ,
1] is continuous.In this case, ( M, ∗ ) is said to be a fuzzy metric on X. Lemma 1. [7] Given a fuzzy metric space (
X, M, ∗ ) , M ( x, y, · ) defines a nondecresingmap on (0 , ∞ ) , ∀ x, y ∈ X. It has been shown in [7] that every fuzzy metric ( M, ∗ ) on X generates a firstcountable topology τ M on X such that { B ( x, r, t ) : x ∈ X, r ∈ (0 , , t > } forms abase for τ M , where B ( x, r, t ) = { y ∈ X : M ( x, y, t ) > − r } , ∀ x ∈ X, r ∈ (0 , , t > . On the other hand, if (
X, d ) is a metric space and M d : X × X × (0 , ∞ ) → [0 , M d ( x, y, t ) = tt + d ( x,y ) , ∀ x, y ∈ X, t > , then ( X, M d , · ) defines a fuzzymetric space ( · being the usual multiplication of real numbers). Moreover, the topology τ ( d ) generated by the metric d coincides with τ M d . Theorem 2.1. [13] Given a fuzzy metric space (
X, M, ∗ ) , ( X, τ M ) is metrizable.Let ( X, M, ∗ ) be a fuzzy metric space and A ⊂ X. If M A = M | A × A × (0 , ∞ ) , then( A, M A , ∗ ) defines a fuzzy metric space called the fuzzy metric subspace of ( X, M, ∗ )on A [15]. Clearly τ M A = ( τ M ) A , ( τ M ) A being the subspace topology on A induced by τ M . ( X, M, ∗ ) is called precompact if for r ∈ (0 ,
1) and t > , there exists a finite subset A of X such that X = S x ∈ A B M ( x, r, t ) [13].Convergence of sequences in ( X, M, ∗ ) is defined with respect to τ M . Thus a sequence( x n ) in ( X, M, ∗ ) is said to be convergent to x ( resp. clusters), if it does so in ( X, τ M )[7]. Theorem 2.2. [7] A sequence ( x n ) in a fuzzy metric space ( X, M, ∗ ) converges to x ∈ X if and only if lim n →∞ M ( x n , x, t ) = 1 , ∀ t > . A sequence ( x n ) in a fuzzy metric space ( X, M, ∗ ) is called Cauchy if for ǫ ∈ (0 , , t > , there exists k ∈ N such that M ( x m , x n , t ) > − ǫ, ∀ m, n ≥ k. It is easy to seethat every convergent sequence in (
X, M, ∗ ) is Cauchy. As usual, ( X, M, ∗ ) is calledcomplete if every Cauchy sequence in it converges [7].The following proposition can be easily deduced. N WEAK G -COMPLETENESS FOR FUZZY METRIC SPACES 3 Proposition 2.1.
Let (
X, d ) be a metric space. Thena) A sequence ( x n ) is Cauchy in ( X, d ) if and only if ( x n ) is Cauchy in ( X, M d , · ) . b) A ⊂ X is complete as a metric subspace of ( X, d ) if and only if A is complete asa fuzzy metric subspace of ( X, M d , · ) . Given two fuzzy metric spaces (
X, M, ∗ ) and ( Y, N, ⋆ ) , a mapping f : X → Y iscalled an isometry if M ( x, y, t ) = N ( f ( x ) , f ( y ) , t ) , ∀ x, y ∈ X, t > . Moreover, if f isonto then ( X, M, ∗ ) and ( Y, N, ⋆ ) are called isometric [12].A fuzzy metric completion [12] of (
X, M, ∗ ) is a complete fuzzy metric space suchthat ( X, M, ∗ ) is isometric to a dense subspace of it.It is interesting to note that unlike metric spaces, a fuzzy metric space may notpossess a fuzzy metric completion [12]. Proposition 2.2. [12] Let (
X, d ) be a metric space having completion ( ˜ X, ˜ d ). Then,( ˜ X, M ˜ d , · ) is the unique (up to isometry) fuzzy metric completion of ( X, M d , · ) . A sequence ( x n ) in a fuzzy metric space ( X, M, ∗ ) is called G -Cauchy if lim n →∞ M ( x n ,x n +1 , t ) = 0 , ∀ t > x n ) in a metric space ( X, d )is called G -Cauchy if lim n →∞ d ( x n , x n +1 ) = 0 [19]. A (fuzzy) metric space in which every G -Cauchy sequence converges is called a G -complete (fuzzy) metric space ([9], [11]).Unfortunately, this new notion of completeness is so strong that even compactnesscannot imply G -completeness. To overcome this drawback, Gregori et. al. [11] intro-duced the following weaker version of completeness. Definition 2.
A (fuzzy) metric space in which every G -Cauchy sequence clusters iscalled a weak G -complete (fuzzy) metric space. Proposition 2.3. [11] Let (
X, d ) be a metric space and ( x n ) a sequence in X. Thena) ( x n ) is G -Cauchy in ( X, d ) if and only if ( x n ) is G -Cauchy in ( X, M d , · ) . b) ( X, d ) is weak G -complete if and only if ( X, M d , · ) is weak G -complete.A sequence ( x n ) in a fuzzy metric space ( X, M, ∗ ) is called fuzzy pseudo-Cauchyif for ǫ ∈ (0 , , t > k ∈ N there exist p, q ( > k ) ∈ N with p = q such that M ( x p , x q , t ) > − ǫ [1]. On the other hand, a sequence ( x n ) in a metric space ( X, d ) iscalled pseudo-Cauchy if for ǫ ∈ (0 ,
1) and k ∈ N there exist p, q ( > k ) ∈ N with p = q such that d ( x p , x q ) < ǫ [5]. Proposition 2.4. [1] Let (
X, d ) be a metric space and ( x n ) a sequence in X. Then( x n ) is pseudo-Cauchy in ( X, d ) if and only if ( x n ) is fuzzy pseudo-Cauchy in ( X, M d , · ) . Main Results
We begin with the characterizations of weak G -complete (fuzzy) metric spaces. Tomeet our requirement, we first extend the notion of cofinally Cauchy sequences in fuzzymetric setting.Howes [14] introduced the notion of cofinally Cauchy sequence by replacing thecondition of residuality with cofinality in the definition of Cauchy sequence. A sequence( x n ) in a metric space ( X, d ) is called cofinally Cauchy if for ǫ >
SUGATA ADHYA AND A. DEB RAY subset N ǫ of N such that d ( x p , x q ) < ǫ, ∀ p, q ∈ N ǫ . If every cofinally Cauchy sequencein (
X, d ) clusters, then (
X, d ) is called cofinally complete.
Definition 3.
A sequence ( x n ) in a fuzzy metric space ( X, M, ∗ ) is said to be fuzzycofinally Cauchy if for ǫ ∈ (0 ,
1) and t > N ǫ of N such that M ( x p , x q , t ) > − ǫ, ∀ p, q ∈ N ǫ . The following is an easy consequence:
Proposition 3.1.
Let (
X, d ) be a metric space and ( x n ) be a sequence in X. Then( x n ) is cofinally Cauchy in ( X, d ) if and only if ( x n ) is cofinally Cauchy in ( X, M d , · ) . Theorem 3.1.
Let (
X, M, ∗ ) be a fuzzy metric space. Then the following conditionsare equivalent:(a) ( X, M, ∗ ) is weak G -complete.(b) Each real-valued continuous function on ( X, τ M ) carries a G -Cauchy sequenceof ( X, M, ∗ ) to a cofinally Cauchy sequence of R (endowed with the usual metric).(c) Each real-valued continuous function on ( X, τ M ) carries a G -Cauchy sequence of( X, M, ∗ ) to a pseudo-Cauchy sequence of R (endowed with the usual metric). Proof. (a) = ⇒ (b): Let f : ( X, τ M ) → R be a continuous function. Choose a G -Cauchy sequence ( x n ) in ( X, M, ∗ ) . Since (
X, M, ∗ ) is weak G -complete, ( x n ) clustersin ( X, τ M ) . Recall that (
X, τ M ) is first countable. So there exists a subsequence ( x r n )of ( x n ) that converges in ( X, τ M ) . Since f is continuous, ( f ( x r n )) is Cauchy in R , andconsequently, ( f ( x n )) is cofinally Cauchy in R . (b) = ⇒ (c): Immediate.(c) = ⇒ (a): Let ( x n ) be a G -Cauchy sequence in ( X, M, ∗ ) . If ( x n ) has a constantsubsequence, then we are done. So, let us assume that ( x n ) has no constant subse-quence. We first prove that ( x n ) has a G -Cauchy subsequence ( x r n ) in ( X, M, ∗ ) ofdistinct terms.Set r = 1 and r n +1 = max { m ∈ N : x m = x r n +1 } , ∀ n ∈ N . Since ( x n ) has noconstant subsequence, r n +1 exists, ∀ n ∈ N . Thus ( x r n ) defines a sequence of distinctterms.Since ( x n ) is G -Cauchy, lim n →∞ M ( x n , x n +1 , t ) = 0 , ∀ t > n →∞ M ( x r n ,x r ( n +1) , t ) = 0 , ∀ t > . Thus ( x r n ) is a G -Cauchy subsequence of ( x n ) having distinctterms in ( X, M, ∗ ).If possible, let ( x r n ) does not cluster in ( X, τ M ). Then A = { x r n : n ∈ N } is aclosed and discrete subset of ( X, τ M ) . Define f : A → R by f ( x r n ) = 2 n , ∀ n ∈ N . Clearly f is continuous on ( X, τ M ). Since A is closed on ( X, τ M ), by Tietze’s extensiontheorem, f extends to a continuous function h on ( X, τ M ) . Note ( x r n ) is G -Cauchyin ( X, M, ∗ ) but ( h ( x r n )) is not pseudo-Cauchy in R , a contradiction. Consequently( x r n ) , and hence ( x n ) , clusters in ( X, τ M ) . Thus (
X, d ) is weak G -complete . (cid:3) In view of Proposition 2.3, the following corollary is obvious:
Corollary 3.1.
Let (
X, d ) be a metric space. Then the following conditions are equiv-alent:
N WEAK G -COMPLETENESS FOR FUZZY METRIC SPACES 5 (a) ( X, d ) is weak G -complete.(b) Each real-valued continuous function on ( X, d ) carries a G -Cauchy sequence of( X, d ) to a cofinally Cauchy sequence of R (endowed with the usual metric).(c) Each real-valued continuous function on ( X, d ) carries a G -Cauchy sequence of( X, d ) to a pseudo-Cauchy sequence of R (endowed with the usual metric). Theorem 3.2.
A closed subspace of a weak G -complete fuzzy metric space is weak G -complete. Proof.
Let A be a closed subset of a weak G -complete fuzzy metric space ( X, M, ∗ ) . Choose a G -Cauchy sequence ( x n ) in ( A, M A , ∗ ) . Then ( x n ) is G -Cauchy in ( X, M, ∗ )and hence has a cluster point c in ( X, τ M ). Since A is closed in ( X, τ M ) , so c ∈ A. Thus c becomes a cluster point of ( x n ) in ( A, M A , ∗ ) . Hence (
A, M A , ∗ ) is weak G -complete. (cid:3) In view of Proposition 2.3, the following corollary is obvious:
Corollary 3.2.
A closed subspace of a weak G -complete metric space is weak G -complete.Since a weak G -complete (fuzzy) metric space is complete, it is natural to ask underwhich conditions the completion of a (fuzzy) metric space is weak G -complete. In whatfollows, we give an answer to this. To establish the main result we require a lemmathat involves the notion of Cauchy-continuous map for fuzzy metric spaces.Recall that given two metric spaces ( X, d ) and (
Y, ρ ) , a mapping f : X → Y isCauchy-continuous if f takes every Cauchy sequence of X to a Cauchy sequence of Y. The natural extension of this notion for fuzzy metric spaces is as follows.
Definition 4.
Let (
X, M, ∗ ) and ( Y, N, ⋆ ) be two fuzzy metric spaces and A ⊂ X. A mapping f : A → Y is called fuzzy Cauchy-continuous if f takes every Cauchysequence of ( A, M A , ∗ ) to a Cauchy sequence of ( Y, N, ⋆ ) . Clearly if f : A → Y is fuzzy Cauchy-continuous, then f is continuous as a mappingfrom ( A, τ M A ) to ( Y, τ N ) . Lemma 2.
Let A be a non-empty subset of a fuzzy metric space ( X, M, ∗ ) having fuzzymetric completion ( ˜ X, ˜ M , ˜ ∗ ) and f : ( A, M A , ∗ ) → R be a fuzzy Cauchy-continuousmap. Then f extends to a fuzzy Cauchy-continuous map f : ( X, M, ∗ ) → R . (Here R is endowed with the standard fuzzy metric induced by the usual metric) Proof.
Let φ : ( X, M, ∗ ) → ( ˜ X, ˜ M , ˜ ∗ ) be an isometry such that φ ( X ) is dense in( ˜ X, τ ˜ M ) . Clearly φ is injective.Define g : φ ( A ) → R such that g = f φ − . Clearly g is fuzzy Cauchy-continuous on( φ ( A ) , ˜ M φ ( A ) , ˜ ∗ ) . We claim that g extends to a fuzzy Cauchy-continuous map g ∗ : φ ( A ) → R . Choose b ∈ φ ( A ) . Since ( ˜
X, τ ˜ M ) is first countable, there exists a sequence ( b n ) in φ ( A ) such that lim n →∞ b n = b in ( ˜ X, τ ˜ M ) . Since ( b n ) is Cauchy in φ ( A ) , so is ( g ( b n )) in R , and consequently, lim n →∞ g ( b n ) exists. SUGATA ADHYA AND A. DEB RAY
Define g ∗ : φ ( A ) → R by g ∗ ( c ) = lim n →∞ g ( c n ) , ∀ c ∈ φ ( A ) where ( c n ) is a sequencein φ ( A ) such that lim n →∞ c n = c in ( ˜ X, τ ˜ M ) . Existence of such a sequence ( c n ) is ensuredfrom the previous argument.Note that g ∗ is well-defined in the sense that for any two sequences ( r n ) and ( s n )in φ ( A ) converging to the same point d ∈ φ ( A ) we have lim n →∞ g ( r n ) = lim n →∞ g ( s n ) . Indeed( r , s , r , s , r , s , · · · ) is Cauchy in φ ( A ) = ⇒ ( g ( r ) , g ( s ) , g ( r ) , g ( s ) , g ( r ) , g ( s ) , · · · )is convergent in R , and consequently, lim n →∞ g ( r n ) = lim n →∞ g ( s n ) . We now show that g ∗ is fuzzy Cauchy-continuous.Let ( y n ) be a Cauchy sequence in φ ( A ) . Then for each n ∈ N , there is a sequence( x nk ) k in φ ( A ) such that y n = lim k →∞ x nk in ( ˜ X, τ ˜ M ) . Consequently g ∗ ( y n ) = lim k →∞ g ( x nk ) in R , ∀ n ∈ N . So for each n ∈ N \{ } , there exists p n ∈ N such that ˜ M ( y n , x nk , n ) > − n , and | g ∗ ( y n ) − g ( x nk ) | < n , ∀ k ≥ p n . Set z n = x np n , ∀ n ∈ N . Choose ǫ ∈ (0 , , t > . Find k ∈ N such that k < min { ǫ , t } . Then ˜ M ( y n , z n , t ) ≥ ˜ M ( y n , z n , n ) > − n > − ǫ , ∀ n ≥ k. Thus ∀ t > , ˜ M ( y n , z n , t ) → | g ∗ ( y n ) − g ( z n ) | → n → ∞ . Choose ǫ ∈ (0 , , t > . Since ˜ ∗ is continuous, there exists δ ∈ (0 ,
1) such that(1 − δ )˜ ∗ (1 − δ )˜ ∗ (1 − δ ) > − ǫ. Find q ∈ N such that ˜ M ( z n , y n , t ) > − δ and ˜ M ( y m , y n , t ) > − δ, ∀ m, n ≥ q. Then ˜ M ( z m , z n , t ) ≥ ˜ M ( z m , y m , t )˜ ∗ ˜ M ( y m , y n , t )˜ ∗ ˜ M ( z n , y n , t ) ≥ (1 − δ )˜ ∗ (1 − δ )˜ ∗ (1 − δ ) > − ǫ, ∀ m, n ≥ q. Thus ( z n ) is Cauchy in φ ( A ) = ⇒ ( g ( z n )) is Cauchy in R = ⇒ ( g ∗ ( y n )) is Cauchyin R . Consequently g ∗ is fuzzy Cauchy-continuous.Since g ∗ | φ ( A ) = g, so g extends to a fuzzy Cauchy-continuous map g ∗ : φ ( A ) → R . Then by Tietze extension theorem, g ∗ extends to a continuous function g : ˜ X → R . Since ˜ X is complete, g is Cauchy-continuous.Let us now define f : X → R by f = gφ. Then f is clearly an extension of f whichis fuzzy Cauchy-continuous. (cid:3) Theorem 3.3.
Let (
X, M, ∗ ) be a fuzzy metric space having a fuzzy metric completion( ˜ X, ˜ M , ˜ ∗ ). Then the following conditions are equivalent:(a) ( ˜ X, ˜ M , ˜ ∗ ) is weak G -complete.(b) Every complete subset (as a fuzzy metric subspace) of X is weak G -complete.(c) Given any fuzzy metric space ( Y, N, ⋆ ) and a fuzzy Cauchy-continuous map f : ( X, M, ∗ ) → ( Y, N, ⋆ ) , f takes a G -Cauchy sequence of ( X, M, ∗ ) to a confinallyCauchy sequence of ( Y, N, ⋆ ).(d) Given a pseudo Cauchy-continuous map f : ( X, M, ∗ ) → R where R is endowedwith the standard fuzzy metric induced by the usual metric, f takes a G -Cauchysequence of ( X, M, ∗ ) to a confinally Cauchy sequence of R .(e) Every G -Cauchy sequence in ( X, M, ∗ ) has a Cauchy subsequence. N WEAK G -COMPLETENESS FOR FUZZY METRIC SPACES 7 Proof. (a) = ⇒ (b): Let Y be a complete subset (as a fuzzy metric subspace) of X and φ : ( X, M, ∗ ) → ( ˜ X, ˜ M , ˜ ∗ ) be an isometry such that φ ( X ) is dense in ( ˜ X, τ ˜ M ) . Choose a G -Cauchy sequence ( y n ) in ( Y, M Y , ∗ ) . Then ( φ ( y n )) , being G -Cauchy in( ˜ X, ˜ M , ˜ ∗ ) , clusters to some point c in ( ˜ X, ˜ M , ˜ ∗ ) . So there is a subsequence ( φ ( y r n )) of( φ ( y n )) such that lim n →∞ φ ( y r n ) = c in ( ˜ X, ˜ M , ˜ ∗ ) , whence lim n →∞ y r n = φ − ( c ) in ( X, M, ∗ ) . Since Y is complete, so is φ ( Y ) (as a fuzzy metric subspace of ˜ X ) whence c ∈ φ ( Y ) . Thus φ − ( c ) ∈ Y. So Y is weak G -complete.(b) = ⇒ (c): Let ( Y, N, ⋆ ) be a fuzzy metric space and ( x n ) be a G -Cauchy sequencein ( X, M, ∗ ) . If possible, let ( x n ) has no Cauchy subsequence.Then A = { x n : n ∈ N } is complete as a fuzzy metric subspace and hence weak G -complete. Consequently, ( x n ) clusters in X, a contradiction. Thus there exists aCauchy subsequence ( x r n ) of ( x n ) in ( X, M, ∗ ) . We first show that, { f ( x r n ) : n ∈ N } is precompact as a fuzzy metric subspace of( Y, N, ⋆ ).Suppose otherwise. Then there exists ǫ ∈ (0 , , t > f ( x m rn ))of ( f ( x r n )) such that N ( f ( x m rp ) , f ( x m rq ) , t ) ≤ − ǫ , ∀ p = q · · · ( ∗ ) . However since( x r n ) is Cauchy in ( X, M, ∗ ), so is ( f ( x r n )) in ( Y, N, ⋆ ) , a contradiction to ( ∗ ) Hence { f ( x r n ) : n ∈ N } is precompact.Choose ǫ ∈ (0 , , t > . Since ∗ is continuous, there exists δ ∈ (0 ,
1) such that(1 − δ ) ∗ (1 − δ ) > − ǫ. Since { f ( x r n ) : n ∈ N } is precompact, there exists y ∈ Y and an infinite subset N of N such that f ( x n ) ∈ B N ( y, δ, t ) , ∀ n ∈ N . Thus ∀ p, q ∈ N , M ( f ( x p ) , f ( x q ) , t ) ≥ M ( f ( x p ) , y, t ) ∗ M ( f ( x q ) , y, t ) ≥ (1 − δ ) ∗ (1 − δ ) > − ǫ. So, ( f ( x n )) is cofinally Cauchy.(c) = ⇒ (d): Immediate.(d) = ⇒ (e): Let ( x n ) be a G -Cauchy sequence in ( X, M, ∗ ) . If ( x n ) has a constantsubsequence, then we are done. So let us assume ( x n ) has no constant subsequence.Then proceeding as in Theorem 3.1, we pass ( x n ) to a G -Cauchy subsequence havingdistinct terms.If possible, let ( x n ) has no Cauchy subsequence in ( X, M, ∗ ) . Let A = { x n : n ∈ N } and f : A → R be such that f ( x n ) = n, ∀ x n ∈ A. We first show that f is fuzzy Cauchy-continuous as a mapping from ( A, M A , ∗ ) to R . Let ( y m ) be a Cauchy sequence in ( A, M A , ∗ ) . If ( y m ) is eventually constant, then( f ( y m )) becomes eventually constant and hence Cauchy. So let us assume ( y m ) is noteventually constant.Choose r = 1 . Since ( y m ) is Cauchy without being eventually constant, so for each m ∈ N there exists r m +1 > r m such that y r ( m +1) = y r , y r , · · · , y r m . Thus ( y r m ) is aCauchy subsequence of ( y m ) having distinct terms. Without loss of generality, let uspass ( y m ) to ( y r m ) . Note that ∃ N ∈ N such that M ( y p , y q , ) > − , ∀ p, q ≥ N and for chosen N r , ∃ N r +1 ( > N r ) ∈ N such that M ( y p , y q , r +2 ) > − r +2 , ∀ p, q ≥ N r +1 . SUGATA ADHYA AND A. DEB RAY
Set A r = { n ∈ N : x n = y j for some j ≥ N r } , ∀ r ∈ N . Then each A r is an infiniteset of positive integers such that A r ⊃ A r +1 , ∀ r ∈ N . Clearly M ( x p , x q , r +1 ) > − r +1 , ∀ p, q ∈ A r . For each r ∈ N , choose n r ∈ A r such that n r < n r +1 . Then ( x n r ) is a Cauchysequence in ( X, M, ∗ ).In fact for chosen ǫ ∈ (0 , , t > r ∈ N such that r +1 < min { ǫ, t } . Then ∀ p, q ≥ r, we have n p , n q ∈ A r , and consequently, M ( x n p , x n q , t ) ≥ M ( x n p , x n q , r +1 ) > − r +1 > − ǫ, ∀ p, q ≥ r. Thus ( x n r ) is Cauchy.But it contradicts our assumption that ( x n ) has no Cauchy subsequence.Hence every Cauchy sequence in ( A, M A , ∗ ) must be eventually constant whence f is fuzzy Cauchy-continuous.Thus, in view of Lemma 2, f extends to a fuzzy Cauchy-continuous function from( X, M, ∗ ) to R . So due to the hypothesis, ( f ( x n )) must be cofinally Cauchy, a contra-diction.Hence the result follows.(e) = ⇒ (a): Let ( y n ) be a G -Cauchy sequence in ( ˜ X, ˜ M , ˜ ∗ ) and φ : ( X, M, ∗ ) → ( ˜ X, ˜ M , ˜ ∗ ) be an isometry such that φ ( X ) is dense in ( ˜ X, τ ˜ M ) . Then ∀ n ∈ N , ∃ x n ∈ X such that ˜ M ( φ ( x n ) , y n , n +1 ) > − n +1 . Choose, ǫ ∈ (0 , , t > . Since ˜ ∗ is continuous, there exists δ ∈ (0 ,
1) such that(1 − δ )˜ ∗ (1 − δ )˜ ∗ (1 − δ ) > − ǫ. Since ( y n ) is G -continuous, there exists a positive integer k > max (cid:8) t , δ (cid:9) such that˜ M ( y n , y n +1 , t ) > − δ, ∀ n ≥ k. Then ∀ n ≥ k, M ( x n , x n +1 , t ) = ˜ M ( φ ( x n ) , φ ( x n +1 ) , t ) ≥ ˜ M ( φ ( x n ) , y n , t )˜ ∗ ˜ M ( y n ,y n +1 , t )˜ ∗ ˜ M ( φ ( x n +1 ) , y n +1 , t ) ≥ ˜ M ( φ ( x n ) , y n , n +1 )˜ ∗ ˜ M ( y n , y n +1 , t )˜ ∗ ˜ M ( φ ( x n +1 ) , y n +1 , n +2 ) ≥ (1 − δ )˜ ∗ (1 − δ )˜ ∗ (1 − δ ) > − ǫ. Thus ( x n ) is G -Cauchy in ( X, M, ∗ ) . Due to the hypothesis, ( x n ) has a Cauchy subsequence ( x r n ) in ( X, M, ∗ ) , and hence( φ ( x r n )) is Cauchy in ( ˜ X, ˜ M , ˜ ∗ ) . Let lim n →∞ φ ( x r n ) = c in ( ˜ X, ˜ M , ˜ ∗ ) . Then for any choice of t > , ∃ p ∈ N such that t > p +1 , and hence ∀ n ≥ p, ˜ M ( φ ( x n ) , y n , t ) ≥ ˜ M ( φ ( x n ) , y n , n +1 ) > − n +1 . Since lim n →∞ (1 − n +1 ) = 1 , it follows that lim n →∞ ˜ M ( φ ( x n ) , y n , t ) = 1 , and hencelim n →∞ ˜ M ( φ ( x r n ) , y r n , t ) = 1 . Thus lim n →∞ h ˜ M ( φ ( x r n ) , c, t )˜ ∗ ˜ M ( φ ( x r n ) , y r n , t ) i = 1 . Since ˜ M ( y r n , c, t ) ≥ ˜ M ( φ ( x r n ) , c, t )˜ ∗ ˜ M ( φ ( x r n ) , y r n , t ) , ∀ n ∈ N , it follows thatlim n →∞ ˜ M ( y r n , c, t ) = 1 . Thus c is a cluster point of ( y n ) in ( ˜ X, ˜ M , ˜ ∗ ) . Hence ( ˜ X, ˜ M , ˜ ∗ ) is weak G -complete. (cid:3) In view of Propositions 2.1 − Corollary 3.3.
Let (
X, d ) be a metric space. Then the followings conditions areequivalent:(a) The completion of (
X, d ) is weak G -complete.(b) Every complete subset (as a metric subspace) of X is weak G -complete. N WEAK G -COMPLETENESS FOR FUZZY METRIC SPACES 9 (c) Given any metric space ( Y, ρ ) and a Cauchy-continuous map f : ( X, d ) → ( Y, ρ ) ,f takes a G -Cauchy sequence of ( X, d ) to a confinally Cauchy sequence of (
Y, ρ ).(d) Given a Cauchy-continuous map f : ( X, d ) → R where R is endowed with theusual metric, f takes a G -Cauchy sequence of ( X, d ) to a confinally Cauchy sequenceof R .(e) Every G -Cauchy sequence in ( X, d ) has a Cauchy subsequence.
Note 1.
In theorem 3.3, it is absolute necessary to assume the existence of fuzzymetric completion of (
X, M, ∗ ) . For otherwise, we may obtain a fuzzy metric spacethat does not have a fuzzy metric completion, however every G -Cauchy sequence in ithas a Cauchy subsequence. For instance, consider the following example:Let ( x n ) ∞ n =3 and ( y n ) ∞ n =3 be two disjoint sequences of distinct points and X = { x n : n ≥ }∪{ y n : n ≥ } . Define M : X × X × (0 , ∞ ) → R by M ( x n , x m , t ) = M ( y n , y m , t ) =1 − h { m,n } − { m,n } i and M ( x n , y m , t ) = M ( y m , x n , t ) = m + n , ∀ m, n ≥ . If ∗ denotes the continuous t -norm defined by a ∗ b = max { , a + b − } , ∀ a, b ∈ [0 ,
1] thenwe know from [12] thati) (
X, M, ∗ ) is a fuzzy metric space without having any fuzzy metric completion;ii) ( x n ) ∞ n =3 and ( y n ) ∞ n =3 are Cauchy sequences in ( X, M, ∗ ) . Since every subsequence of a Cauchy sequence is Cauchy, it is immediate to realizethat every G -sequence in X has a Cauchy subsequence, though ( X, M, ∗ ) has no fuzzymetric completion. Corollary 3.4.
Let X be a (fuzzy) metric space having a (fuzzy) metric completionwhich is weak G -complete. Then every G -Cauchy sequence in X is (fuzzy) cofinallyCauchy. Proof.
Immediate from the third conditions of Theorem 3.3 and Corollary 3.3 by con-sidering Y to be the space X itself and f to be the identity mapping on X . (cid:3) Note 2. ( P ni =1 1 i ) is a G -Cauchy sequence in R (endowed with the usual metric)which is not cofinally Cauchy. Hence R is not weak G -complete. Thus unlike cofinallycomplete metric spaces [4] a finite dimensional normed linear space may not be weak G -complete. References [1] Adhya, S., & Ray, A. D. Some Properties of Lebesgue Fuzzy Metric Spaces. arXiv preprintarXiv:2001.09840 (2020).[2] Alaca, C., Turkoglu, D., & Yildiz, C. Fixed Points in Intuitionistic Fuzzy Metric Spaces. Chaos,Solitons & Fractals, 29(5), 1073-1078 (2006).[3] Atsuji, M. Uniform Continuity of Continuous Functions of Metric Spaces. Pacific Journal of Math-ematics, 8(1), 11-16 (1958).[4] Beer, G. Between Compactness and Completeness. Topology and its Applications, 155(6), 503-514(2008).[5] Beer, G. More about Metric Spaces on which Continuous Functions are Uniformly Continuous.Bulletin of the Australian Mathematical Society, 33(3), 397-406 (1986).[6] Fang, J. X. On Fixed Point Theorems in Fuzzy Metric Spaces. Fuzzy Sets and Systems, 46(1),107-113 (1992).[7] George, A., & Veeramani, P. On Some Results in Fuzzy Metric Spaces. Fuzzy Sets and Systems,64(3), 395-399 (1994). [8] George, A., & Veeramani, P. On Some Results of Analysis for Fuzzy Metric Spaces. Fuzzy Setsand Systems, 90(3), 365-368 (1997).[9] Grabiec, M. Fixed Points in Fuzzy Metric Spaces. Fuzzy Sets and Systems, 27(3), 385-389 (1988).[10] Gregori, V., Mi˜nana, J. J., Roig, B., & Sapena, A. On Completeness in Metric Spaces and FixedPoint Theorems. Results in Mathematics, 73(4), 142 (2018).[11] Gregori, V., Mi˜nana, J. J., & Sapena, A. Banach Contraction Principles in Fuzzy Metric Spaces.Fixed Point Theory, 19(1), 235-248 (2018).[12] Gregori, V., & Romaguera, S. On Completion of Fuzzy Metric Spaces. Fuzzy Sets and Systems,130(3), 399-404 (2002).[13] Gregori, V., & Romaguera, S. Some Properties of Fuzzy Metric Spaces. Fuzzy Sets and Systems,115(3), 485-489 (2000).[14] Howes, N. On Completeness. Pacific Journal of Mathematics, 38(2), 431-440 (1971).[15] Ko˘cinac, L. D. Selection Properties in Fuzzy Metric Spaces. Filomat, 26(2), 305-312 (2012).[16] Kramosil, I., & Michalek, J. Fuzzy Metrics and Statistical Metric Spaces. Kybernetika, 11, 326-334(1975).[17] Mihet, D. A Banach Contraction Theorem in Fuzzy Metric Spaces. Fuzzy Sets and Systems,144(3), 431-439 (2004).[18] Mishra, S. N., Sharma, N., & Singh, S. L. Common Fixed Points of Maps on Fuzzy Metric Spaces.International Journal of Mathematics and Mathematical Sciences, 17(2), 253-258 (1994).[19] Tirado, P. On Compactness and G -Completeness in Fuzzy Metric Spaces. Iranian Journal of FuzzySystems, 9(4), 151-158 (2012). Department of Mathematics, The Bhawanipur Education Society College. 5, LalaLajpat Rai Sarani, Kolkata 700020, West Bengal, India.
E-mail address : [email protected] Department of Pure Mathematics, University of Calcutta. 35, Ballygunge CircularRoad, Kolkata 700019, West Bengal, India.
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