Some remarks on the metrizability of some well known generalized metric-like structures
aa r X i v : . [ m a t h . GN ] M a y SOME REMARKS ON THE METRIZABILITY OF SOME WELLKNOWN GENERALIZED METRIC-LIKE STRUCTURES
SUMIT SOM , ADRIAN PETRUS¸EL , LAKSHMI KANTA DEY Abstract.
In [2, An, V.T., Tuyen, Q.L. and Dung, V.N., Stone-type theorem on b -metric spaces and applications, Topology Appl. 185-186 (2015), 50-64.], An et al. hadprovided a sufficient condition for b -metric spaces to be metrizable. However, theirproof of metrizability relied on an assumption that the distance function is continuousin one variable. In this short note, we improve upon this result in a more simplifiedway without considering any assumption on the distance function. Moreover, weprovide two shorter proofs of the metrizability of F -metric spaces recently introducedby Jleli and Samet in [8, Jleli, M. and Samet, B., On a new generalization of metricspaces, J. Fixed Point Theory Appl. (2018) 20:128]. Lastly, in this short note, we givean alternative proof of the metrizability of θ -metric spaces introduced by Khojastehet al. in [10, Khojasteh, F., Karapinar, E. and Radenovic, S., θ -metric space: AGeneralization, Math. Probl. Eng. Volume 2013, Article 504609, 7 pages]. Metrizability of b -metric spaces In the year 1993, Czerwik [5] had introduced the notion of a b -metric as a general-ization of a metric and further, in 1998, Czerwik [6] had modified this notion where thecoefficient 2 was replaced by coefficient K ≥ . Surprisingly in the year 1998, Aimaret al. [1] proved the metrizability of such spaces. In this sequel, intendant readers cansee [4] for some more results. In the year 2010, Khamsi and Hussain [9] defined theconcept of a b -metric under the name metric-type spaces where they had consideredthe coefficient to be K > . To avoid confusion, the metric-type in the sense of Khamsiand Hussain [9] will be called b -metric in this short note. Before going further, we liketo recall the definition of a b -metric space from [9] as follows: Definition 1.1. [9, Definition 6.] Let X be a non-empty set and K > . A distancefunction D : X × X → [0 , ∞ ) is said to be a b -metric on X if it satisfies the followingconditions:(i) D ( x, y ) = 0 ⇐⇒ x = y for all ( x, y ) ∈ X × X ;(ii) D ( x, y ) = D ( y, x ) for all ( x, y ) ∈ X × X ;(iii) D ( x, z ) ≤ K [ D ( x, y ) + D ( y, z )] for all x, y, z ∈ X .Then the triple ( X, D, K ) is called a b -metric space. If we take K = 1 , then X becomes a metric space. So b -metric spaces are more general than the standard metricspaces. Again, in the year 2015, An et al. [2] presented a proof for the metrizability of b -metric spaces with coefficient K >
0. However, they proved the metrizability result
Key words and phrases. b -metric space, F -metric space, θ -metric space, metrizability.2010 Mathematics Subject Classification . 54E35, 54H99. on an assumption that the distance function is continuous in one variable. We willstate first the main theorem and its corollary due to An et al.
Theorem 1.2. [2, Theorem 3.15.] Let ( X, D, K ) be a b -metric space. If D is contin-uous in one variable then every open cover of X has an open refinement which is bothlocally finite and σ -discrete. Corollary 1.3. [2, Corollary 3.17.] Let ( X, D, K ) be a b -metric space. If D iscontinuous in one variable then X is metrizable. One of the main motivation of this short note is to give a simple proof of the metriz-ability of b -metric spaces with coefficient K >
Theorem 1.4. [7, Page 137.] Let X be a topological space and F : X × X → [0 , ∞ ) be a distance function on X . If the distance function F satisfies (i) F ( x, y ) = 0 ⇐⇒ x = y for all ( x, y ) ∈ X × X ; (ii) F ( x, y ) = F ( y, x ) for all ( x, y ) ∈ X × X and one of the following conditions: (iii-A) Given a point a ∈ X and a number ε > , there exists φ ( a, ε ) > such that if F ( a, b ) < φ ( a, ε ) and F ( b, c ) < φ ( a, ε ) then F ( a, c ) < ε ; (iii-B) if a ∈ X and { a n } n ∈ N , { b n } n ∈ N are two sequences in X such that F ( a n , a ) → and F ( a n , b n ) → as n → ∞ then F ( b n , a ) → as n → ∞ ; (iii-C) for each point a ∈ X and positive number k, there is a positive number r suchthat if b ∈ X for which F ( a, b ) ≥ k, and c is any point then F ( a, c )+ F ( b, c ) ≥ r ,then the topological space X is metrizable. Niemytski and Wilson showed that the three conditions (iii-A), (iii-B), (iii-C) areequivalent. Any distance function which satisfies any one of the three conditions, iscalled locally regular. Now in the upcoming theorem we present a shorter proof of themetrizability of b -metric spaces. Theorem 1.5.
Let ( X, D, K ) , K > be a b -metric space. Then X is metrizable.Proof. Let (
X, D, K ) be a b -metric space. By the definition of a b -metric space, thedistance function D : X × X → [0 , ∞ ) on X satisfies the first two conditions of Niemytskiand Wilson’s metrization result, i.e,(i) D ( x, y ) = 0 ⇐⇒ x = y for all ( x, y ) ∈ X × X ;(ii) D ( x, y ) = D ( y, x ) for all ( x, y ) ∈ X × X .Now we prove the third condition, i.e., the ”locally regular” condition and for that, weprove the condition (iii-C) of Theorem 1.4. Let a ∈ X and t be a positive real number. OME REMARKS ON THE METRIZABILITY OF SOME METRIC-LIKE STRUCTURES 3
Assume that b ∈ X such that D ( a, b ) ≥ t. If c is any point in X then by the definitionof a b -metric space we have, D ( a, b ) ≤ K (cid:16) D ( a, c ) + D ( c, b ) (cid:17) = ⇒ (cid:16) D ( a, c ) + D ( c, b ) (cid:17) ≥ tK = r > . This shows that the distance function D : X × X → [0 , ∞ ) of a b -metric space satisfiesthe locally regular condition. Similarly conditions (iii-A) and (iii-B) of Theorem 1.4 areeasily satisfied by any b -metric. Consequently, by Niemytski and Wilson’s metrizationtheorem we can conclude that the b -metric space X is metrizable. (cid:3) Remark . Certainly the above metrizability result is superior, in some sense, to theones in [1, 2, 4].
Remark . From Theorem 1.5, we can conclude that if (
X, D, K ) , K > b -metric space, then there exists a metric d : X × X → [0 , ∞ ) on X such that X ismetrizable with respect to the metric d. Thus, the topological properties of b -metricspaces discussed in [9, Proposition 2, Proposition 3] are equivalent to those of thestandard metric spaces.2. Metrizability of F -metric spaces Recently, Jleli and Samet [8] proposed a new generalization of the usual notion ofmetric spaces. By means of a certain class of functions, the authors defined the notionof an F -metric space. Let us first recall the definition of such spaces. Let F denote theclass of functions f : (0 , ∞ ) → R which satisfy the following conditions:( F ) f is non-decreasing, i.e., 0 < s < t ⇒ f ( s ) ≤ f ( t ).( F ) For every sequence { t n } n ∈ N ⊆ (0 , + ∞ ), we havelim n → + ∞ t n = 0 ⇐⇒ lim n → + ∞ f ( t n ) = −∞ . The definition of an F -metric space has been introduced as follows. Definition 2.1. [8, Definition 2.1.] Let X be a non-empty set and D : X × X → [0 , ∞ )be a given mapping. Suppose there exists ( f, α ) ∈ F × [0 , ∞ ) such that:(D1) D ( x, y ) = 0 ⇐⇒ x = y for all ( x, y ) ∈ X × X .(D2) D ( x, y ) = D ( y, x ) for all ( x, y ) ∈ X × X .(D3) For every ( x, y ) ∈ X × X , for each N ∈ N , N ≥ u i ) Ni =1 ⊆ X with ( u , u N ) = ( x, y ), we have D ( x, y ) > ⇒ f ( D ( x, y )) ≤ f N − X i =1 D ( u i , u i +1 ) ! + α. Then D is said to be an F -metric on X and the pair ( X, D ) is said to be an F -metricspace. Hence, the class of all F -metric spaces contain the class of all metric spaces forany f ∈ F and α = 0 . The following definitions and propositions from [8] will beneeded.
S. SOM, A. PETRUS¸EL, L.K. DEY
Definition 2.2. [8, Definition 4.1.] Let (
X, D ) be an F -metric space. A subset C of X is said to be F -open if for every x ∈ C , there is some r > B ( x, r ) ⊂ C where B ( x, r ) = { y ∈ X : D ( y, x ) < r } . We say that a subset C of X is F -closed if X \ C is F -open. The family of all F -opensubsets of X is denoted by τ F . Definition 2.3. [8, Definition 4.3.] Let (
X, D ) be an F -metric space. Let { x n } n ∈ N be a sequence in X. We say that { x n } n ∈ N is F -convergent to x ∈ X if { x n } n ∈ N isconvergent to x ∈ X with respect to the topology τ F . Proposition 2.4. [8, Proposition 4.4.] Let ( X, D ) be an F -metric space. Then, forany nonempty subset A of X , we have x ∈ ¯ A, r > ⇒ B ( x, r ) ∩ A = φ. Proposition 2.5. [8, Proposition 4.5.] Let ( X, D ) be an F -metric space. Let { x n } n ∈ N be a sequence in X and x ∈ X. Then the following are equivalent: (i) { x n } n ∈ N is F -convergent to x. (ii) D ( x n , x ) → as n → ∞ . Proposition 2.6. [8, Proposition 4.6.] Let ( X, D ) be an F -metric space and { x n } n ∈ N be a sequence in X. Then ( x, y ) ∈ X × X, lim n →∞ D ( x n , x ) = lim n →∞ D ( x n , y ) = 0 = ⇒ x = y. Very recently Som et. all. [11] proved that this newly defined structure is metrizableby using the definition of metrizability. However, their proof is technical and a bitlengthy. In this short note, we give two alternative proofs of metrizability of thisstructure using Chittenden’s metrization theorem [3] and metrization theorem due toNiemytski and Wilson (discussed in Theorem 1.4). It may be noted that these proofsare very simple. Before proceeding to the metrizability result for F -metric spaces, werecall the metrization result due to Chittenden [3]. Theorem 2.7. [3] Let X be a topological space and F : X × X → [0 , ∞ ) be a distancefunction on X . If the distance function F satisfies the following conditions: (i) F ( x, y ) = 0 ⇐⇒ x = y for all ( x, y ) ∈ X × X ; (ii) F ( x, y ) = F ( y, x ) , for all ( x, y ) ∈ X × X ; (iii) (Uniformly regular) For every ε > and x, y, z ∈ X there exists φ ( ε ) > suchthat if F ( x, y ) < φ ( ε ) and F ( y, z ) < φ ( ε ) then F ( x, z ) < ε, then the topological space X is metrizable. Now in the upcoming theorem, we present, by two different approaches, two shortproofs of the metrizability of F -metric spaces. The first approach is by using Chitten-den’s metrization theorem, while the second one is by using Niemytski and Wilson’smetrization theorem. OME REMARKS ON THE METRIZABILITY OF SOME METRIC-LIKE STRUCTURES 5
Theorem 2.8.
Let ( X, D ) be an F -metric space with ( f, α ) ∈ F × [0 , ∞ ) . Then X ismetrizable.Proof. Approach I.
Let X be an F -metric space with ( f, α ) ∈ F × [0 , ∞ ). By the definition of an F -metric space, the distance function D : X × X → [0 , ∞ ) satisfies the first two conditionsof Chittenden’s metrization result, i.e,(i) D ( x, y ) = 0 ⇐⇒ x = y for all ( x, y ) ∈ X × X .(ii) D ( x, y ) = D ( y, x ) for all ( x, y ) ∈ X × X .Now we prove the third condition, i.e., the ”uniformly regular” condition. Let ε > x, y, z ∈ X. If x = z , then D ( x, z ) = 0. So in this case φ ( ε ) = c where c is anypositive real number will serve the purpose. Let x = z. Then D ( x, z ) > . So by thedefinition of an F -metric space we have, f ( D ( x, z )) ≤ f ( D ( x, y ) + D ( y, z )) + α. (2.1)By the F condition, for ( f ( ε ) − α ) ∈ R there exists δ > < t < δ = ⇒ f ( t ) < f ( ε ) − α. Let us choose φ ( ε ) = δ . If D ( x, y ) < δ and D ( y, z ) < δ then D ( x, y ) + D ( y, z ) < δ. So by the equation 2.1, we have f ( D ( x, z )) < f ( ε )= ⇒ D ( x, z ) < ε. This shows that the distance function D of an F -metric space satisfies the uniformlyregular condition. Consequently, by Chittenden’s metrization result we can concludethat the F -metric space X is metrizable. Approach II.
In this part we show that any F -metric D : X × X → [0 , ∞ ) satisfies condition (iii-B)of Theorem 1.4. Interesting reader can also check that, the F -metric D : X × X → [0 , ∞ ) satisfies condition (iii-A) of Theorem 1.4, by proceeding similarly as the proofof “uniformly regular” condition in Theorem 2.8 under approach I. Let a ∈ X and { a n } n ∈ N , { b n } n ∈ N are two sequences in X such that D ( a n , a ) → D ( a n , b n ) → n → ∞ . Let ε > . By F condition, for ( f ( ε ) − α ) ∈ R there exists δ > < t < δ = ⇒ f ( t ) < f ( ε ) − α. For δ > , there exists k , k ∈ N such that D ( a n , a ) < δ ∀ n ≥ k and D ( a n , b n ) < δ ∀ n ≥ k . Now if n ≥ max { k , k } and a = b n , then by the definition of an F -metric space, wehave f ( D ( a, b n )) ≤ f ( D ( a, a n ) + D ( a n , b n )) + α = ⇒ f ( D ( a, b n )) < f ( ε ) = ⇒ D ( a, b n ) < ε. This shows that D ( b n , a ) → n → ∞ . Thus, by the metrization criterion due to Niemytski and Wilson, we can concludethat the F -metric space X is metrizable. (cid:3) S. SOM, A. PETRUS¸EL, L.K. DEY
Remark . Let us show now that any F -metric D [8] satisfies condition (iii-C) ofTheorem 1.4. Let a ∈ X and k > . Also, assume that b ∈ X such that D ( a, b ) ≥ k. We have to find r > a ∈ X and k > c ∈ X, thecondition D ( a, c ) + D ( b, c ) ≥ r is satisfied. By F condition, for ( f ( k ) − α ) ∈ R there exists r >
X, D ) be an F -metric spacethen there exists a metric d : X × X → [0 , ∞ ) on X such that X is metrizable withrespect to the metric d. So, the topological properties of F -metric spaces discussed inProposition 2.4-2.6 are equivalent to those of the standard metric counterparts.3. Metrizability of θ -metric spaces In 2013, Khojasteh et all. [10] introduced the notion of a θ -metric space by using theconcept of an B -action on the set [0 , ∞ ) × [0 , ∞ ) . Before proceeding to the definitionof θ -metric space, we recall the definition of an B -action (see [10]), as follows. Definition 3.1. [10, Definition 4.] Let θ : [0 , ∞ ) × [0 , ∞ ) → [0 , ∞ ) be a continuousmapping with respect to each variable. Let Im ( θ ) = { θ ( s, t ) : s, t ≥ } . Then θ is calledan B -action if and only if the following conditions are satisfied :(i) θ (0 ,
0) = 0 and θ ( s, t ) = θ ( t, s ) for all s, t ≥ θ ( x, y ) < θ ( s, t ) if either x ≤ s, y < t or x < s, y ≤ t ;(iii) For each m ∈ Im ( θ ) and for each t ∈ [0 , m ] , there exists s ∈ [0 , m ] such that θ ( s, t ) = m ;(iv) θ ( s, ≤ s for all s > B -actions by Y . Now, we will recall(see [10]) the definition of a θ -metric space, as follows. Definition 3.2. [10, Definition 11.] Let X be a non-empty set. A distance function d : X × X → [0 , ∞ ) is said to be a θ -metric on X with respect to an B -action θ ∈ Y ifthe following conditions are satisfied:(i) d ( x, y ) = 0 ⇐⇒ x = y for all ( x, y ) ∈ X × X ; OME REMARKS ON THE METRIZABILITY OF SOME METRIC-LIKE STRUCTURES 7 (ii) d ( x, y ) = d ( y, x ) for all ( x, y ) ∈ X × X ;(iii) d ( x, z ) ≤ θ ( d ( x, y ) , d ( y, z )) for all x, y, z ∈ X .The triple ( X, d, θ ) is called a θ -metric space. If we take θ ( s, t ) = s + t, s, t ≥ θ -metric space reduce to metric space. In the same paper, Khojasteh et all. [10]also developed some topological structure induced by the θ -metric and concluded thatit is a metrizable topological space. However their proof of metrizability relies onthe prior knowledge of the uniformity of an uniform space X. In our paper, we provethe metrizability of θ -metric spaces by using the well-known Niemytski and Wilson’smetrization theorem. Theorem 3.3.
Let ( X, d, θ ) be a θ -metric space where θ is an B -action on [0 , ∞ ) × [0 , ∞ ) . Then X is metrizable.Proof. Throughout this proof, we will use the standard norm on the set [0 , ∞ ) × [0 , ∞ )as k ( x, y ) k = p x + y , x, y ≥ . First of all, we show that the B -action θ is continuousat the point (0 , . Suppose that { ( s n , t n ) } n ∈ N is a sequence in [0 , ∞ ) × [0 , ∞ ), suchthat ( s n , t n ) → (0 ,
0) as n → ∞ . This implies s n → t n → n → ∞ in thestandard norm in [0 , ∞ ) × [0 , ∞ ) . Now, as the B -action θ is continuous in both of thevariables, we get that θ ( s n , t n ) → θ (0 ,
0) = 0 as n → ∞ . This shows that the B -action θ is continuous at the point (0 , . Now we prove that X is metrizable. By the definitionof a θ -metric space, the distance function d : X × X → [0 , ∞ ) on X satisfies the firsttwo conditions of Niemytski and Wilson’s metrization result, i.e,(i) d ( x, y ) = 0 ⇐⇒ x = y for all ( x, y ) ∈ X × X ;(ii) d ( x, y ) = d ( y, x ) for all ( x, y ) ∈ X × X .Now we show that any θ -metric d : X × X → [0 , ∞ ) satisfies the condition (iii-B)and (iii-C) of Theorem 1.4. Interesting reader can also check that, the θ -metric d : X × X → [0 , ∞ ) also satisfies the condition (iii-A) of Theorem 1.4. Let a ∈ X and { a n } n ∈ N , { b n } n ∈ N are two sequences in X such that d ( a n , a ) → d ( a n , b n ) → n → ∞ . We show that d ( b n , a ) → n → ∞ . Now ( d ( a n , a ) , d ( a n , b n )) → (0 ,
0) as n → ∞ in the standard norm on [0 , ∞ ) × [0 , ∞ ) . As the B -action θ is continuous at thepoint (0 ,
0) so θ ( d ( a n , a ) , d ( a n , b n )) → θ (0 ,
0) = 0 as n → ∞ . Now from the definitionof θ -metric space we have, d ( a, b n ) ≤ θ ( d ( a n , a ) , d ( a n , b n ))= ⇒ d ( a, b n ) → n → ∞ . So the θ -metric d : X × X → [0 , ∞ ) satisfies the condition (iii-B) of Theorem 1.4. Nowwe check for condition (iii-C). Let a ∈ X and k > . Let b ∈ X such that d ( a, b ) ≥ k. As the B -action θ is continuous at the point (0 , k > δ > θ ( x, y ) < k whenever ( x, y ) ∈ B (cid:16) (0 , , δ (cid:17) \ (cid:16) [0 , ∞ ) × [0 , ∞ ) (cid:17) . S. SOM, A. PETRUS¸EL, L.K. DEY
Here B (cid:16) (0 , , δ (cid:17) denotes the open ball centered at (0 ,
0) and radius δ in the standardnorm, i.e, B (cid:16) (0 , , δ (cid:17) = n ( x, y ) ∈ R : k ( x, y ) k < δ o . Let c ∈ X. From the definitionof θ -metric space we have d ( a, b ) ≤ θ ( d ( a, c ) , d ( c, b ))= ⇒ θ ( d ( a, c ) , d ( c, b )) ≥ k = ⇒ ( d ( a, c ) , d ( c, b )) / ∈ B (cid:16) (0 , , δ (cid:17) \ (cid:16) [0 , ∞ ) × [0 , ∞ ) (cid:17) = ⇒ d ( a, c ) + d ( c, b ) ≥ δ as ( d ( a, c ) , d ( c, b )) ∈ [0 , ∞ ) × [0 , ∞ ) , so, ( d ( a, c ) , d ( c, b )) / ∈ B (cid:16) (0 , , δ (cid:17) . So either d ( a, c ) ≥ δ √ or d ( c, b ) ≥ δ √ . So we have d ( a, c ) + d ( c, b ) ≥ δ √ . Thisshows that the θ -metric on X satisfies condition (iii-C) of Theorem 1.4. Thus, by themetrization criterion due to Niemytski and Wilson, we can conclude that, the θ -metricspace X is metrizable. (cid:3) Open question.
Can an explicit metric d separately be constructed with respectto which b -metric spaces with coefficient K > θ -metric spaces are metrizable ? Acknowledgement.
The Research is funded by the Council of Scientific and IndustrialResearch (CSIR), Government of India under the Grant Number: / /EM R − II . We express our deep gratitude to Professor Pratulananda Das for his valuablesuggestions during the preparation of the draft. References [1] Aimar, H., Iaffei, B. and Nitti, L. On the Macias-Segovia metrization of quasi-metric spaces,
Rev.Un. Mat. Argentina 41 (2) (1998), 67-75. [2] An, V.T., Tuyen, Q.L. and Dung, V.N. Stone-type theorem on b -metric spaces and applications, Topology Appl. 185-186 (2015), 50-64. [3] Chittenden, E.W. On the equivalence of ´Ecart and voisinage,
Trans. Amer. Math. Soc. 18 (2)(1917), 161-166. [4] Cobza¸s, S. B -metric spaces, fixed points and Lipschitz functions, arXiv:1802.02722v3 [math.FA]25 Mar 2019.[5] Czerwik, S. Contraction mappings in b -metric spaces, Acta Math.Univ.Osstrav. 1 (1) (1993), 5-11. [6] Czerwik, S. Nonlinear set-valued contraction mappings in b -metric spaces, Atti Semin. Mat. Fis.Univ. Modena 46 (1998), 263-276. [7] Frink, A.H. Distance functions and the metrization problem,
Bull. Amer. Math. Soc. 43 (2) (1937),133-142. [8] Jleli, M. and Samet, B. On a new generalization of metric spaces,
J. Fixed Point Theory Appl.(2018), 20:128. [9] Khamsi, M.A. and Hussain, N. KKM mappings in metric type spaces,
Nonlinear Anal. 73 (9)(2010), 3123-3129. [10] Khojasteh, F., Karapinar, E. and Radenovic, S. θ -metric space: A generalization, Math. Probl.Eng. Volume 2013, Article ID 504609, 7 pages. [11] Som, S., Bera, A. and Dey, L.K. Some remarks on the metrizability of F -metric spaces, J. FixedPoint Theory Appl. (2020), 22:17. Sumit Som, Department of Mathematics, National Institute of Technology Durga-pur, India.
E-mail address : [email protected] OME REMARKS ON THE METRIZABILITY OF SOME METRIC-LIKE STRUCTURES 9 Adrian Petrus¸el, Department of Mathematics, Babes¸-Bolyai University Cluj-Napoca,Romania and Academy of Romanian Scientists Bucharest, Romania
E-mail address : [email protected] Lakshmi Kanta Dey, Department of Mathematics, National Institute of TechnologyDurgapur, India.
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