A Latif

C Zhu

D Turkoglu

LG Huang

M Khani

S Shukla

Shou Lin

Xun Ge

Z Mustafa

Journal of Inequalities and Applications

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Ge and Lin Journal of Inequalities and Applications  (2016) 2016:16 DOI 10.1186/s13660-016-0962-yRESEARCH Open AccessBalls in generalizations of metric spacesXun Ge1* and Shou Lin2*Correspondence:gexun@suda.edu.cn1School of Mathematical Sciences,Soochow University, Suzhou,215006, P.R. ChinaFull list of author information isavailable at the end of the articleAbstractThis paper discusses balls in partial b-metric spaces and cone metric spaces,respectively. Let (X ,pb) be a partial b-metric space in the sense of Mustafa et al. For thefamily  of all pb-open balls in (X ,pb), this paper proves that there are x, y ∈ B ∈ such that B′  B for all B′ ∈ , where B and B′ are with centers x and y, respectively.This result shows that  is not a base of any topology on X , which shows that aproposition and a claim on partial b-metric spaces are not true. By some relationsamong , <, and ≤ in cone metric spaces, this paper also constructs a cone metricspace (X ,d) and shows that {y ∈ X : d(x, y) ε} = {y ∈ X : d(x, y)≤ ε} in general, whichcorrects an error on cone metric spaces. However, it must be emphasized that thesecorrections do not aﬀect the rest of the results in the relevant papers.MSC: 54A10; 54E35Keywords: ball; partial b-metric space; cone metric space1 IntroductionPartial b-metric spaces and cone metric spaces are important generalizations of metricspaces, which were introduced and investigated by Shukla in [] and Huang-Zhang in [],respectively.Recently, Mustafa et al. introduced a new concept of partial b-metric by modifying par-tial b-metric in the sense of [] in order to guarantee that each partial b-metric pb caninduce a b-metric ([]). Furthermore, they proved the following proposition.Proposition . ([]) Let (X,pb) be a partial b-metric space (in the sense of []). For eachx ∈ X and ε > , the pb-open ball with center x and radius ε isBpb (x, ε) ={y ∈ X : pb(x, y) < pb(x,x) + ε}.Then for each Bpb (x, ε) and each y ∈ Bpb (x, ε), there is δ >  such that Bpb (y, δ)⊆ Bpb (x, ε).Thus, from Proposition ., the following claim arose naturally ([]).Claim . ([]) Let (X,pb) be a partial b-metric space (in the sense of []). Put  ={Bpb (x, ε) : x ∈ X and ε > }, i.e.,  is the family of all pb-open balls. Then  is a base ofsome topology on X.It is also worthy noting that Proposition . and Claim . were cited in [].© 2016 Ge and Lin. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in anymedium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, andindicate if changes were made.Ge and Lin Journal of Inequalities and Applications  (2016) 2016:16 Page 2 of 7For balls in a conemetric space (X,d), Turkoglu andAbuloha gave the following equality(see [], Proposition ), where x ∈ X and ε 	 θ :Equality . {y ∈ X : d(x, y) ε} = {y ∈ X : d(x, y)≤ ε}.In this paper, we discuss Proposition ., Claim ., and Equality .. For Proposition .and Claim ., we construct a partial b-metric space (X,pb) in the sense of [], and showthat there are a pb-open ball Bpb (x, ε) and y ∈ Bpb (x, ε) such that Bpb (y, δ) Bpb (x, ε) for allδ > , and hence  is not a base of any topology on X, which shows that Proposition .(including its proof ) and Claim . are not true. For Equality ., we establish some rela-tions between balls and their closures in conemetric spaces by, <, and≤, andwe give anexample to show that Equality . is not true. However, it must be emphasized that thesecorrections do not aﬀect the rest of the results in [, ].Throughout this paper, N, R, and R+ denote the set of all natural numbers, the set of allreal numbers and the set of all nonnegative real numbers, respectively. For a subset A ofa space X, F denotes the closure of F in X. For undeﬁned notations and terminology, onecan refer to [, ].2 Results and discussionWe give the main results of this paper by the following two subsections.2.1 pb-Open balls in partial b-metric spacesThe following partial b-metric spaces were introduced by Shukla in [].Deﬁnition . [] Let X be a non-empty set. A mapping pb : X × X −→ R+ is called apartial b-metric with coeﬃcient s ≥  and (X,pb) is called a partial b-metric space withcoeﬃcient s≥  if the following are satisﬁed for all x, y, z ∈ X:() x = y⇐⇒ pb(x,x) = pb(y, y) = pb(x, y).() pb(x, y) = pb(y,x).() pb(x,x)≤ pb(x, y).() pb(x, y)≤ s(pb(x, z) + pb(z, y)) – pb(z, z).Remark . If s =  in Deﬁnition ., then (X,pb) is a partial metric space, which was in-troduced byMatthews (for example, see []). Further, put dpb : X×X −→R+ by dpb (x, y) =pb(x, y) – pb(x,x) – pb(y, y) for all x, y ∈ X, then dpb is a metric on X and (X,dp) is a metricspace.However, if s > , then we cannot guarantee that each partial b-metric can induce a b-metric by the method in Remark .. So Mustafa et al. gave the following partial b-metricpb bymodifying Deﬁnition .() and proved that the pb induces a b-metric by themethodin Remark ..Deﬁnition . ([]) Let X be a non-empty set. A mapping pb : X × X −→ R+ is called apartial b-metric with coeﬃcient s ≥  and (X,pb) is called a partial b-metric space withcoeﬃcient s≥  if the following are satisﬁed for all x, y, z ∈ X:() x = y⇐⇒ pb(x,x) = pb(y, y) = pb(x, y).() pb(x, y) = pb(y,x).Ge and Lin Journal of Inequalities and Applications  (2016) 2016:16 Page 3 of 7() pb(x,x)≤ pb(x, y).() pb(x, y)≤ s(pb(x, z) + pb(z, y) – pb(z, z)) + –s (pb(x,x) + pb(y, y)).Remark . If x, y, z satisfy Deﬁnition .(), (), () and are diﬀerent from each other,then it is easy to check that x, y, z Deﬁnition .() holds.As a known fact, Proposition . and Claim . are not true if (X,pb) is a partial b-metricspace in the sense ofDeﬁnition . ([]). So it is important to checkwhether Proposition .and Claim . are true if (X,pb) is a partial b-metric space in the sense of Deﬁnition ..The following example shows that the result of the check is negative, which comes from[]. In the following, all partial b-metric spaces are in the sense of Deﬁnition ..Example . Let X = {u, v,w} and put pb : X ×X −→R+ as follows:(i) pb(u,u) = pb(w,w) =  and pb(v, v) = ..(ii) pb(u,w) = pb(w,u) = ..(iii) pb(v,w) = pb(w, v) = .(iv) pb(u, v) = pb(v,u) = .Let Bpb (u, ε) be described in Proposition. Then the following hold:() pb is a partial b-metric with coeﬃcient s = .() w ∈ Bpb (u, ) and for any ε > , Bpb (w, ε) Bpb (u, ).Proof () It is not diﬃcult to check that pb satisﬁes Deﬁnition .(), (), (). In order tocheck that pb satisﬁes Deﬁnition .(), we only need to consider the following three casesby Remark ..() x = u, y = v, z = w:pb(u, v) = ,(pb(u,w) + pb(w, v) – pb(w,w))+  – (pb(u,u) + pb(v, v))= .So pb(u, v)≤ (pb(u,w) + pb(w, v) – pb(w,w)) + – (pb(u,u) + pb(v, v)).() x = u, y = w, z = v:pb(u,w) = .,(pb(u, v) + pb(v,w) – pb(v, v))+  – (pb(u,u) + pb(w,w))= ..So pb(u,w)≤ (pb(u, v) + pb(v,w) – pb(v, v)) + – (pb(u,u) + pb(w,w)).() x = v, y = w, z = u:pb(v,w) = ,(pb(v,u) + pb(u,w) – pb(u,u))+  – (pb(v, v) + pb(w,w))= .So pb(v,w)≤ (pb(v,u) + pb(u,w) – pb(u,u)) + – (pb(v, v) + pb(w,w)).Thus, pb is a partial b-metric with coeﬃcient s = .Ge and Lin Journal of Inequalities and Applications  (2016) 2016:16 Page 4 of 7() Since pb(u,w) = . <  +  = pb(u,u) + , w ∈ Bpb (u, ). In addition, for any ε > ,pb(w, v) =  <  + ε = pb(w,w) + ε, so v ∈ Bpb (w, ε). On the other hand, pb(u, v) =  ≮  = +  = pb(u,u) + , so v /∈ Bpb (u, ). This shows that Bpb (w, ε) Bpb (u, ). Remark . Example . shows that Proposition . and Claim . are not true if (X,pb)is a partial b-metric space.However, we have the following.Proposition . ([]) Let (X,pb) be a partial b-metric space and  be described inClaim .. Then  is a subbase for some topology on X.We denote the topology by Tpb .It is well known that the space (X,Tpb ) is T but does not need to be T ([]). The fol-lowing proposition give a suﬃcient and necessary such that (X,Tpb ) is a T-space.Proposition . Let (X,pb) be a partial b-metric space in the sense of Deﬁnition .. Thenthe following are equivalent:() (X,Tpb ) is a T-space.() pb(x, y) > max{pb(x,x),pb(y, y)} for each pair of distinct points x, y ∈ X .Proof () ⇒ (): Let (X,Tpb ) be a T-space. If x, y ∈ X and x = y, then there is a neigh-borhood U of x such that y /∈ U . Since  is a subbase of (X,Tpb ) from Proposition .,there are ε, ε, . . . , εk >  such that y /∈ ⋂{Bpb (x, εi) : i = , , . . . ,k}, and hence there isi ∈ {, , . . . ,k} such that y /∈ Bpb (x, εi ). So pb(x, y) ≥ pb(x,x) + εi > pb(x,x). In the sameway, pb(x, y) > pb(y, y). So pb(x, y) > max{pb(x,x),pb(y, y)}.() ⇒ (): Let x, y ∈ X and x = y. If pb(x, y) > max{pb(x,x),pb(y, y)}. Then pb(x, y) >pb(x,x). Put ε = pb(x, y) –pb(x,x) > , then pb(x, y) = pb(x,x) + ε, and so y /∈ Bpb (X, ε). In thesame way, there is ε′ >  such that x /∈ Bpb (y, ε′). Consequently, (X,Tpb ) is a T-space. 2.2 Balls in cone metric spacesDeﬁnition . ([, ]) Let E be a real Banach space. A subset P of E is called a cone of Eand (E,P) is called a cone space if the following are satisﬁed: where θ is zero element in E.() P is closed, P = ∅, and P = {θ}.() a,b ∈R+ and α,β ∈ P ⇒ aα + bβ ∈ P.() α, –α ∈ P ⇒ α = θ .Deﬁnition . ([, ]) Let (E,P) be a cone space. Some partial orderings ≤, <, and  onE with respect to P are deﬁned as follows, respectively, where P◦ denotes the interior of P.Let α,β ∈ E.() α ≤ β if β – α ∈ P.() α < β if α ≤ β and α = β .() α  β if β – α ∈ P◦.Remark . Let (E,P) be a cone space. For the sake of conveniences, we also use nota-tions ‘≥’, ‘>’, and ‘	′’ on E with respect to P. The meanings of these notations are clear andthe following hold:() α ≥ θ if and only if α ∈ P.Ge and Lin Journal of Inequalities and Applications  (2016) 2016:16 Page 5 of 7() α 	 θ if and only if α ∈ P◦.() α – β 	 θ if and only if α 	 β .() α – β ≥ θ if and only if α ≥ β .() α 	 β ⇒ α > β ⇒ α ≥ β .In addition, in order to guarantee the existence of elements ε 	 θ , we always assumethat the cone P has non-empty interior ([]).Deﬁnition . ([, ]) LetX be a non-empty set and let (E,P) be a cone space. Amappingd : X ×X −→ E is called a cone metric on X, and (X,d) is called a cone metric space if thefollowing are satisﬁed:() d(x, y)≥ θ for all x, y ∈ X and d(x, y) = θ if and only if x = y.() d(x, y) = d(y,x) for all x, y ∈ X .() d(x, y)≤ d(x, z) + d(z, y) for all x, y, z ∈ X .Notation . Let (X,d) be a cone metric space, x ∈ X, and ε 	 θ . In this section, we usethe following notations for balls in (X,d):() B(x, ε) = {y ∈ X : d(x, y) ε}.() B(x, ε) = {y ∈ X : d(x, y) < ε}.() B(x, ε) = {y ∈ X : d(x, y)≤ ε}.Proposition . ([]) Let (X,d) be a cone metric space. Put B = {B(x, ε) : x ∈ X andε 	 θ}. Then B is a base for some topology T on X.In this section, just as the investigation in [], we always suppose that each cone metricspace is a topological space described in Proposition ..Proposition . Let (X,d) be a cone metric space. For each x ∈ X and each ε 	 θ , thefollowing hold:() B(x, ε)⊆ B(x, ε)⊆ B(x, ε).() B(x, ε)⊆ B(x, ε)⊆ B(x, ε).() B(x, ε) = B(x, ε).Proof () It holds by Remark .().() It holds by the above item ().() Let y ∈ B(x, ε). Then, whenever η 	 θ ,B(y,η)∩B(x, ε) = ∅. Pick z ∈ B(y,η)∩B(x, ε).Then d(x, z)≤ ε and d(z, y) η. It follows that d(x, y)≤ d(x, z)+d(z, y) ε+η. Let η → θ .Then d(x, y)≤ ε. So y ∈ B(x, ε). This proves that B(x, ε)⊆ B(x, ε). On the other hand, itis clear that B(x, ε)⊇ B(x, ε). So B(x, ε) = B(x, ε). The following example shows that any ‘⊆’ in Proposition .(), () cannot be replacedby ‘=’.Example . Let the cone space (E,P) be deﬁned as in [], Example , i.e., E =R = {(r, s) :r, s ∈ R} is the Euclidean plane and P = {(r, s) ∈ E : r, s ≥ }. Let X = {x, y, z}. Deﬁne d :X ×X −→ E as follows: d(x,x) = d(y, y) = d(z, z) = (, ), d(x, y) = d(y,x) = d(y, z) = d(z, y) =(, ), and d(x, z) = d(z,x) = (, ). It is not diﬃcult to check that (X,d) is a cone metricspace. Let ε = (, )	 θ .Ge and Lin Journal of Inequalities and Applications  (2016) 2016:16 Page 6 of 7() Note that d(x, y) = (, ) = ε. By Remark ., d(x, y)≤ ε, d(x, y)≮ ε, and d(x, y)  ε.So y /∈ B(x, ε), y /∈ B(x, ε), and y ∈ B(x, ε). Also, ε – d(x, z) = (, ) – (, ) = (, ) ∈P – ({θ} ∪ P◦). By Remark ., ε – d(x, z) > θ , and ε – d(x, z) 	 θ , hence d(x, z) < εand d(x, z)  ε. So z /∈ B(x, ε), z ∈ B(x, ε), and y ∈ B(x, ε). It follows that B(x, ε) ={x}, B(x, ε) = {x, z}, and B(x, ε) = {x, y, z}. So any ‘⊆’ in Proposition .() cannotbe replaced by ‘=’.() Note that (X,d) is Hausdorﬀ ([]). In fact, each cone metric space is metrizable ([]).So B(x, ε) = B(x, ε), B(x, ε) = B(x, ε), and B(x, ε) = B(x, ε). By the above item (),any ‘⊆’ in Proposition .() cannot be replaced by ‘=’.Remark .() By Example ., Equality . is not true. Indeed, in Example ., B(x, ε) = {x} andB(x, ε) = {x, y, z}. So B(x, ε) = B(x, ε).() Let (X,d) be a cone metric space. In [], the authors showed that B(x, ε) and B(x, ε)are sequentially closed in (X,d) ([], Proposition ). In fact, (X,d) is metrizable, andhence (X,d) is a sequential space, i.e., closed and sequentially closed in (X,d) areequivalent. On the other hand, indeed, the closure B(x, ε) of B(x, ε) is closed andB(x, ε) is closed by Proposition .().3 ConclusionsThis paper discusses balls in partial b-metric spaces and cone metric spaces, respectively.Conclusion . Let (X,pb) be a partial b-metric space in the sense of []. For the familyof all pb-open balls in (X,pb), this paper proves that there are x, y ∈ B ∈  such that B′  Bfor all B′ ∈ , where B and B′ are with centers x and y, respectively. This result shows that is not a base of any topology on X, which shows that [], Proposition  and the claimfollowing [], Proposition , are not true.Conclusion . Let (X,d) be a cone metric space. By , <, and ≤ in (X,d), this paperestablishes some relations among {y ∈ X : d(x, y)  ε}, {y ∈ X : d(x, y) < ε}, and {y ∈ X :d(x, y) ≤ ε}. Furthermore, this paper also constructs a cone metric space (X,d) such that{y ∈ X : d(x, y) ε} = {y ∈ X : d(x, y) ≤ ε} for some x, y ∈ X and ε 	 , which shows thatthe equality in [], Proposition , is not true.However, it must be emphasized that these corrections in this paper do not aﬀect therest of the results in the relevant papers.Competing interestsThe authors declare that they have no competing interests.Authors’ contributionsAll authors contributed equally and signiﬁcantly in writing this paper. All authors read and approved the ﬁnal manuscript.Author details1School of Mathematical Sciences, Soochow University, Suzhou, 215006, P.R. China. 2Department of Mathematics,Ningde Normal University, Fujian, 352100, P.R. China.AcknowledgementsThe authors wish to thank the reviewers for reviewing this paper and oﬀering many valuable comments and suggestions.This project is supported by the National Natural Science Foundation of China (no. 11471153, 11301367, 61472469,11461005), the Doctoral Fund of Ministry of Education of China (no. 20123201120001), the China Postdoctoral ScienceGe and Lin Journal of Inequalities and Applications  (2016) 2016:16 Page 7 of 7Foundation (no. 2013M541710, 2014T70537), the Jiangsu Province Natural Science Foundation (no. BK20140583), theJiangsu Province Postdoctoral Science Foundation (no. 1302156C) and the Priority Academic Program Development ofJiangsu Higher Education Institutions.Received: 26 September 2015 Accepted: 7 January 2016References1. Shukla, S: Partial b-metric spaces and ﬁxed point theorems. Mediterr. J. Math. 11, 703-711 (2014)2. Huang, LG, Zhang, X: Cone metric spaces and ﬁxed point theorems of contractive mappings. J. Math. Anal. Appl. 332,1468-1476 (2007)3. Mustafa, Z, Roshan, JR, Parvaneh, V, Kadelburg, Z: Some common ﬁxed point results in ordered partial b-metric spaces.J. Inequal. Appl. 2013, 562 (2013)4. Latif, A, Roshan, JR, Parvaneh, V, Hussain, N: Fixed point results via α-admissible mappings and cyclic contractivemappings in partial b-metric spaces. J. Inequal. Appl. 2014, 345 (2014)5. Turkoglu, D, Abuloha, M: Cone metric spaces and ﬁxed point theorems in diametrically contractive mappings. ActaMath. Sin. 26, 489-496 (2010)6. Ge, X, Lin, S: A note on partial b-metric spaces. Mediterr. J. Math. (2016, in press). doi:10.1007/s00009-015-0548-97. Zhu, C, Xu, W, Chen, C, Zhang, X: Common ﬁxed point theorems for generalized expansive mappings in partialb-metric spaces and an application. J. Inequal. Appl. 2014, 475 (2014)8. Khani, M, Pourmahdian, M: On the metrizability of cone metric space. Topol. Appl. 158, 190-193 (2011)

Balls in generalizations of metric spaces

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Balls in generalizations of metric spaces

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