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Balls in generalizations of metric spaces
Journal of Inequalities and Applications volume 2016, Article number: 16 (2016)
Abstract
This paper discusses balls in partial bmetric spaces and cone metric spaces, respectively. Let \((X,p_{b})\) be a partial bmetric space in the sense of Mustafa et al. For the family △ of all \(p_{b}\)open balls in \((X,p_{b})\), this paper proves that there are \(x,y\in B\in\triangle\) such that \(B'\nsubseteq B\) for all \(B'\in\triangle\), 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 a proposition and a claim on partial bmetric spaces are not true. By some relations among ≪, <, and ≤ in cone metric spaces, this paper also constructs a cone metric space \((X,d)\) and shows that \(\overline{\{y\in X:d(x,y)\ll\varepsilon\}}\ne\{y\in X:d(x,y)\le\varepsilon\}\) in general, which corrects an error on cone metric spaces. However, it must be emphasized that these corrections do not affect the rest of the results in the relevant papers.
Introduction
Partial bmetric spaces and cone metric spaces are important generalizations of metric spaces, which were introduced and investigated by Shukla in [1] and HuangZhang in [2], respectively.
Recently, Mustafa et al. introduced a new concept of partial bmetric by modifying partial bmetric in the sense of [1] in order to guarantee that each partial bmetric \(p_{b}\) can induce a bmetric ([3]). Furthermore, they proved the following proposition.
Proposition 1.1
([3])
Let \((X,p_{b})\) be a partial bmetric space (in the sense of [3]). For each \(x\in X\) and \(\varepsilon>0\), the \(p_{b}\)open ball with center x and radius ε is
Then for each \(B_{p_{b}}(x,\varepsilon)\) and each \(y\in B_{p_{b}}(x,\varepsilon)\), there is \(\delta>0\) such that \(B_{p_{b}}(y,\delta)\subseteq B_{p_{b}}(x,\varepsilon)\).
Thus, from Proposition 1.1, the following claim arose naturally ([3]).
Claim 1.2
([3])
Let \((X,p_{b})\) be a partial bmetric space (in the sense of [3]). Put \(\triangle=\{B_{p_{b}}(x,\varepsilon):x\in X\textit{ and }\varepsilon>0\}\), i.e., △ is the family of all \(p_{b}\)open balls. Then △ is a base of some topology on X.
It is also worthy noting that Proposition 1.1 and Claim 1.2 were cited in [4].
For balls in a cone metric space \((X,d)\), Turkoglu and Abuloha gave the following equality (see [5], Proposition 2), where \(x\in X\) and \(\varepsilon\gg\theta\):
Equality 1.3
\(\overline{\{y\in X:d(x,y)\ll\varepsilon\}}=\{ y\in X:d(x,y)\le\varepsilon\}\).
In this paper, we discuss Proposition 1.1, Claim 1.2, and Equality 1.3. For Proposition 1.1 and Claim 1.2, we construct a partial bmetric space \((X,p_{b})\) in the sense of [3], and show that there are a \(p_{b}\)open ball \(B_{p_{b}}(x,\varepsilon)\) and \(y\in B_{p_{b}}(x,\varepsilon)\) such that \(B_{p_{b}}(y,\delta)\nsubseteq B_{p_{b}}(x,\varepsilon)\) for all \(\delta>0\), and hence △ is not a base of any topology on X, which shows that Proposition 1.1 (including its proof) and Claim 1.2 are not true. For Equality 1.3, we establish some relations between balls and their closures in cone metric spaces by ≪, <, and ≤, and we give an example to show that Equality 1.3 is not true. However, it must be emphasized that these corrections do not affect the rest of the results in [3, 5].
Throughout this paper, \(\mathbb {N}\), \(\mathbb {R}\), and \(\mathbb {R}^{+}\) denote the set of all natural numbers, the set of all real numbers and the set of all nonnegative real numbers, respectively. For a subset A of a space X, F̅ denotes the closure of F in X. For undefined notations and terminology, one can refer to [3, 5].
Results and discussion
We give the main results of this paper by the following two subsections.
\(p_{b}\)Open balls in partial bmetric spaces
The following partial bmetric spaces were introduced by Shukla in [1].
Definition 2.1
[1]
Let X be a nonempty set. A mapping \(p_{b}:X\times X\longrightarrow \mathbb {R}^{+}\) is called a partial bmetric with coefficient \(s\ge1\) and \((X,p_{b})\) is called a partial bmetric space with coefficient \(s\ge1\) if the following are satisfied for all \(x,y,z\in X\):

(1)
\(x=y \Longleftrightarrow p_{b}(x,x)=p_{b}(y,y)=p_{b}(x,y)\).

(2)
\(p_{b}(x,y)=p_{b}(y,x)\).

(3)
\(p_{b}(x,x)\le p_{b}(x,y)\).

(4)
\(p_{b}(x,y)\le s(p_{b}(x,z)+p_{b}(z,y))p_{b}(z,z)\).
Remark 2.2
If \(s=1\) in Definition 2.1, then \((X,p_{b})\) is a partial metric space, which was introduced by Matthews (for example, see [3]). Further, put \(d_{p_{b}}:X\times X\longrightarrow \mathbb {R}^{+}\) by \(d_{p_{b}}(x,y)=2p_{b}(x,y)p_{b}(x,x)p_{b}(y,y)\) for all \(x,y\in X\), then \(d_{p_{b}}\) is a metric on X and \((X,d_{p})\) is a metric space.
However, if \(s>1\), then we cannot guarantee that each partial bmetric can induce a bmetric by the method in Remark 2.2. So Mustafa et al. gave the following partial bmetric \(p_{b}\) by modifying Definition 2.1(4) and proved that the \(p_{b}\) induces a bmetric by the method in Remark 2.2.
Definition 2.3
([3])
Let X be a nonempty set. A mapping \(p_{b}:X\times X\longrightarrow \mathbb {R}^{+}\) is called a partial bmetric with coefficient \(s\ge1\) and \((X,p_{b})\) is called a partial bmetric space with coefficient \(s\ge1\) if the following are satisfied for all \(x,y,z\in X\):

(1)
\(x=y \Longleftrightarrow p_{b}(x,x)=p_{b}(y,y)=p_{b}(x,y)\).

(2)
\(p_{b}(x,y)=p_{b}(y,x)\).

(3)
\(p_{b}(x,x)\le p_{b}(x,y)\).

(4)
\(p_{b}(x,y)\le s(p_{b}(x,z)+p_{b}(z,y)p_{b}(z,z))+\frac {1s}{2}(p_{b}(x,x)+p_{b}(y,y))\).
Remark 2.4
If x, y, z satisfy Definition 2.3(1), (2), (3) and are different from each other, then it is easy to check that x, y, z Definition 2.3(4) holds.
As a known fact, Proposition 1.1 and Claim 1.2 are not true if \((X,p_{b})\) is a partial bmetric space in the sense of Definition 2.1 ([6]). So it is important to check whether Proposition 1.1 and Claim 1.2 are true if \((X,p_{b})\) is a partial bmetric space in the sense of Definition 2.3. The following example shows that the result of the check is negative, which comes from [6]. In the following, all partial bmetric spaces are in the sense of Definition 2.3.
Example 2.5
Let \(X=\{u,v,w\}\) and put \(p_{b}:X\times X\longrightarrow \mathbb {R}^{+}\) as follows:

(i)
\(p_{b}(u,u)=p_{b}(w,w)=1\) and \(p_{b}(v,v)=0.5\).

(ii)
\(p_{b}(u,w)=p_{b}(w,u)=1.5\).

(iii)
\(p_{b}(v,w)=p_{b}(w,v)=1\).

(iv)
\(p_{b}(u,v)=p_{b}(v,u)=3\).
Let \(B_{p_{b}}(u,\varepsilon)\) be described in Proposition. Then the following hold:

(1)
\(p_{b}\) is a partial bmetric with coefficient \(s=3\).

(2)
\(w\in B_{p_{b}}(u,1)\) and for any \(\varepsilon>0\), \(B_{p_{b}}(w,\varepsilon)\nsubseteq B_{p_{b}}(u,1)\).
Proof
(1) It is not difficult to check that \(p_{b}\) satisfies Definition 2.3(1), (2), (3). In order to check that \(p_{b}\) satisfies Definition 2.3(4), we only need to consider the following three cases by Remark 2.4.

(1)
\(x=u\), \(y=v\), \(z=w\):
$$\begin{aligned}& p_{b}(u,v)=3, \\& 3\bigl(p_{b}(u,w)+p_{b}(w,v)p_{b}(w,w)\bigr)+ \frac{13}{2}\bigl(p_{b}(u,u)+p_{b}(v,v)\bigr)=3. \end{aligned}$$So \(p_{b}(u,v)\le3(p_{b}(u,w)+p_{b}(w,v)p_{b}(w,w))+ \frac{13}{2}(p_{b}(u,u)+p_{b}(v,v))\).

(2)
\(x=u\), \(y=w\), \(z=v\):
$$\begin{aligned}& p_{b}(u,w)=1.5 , \\& 3\bigl(p_{b}(u,v)+p_{b}(v,w)p_{b}(v,v)\bigr)+ \frac{13}{2}\bigl(p_{b}(u,u)+p_{b}(w,w)\bigr)=8.5 . \end{aligned}$$So \(p_{b}(u,w)\le3(p_{b}(u,v)+p_{b}(v,w)p_{b}(v,v))+ \frac{13}{2}(p_{b}(u,u)+p_{b}(w,w))\).

(3)
\(x=v\), \(y=w\), \(z=u\):
$$\begin{aligned}& p_{b}(v,w)=1, \\& 3\bigl(p_{b}(v,u)+p_{b}(u,w)p_{b}(u,u)\bigr)+ \frac{13}{2}\bigl(p_{b}(v,v)+p_{b}(w,w)\bigr)=9. \end{aligned}$$So \(p_{b}(v,w)\le3(p_{b}(v,u)+p_{b}(u,w)p_{b}(u,u))+ \frac{13}{2}(p_{b}(v,v)+p_{b}(w,w))\).
Thus, \(p_{b}\) is a partial bmetric with coefficient \(s=3\).
(2) Since \(p_{b}(u,w)=1.5<1+1=p_{b}(u,u)+1\), \(w\in B_{p_{b}}(u,1)\). In addition, for any \(\varepsilon>0\), \(p_{b}(w,v)=1<1+\varepsilon =p_{b}(w,w)+\varepsilon\), so \(v\in B_{p_{b}}(w,\varepsilon)\). On the other hand, \(p_{b}(u,v)=3\nless2=1+1=p_{b}(u,u)+1\), so \(v\notin B_{p_{b}}(u,1)\). This shows that \(B_{p_{b}}(w,\varepsilon)\nsubseteq B_{p_{b}}(u,1)\). □
Remark 2.6
Example 2.5 shows that Proposition 1.1 and Claim 1.2 are not true if \((X,p_{b})\) is a partial bmetric space.
However, we have the following.
Proposition 2.7
([7])
Let \((X,p_{b})\) be a partial bmetric space and △ be described in Claim 1.2. Then △ is a subbase for some topology on X. We denote the topology by \(\mathscr {T}_{p_{b}}\).
It is well known that the space \((X,\mathscr {T}_{p_{b}})\) is \(T_{0}\) but does not need to be \(T_{1}\) ([7]). The following proposition give a sufficient and necessary such that \((X,\mathscr {T}_{p_{b}})\) is a \(T_{1}\)space.
Proposition 2.8
Let \((X,p_{b})\) be a partial bmetric space in the sense of Definition 2.3. Then the following are equivalent:

(1)
\((X,\mathscr {T}_{p_{b}})\) is a \(T_{1}\)space.

(2)
\(p_{b}(x,y)>\max\{p_{b}(x,x),p_{b}(y,y)\}\) for each pair of distinct points \(x,y\in X\).
Proof
(1) ⟹ (2): Let \((X,\mathscr {T}_{p_{b}})\) be a \(T_{1}\)space. If \(x,y\in X\) and \(x\ne y\), then there is a neighborhood U of x such that \(y\notin U\). Since △ is a subbase of \((X,\mathscr {T}_{p_{b}})\) from Proposition 2.7, there are \(\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{k}>0\) such that \(y\notin\bigcap\{B_{p_{b}}(x,\varepsilon_{i}):i=1,2,\ldots,k\}\), and hence there is \(i_{0}\in\{1,2,\ldots,k\}\) such that \(y\notin B_{p_{b}}(x,\varepsilon_{i_{0}})\). So \(p_{b}(x,y)\ge p_{b}(x,x)+\varepsilon_{i_{0}}>p_{b}(x,x)\). In the same way, \(p_{b}(x,y)>p_{b}(y,y)\). So \(p_{b}(x,y)>\max\{ p_{b}(x,x),p_{b}(y,y)\}\).
(2) ⟹ (1): Let \(x,y\in X\) and \(x\ne y\). If \(p_{b}(x,y)>\max\{p_{b}(x,x),p_{b}(y,y)\}\). Then \(p_{b}(x,y)>p_{b}(x,x)\). Put \(\varepsilon=p_{b}(x,y)p_{b}(x,x)>0\), then \(p_{b}(x,y)=p_{b}(x,x)+\varepsilon\), and so \(y\notin B_{p_{b}}(X,\varepsilon)\). In the same way, there is \(\varepsilon'>0\) such that \(x\notin B_{p_{b}}(y,\varepsilon')\). Consequently, \((X,\mathscr {T}_{p_{b}})\) is a \(T_{1}\)space. □
Balls in cone metric spaces
Definition 2.9
Let E be a real Banach space. A subset P of E is called a cone of E and \((E,P)\) is called a cone space if the following are satisfied: where θ is zero element in E.

(1)
P is closed, \(P\ne\emptyset\), and \(P\ne\{\theta\}\).

(2)
\(a,b\in \mathbb {R}^{+}\) and \(\alpha,\beta\in P \Longrightarrow a\alpha +b\beta\in P\).

(3)
\(\alpha,\alpha\in P \Longrightarrow \alpha=\theta\).
Definition 2.10
Let \((E,P)\) be a cone space. Some partial orderings ≤, <, and ≪ on E with respect to P are defined as follows, respectively, where \(P^{\circ}\) denotes the interior of P. Let \(\alpha,\beta\in E\).

(1)
\(\alpha\le\beta\) if \(\beta\alpha\in P\).

(2)
\(\alpha<\beta\) if \(\alpha\le\beta\) and \(\alpha\ne\beta\).

(3)
\(\alpha\ll\beta\) if \(\beta\alpha\in P^{\circ}\).
Remark 2.11
Let \((E,P)\) be a cone space. For the sake of conveniences, we also use notations ‘≥’, ‘>’, and ‘≫′’ on E with respect to P. The meanings of these notations are clear and the following hold:

(1)
\(\alpha\ge\theta\) if and only if \(\alpha\in P\).

(2)
\(\alpha\gg\theta\) if and only if \(\alpha\in P^{\circ}\).

(3)
\(\alpha\beta\gg\theta\) if and only if \(\alpha\gg\beta\).

(4)
\(\alpha\beta\ge\theta\) if and only if \(\alpha\ge\beta\).

(5)
\(\alpha\gg\beta \Longrightarrow \alpha>\beta \Longrightarrow \alpha\ge\beta\).
In addition, in order to guarantee the existence of elements \(\varepsilon\gg\theta\), we always assume that the cone P has nonempty interior ([5]).
Definition 2.12
Let X be a nonempty set and let \((E,P)\) be a cone space. A mapping \(d:X\times X\longrightarrow E\) is called a cone metric on X, and \((X,d)\) is called a cone metric space if the following are satisfied:

(1)
\(d(x,y)\ge\theta\) for all \(x,y\in X\) and \(d(x,y)=\theta\) if and only if \(x=y\).

(2)
\(d(x,y)=d(y,x)\) for all \(x,y\in X\).

(3)
\(d(x,y)\le d(x,z)+d(z,y)\) for all \(x,y,z\in X\).
Notation 2.13
Let \((X,d)\) be a cone metric space, \(x\in X\), and \(\varepsilon\gg\theta\). In this section, we use the following notations for balls in \((X,d)\):

(1)
\(B(x,\varepsilon)=\{y\in X:d(x,y)\ll\varepsilon\}\).

(2)
\(B_{1}(x,\varepsilon)=\{y\in X:d(x,y)<\varepsilon\}\).

(3)
\(B_{2}(x,\varepsilon)=\{y\in X:d(x,y)\le\varepsilon\}\).
Proposition 2.14
([5])
Let \((X,d)\) be a cone metric space. Put \(\mathscr {B}=\{B(x,\varepsilon):x\in X\textit{ and}\ \varepsilon\gg\theta\}\). Then \(\mathscr {B}\) is a base for some topology \(\mathscr {T}\) on X.
In this section, just as the investigation in [5], we always suppose that each cone metric space is a topological space described in Proposition 2.14.
Proposition 2.15
Let \((X,d)\) be a cone metric space. For each \(x\in X\) and each \(\varepsilon\gg\theta\), the following hold:

(1)
\(B(x,\varepsilon)\subseteq B_{1}(x,\varepsilon)\subseteq B_{2}(x,\varepsilon)\).

(2)
\(\overline{B(x,\varepsilon)}\subseteq\overline{B_{1}(x,\varepsilon )}\subseteq\overline{B_{2}(x,\varepsilon)}\).

(3)
\(\overline{B_{2}(x,\varepsilon)}=B_{2}(x,\varepsilon)\).
Proof
(1) It holds by Remark 2.11(5).
(2) It holds by the above item (1).
(3) Let \(y\in\overline{B_{2}(x,\varepsilon)}\). Then, whenever \(\eta\gg \theta\), \(B(y,\eta)\cap B_{2}(x,\varepsilon)\ne\emptyset\). Pick \(z\in B(y,\eta)\cap B_{2}(x,\varepsilon)\). Then \(d(x,z)\le\varepsilon\) and \(d(z,y)\ll\eta\). It follows that \(d(x,y)\le d(x,z)+d(z,y)\ll\varepsilon+\eta\). Let \(\eta\rightarrow\theta\). Then \(d(x,y)\le\varepsilon\). So \(y\in B_{2}(x,\varepsilon)\). This proves that \(\overline{B_{2}(x,\varepsilon)}\subseteq B_{2}(x,\varepsilon)\). On the other hand, it is clear that \(\overline{B_{2}(x,\varepsilon )}\supseteq B_{2}(x,\varepsilon)\). So \(\overline{B_{2}(x,\varepsilon)}=B_{2}(x,\varepsilon)\). □
The following example shows that any ‘⊆’ in Proposition 2.15(1), (2) cannot be replaced by ‘=’.
Example 2.16
Let the cone space \((E,P)\) be defined as in [5], Example 1, i.e., \(E=\mathbb {R}^{2}=\{(r,s):r,s\in \mathbb {R}\}\) is the Euclidean plane and \(P=\{(r,s)\in E:r,s\ge0\}\). Let \(X=\{x,y,z\}\). Define \(d:X\times X\longrightarrow E\) as follows: \(d(x,x)=d(y,y)=d(z,z)=(0,0)\), \(d(x,y)=d(y,x)=d(y,z)=d(z,y)=(1,1)\), and \(d(x,z)=d(z,x)=(1,0)\). It is not difficult to check that \((X,d)\) is a cone metric space. Let \(\varepsilon=(1,1)\gg\theta\).

(1)
Note that \(d(x,y)=(1,1)=\varepsilon\). By Remark 2.11, \(d(x,y)\le\varepsilon\), \(d(x,y)\nless\varepsilon \), and \(d(x,y)\not\ll\varepsilon\). So \(y\notin B(x,\varepsilon)\), \(y\notin B_{1}(x,\varepsilon)\), and \(y\in B_{2}(x,\varepsilon)\). Also, \(\varepsilond(x,z)=(1,1)(1,0)=(0,1)\in P(\{\theta\}\cup P^{\circ})\). By Remark 2.11, \(\varepsilond(x,z)>\theta\), and \(\varepsilon d(x,z)\not\gg\theta\), hence \(d(x,z)<\varepsilon\) and \(d(x,z)\not\ll\varepsilon\). So \(z\notin B(x,\varepsilon)\), \(z\in B_{1}(x,\varepsilon)\), and \(y\in B_{2}(x,\varepsilon)\). It follows that \(B(x,\varepsilon)= \{x\}\), \(B_{1}(x,\varepsilon)=\{x,z\}\), and \(B_{2}(x,\varepsilon)=\{x,y,z\}\). So any ‘⊆’ in Proposition 2.15(1) cannot be replaced by ‘=’.

(2)
Note that \((X,d)\) is Hausdorff ([5]). In fact, each cone metric space is metrizable ([8]). So \(\overline{B(x,\varepsilon)}=B(x,\varepsilon)\), \(\overline{B_{1}(x,\varepsilon)}=B_{1}(x,\varepsilon)\), and \(\overline{B_{2}(x,\varepsilon)}=B_{2}(x,\varepsilon)\). By the above item (1), any ‘⊆’ in Proposition 2.15(2) cannot be replaced by ‘=’.
Remark 2.17

(1)
By Example 2.16, Equality 1.3 is not true. Indeed, in Example 2.16, \(\overline{B(x,\varepsilon)}=\{x\}\) and \(B_{2}(x,\varepsilon)=\{x,y,z\}\). So \(\overline{B(x,\varepsilon)}\ne B_{2}(x,\varepsilon)\).

(2)
Let \((X,d)\) be a cone metric space. In [5], the authors showed that \(\overline{B(x,\varepsilon)}\) and \(B_{2}(x,\varepsilon)\) are sequentially closed in \((X,d)\) ([5], Proposition 2). In fact, \((X,d)\) is metrizable, and hence \((X,d)\) is a sequential space, i.e., closed and sequentially closed in \((X,d)\) are equivalent. On the other hand, indeed, the closure \(\overline{B(x,\varepsilon)}\) of \(B(x,\varepsilon)\) is closed and \(B_{2}(x,\varepsilon)\) is closed by Proposition 2.15(2).
Conclusions
This paper discusses balls in partial bmetric spaces and cone metric spaces, respectively.
Conclusion 3.1
Let \((X,p_{b})\) be a partial bmetric space in the sense of [3]. For the family △ of all \(p_{b}\)open balls in \((X,p_{b})\), this paper proves that there are \(x,y\in B\in\triangle\) such that \(B'\nsubseteq B\) for all \(B'\in\triangle\), 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 [3], Proposition 4 and the claim following [3], Proposition 4, are not true.
Conclusion 3.2
Let \((X,d)\) be a cone metric space. By ≪, <, and ≤ in \((X,d)\), this paper establishes some relations among \(\{y\in X:d(x,y)\ll\varepsilon\}\), \(\{y\in X:d(x,y)<\varepsilon\}\), and \(\{y\in X:d(x,y)\le\varepsilon\}\). Furthermore, this paper also constructs a cone metric space \((X,d)\) such that \(\overline{\{y\in X:d(x,y)\ll\varepsilon\}}\ne\{y\in X:d(x,y)\le \varepsilon\}\) for some \(x,y\in X\) and \(\varepsilon\gg0\), which shows that the equality in [5], Proposition 2, is not true.
However, it must be emphasized that these corrections in this paper do not affect the rest of the results in the relevant papers.
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Acknowledgements
The 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 Science Foundation (no. 2013M541710, 2014T70537), the Jiangsu Province Natural Science Foundation (no. BK20140583), the Jiangsu Province Postdoctoral Science Foundation (no. 1302156C) and the Priority Academic Program Development of Jiangsu Higher Education Institutions.
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Ge, X., Lin, S. Balls in generalizations of metric spaces. J Inequal Appl 2016, 16 (2016). https://doi.org/10.1186/s136600160962y
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MSC
 54A10
 54E35
Keywords
 ball
 partial bmetric space
 cone metric space