Balls in generalizations of metric spaces

Xun Ge,Shou Lin

doi:10.1186/s13660-016-0962-y

Abstract

This paper discusses balls in partial b-metric spaces and cone metric spaces, respectively. Let $(X,p_{b})$ be a partial b-metric 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 b-metric 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.

Highlights

Partial b-metric spaces and cone metric spaces are important generalizations of metric spaces, 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 partial b-metric in the sense of [ ] in order to guarantee that each partial b-metric pb can induce a b-metric ([ ])
2.1 pb-Open balls in partial b-metric spaces The following partial b-metric spaces were introduced by Shukla in [ ]
A mapping pb : X × X −→ R+ is called a partial b-metric with coefficient s ≥ and (X, pb) is called a partial b-metric space with coefficient 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)

Summary

Introduction

Partial b-metric spaces and cone metric spaces are important generalizations of metric spaces, 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 partial b-metric in the sense of [ ] in order to guarantee that each partial b-metric pb can induce a b-metric ([ ]). For balls in a cone metric space (X, d), Turkoglu and Abuloha gave the following equality (see [ ], Proposition ), where x ∈ X and ε θ : Equality . We construct a partial b-metric space (X, pb) in the sense of [ ], and show that there are a pb-open ball Bpb (x, ε) and y ∈ Bpb (x, ε) such that Bpb (y, δ) Bpb (x, ε) for all δ > , and is not a base of any topology on X, which shows that Proposition .

Results

Conclusion