Balls in generalizations of metric spaces

Abstract

Abstract This paper discusses balls in partial b-metric spaces and cone metric spaces, respectively. Let ( X , p b ) $(X,p_{b})$ be a partial b-metric space in the sense of Mustafa et al. For the family △ of all p b $p_{b}$ -open balls in ( X , p b ) $(X,p_{b})$ , this paper proves that there are x , y ∈ B ∈ △ $x,y\in B\in\triangle$ such that B ′ ⊈ B $B'\nsubseteq B$ for all B ′ ∈ △ $B'\in\triangle$ , where B and B ′ $B'$ 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 ) $(X,d)$ and shows that { y ∈ X : d ( x , y ) ≪ ε } ‾ ≠ { y ∈ X : d ( x , y ) ≤ ε } $\overline{\{y\in X:d(x,y)\ll\varepsilon\}}\ne\{y\in X:d(x,y)\le\varepsilon\}$ 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.

Keywords

• ball
• partial b-metric space
• cone metric space