Balls in generalizations of metric spaces

This 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 the family 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 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 {y ∈ X : d(x, y) ε} = {y ∈ X : d(x, y)≤ ε} 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.