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 ∈ 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.

So -metrizable spaces are a class of important generalized metric spaces between metric spaces and s n sn -metrizable spaces where a space is called an so -metrizable space if it has a σ \sigma -locally finite so -network. As the further work that attaches to the celebrated Alexandrov conjecture, it is interesting to characterize so -metrizable spaces by images of metric spaces. This paper gives such characterizations for so -metrizable spaces. More precisely, this paper introduces so -open mappings and uses the “Pomomarev’s method” to prove that a space X X is an so -metrizable space if and only if it is an so -open, compact-covering, σ \sigma -image of a metric space, if and only if it is an so -open, σ \sigma -image of a metric space. In addition, it is shown that so -open mapping is a simplified form of s n sn -open mapping (resp. 2-sequence-covering mapping if the domain is metrizable). Results of this paper give some new characterizations of so -metrizable spaces and establish some equivalent relations among so -open mapping, s n sn -open mapping and 2-sequence-covering mapping, which further enrich and deepen generalized metric space theory.