In this paper, generalized metric spaces are introduced as a common generalization of tvs-cone metric spaces, partial metric spaces and b-metric spaces, and a unified approach is proposed to some fixed point results by using generalized metric spaces. Specifically, Banach's contraction principle and Kannan type fixed point theorem, as well as other types fixed point results on generalized metric spaces are given, respectively.

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.

Songlin Yang;, Ting Xu, and Yongcun Shao. International Journal of Fuzzy Logic and Intelligent Systems 2022;22:202-12. https://doi.org/10.5391/IJFIS.2022.22.2.202