so-metrizable spaces and images of metric spaces

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