so-metrizable spaces and images of metric spaces

so-metrizable spaces are a class of important generalized metric spaces between metric spaces and sn-metrizable spaces where a space is called an so-metrizable space if it has a 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 is an so-metrizable space if and only if it is an so-open, compact-covering, sigma-image of a metric space, if and only if it is an so-open, sigma-image of a metric space. In addition, it is shown that so-open mapping is a simplified form of 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, sn-open mapping and 2-sequence-covering mapping, which further enrich and deepen generalized metric space theory.