ℐ-sn-metrizable spaces and the images of semi-metric spaces

The theory of generalized metric spaces is an active topic in general topology. In this article, we utilize the concepts of ideal convergence and networks to discuss the metrization problem and the mutual classification problem between spaces and mappings in topological spaces. We define ℐ {\mathcal{ {\mathcal I} }} - s n sn -metrizable spaces, obtain several characterizations of ℐ {\mathcal{ {\mathcal I} }} - s n sn -metrizable spaces, and establish some mapping relations between ℐ {\mathcal{ {\mathcal I} }} - s n sn -metrizable spaces and semi-metric spaces. These not only generalize some theorems in generalized metric theory, but also find further applications of ideal convergence in general topology.