A space is called an $so$-metrizable space if it is a regular space with a $\sigma$-locally finite sequentially open network. This paper proves that $so$-metrizable spaces are preserved under perfect mappings and under closed sequence-covering mappings, which give an affirmative answer to a question on preservations of so-metrizable spaces under some closed mappings. Also, we prove that the closed image of an $so$-metrizable space is an $so$-metrizable space if it is a topological group.