How can I prove that the perfect image of a metrizable space is metrizable? I know the following three theorems about equivalent conditions of metrizability.
A space $X$ is metrizable if and only if it is regular and has a basis that is countably locally finite.
A space $X$ is metrizable if and only if it is regular and has a basis that is countably locally discrete.
A space $X$ is metrizable if and only if it is a paracompact Hausdorff space that is locally metrizable.
Also I know that Hausdorffness, regularity, local compactness, second-countability, and paracompactness is preserved by perfect maps. How can I prove that metrizability is preserved by a perfect map using these facts(or +@ if needed)?