On symmetrizability and perfectness of second-countable spaces

A symmetrizability criterion of Arhangelskii implies that a second-countable Hausdorff space is symmetrizable if and only if it is perfect. We present an example of a non-symmetrizable second-countable submetrizable space of cardinality q₀ and study the smallest possible cardinality qᵢ of a non-symmetrizable second-countable Tᵢ-space for i∈\1,2\.