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 ${\mathfrak{q}}_0$ and study the smallest possible cardinality ${\mathfrak{q}}_i$ of a non-symmetrizable second-countable $T_i$-space for $i\in \{1,2\}$.