On symmetrizability and perfectness of second-countable spaces - Banakh, Iryna; Banakh, Taras; Bazylevych, Lidiya (Topology Proceedings, 2022)
A symmetrizability criterion of Arhangelskǐ 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 $q_0$ and study the smallest possible cardinality $q_i$ of a non-symmetrizable secondcountable $T_i$-space for $i \in\{1,2\}$.